Multiphasic Intervertebral Disc Mechanics: Theory and Application
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DOI: 10.1007/s11831-012-9073-1
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- Karajan, N. Arch Computat Methods Eng (2012) 19: 261. doi:10.1007/s11831-012-9073-1
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Abstract
The human spine is the flexible support structure of our body. Its geometric shape is a result of the human evolutionary history, where especially the lumbar spine area (L1–L5) is most at risk of causing discomfort resulting from mechanical stresses. Herein, the intervertebral discs (IVD) between the vertebral bodies are the most susceptible elements, as these avascular structures have to provide the flexibility. It is widely accepted that the IVD are often the trigger for back pain. In the context of biomechanical research, it is therefore important to develop a model for the human lumbar spine with particular focus on the IVD.
The objective of the presented work is to subject the IVD of the lumbar spine to continuum-biomechanical research. Herein, a three-dimensional (3-d) finite-element model is developed that allows to estimate the influence of lumbar spine motion, i.e., bending, torsion and compression, on the resulting stress-field inside the IVD. Following this, the model can be utilised to detect inappropriate or excessive loading of the spine. The theoretical description of the IVD is based on a multi-phase continuum approach in the framework of the well-known “Theory of Porous Media” (TPM). This is a natural choice resulting from the avascular composition of the IVD. In general, IVD tissue is categorised as charged hydrated material with mechanical and electro-chemical internal coupling mechanisms. In order to capture these couplings, the underlying model incorporates an extracellular matrix (ECM) with fixed negative charges, which is saturated by a mixture of a liquid solvent and ions. Following the basic concept of the TPM, a volumetric averaging process is prescribed leading to volume fractions for the pore space and the solid skeleton as well as molar concentrations for the ion species in the pore fluid.
In detail, the IVD exhibits a gelatinous core known as the nucleus pulposus (NP), and an onion-like surrounding structure consisting of anisotropic crosswise fibre-reinforced lamellae, the annulus fibrosus (AF). Both regions are seamlessly merging into each other and consist of mostly collagen fibres of varying strength and direction as well as proteoglycans with adhering negative charges. As a result of these fixed negative charges and the fact that the interstitial fluid carries positively and negatively charged ions, a model is created that describes the mutually coupled behaviour of solid deformation and fluid flow. To illustrate the resulting coupled swelling and shrinkage process, it is sufficient to recall that the size of the human body is reduced by roughly 2 centimetres during the waking phase of the day. This change in height is triggered by mechanical loads stemming from the body weight, which squeezes the interstitial fluid out of the IVD, thereby losing altitude. Simultaneously, an electro-chemical imbalance is generated, which is compensated during the nocturnal resting phase, where the interstitial fluid is driven back into the IVD due to osmotic effects.
In summary, the presented paper provides a review on IVD mechanics and can be understood as a compilation of relevant information giving valuable guidance to researchers starting to work in this challenging field. In this regard, the paper opens with anatomical and chemical fundamentals of IVD tissue with a particular focus on the inherent inhomogeneities. This is followed by an introduction to the continuum-mechanical fundamentals including an overview of the TPM as well as the inelastic non-linear kinematics and the balance equations of the resulting porous continua. Thereafter, the constitutive modelling process is illustrated with a particular focus on the thermo-dynamically consistent modelling process of the intrinsic viscoelasticity of the ECM as well as the inhomogeneous characteristics of the embedded collagen fibres and the viscous pore fluid. The resulting system of coupled partial differential equations is then numerically discretised using the finite-element method which finally allows the simulation of deformation processes of the IVD using either single- or multi-processor machines. Moreover, an automated computation scheme is presented to systematically capture the inhomogeneities of the IVD. The paper is closed with several sample applications, which embrace the capabilities of the presented computational model on the one hand and give advice for further validation in terms of the applied material parameters.
1 Introduction and Overview
1.1 Motivation
Clinical Point of View
In industrialised western countries, almost everybody has been suffering from low back pain (LBP) at least once in his lifetime, for example, in a representative survey carried out in the early nineties and another one conducted ten years later, about 85 % of the German population had already experienced this widespread ailment, cf., e.g., Schmidt et al. [132]. Regarding the same surveys, current LBP was reported from about 40 % of the people asked, while 21 % of the asked probands were having severe pain. Moreover, low back pain is the number one cause of work-related health problems in many European countries. The second European survey on working conditions reported an overall average of 30 % of the working people in Europe suffering from LBP, while the fourth survey of 2007 recorded a slight decrease of the average to 25 %, cf. Parent-Thirion et al. [121]. According to European Foundation for the Improvement of Living and Working Conditions [54], these work related LBP problems vary by occupation ranging from 16 % for the administrative staff to 58 % for skilled agricultural workers. Since both occupations work in a seated position, this tremendous difference can only be explained by the enormous vibrations an agricultural worker is exposed to while sitting on farming machines (Lis et al. [95]).
This significant number causes not only a substantial amount of direct health-care costs but is also responsible for a significant socioeconomic impact. In Germany for instance, 4 % of all work force is lost every year due to LBP. But besides the monetary factors, there is of course the pain so many people are suffering from. Although the cause of low back pain remains unclear in many patients, it is generally accepted that a deterioration of the intervertebral disc (IVD) plays an important part in the cause of the complaints, cf. Vanharanta et al. [170]. More detailed information on traumatic and degenerated IVD, as well as on its medical treatment, can be found in Wilke and Claes [185]. In this regard, many different ways of treatment are possible, but in very severe cases it is best to fusion the involved vertebral bodies or replace the deteriorated IVD by an implant, which should similarly respond to mechanical loading like the original.
These issues were motivation enough to initiate a remarkable research interest in intervertebral disc mechanics over the past three decades. Regarding the steady increase of computing power in the same period of time, more and more researchers are confident that a possible explanation and remedy for LBP lies in numerical simulation techniques.
Biomechanical and Computational Point of View
The biomechanical challenge lies within the structural complexity of the spine itself arising from a great individuality of the patient data with an irregular and hard to determine 3-d anatomical shape. As the name implies, the intervertebral disc is embedded in-between two vertebral bodies, thereby forming a motion segment of the spine. Two main regions can be distinguished in an axial cut through the IVD, a gelatinous core, the nucleus pulposus (NP) enclosed by a fibrous, lamellar structure, the anulus fibrosus (AF), having aligned collagen fibres as dominant structural elements. Both regions are composed of a porous multi-component microstructure consisting of a charged hydrated extracellular matrix (ECM) carrying fixed negative charges, as well as an ionised interstitial fluid yielding a swelling-active material. Additionally, the inner structure and thus, the associated physical properties are inhomogeneously distributed over the 3-d anatomic shape of the tissue. For a more detailed description, the reader is referred to Sect. 2 or to Ayad and Weiss [7], Ehlers et al. [46], Marchand and Ahmed [99], Mow and Hayes [110], Urban and Roberts [168] and references therein. Thus, all the properties characterising soft biological tissues in general are unified within the IVD. Moreover, in the context of spine mechanics, the IVD plays an important key role on the overall performance of the spine.
Regarding numerical simulations of the spine or soft biological tissues in general, the finite element method (FEM) has been proven to be a well-suited numerical approximation method of the resulting differential equations describing the overall problem. In the past, numerous finite element models have already been applied to the broad field of computational biomechanics, but often exhibit certain essential deficiencies concerning the reproduction of the relevant material properties occurring in a hydrated soft biological tissue. As a matter of fact, singlephasic models can never predict the interstitial fluid flow or related effects like osmosis, but are, on the other hand, simpler to implement and numerically cheaper than biphasic models, when regarded from an computational point of view. However, quite a few models have more than one phase but are often restricted to the small-strain domain or are not capable of capturing the intrinsic viscoelasticity of the solid skeleton, which might be dominant in some applications. Moreover, the reinforcement of collagen fibres is often captured by 1-d elements spanning between the corner nodes of the respective 3-d finite elements used to discretise the biological soft tissue, thereby causing a mesh dependency of the solution.
Keeping this in mind, computational biomechanics applied to the IVD and especially to one or even several motion segments is still a challenging task, because it is still not solved in detail, to what extent the several tissue properties of the IVD influence the overall behaviour of the spine. Moreover, concerning numerical simulations of the spine, several complex and coupled problems have to be solved simultaneously. Neglecting the difficulties resulting from the contact of the spinous processes via the facet joints or the enormous difference in stiffness between the IVD and the adjacent vertebrae, the IVD exhibits several other coupled effects. In particular, there is the dissipative behaviour resulting from the viscous “fluid flow” inside the tissue, which can either be mechanically or electro-chemically driven, as well as flow-independent (intrinsic) viscoelastic properties of the ECM. Moreover, the tissue is partly reinforced by fibres yielding an anisotropic behaviour and is exposed to large deformations, when computing real-life problems.
Another difficulty arises from the application of the developed numerical model. It is often cumbersome to obtain reliable experimental data to identify the theoretically introduced material parameters. In this regard, it is also difficult to define the boundary conditions for realistic boundary-value problems in the framework of the finite-element method (FEM).
Moreover, concerning the FEM, such sophisticated and complex models exhibit many degrees of freedom when applied to a general 3-d discretisation of the spine. This effect is not only due to the numerical expensiveness of the mixed finite elements in use, but also due to the complex geometries involved which have to be approximated in a reasonable way. Following this leads to rapidly growing systems of equations which have to be solved during such a numerical approximation, and thus, to an enormous numerical effort. Considering the fact that a reliable solution depends on the accuracy of the numerical approximation, and thus, relies upon a reasonably fine mesh, the computational limits of a single personal computer (PC) available at present are reached very easily. Hence, parallel computation strategies come into play, in order to combine the power of several PCs to solve the problem simultaneously.
1.2 Scope, Aims and State of the Art
Ranging from the continuum-mechanical modelling to its numerical realisation, it is the aim of this contribution to develop a finite element model, which is as simple as possible, but at the same time complex enough, to capture many of the relevant properties of the IVD influencing the overall behaviour of the spine. As the needed level of complexity as well as the relevant properties are not known a priori, the presented model includes simple but also more sophisticated approaches, thereby still providing the possibility to easily switch between them. The needed complexity level and the respective material parameters involved will be identified using experimental results taken from the related literature. Moreover, a parameter sensitivity analysis is carried out for compression-bending problems of a non-degenerated motion segment of the lumber spine in order to obtain the influence of certain parameters on the overall deformation behaviour. Finally, two simulations of the lumbar spine will be carried out in parallel using a multi-processor machine, thereby illustrating the efficiency of the proposed finite element model.
To be more precise, a detailed and thermodynamically consistent continuum-mechanical modelling approach is followed and formulated in a very general way in order to reproduce the behaviour of almost any charged hydrated tissue. Herein, the Theory of Porous Media (TPM), cf. de Boer [26], Bowen [19], Ehlers [34, 38] or Mow et al. [113] is applied, which allows for a convenient and modular treatment of the necessary constitutive assumptions covering all (at first sight) relevant tissue properties. Moreover, polyconvex material laws describing the partly viscoelastic and anisotropic behaviour of the ECM are included, where the form of the respective constitutive functions is chosen to be of polynomial character, thereby including several complexity levels within the same implementation (Markert et al. [102] and Ogden [120]). In this regard, it is still not clear, to what extent the intrinsic viscoelastic effects stem from the isotropic part of the ECM or the structural collagen found in the AF. In this contribution, only the isotropic part of the ECM is modelled to behave viscoelastic, while the structural collagen fibres of type I remain hyper-elastic. The reason for this assumption becomes clear when the findings of Holzapfel et al. [71] are observed, stating that there was little to no rate dependency when a lamella of the AF is pulled in fibre direction. Following this, the intrinsic viscoelastic formulation of the isotropic part of the ECM is based on a generalised Maxwell model as it is described in Ehlers and Markert [42] or Markert [100], whereas the collagen fibres are assumed to behave purely elastic and are modelled in a continuum-mechanical way using structural tensors, cf. Boehler [17] or Spencer [150, 151]. Besides other features like the deformation-dependent permeability (Eipper [50], Markert [101]) and the possibility to include inhomogeneities, e.g., varying fibre alignment or location-dependent mechanical and chemical behaviour (Ehlers et al. [45, 46]), osmotic effects are also included. Herein, the simplifying assumption of Lanir [90] is applied stating that a consideration of the free movable ions is needless if no sudden concentration changes of the surrounding fluid occur, which is the case for living soft biological tissues. In this regard, an osmotic pressure contribution is formulated solely depending on the fixed negative charges which can be expressed in terms of the movement of the ECM.
Following this, the scope is to model the IVD in a non-degenerated condition, in order to understand its complex behaviour regarding the interplay of its structural, mechanical, and electro-chemical components as well as their influence on the overall response of the spine. Thus, the model can a priori serve as a numerical laboratory, where the influence of different scenarios concerning the composition of the disc are easily obtained, cf. Ehlers et al. [46]. Moreover, the influence of the IVD on the overall performance of the spine is investigated via computations on motion or spine segments, cf. Ehlers et al. [47]. Regarding the design of new IVD implants, this knowledge is of great benefit, as the implant must behave like the non-degenerated original in a best possible manner. After the approximate identification of the involved material parameters, the presented model may then be able to compute reference solutions for the overall design as well as the fine tuning of IVD implants.
In order to perform these computations, the developed model is numerically approximated using the FEM. Herein, the respective governing equations are implemented into the FE-tool PANDAS which is going back to Ehlers and Ellsiepen [41] and Ellsiepen [52]. Moreover, an interface is developed to couple the sequential research code PANDAS to the parallel solver M++ of Wieners [181], whereas the coupling is based on an already existing interface developed by Ammann [2] and Wieners et al. [182, 184]. Finally, the numerical solution of even large 3-d problems at suitable computational costs is possible, cf. Ehlers et al. [47] or Wieners et al. [183].
Currently, there are several models and modules available, which are used to describe soft biological tissues in general. However, most of them do not cover all the requirements needed for the continuum-mechanical modelling of the IVD. For example, the models used for state-of-the-art simulations of the spine are often restricted to singlephasic materials, thereby a priori excluding effects resulting from interstitial fluid flow and osmosis. The first spine simulations with an advanced IVD model were carried out by Shirazi-Adl [140, 141] or Shirazi-Adl et al. [142, 143] using a deformation-dependent Hooke-type material tangent and nonlinear spring elements to capture the collagen fibres in the AF. A similar model but using more sophisticated (singlephasic) material models and a detailed geometry approximation can be found in Rohlmann et al. [128], Schmidt et al. [133, 134]. In order to overcome the resulting grid dependency of the alignment of the spring elements, the anisotropic continuum theory based on Boehler [17] and Spencer [150, 151] was adopted and applied to the unique behaviour of the collagen fibres by, e.g., Elliott and Setton [51] Holzapfel et al. [70] or Klisch and Lotz [85], whereas Balzani [9], Balzani et al. [10], Markert et al. [102] or Schröder and Neff [137] developed the theory further towards a polyconvex framework. This knowledge was then incorporated in the lumbar spine simulations of Eberlein et al. [31, 32].
Since these models are singlephasic, they do not account for the fluid flow in the disc which is important for nutritional purposes and maintaining the biological composition, see, e.g., Holm and Nachemson [69] or Urban and Holm [166]. Hence, at least a second phase, the interstitial fluid, has to be introduced to overcome this deficiency. In this context, several biphasic models, based on the works of Biot [13], Bowen [18] and Mow et al. [112], are applied to describe soft biological tissues in general or with application to the IVD, see, e.g., Argoubi and Shirazi-Adl [5], Iatridis et al. [77], Klisch and Lotz [86], Li et al. [93] or Riches et al. [125]. However, as the occurring osmotic effects influence the fluid flow in the tissue, it also has to be considered. In order to include these electro-chemomechanically driven swelling phenomena, the biphasic models need to be extended, thereby emerging two different approaches. On the one hand, there is the independent description of the freely movable ions of the pore fluid yielding a complex and strongly coupled system of partial differential equations (PDE) leading to a sometimes difficult numerical treatment, cf. Chen et al. [22], Ehlers and Acartürk [39], Ehlers et al. [44, 49], Frijns et al. [57, 58], Huyghe et al. [74], Iatridis et al. [78], Kaasschieter et al. [80], van Loon et al. [171] or Mow et al. [114]. On the other hand, a much simpler algorithm capable of describing osmotic effects is based on the assumption of an instantaneous chemical equilibrium throughout the domain of the IVD, cf. Lanir [90]. Following this, only the solid skeleton is extended by almost volume-free, fixed negative charges and a constitutively computed osmotic pressure is added to the hydraulic pressure. This procedure is adopted in the context of a two-phase model by Ehlers et al. [45, 46, 48], Hsieh et al. [72] or Laible et al. [88]. A comparison of the two approaches yields a good approximation of the exact solution for the simplified model, even when simulating sudden concentration changes in the external solution, cf. Wilson et al. [187]. However, as sudden changes in the surrounding concentration are usually not applicable for living organisms, Lanir’s assumption is suitable for the numerical simulation of the IVD for instance.
1.3 Overview
Starting with Sect. 2, the needed basic anatomical knowledge is reviewed and summarised in the context of the intervertebral disc and its material properties. In particular, the microscopic composition of the IVD is extensively discussed, which finally leads to an electro-chemically active material with anisotropic, inhomogeneous and strongly dissipative behaviour. Since the goal of this thesis is to perform realistic computations on a motion segment as well as a section of the spine, the neighbouring components like the cartilaginous endplates as well as the vertebrae are briefly introduced. Moreover, some fundamental terms of basic chemistry are reviewed, in order to get a better understanding of the involved osmotic process.
Section 3 is concerned with the continuum-mechanical fundamentals needed to model the porous IVD. After a brief introduction into the Theory of Porous Media with a direct application to electro-chemically active materials, the IVD is, in a first step, decomposed into its five main components, i.e., the solid, the pore liquid, the fixed negative charges and the dissolved cations and anions. Thereafter, the involved kinematical relations are illustrated, where a particular focus is placed on the multiplicative decomposition of the solid deformation gradient, needed to describe finite inelastic kinematics of a viscoelastic solid skeleton. As a next step, the five classical balance equations as well as a balance for the charges are introduced via the universal master balance concept.
The objective of Sect. 4 is to characterise the universally valid balance equations with characteristic constitutive assumptions and response functions in order to obtain the desired material behaviour of the IVD. Following this, the introduced balance relations are firstly reduced for a simpler treatment, but without cancelling out the primary effects. In this regard, Lanir’s assumption is introduced and the resulting description of the swelling-active material using only a binary model with attached fixed negative charges is discussed. Thereafter, attention is drawn to the fundamentals of material theory. In this regard, the general concept of material symmetry is discussed with a particular focus on fibre-reinforced materials as well as the closely related concept of material frame indifference. Following this will then lead to the concept of isotropic tensor functions, which is needed to specify the respective constitutive equations. Herein, thermodynamically admissible response functions are derived to describe the osmotic influence, the viscoelastic and anisotropic solid skeleton as well as the percolation process of an incompressible viscous pore fluid.
In Sect. 5, the derived governing equations are numerically approximated using the framework of the mixed finite element method in space and the finite difference method in time. In this regard, the resulting discrete system of coupled nonlinear partial differential equations is presented and a step within the multilevel Newton method is described using the simplest possible but stable time integration, i.e., the implicit Euler method. The last part of this section describes how the resulting discrete problem is prepared for parallel computations using several CPU at the same time. In this regard, the basic idea of the underlying interface between the two programs PANDAS and M++ is illustrated.
The application of the developed model to realistic problems of the IVD is carried out in Sect. 6. Herein, a realistic computation of a motion segment is only possible, when the structural properties are taken into account. Hence, the problem of geometry acquisition and finite element mesh generation is presented in combination with the numerical computation of the respective collagen fibre directions as well as the distribution of the location-dependent material parameters needed to capture the inhomogeneous properties. In this context, an algorithm is derived, which is capable of computing these quantities totally independent of the underlying finite element mesh.
Moreover, the biggest problem of identifying the involved material parameters is discussed and solved using, where available, experimental data as well as material parameters obtained from a vast collection of related literature sources. Since the parameters often appear in a coupled manner, their identification is only possible via inverse computations. Following this, a numerical sensitivity analysis is carried out yielding an indication for the relevant parameters in experiments concerning a motion segment in a short-duration compression-flexion experiment as well as in long-term loading situations. Subsequently, two parallel simulations of a lumbar spine segment are presented. Herein, the healthy state is qualitatively compared to the case of a stiffened L4–L5 motion segment.
A final conclusion and discussion is given in Sect. 7 including an illustration of further possible developments based on the presented work.
2 Anatomical and Chemical Fundamentals
Before going into details of the continuum-mechanical modelling procedure, some anatomical, biological and chemical fundamentals regarding the IVD are presented in this section. As many engineers usually have little knowledge in mechanobiology, this addresses especially the basic anatomical terminology as well as the biological composition and the resulting mechanical behaviour of the IVD. Concerning the biomechanics of the human spine, a comprehensive overview can be found in, e.g., White and Panjabi [179], whereas for detailed information on the functioning and microscopic composition of the IVD in particular, the reader is referred to the standard works by Ayad and Weiss [7], Eyre [55], Hukins [73], Marchand and Ahmed [99], Szirmai [157], or Urban and Roberts [168] and references therein.
For a more comprehensive introduction into the peculiarities of the electro-chemical background, the reader is encouraged to refer to the standard text of Lide [94] among many others. Concerning biological tissues and membranes, a comprehensive overview can be found in, e.g., Sten-Knudsen [155].
2.1 Tissue Properties of the Lumbar IVD
Compared to the other components of the spine, the IVD has a relatively simple geometry which is more or less adjusted to the adjacent vertebrae. However, from a continuum-mechanical modelling point of view, the IVD is the most challenging material to be described within the spine. This is not only due to its complex and strongly coupled material behaviour, but also due to its inherent inhomogeneous structure leading to an inhomogeneous distribution of the mechanical properties.
2.1.1 Biochemical Composition
Proceeding from a macroscopic examination in the context of a dissection procedure, the geometry of the IVD evolves after its detachment from the rest of the spine. Herein, the overall shape is predetermined by the adjacent vertebrae, i.e., it has a similar axial cross section as well as convex superior and inferior sides. However, while the side surfaces of the vertebrae are concave, the surrounding side surface of the IVD is slightly convex. In general, the size of the discs follows the same tendency as the vertebral bodies, i.e., they are smallest in the cervical and largest in the lumbar region. Regarding the IVD of an adult human between the vertebrae L1 and L5, they exhibit an average height of about 15–17 mm as well as a “diameter” of about 50 mm [168].
When the IVD is cut through the midsagittal plane, another structural element can be seen which forms the superior and inferior connection to the adjacent vertebrae. These roughly 1 mm thick layers consist of hyaline cartilage and are known as the cartilaginous endplate (CEP) which completely cover the NP and parts of the inner AF [55].
Following Ayad and Weiss [7], Eyre [55] or Urban and Roberts [168] among others, the IVD appears as a porous multi-component microstructure composed of a charged hydrated extracellular matrix (ECM). In this regard, 95 % of the tissue’s wet weight consists of the ECM, i.e., structural macromolecules like proteoglycans (PG) and collagen, as well as water containing dissolved solutes. To be more precise, PG are large complex biomolecules and are composed of a protein core to which one or more glycosaminoglycan (GAG) chains are covalently attached. Following this, the GAG chains are linear polymers which have a large number of sulphate and carboxylate groups [55], i.e. they are polyanions. Thus, the tissue can be considered to contain a polymer network with fixed negative charges which smaller ions, e.g., the dissolved solutes, can diffuse to. Following this, the fixed charge density (FCD) mainly determines the distribution of the charged solutes.
Out of the at least 18 known different types of collagen [111], there are two main groups to be found within the IVD. Firstly, this is collagen of type I, which is well organised and forms parallel fibre bundles with a diameter ranging from 40 to approximately 330 μm [99]. Thus, these fibre bundles can be seen by the naked eye. In contrast, collagen fibres of type II have a much smaller diameter ranging from 10 to 300 nm [111], and form a loose and randomly oriented network of fibrils. Herein, both types of collagen are found in the AF, while the NP contains only type-II fibrils. However, there is no sharp border in the occurrence of the two types of collagen between the NP and AF, but rather a smooth transition.
In general, the water content of the IVD is highest in the NP reaching about 80 to 90 % of the wet weight and decreases in the AF to an average of 65 % of the wet weight [116, 168]. This water content is by no means stable throughout the life of an individual. For instance, the diurnal loss of water due to a regular loading of the disc is approximately 10 to 20 % of the total water content which is re-imbibed during rest at night due to the water attracting characteristic of the GAG chains in the ECM.^{1} Note in passing that the main supply of nutrients is not due to the fluid exchange resulting from diurnal interplay of loading and unloading, but is mainly due to diffusion in the pore fluid, cf. Urban et al. [169]. In contrast, there is a steady decrease of the water content with age which is an irreversible process due to the degeneration of the tissue.
Moreover, not all of the water content of the disc is found in-between the pores of the ECM. The collagen fibres and fibrils also contribute a considerable amount to the overall water content, as they contain trapped intrafibrillar water which is not able to flow freely inside the disc, but is still able to communicate with the surrounding extrafibrillar water via diffusion [168]. Herein, it is possible to verify the existence of the intrafibrillar water in vivo^{2} via an MRI analysis but it is not possible yet to quantitatively measure the ratio between the two water compartments, cf. Effelsberg [33]. In this regard, the quantitative diurnal loss of extrafibrillar water has also been proved, while the intrafibrillar water content was constant.
Similarly, the CEP is made of hyaline cartilage, where water contributes approximately 72 % of the wet weight, while collagen of type II and PG account for about 66 % and 18 % of the remaining dry weight, respectively. Herein, the collagen fibrils in the central region of the CEP run tangential to the surface of the vertebrae, while the fibrils in the region of the AF emerge perpendicular, thereby directly reaching into the lamellae of the AF. However, in a band of about 1 cm, the coarse fibres of the outermost AF are anchored directly to the vertebral bodies via the so-called Sharpey’s fibres [55].
2.1.2 Inhomogeneous Structure
The IVD exhibits several inhomogeneities regarding the distribution and alignment of its components. According to Eyre [55], Urban and Roberts [168], and Ayad and Weiss [7], the PG account for approximately 60 % of the dry weight throughout the NP, while the portion in the AF is linearly decreasing in radial direction reaching a minimal value of about 10 % of the dry weight. Hence, the FCD resulting from the attached GAG chains is also highest in the centre of the IVD and decreases towards the periphery, which corresponds to measurements performed by Urban and Holm [166] or Urban and Maroudas [167]. As water is attracted by the fixed negative charges, the water content follows a similar progression, i.e., it is highest in the NP reaching about 85 % of the wet weight and gradually decreases within the AF to the lowest percentage of approximately 55 % of the wet weight in the outermost layer [116, 168]. Note in passing that the PG content degenerates with age which causes a decrease of the water content. In particular, Ayad and Weiss [7] and Antoniou et al. [3] describe the NP of a 60-year-old subject having a water content of 70 % of the wet weight and a PG content of only 30 % of the dry weight.
A reversed distribution is observed regarding the ubiquitous collagen, where its fraction of the dry weight is highest in the outer AF, while it gradually declines towards the NP. Quantitative measurements in [3, 7, 55] reveal a collagen content of 50 to 60 % of the dry weight in the peripheral lamellae of the AF, while it decreases more or less linearly towards the NP having a collagen content of 15 to 30 % of the dry weight. Moreover, Eyre and Muir [56] measured the division of the two main types of collagen occurring in the IVD. Herein, more than 85 % of the collagen in the NP is of type II, while the remainder is of type-I collagen. Regarding the overall AF, the collagen division is 44 % type I and 56 % type II having a general tendency of a higher type-I content in the outer AF which is decreasing towards the NP in favour of a rising content of type II [3]. Note in passing that after the age of 25, the fraction of collagen remains essentially constant. Nevertheless, the relative rates of deposition of type I and type II collagen may change with time if, for instance, the diffusion of nutrients into the IVD becomes abnormal [56]. The remaining percentage of the overall dry weight is made of minor non-collageneous proteins which will not be discussed in further detail in this contribution.
Moreover, the AF is not only a complex aggregate due to the inhomogeneous arrangement of its components, but also due to its highly irregular structure, i.e., its morphology. In this regard, it was a common belief that the lamellae of the AF become narrower and less distinct as the AF merges into the NP in what is often termed the transition zone [55]. About ten years later, Marchand and Ahmed [99] proved the opposite by a layer-by-layer peeling of several anuli fibrosi at different ages resulting in thinner lamellae in the periphery compared with the inner regions. The overall trend, however, was captured in an identical manner, i.e., anterior layers are, in general, thicker than posterior ones. Furthermore, adjacent lamellae are sometimes partly interwoven leading to incomplete fibre bundles which do not run all the way from the superior to the inferior vertebrae. In this regard, incomplete lamellae primarily occur throughout the posterior AF as well as in the inner regions of the lateral and anterior AF. In contrast, most of the lamellae in the peripheral lateral and anterior AF are continuous. For more detailed information on these measurement, the interested reader is referred to Marchand and Ahmed [99] or Tsuji et al. [164]. Thus, regarding the collagen content as well as the continuity of the lamellae, there is a general radial variation throughout the AF, while in circumferential direction only the postero-lateral part is exposed to changes in continuity of the adjacent lamellae.
Another morphological inhomogeneity concerns the alignment of the large type-I collagen fibre bundles within the lamellae of the AF. In this regard, Holzapfel et al. [71] performed an extensive study on the IVD of the upper lumbar spine being taken from several age groups. Herein, also a layer-by-layer peeling technique was used, where the inclination of the fibres with respect to the transverse plane was measured at seven circumferentially positioned measuring points. Following this, the inclination angle at the midsagittal ventral position is about 20^{∘} and 50^{∘} at the midsagittal dorsal position having an almost linear development of the angle in the interjacent circumferential direction. However, there is neither an indication for a variation in radial direction nor in age.
2.1.3 Characteristic Mechanical Behaviour
Biological soft tissues like cartilage in general or the IVD in particular can withstand an enormous amount of load. This is mainly due to the highly hydrated properties of the tissue and the fact of a very low permeability of the ECM. In this context, Szirmai [157] or Urban and Roberts [168] among others suggest that the rather large PG are entangled and trapped by a type-II collagen network and imbibe water, thereby “inflating” the fine network of fibrils. Thus, the fixed negative charges of the attached GAG chains underlie the same movement as the whole ECM. Following this, the resulting microstructure is able to carry loads via hydrostatic pressure in the interstitial fluid and tension in the collagen fibrils, though none of the components could do so alone. In this context, the ECM of the tissue exhibits a very low permeability to water, which maintains the load-carrying mechanism until the water is expressed from the tissue. Note in passing that IVD tissue is approximately three decades less permeable than clayey silt,^{3} which is frequently used as a seal unit in embankments and is often regarded as almost impermeable in geomechanics.
On the other hand, this characteristic makes it rather difficult to experimentally determine the mechanical behaviour of the ECM alone, as it always needs to be in a hydrated state in order to obtain reliable results. Thus, volumetric deformations of a soft biological tissue are always coupled to the movement of the pore fluid and therefore, are strongly governed by the permeability of the ECM. Different however is the case of moderate shear deformations in the linear strain regime, where no volumetric changes occur. In fact, this is the most convenient possibility to measure the mechanical characteristics of the ECM, as it is decoupled from volumetric deformations and thus, from the permeability.
Moreover, the loose network of type-II collagen fibrils within the NP is distributed statistically equal, thereby leading to an isotropic behaviour of the NP. Regarding the aligned collagen in the AF, it is obvious that the respective lamellae exhibit a transversely isotropic behaviour. Following this, the mechanical behaviour of the structural collagen fibres of type I can be identified relatively straightforward, when a single lamella of the AF is dissected and tested in fibre direction by a tension apparatus. The prominent stress-strain characteristic of the aligned collagen evolves at first sight as it follows an exponential progression in the tension regime. In this regard, the flat toe region of the characteristic curve is due to the straightening of the coiled collagen fibres which quickly exhibit a locking behaviour with a very steep increase of stress when further strain is applied. This behaviour can be experienced very simply by pulling ones earlobes, i.e., for small elongations there is hardly any resistance, but once a certain point is reached, the required force to elongate any further rises quickly. Note in passing that in analogy to the theory of ropes, the collagen fibres alone are not able to withstand any compressive or shear forces.
Finally, the IVD tissue is characterised by a strongly coupled dissipative behaviour, which is, to a big extent, due to the interstitial fluid flow. In this regard, fluid flow can be deformation-driven due to applied external mechanical forces or electro-chemically driven, when water is attracted by the fixed negative charges. Thus, fluid flow is coupled to volumetric deformations, which again lead to a electro-chemical imbalance of fixed negative charges and the surrounding dissolved ions.
Another dissipative effect results from the so-called intrinsic viscoelasticity of the ECM, which can only be proved in shear experiments with moderate deformations, in order to decouple the intrinsic effects from the dissipation of the viscous pore fluid. In this regard, it is proven that cartilage and other isotropic tissues exhibit intrinsic viscoelastic behaviour [42, 68]. Torsional shear experiments on cylindrical specimens harvested from the NP show a slight viscoelastic behaviour, which allows for the conclusion of an intrinsically viscoelastic ECM [76]. Herein, most of the stress relaxation occurs rather rapid within approximately three seconds after the shear load is applied. Thus, the time frame of the intrinsic viscoelasticity is much smaller compared to the time needed for fluid flow, which takes place over a period of hours. However, it is still not clear, to which extent the intrinsic viscoelastic effects stem from proteoglycans or collagen. Concerning the AF, the measurements of Holzapfel et al. [71] showed only little rate dependency of the type-I collagen fibres in the AF. Following this, it seems that only the loose network of type-II collagen as well as the PG exhibit a viscoelastic behaviour, which contributes to the intrinsic viscoelasticity of the isotropic ECM of cartilage or NP tissue.
2.1.4 Load Transmission Mechanism
The entire IVD has to transmit the majority of loads down the spinal column and due to its inherent flexibility, the load transmission frequently takes place under awkward deformations. Herein, the applied loads do not only result from the body weight and the lifted weights, but also from adjacent muscle activity, which is needed to hold the body in a desired position, cf. Urban and Roberts [168] and references therein. In general, the load can be classified by its duration, i.e., short duration and high-amplitude loads (e.g. jerk lifting) and long duration loads due to normal physiological actions (e.g. standing erect, carrying minor loads). In this regard, Nachemson started to extensively quantify these loads on the IVD in the 1960s and published in his later work [115] that a person having a mass of 70 kg experiences a load of approximately 500 N on the third lumbar IVD, while standing at ease. This magnitude is almost quadrupled, when the same person lifts a weight of only 10 kg, thereby having the knees straight and the back bent over. As a rule of thumb, the least loads are applied when a person is lying supine, i.e., in a horizontal position on its back, the loads increase while standing and sitting, and are highest, when lifting a weight while the upper body is bent and twisted.
Following this, the IVD is often referred to as a hydrodynamic ball bearing [157], where its enormous load-bearing capacity is mainly due to a very ingenious interplay between its highly hydrated NP and its strongly fibre-reinforced AF. In fact, when the disc is mechanically loaded, it deforms and the NP is pressurised because of its considerable resistance against volumetric compression. Thus, the NP bulges into the CEP as well as radially into the AF, thereby setting its fibres into tension. Note in passing that an intradiscal pressure of 0.5 MPa can be measured in vivo in the fourth lumbar IVD when a person of 70 kg is standing at ease, while a pressure of 2.3 MPa arises when the same person lifts a weight of 20 kg having the knees straight and the back bent [186]. Because of the unique arrangement of the durable type-I collagen fibres, the AF is able to contain the high bulging-pressure from the NP in analogy to a thick-walled pressure vessel, while still maintaining the ability to be compressed, rotated or bent.
Moreover, due to the water attracting characteristic of the GAG chains, the NP appears to be under constant pressure [157] known as osmotic pressure. Thus, the fixed negative charges additionally carry parts of the applied load [73]. In fact, the more water is driven out of the tissue through a mechanical volumetric compression, the greater the water-attracting capacity of the fixed negative charges inside the tissue gets. For more information on osmotic effects, please refer to Sect. 2.2.2 or to any text on basic biochemistry. Note in passing that this effect can be observed in the diurnal variation of the body height which reaches up to 2 cm, because some of the interstitial fluid is expressed from the tissue during the high loadings of normal daily activities and is re-imbibed at rest during sleep at night [21, 98]. In this regard, the in vivo observations of Wilke et al. [186] support this statement. In an over-night measurement, the pressure inside the NP increased from about 0.1 MPa in the evening to about 0.24 MPa in the morning yielding a total increase of about 0.14 MPa over the whole night.
2.2 Basic Chemistry
The classical field of continuum mechanics usually includes the description of the response of arbitrary bodies being exposed to mechanical or thermal loading situations. Following this, many engineers working in this field have little knowledge about possible electro-chemical effects which should be included generally in the case of soft biological tissues. In this regard, the following section is by no means complete, rather it should give a brief overview of the necessary relations in order to have them handy when they are needed in the following sections.
2.2.1 Concentration Measures
Whenever electro-chemically active materials (such as soft biological tissues) are discussed and described, the fixed negative charges mentioned in Sect. 2.1.1 as well as the ions of at least one dissolved salt in the pore fluid have to be included in the simulation process. Following this, the amount of the chemically active substances needs to quantified. In this regard, a very important quantity is known as Avogadro’s number which is frequently used to convert measures from the atomic scale of substances like, for instance, atoms, ions, electrons or molecules to the physical macro scale. To be more precise, in 1971 Avogadro’s number entered the International System of Units (SI) and its numerical value is formally defined to be the number of carbon-12 atoms in 12 grams of unbound carbon-12 in its rest-energy electronic state. According to the Committee on Data for Science and Technology (CODATA), the current best approximated and accepted value for the Avogadro constant is N_{A}=6.02214179×10^{23} mol^{−1} [107]. Thus, one mol of carbon-12 atoms weighs 12 grams and N_{A} accounts for the number of entities in a mol.
However, regarding classical field theories like, for instance, continuum mechanics, one essential drawback is the fact that the extensive measurement of entities is always leading to an absolute quantity. In this regard, an extensive quantity explicitly depends on the size of the described system, i.e., if a homogeneous system that contains a certain number of entities is divided in two equal parts, the overall number of entities is also divided by two. Thus, it is not possible to assign a unique number of entities to every point of the system. Following this, extensive quantities need to be intensified by relating them to another extensive quantity like mass or volume. One famous representative for intensive quantities is the mass density, where the mass is divided by the volume. Thus, in a homogeneous system, the mass density is constant at every point, no matter how often the overall system is divided into subsystems.
Several other intensive concentration measures can be given like, for instance, the molality, the osmolarity, or the mole and the mass fraction. For example, the molality is computed, when the number of charges is divided by the mass of the solvent, i.e., the liquid L. However, these quantities will not be needed in this contribution and are therefore not discussed any further.
2.2.2 Osmosis
Naturally, the two communicating compartments have the desire to equalise their concentrations of solutes, i.e., if the compartments were infinitely large and no counter pressure was present, an infinite amount of distilled water would move through the semi-permeable membrane in order to dilute the solution in the left chamber. Regarding the situation in Fig. 3, a mechanical counter pressure in form of a hydrostatic pressure evolves in the left compartment, whenever the solvent is moving in from the right. This mechanical counter pressure forces the solvent to be driven out again, thereby moving back into the right chamber. In this context, a water movement takes place, until the mechanical counter pressure resulting from the higher fluid level in the left compartment equals out the chemically driven desire of the distilled water to enter the left compartment in order to dilute the solution. Following this, osmosis is defined as the net flow of solvent through a semi-permeable membrane that is driven by the difference between the molar solute concentrations of the communicating compartments. Moreover, the osmotic pressure difference is then defined as the measurable hydrostatic pressure difference between the compartments after the net fluid movement comes to a rest at the equilibrium state.
Finally, note that the natural state of soft biological tissues is in equilibrium, when the tissue is surrounded by a physiological sodium chloride solution. Whenever the surrounding concentration is changed, the tissue dimensions (volume) change also, but in the opposite direction, i.e., if the external ion concentration is increased, the tissue dimensions decrease. Thus, the natural state is always a swollen state, as the ingrown fixed charges of the GAG chains have already attracted all the counter ions and an excess of solvent, needed to fulfil the electro-neutrality condition and the chemical equilibrium, respectively. Applying this concept to a binary aggregate of materially incompressible solid and fluid constituents means that the ECM is permanently pre-stressed with a hydrostatic stress state resulting from the Donnan osmotic pressure.
3 Continuum-Mechanical Fundamentals
The following chapter gives a brief overview of the continuum-mechanical fundamentals needed to understand the modelling process using the framework of the Theory of Porous Media (TPM). In particular, this addresses the concept of volume fractions, the finite kinematical relations, the five balance equations, and the fundamental rules of material theory needed for the constitutive modelling process described in Sect. 4.
3.1 The Theory of Porous Media
Regarding the historical evolution of the TPM, the first continuum-mechanical considerations describing the consolidation problem of biphasic geomaterials trace back to the phenomenological approach by Biot [13]. Shortly after, the Theory of Mixtures (TM) was developed using the framework of general thermodynamical considerations, where the most important works trace back to Truesdell and Toupin [163] and Bowen [18]. The TM was then extended by the concept of volume fractions in the publications of Mills [106] and Bowen [19], which in turn was continuously improved and further developed to the current understanding of the TPM by de Boer [26] and Ehlers [34, 36, 38].
Following this would lead to a quadriphasic model describing the solid skeleton φ^{S} (including the fixed negative charges φ^{fc}) and each of the components φ^{β} independently, see, e.g., Frijns et al. [57], Lai et al. [87] and Acartürk [1]. Exploiting the electro-neutrality condition (5)_{2} allows for a reduction to a triphasic model. This causes the number of independent primary variables, which are needed to describe the remaining constituents in the numerical modelling process, to decrease from four to three, i.e., the solid displacement, the pressure and the cation concentration [44]. As already mentioned in Sect. 2.2.2, it is possible to further reduce this model by applying the assumption of Lanir [90] stating that the tiny mobile ions are assumed to diffuse rapidly through the liquid and by themselves, do not give rise to concentration gradients. In the context of living soft biological tissues, such as the IVD, this simplification makes sense, because sudden concentration changes of the surrounding fluid yielding large perturbations of the chemical equilibrium do not occur. Hence, the soft biological tissue is always immediately in electro-chemical equilibrium, which allows the application of the Donnan [29] equilibrium not only at the domain boundary, but also in the inside. The arising osmotic properties can therefore be sufficiently described without considering the independent movement of the ions. Following this, the respective ion concentrations may be computed inside the tissue using (6), and no independent variables need to be introduced for the solutes. Thus, the following chapters will derive an extended biphasic formulation consisting of a solid skeleton carrying volume-free fixed negative charges which interact with a saturating pore fluid. Note in passing that it is common to use the terminology “phase” instead of constituent or component, even though the usual terminology of phase transitions, i.e., solid to liquid to gas, is not meant here.
Remark
First benchmark computations on simple geometries using a triphasic displacement-pressure-concentration formulation were accompanied by oscillations and a general numerical instability, cf. Ehlers and Acartürk [39]. The cause of the oscillations can be traced back to (6)_{1}, as for this model, it is only valid at the domain boundary. Thus, the concentration boundary condition (BC) implicitly depends on the solution of the solid displacement field inside the domain and a numerical stabilisation is only possible if the BC is weakly fulfilled and included in the iteration process, cf. Ehlers et al. [44]. Bearing in mind that future simulations of the intensely inhomogeneous, anisotropic IVD (including the much stiffer, adjacent vertebrae) already evolve several other numerical difficulties, the simpler approach using the extended biphasic formulation is used here. Note in passing that other theories, which exhibit four primary degrees of freedom [87], i.e., solid displacement and three modified chemical potentials for water, cations and anions, do not suffer from these oscillations, but are from a computational point of view still a lot more ‘expensive’.
3.2 Kinematical Relations
The following sections offer a brief overview of the kinematical relations which are needed to describe the nonlinear deformation process of a porous material. Herein, the overall aggregate body \({\mbox {$\mathcal {B}$}}\) is defined as the connected manifold of material points P^{α} which may follow independent motions that can be described with respect to a fixed origin \({\mbox {$\mathcal {O}$}}\). Note that all quantities related to the motion of a constituent are indicated in the subscript ( ⋅ )_{α}, while all other quantities indicate their affiliation in the superscript ( ⋅ )^{α}. Moreover, note that due to the reasons given in the preceding section, the introduction of independent motions for the solvents of the pore fluid is omitted here. For a comprehensive introduction of these kinematical quantities, the reader is referred to [1, 39, 49].
3.2.1 Motion of a Porous Material
3.2.2 Inelastic Solid Kinematics
Remark
3.2.3 Solid Deformation and Strain Measures
The intention of this section is to give a brief introduction into the particulars of finite deformation and strain measures, which result from the consequences of a multiplicative split of the solid deformation gradient. In this regard, the respective quantities will be exemplary derived between the reference and actual configuration and logically extended for the inelastic and elastic parts of one Maxwell element, thereafter. For a more comprehensive introduction please consult the works of Lee [92], Ehlers [34–36] or Markert [100] as well as the respective references therein.
Furthermore, it can be noted that the right Cauchy-Green deformation tensors describe the overall as well as the inelastic part of the deformation from the referential frame, while the intermediate configuration functions act as a “reference configuration” for the description of the elastic process. The opposite holds for the left Cauchy-Green deformation tensors, which describe the overall and elastic process looking back from the actual configuration and the inelastic part of the deformation from the intermediate frame.
Finally note that there are several other possibilities to define strain measures which are different to the ones mentioned above, but will have no relevance in this contribution. For more information, please refer to Truesdell and Noll [162].
3.2.4 Spectral Representation of the Deformation Tensors
For the postulation of finite material laws during the constitutive modelling process, it is often convenient to make use of the spectral representation of the deformation tensors. However, the purpose of the present contribution is to give a brief introduction into the particulars of the spectral representation, whereas more detailed information can be found in Lambrecht [89], Markert [100] and references therein.
3.2.5 Solid Deformation and Strain Rates
3.3 Stress Measures and Dual Variables
3.4 Balance Relations for Porous Media
In this context, the well-known master balance principle from single-phase materials (see, e.g., Haupt [67]) can be applied for the constituents as well as the aggregate. The following two sections contain a brief representation taken from Ehlers [38], where the balances of the overall aggregate are discussed firstly, followed by the balances of the constituents. A more comprehensive description can be found in Diebels [27].
3.4.1 Aggregate Balance Relations
3.4.2 Component Balance Relations
3.4.3 Entropy Principle
4 Constitutive Modelling
Together with the presented continuum-mechanical fundamentals of the preceding section, the development of several distinct sets of governing equations is possible, thereby representing a manifold of multiphasic models. However, for the purpose of this contribution, the focus lies in the description of the charged hydrated anisotropic inhomogeneous IVD tissue as it is described in Sect. 2.1. In this regard, the materially independent balance equations and the missing constitutive equations for the involved constituents need to be adjusted to describe the physical response of the soft biological tissue.
4.1 Extended Biphasic TPM Model
The purpose of this section is to derive the set of balance equations for an extended incompressible biphasic model with the aid of simplifying assumptions as well as the general balance relations given in the preceding Sect. 3.4. Moreover, the entropy principle will be evaluated leading to concrete restrictions for the postulation of constitutive equations discussed in the sections thereafter.
4.1.1 Preliminary Assumptions and Balance Relations
As a first step, the general forms of the local constituent balance relations (101) are adapted to the special case of soft biological tissues, where an intrinsically anisotropic dissipative charged ECM is coupled with a viscous pore fluid. The involved ions lead to a coupling between the mechanical and the electro-chemical properties, thereby characterising the tissue as a swelling-active material. However, with the focus on the application to the inhomogeneous IVD, the model under consideration needs to be complex enough to represent the relevant material properties, while simultaneously being simple enough, to allow for a computability of real life spine experiments on today’s hardware.
The starting point is the a priori assumption made in Sect. 3.1, where the number of unknown motion functions for the involved charged solid as well as the fluid components (L,+,−) has been reduced to a minimum. Herein, use was made of Lanir’s assumption [90] stating that without extreme concentration jumps of the surrounding fluid, the tissue is always in electro-chemical equilibrium. Thus, the free movable ions do not have to be considered separately, but are assumed to be always immediately in a position, where they are needed to fulfil the electro-neutrality condition. The remaining kinematical quantities outline the extended binary model having volume-free fixed negative charges attached to the saturated solid skeleton. In this regard, the fixed charges are kinematically bound to the solid movement, but with regard to the surrounding fluid components, they contribute to an internal ion concentration and thus, almost behave like dissolved ions.
materially incompressible constituents (ρ^{αR}=const.)
no mass exchange between the constituents (\(\hat{\rho}^{\alpha}\equiv0\))
fully saturated conditions for the aggregate (n^{S}+n^{F}=1)
quasi-static conditions (\({\stackrel {\prime \prime }{\mathbf{x}}_{\alpha}}\equiv \mbox {$\mathbf {0}$}\), \(\ddot{\mathbf {x}}\equiv \mbox {$\mathbf {0}$}\))
uniform gravitation for all constituents (b^{α}=g)
non-polar constituents (\(\hat{\mathbf {m}}{}^{\alpha}\equiv \mbox {$\mathbf {0}$}\))
isothermal conditions (Θ^{α}≡Θ=const.)
Moreover, in order to include the electro-chemical effects while the tissue is deforming, it is necessary to derive an equation describing the extensive change of the number of fixed negative charges via an intensive concentration measure. As the fixed charges almost behave like dissolved ions when they are surrounded by a pore fluid, the molar concentration \(c^{fc}_{m}\) of the fixed charges is introduced relating the molar number of charges to the surrounding fluid volume, cf. Sect. 2.2.1. In this regard, there is no distinction between the intra- and extrafibrillar fluid compartments as is described in Sect. 2.1.1. Thus, the fixed charges φ^{fc} are related to the total local fluid volume knowing well that this may lead to an overestimate in the AF, where most of the intrafibrillar water is located in the structural collagen, cf., e.g., Schröder et al. [138].
4.2 Basic Thermodynamical Principles
Having adapted the materially independent balance equations for the special case of charged hydrated IVD tissue under quasi static conditions, further constitutive assumptions have to be derived; not only to close the resulting set of governing equations but also to further characterise the material behaviour. This is of particular need, as a sample taken from the isotropic NP will react different to mechanical loading compared to a fibre-reinforced sample of the AF. Thus, the constitutive equations to be derived in the proceeding sections must characterise the physical behaviour of the involved constituents as well as their interaction.
In this regard, care must be taken that none of the constitutive relations violates any of the basic thermodynamical principles of constitutive modelling, which basically trace back to the works of Truesdell [160], Noll [117, 118] and Coleman and Noll [24]. In particular, these are the principles of determinism, equipresence, and local action as well as material frame indifference, universal dissipation, and material symmetry. These can be found in a comprehensive summary with application to the constitutive modelling process in the work of Wang and Truesdell [177]. Note in passing that the satisfaction of these principles is an important premise in order to claim for thermodynamical consistency of the proposed model. For the purpose of this contribution, these principles will be briefly revised and applied in the proceeding sections.
4.2.1 Determinism, Equipresence and Local Action
4.2.2 Material Frame Indifference
4.2.3 Universal Dissipation
Following the constitutive modelling process, the principle of dissipation has to be satisfied stating that every admissible thermodynamical process has to fulfil the entropy inequality of the overall aggregate. In this regard, the Clausius-Planck inequality in the form of (127) is evaluated following the approved procedure of Coleman and Noll [24]. Herein, convenient a priori constitutive assumptions, which led to a reduction of the set of process variables, allow for a quick evaluation, where each part of the inequality has to be greater or equal to zero separately. Note in passing that the more general, but also lengthy evaluation procedure by Liu [96] and Liu and Müller [97] involves a Lagrange multiplier for each balance relation. However, the final result for the simple case of the present incompressible binary aggregate is identical.
4.2.4 Material Symmetry
The purpose of this section is to deliver a brief understanding of the vast matter on material symmetries. For more comprehensive information on this topic, the reader is referred to the representative works of Truesdell and Noll [162], Boehler [17], Spencer [151, 152], Zheng and Boehler [191], Schröder [135] or Apel [4] and references therein.
The principle of material symmetry is based on observations in the laboratory and can be illustrated by a simple experiment. Imagine a wooden cube which is compressed in fibre direction. In a second test, the cube is brought into a new reference position before being tested again. In this context, the testing device will always record the same response, as long as the cube is only rotated about its fibre direction. A different response will be recorded if the wooden specimen is rotated about any other axis. Thus, in the case of fibre-reinforced or generally anisotropic materials, the amount of stored elastic energy does not only depend on the applied deformation but also on the orientation of the material.
However, the allowed symmetry transformations H between the neighbouring reference configurations are by no means arbitrary, as there is no known physical object, in which the stress remains unaltered, when the body is compressed or expanded. Thus, the transformations of the material symmetry group \(\mathcal {M}\mathcal {G}_{3}\subset {\mbox {$\mathcal {U}$}}_{3}\) must be at least a subset of the unimodular group \({\mbox {$\mathcal {U}$}}_{3}\), cf. Gurtin and Williams [63]. Note in passing that \(\mathcal {M}\mathcal {G}_{3}={\mbox {$\mathcal {U}$}}_{3}\) is the characteristic of an isotropic fluid, cf. Noll [118]. Here, for the purpose of the underlying solid skeleton, further restrictions must hold. According to Wang and Truesdell [177], a solid is a material, where only rigid-body rotations are indiscernible and thus, the symmetry group \(\mathcal {M}\mathcal {G}_{3}\subset {\mbox {$\mathcal {O}$}}_{3}\) must be a subset of the orthogonal group \({\mbox {$\mathcal {O}$}}_{3}\) containing all shape-preserving rotations. Consequently, if \(\mathcal {M}\mathcal {G}_{3}={\mbox {$\mathcal {O}$}}_{3}\) the solid material has an isotropic behaviour, while anisotropic effects occur for all other cases. The different possible types of anisotropy are numerous and can be divided into two continuous groups as well as eleven finite sub groups describing a total of thirty-two crystal classes. For more general information on these groups, please refer to Spencer [150], Truesdell and Noll [162] or to any text on crystal physics.
Remark
From a physical point of view, some symmetry transformations \(\mathbf {H}\in {\mbox {$\mathcal {O}$}}_{3}\) make not much sense, as they are impossible to physically apply or accomplish. A material body, for instance, may not be compressed to a single point and recovered on the other side, like in the case of the central inversion with H=−I. Thus, the interpretation of the material symmetry group \(\mathcal {M}\mathcal {G}_{3}\subset {\mbox {$\mathcal {S}$}}{\mbox {$\mathcal {O}$}}_{3}\) being restricted to a sub-group of the rotational group \({\mbox {$\mathcal {S}$}}{\mbox {$\mathcal {O}$}}_{3}\) is also possible. However, from a purely geometrical point of view, \(\mathcal {M}\mathcal {G}_{3}\) may be a subset of the full orthogonal group \({\mbox {$\mathcal {O}$}}_{3}\).
4.3 Consequences from the Basic Principles
4.3.1 Selected Symmetry Groups and Their Structural Tensors
The following section offers a brief introduction into the symmetry groups of interest concerning the modelling of fibre-reinforced materials. Note that these materials play a rather small role in the many possible symmetry groups found within the thirty-two crystal classes for instance. For more information on the characterisation of all symmetry groups, please refer to the comprehensive overview in, e.g., Apel [4], while in the case of fibre-reinforced materials please consult Spencer [151, 152] among others. An overview of the corresponding structural tensors can then be taken from Zheng and Spencer [192].
4.3.2 Representation with Isotropic Tensor Functions
In the context of finite continuum mechanics, representation theorems play a major role in the problem of defining mathematically and physically sound constitutive equations. For instance, the scalar-valued solid strain energy function \(W^{S}_{\mathrm{EQ}}\) subjected to condition (159)_{1} is known as an anisotropic tensor function with respect to the argument C_{S} under transformations of the symmetry group \(\mathcal {M}\mathcal {G}_{3}\). Following this approach leads to dependencies on the basis system in which \(\pmb {\mathcal {M}}^{S}\) is defined. However, in the context of finite continuum mechanics, and especially with respect to the general numerical treatment, such a restriction is undesirable. Therefore, an alternative formulation is sought after, where the representation of anisotropic materials can be accomplished under all orthogonal transformations \({\mbox {$\mathcal {O}$}}_{3}\), thereby automatically including the material symmetry group. In this regard, it is necessary to express the physical properties of an actually anisotropic material element using formally isotropic relations. Exactly this is accomplished by applying the concept of isotropic tensor functions.
In general, two different approaches for the representation with isotropic tensor functions can be distinguished. One is concerned with non-polynomial tensor functions while the other one is restricted to polynomial tensor functions due to mathematical convenience. The former one traces back to the works of Smith [148], Wang [175, 176], and Boehler [15], while important work for the latter case was done by Rivlin and Ericksen [127], Spencer [150, 153], Spencer and Rivlin [154] or Truesdell and Noll [162]. A detailed overview and a slightly different method can be found in Zheng [190]. However, a difference between these approaches is only noticeable for vector- and tensor-valued tensor functions. In this regard, the main problem is to find general, complete and irreducible representations for tensor functions, thereby keeping the resulting parameters at a minimum. Herein, the non-polynomial approach seems to be slightly in favour regarding the involved terms.
For the purpose of this contribution, only scalar-valued tensor functions are needed to formulate the solid strain energy \(W^{S}_{\mathrm{EQ}}\). The resulting tensor-valued stress-strain relation is then obtained from the evaluation of the entropy inequality using relation (151). In this regard, the solution to the representation problem of scalar-valued isotropic tensor functions traces back to the classical theory of invariants, which is described in a rather mathematical way in, e.g., Grace and Young [61], Weyl [178]. An application to continuum-mechanical problems can be found in Boehler [17] or Spencer [150, 153].
4.3.3 The Idea of the Theory of Invariants
The problem is then to find invariants which depend on the argument tensors and form a minimal integrity basis \(\mathcal {I}_{S}\). The existence of such an integrity basis is assured by Hilbert’s theorem stating that for any finite set of tensors of any order there exists a finite integrity base. A set \(\mathcal {I}_{S}\) is called minimal, if it contains only irreducible invariants and an invariant is irreducible, if it cannot be expressed as a polynomial of the remaining invariants.
Minimal integrity basis for general tensors exposed to proper orthogonal rotations
Arguments | Basic and mixed invariants J_{Sn} |
---|---|
A_{1} | \(\operatorname {tr}{\mathbf {A}_{1}},\operatorname {tr}{\mathbf {A}_{1}^{2}},\operatorname {tr}{\mathbf {A}_{1}^{3}}\) |
A_{1} | \(\operatorname {tr}{\mathbf {A}_{1}},\operatorname {tr}{\mathbf {A}_{1}^{2}},\operatorname {tr}{\mathbf {A}_{1}^{3}}\), \(\operatorname {tr}{\mathbf {A}_{2}},\operatorname {tr}{\mathbf {A}_{2}^{2}},\operatorname {tr}{\mathbf {A}_{2}^{3}}\), |
A_{2} | \(\operatorname {tr}{(\mathbf {A}_{1}\mathbf {A}_{2})},\operatorname {tr}{(\mathbf {A}_{1}\mathbf {A}_{2}^{2})},\operatorname {tr}{(\mathbf {A}_{1}^{2}\mathbf {A}_{2})}, \operatorname {tr}{(\mathbf {A}_{1}^{2}\mathbf {A}_{2}^{2})}\) |
A_{1} | \(\operatorname {tr}{\mathbf {A}_{1}},\operatorname {tr}{\mathbf {A}_{1}^{2}},\operatorname {tr}{\mathbf {A}_{1}^{3}}\), \(\operatorname {tr}{\mathbf {A}_{2}},\operatorname {tr}{\mathbf {A}_{2}^{2}},\operatorname {tr}{\mathbf {A}_{2}^{3}}\), \(\operatorname {tr}{\mathbf {A}_{3}},\operatorname {tr}{\mathbf {A}_{3}^{2}},\operatorname {tr}{\mathbf {A}_{3}^{3}}\), |
A_{2} | \(\operatorname {tr}{(\mathbf {A}_{1}\mathbf {A}_{2})},\operatorname {tr}{(\mathbf {A}_{1}\mathbf {A}_{2}^{2})},\operatorname {tr}{(\mathbf {A}_{1}^{2}\mathbf {A}_{2})}, \operatorname {tr}{(\mathbf {A}_{1}^{2}\mathbf {A}_{2}^{2})}\), |
A_{3} | \(\operatorname {tr}{(\mathbf {A}_{1}\mathbf {A}_{3})},\operatorname {tr}{(\mathbf {A}_{1}\mathbf {A}_{3}^{2})},\operatorname {tr}{(\mathbf {A}_{1}^{2}\mathbf {A}_{3})}, \operatorname {tr}{(\mathbf {A}_{1}^{2}\mathbf {A}_{3}^{2})}\), |
\(\operatorname {tr}{(\mathbf {A}_{2}\mathbf {A}_{3})},\operatorname {tr}{(\mathbf {A}_{2}\mathbf {A}_{3}^{2})},\operatorname {tr}{(\mathbf {A}_{2}^{2}\mathbf {A}_{3})}, \operatorname {tr}{(\mathbf {A}_{2}^{2}\mathbf {A}_{3}^{2})}\), | |
\(\operatorname {tr}{(\mathbf {A}_{1}\mathbf {A}_{2}\mathbf {A}_{3})},\operatorname {tr}{(\mathbf {A}_{1}^{2}\mathbf {A}_{2}\mathbf {A}_{3})}, \operatorname {tr}{(\mathbf {A}_{1}\mathbf {A}_{2}^{2}\mathbf {A}_{3})},\operatorname {tr}{(\mathbf {A}_{1}\mathbf {A}_{2}\mathbf {A}_{3}^{2})}\), | |
\(\operatorname {tr}{(\mathbf {A}_{1}^{2}\mathbf {A}_{2}^{2}\mathbf {A}_{3})},\operatorname {tr}{(\mathbf {A}_{1}^{2}\mathbf {A}_{2}\mathbf {A}_{3}^{2})}, \operatorname {tr}{(\mathbf {A}_{1}\mathbf {A}_{2}^{2}\mathbf {A}_{3}^{2})}\) |
Note that the expression \(\mathbf {A}_{i}^{2}\) denotes the tensor product A_{i}A_{i}. Moreover, the traces of a single tensor represent basic invariants, while traces of more than one tensor are known as mixed invariants. Another characteristic worth mentioning is that the mixed invariants are also invariant with respect to cyclic permutations of their tensor arguments.
4.3.4 Integrity Bases for Fibre-Reinforced Materials
Minimal integrity basis for fibre-reinforced materials
Arguments | Basic and mixed invariants J_{Sn} | Usage |
---|---|---|
C_{S} | \(\operatorname {tr}{\mathbf {C}_{S}},\operatorname {tr}{\mathbf {C}_{S}^{2}},\operatorname {tr}{\mathbf {C}_{S}^{3}}\) | isotropy |
C_{S} | \(\operatorname {tr}{\mathbf {C}_{S}},\operatorname {tr}{\mathbf {C}_{S}^{2}},\operatorname {tr}{\mathbf {C}_{S}^{3}}\), | transverse isotropy |
\(\pmb {\mathcal {M}}^{S}_{a}\) | \(\operatorname {tr}{(\pmb {\mathcal {M}}^{S}_{a}\mathbf {C}_{S})},\operatorname {tr}{(\pmb {\mathcal {M}}^{S}_{a}\mathbf {C}_{S}^{2})}\), | |
C_{S} | \(\operatorname {tr}{\mathbf {C}_{S}},\operatorname {tr}{\mathbf {C}_{S}^{2}},\operatorname {tr}{\mathbf {C}_{S}^{3}}\), | orthotropy/ prismatic symmetry |
\(\pmb {\mathcal {M}}^{S}_{a}\) | \(\operatorname {tr}{(\pmb {\mathcal {M}}^{S}_{a}\mathbf {C}_{S})},\operatorname {tr}{(\pmb {\mathcal {M}}^{S}_{a}\mathbf {C}_{S}^{2})}\), | |
\(\pmb {\mathcal {M}}^{S}_{b}\) | \(\operatorname {tr}{(\pmb {\mathcal {M}}^{S}_{b}\mathbf {C}_{S})},\operatorname {tr}{(\pmb {\mathcal {M}}^{S}_{b}\mathbf {C}_{S}^{2})}\), | |
\(\operatorname {tr}{(\pmb {\mathcal {M}}^{S}_{a}\pmb {\mathcal {M}}^{S}_{b}\mathbf {C}_{S})}, \operatorname {tr}{(\pmb {\mathcal {M}}^{S}_{a}\pmb {\mathcal {M}}^{S}_{b}\mathbf {C}_{S}^{2})}\), | ||
\(\operatorname {tr}{(\pmb {\mathcal {M}}^{S}_{a}\pmb {\mathcal {M}}^{S}_{b})}\) |
Remark
4.3.5 Further Requirements for Strain Energy Functions
Having defined a list of suitable scalar arguments for the solid strain energy function W^{S}, some further restrictions of physical and mathematical nature must be applied, which are briefly discussed in the following.
Mathematical limitations stem from considerations regarding the numerical solution of variational problems, as it will be discussed in Sect. 5.3. In this regard, the essential requirement is the existence of minimisers in order to obtain solutions for general initial boundary value problems. Following the argumentation of Ball [8] and references therein, the existence of minimisers is guaranteed, if W^{S} is a convex function in F_{S} and F_{Se}, respectively. From a mathematical point of view this requirement comes handy, as the solutions obtained are also unique. However, from a physical point of view, convexity conflicts with the requirement of material frame indifference. Moreover, the description of material instabilities is a priori ruled out, as a strictly convex strain energy is unique, whereas especially its non-uniqueness is essential for the description of buckling.
For a more detailed mathematical description of this issue, please refer to the classical works of Ciarlet [23], Marsden and Hughes [103], Šilhavý [144], as well as to the comprehensive summaries found in Schröder [136] or Balzani [9].
4.4 The Solid Skeleton
The purpose of this section is to apply all the restrictions of the preceding subsections in order to finally postulate a suitable strain energy function for the solid skeleton of a swelling-active and fibre-reinforced biological soft tissue. In particular, special attention is drawn on the modular character of W^{S}, where it is advantageous to a priori define a general structure to be used. The actual postulation of the respective strain energies is then carried out thereafter.
4.4.1 Structure of the Strain Energy and Its Derivatives
As already discussed in Sect. 4.2.3, the strain energy W^{S} consists of at least two main parts. An equilibrium part \(W^{S}_{\mathrm{EQ}}\), which is associated with the overall deformation of the underlying rheological model, as well as a non-equilibrium part \(W^{S}_{\mathrm{NEQ}}\) characterising the elastic springs in the Maxwell elements. Recalling the characteristic properties of the IVD as well as its mechanical behaviour of Sect. 2.1, the equilibrium part must be capable of capturing the elastic collagen fibres of type I, while the non-equilibrium part is responsible for the viscous over stresses in the isotropic portion of the tissue.
Moreover, osmotic effects must be included regarding the water attracting characteristic of the PG’s in the ECM. At first sight, however, it is not comprehensible why the osmotic pressure, which is usually measured in the pore fluid, is computed from the solid strain energy. This fact becomes evident in the context of the underlying extended binary model, where osmosis triggers a diffusion process, thereby causing an influx of the surrounding fluid. In turn, the additional fluid inside the tissue leads to a variation of the volume which is occupied by the incompressible pore fluid, cf. Sect. 2.2.2. Thus, a purely volumetric deformation (dilatation) is initiated in the case of isotropic material behaviour, which may be accompanied by shape changes in cases where type-I collagen fibres are present. Following this, the osmotic pressure is actually a reaction force of hydrostatic character in the solid skeleton which is caused by a permanent “internal displacement boundary condition”. As a consequence from the evaluation of the Clausius-Planck inequality, the osmotic pressure contribution must be derivable from a solid strain energy function, thereby contributing to a thermodynamically consistent model.
4.4.2 Osmotic Contribution
4.4.3 Isotropic Contribution
4.4.4 Anisotropic Contribution
According to the argumentation of Sect. 4.4.1, the anisotropic part of the equilibrium strain energy may depend on the mixed invariants J_{S4} to J_{S8} of the corresponding integrity base used in (183), which is used for prismatic symmetry or local orthotropy. However, the choice of how many mixed invariants are incorporated is of constitutive nature and thus, \(W^{S}_{\mathrm{ANISO}}\) may only depend on a subset of selected mixed invariants.
Finally note that there are numerous other possibilities to model the mechanical behaviour of collagen fibres using polyconvex strain energy functions. For more detailed information on this topic, the reader is referred to Balzani [9], Balzani et al. [10], Holzapfel et al. [70], Itskov and Aksel [79] or Schröder and Neff [137].
4.4.5 Non-equilibrium Contribution
4.5 The Viscous Interstitial Fluid
Since the set of process variables is empty for the incompressible fluid, there is no energy to be determined. However, regarding the dissipation inequality (150), there are still two proportionalities to be evaluated, i.e., \(\mathbf {T}^{F}_{E}\propto \mathbf {D}_{F}\) and \(\hat{\mathbf {p}}{}^{F}_{E}\propto-\mathbf {w}_{F}\), respectively.
4.5.1 The Darcy Filter Law
the higher the filter velocity n^{F}w_{F},
the greater the absolute momentum exchange,
the higher the partial viscosity n^{F}μ^{FR},
the greater the absolute momentum exchange,
the smaller the permeability (poresize),
the greater the absolute momentum exchange.
5 Numerical Treatment
The purpose of this chapter is to briefly review the numerical tools which are necessary to perform realistic computations of the lumbar spine in Sect. 6. Herein, the first objective is to rearrange the presented set of governing equations such that they are approachable with a suitable numerical solution scheme. In particular, the presented model consists of a set of coupled partial differential equations (PDE) of first order in time and second order in space, respectively. Thus, a suitable temporal and spatial discretisation is needed. For the purpose of this contribution, the spatial discretisation will be carried out using finite elements, while the discretisation in time is accomplished with the finite difference method. Thereafter, a reliable solution scheme is needed to solve for the unknown quantities of the discretised system of equations. However, as the overall complexity of the problem quickly reaches the capacity of a single central processing unit (CPU), a parallelisation technique is additionally discussed in a brief review.
5.1 Finite Element Method in Space
Concerning the numerical treatment of continuum-mechanical problems involving solids as well as porous media, the variational approach offered by the Finite Element Method (FEM) has been proven to provide a suitable framework. In this regard, a vast selection of references on the FEM can be given, i.e., Bathe [11], Schwarz [139] or Zienkiewicz and Taylor [193, 194] among others, whereas Ellsiepen [52], Ehlers and Ellsiepen [41] or Ammann [2] particularly consider the coupled numerical approximation of porous media models using mixed finite elements.
5.1.1 Weak Formulation
As the Finite Element Method (FEM) is a numerical approximation method, the governing equations need to be brought into a form, which is suitable for a numerical treatment. In this regard, the local balances of the overall aggregate need to be converted from a local (strong) form to an integral (weak) form. This crucial step is important, as it is often impossible to determine a closed form solution of the unknown field quantities at every point x of the underlying spatial domain Ω of the aggregate body \({\mbox {$\mathcal {B}$}}\), especially in the case of complicated geometries like they often occur in biomechanics.
According to Ehlers and Acartürk [39], an approach using p as unknown field quantity would lead to unstable numerical solutions with oscillations in the primary variables {p,u_{S}}. Note that these numerical instabilities have also been observed by other authors who use a triphasic swelling model, cf. Snijders et al. [149]. Moreover, such instability problems are also known in other fields addressing fluid-structure interactions, moving boundaries or free surfaces, cf. Wall [174] among others. Following Ehlers and Acartürk [39] and Ehlers et al. [49], the best procedure to overcome these oscillation problems is to weakly impose the corresponding Dirichlet boundary conditions in the governing set of equations by a penalty-like method [195]. However, the quickest and simplest way to overcome these problems is to chose the pair as primary variables, which will be the choice for the remaining part of this contribution.
5.1.2 Spatial Discretisation Using Mixed Finite Elements
Moreover, for a convenient implementation of the trial and test functions (243), the finite elements also serve as discrete local carriers for the respective basis functions. Herein, the basis functions are defined on a single reference element and are often referred to as shape or interpolation functions. This leads to an efficient applicability of numerical integration techniques like the Gauß quadrature, for instance, which is important regarding the numerical integration of the weak forms (240). Following this, the standard reference element is expressed in local coordinates \(\pmb {\xi }\), while the global basis functions are obtained from a geometry transformation to the global position x. As is usual in the FEM, the geometry transformation is carried out using an isoparametric concept, where geometry and displacements are expressed by the same set of basis functions.
Regarding numerical accuracy, stability and numerical costs, the suitable choice of mixed finite element formulations is by no means phenomenological, but rather strictly mathematical. In this regard, the inf-sup condition, often referred to as Ladyzhenskaya-Babuška-Brezzi (LBB) condition, needs to be fulfilled, cf., e.g., Brezzi and Fortin [20] or Wieners [180]. Following this, the Taylor-Hood elements fulfil the LBB condition and are the best possible choice from a stability and accuracy point of view. In regard of the complex geometries frequently involved in computational biomechanics, the quadratic discretisation also leads to a good geometry approximation, even with a relatively small number of elements. However, concerning the general 3-d case, the enormous number of mid nodes in finer meshes causes quite large systems of equations, which often have to be solved in parallel, cf. Wieners et al. [183] or Ehlers et al. [47]. One possibility to overcome this problem is to use the so-called MINI element suggested by Arnold et al. [6], which has an enriched linear shape function for \(\mathbf {u}_{S}^{h}\) using a bubble node. This leads to stable results but the obtained solution is not as accurate anymore, especially concerning the pressure field. Note that a linear-linear approximation of \(\mathbf {u}_{S}^{h}\) and leads to strange instabilities causing mesh dependent solutions due to the so-called spurious pressure modes, see, e.g., Brezzi and Fortin [20].
5.1.3 Semi Discrete Initial-Value Problem
Next, the abstract system (247) needs to be classified with regard to the type of differential equations it contains. This is accomplished by a formal comparison of (247) with the corresponding weak forms and evolution equations, respectively. Following this, the aggregate momentum balance (240)_{1} has no explicit dependence on the temporal change of the primary variables in u, while the aggregate volume balance (240)_{2} has only a dependence on the temporal change of the displacements \((\boldsymbol {u}_{S}^{j})'\) but not the temporal change of the hydraulic pressure . This characteristic leads to a singular generalised mass matrix M and thus, the global system \(\pmb {\mathcal {G}}\) can be classified as an index-1 system of differential-algebraic equations (DAE) of first order in time, cf. Ellsiepen [52]. A comparison of (246) and (247)_{2} quickly reveals that A is a regular identity matrix, which classifies the local system \(\pmb {\mathcal {L}}\) to be a system of ordinary differential equations (ODE) of first order in time.
5.2 Finite Difference Method in Time
In order to solve the spatially discretised initial value problem (247), the system has to be discretised in time using a suitable numerical time integration method. Herein, the most important class of single step methods for differential equations of first order in time is collected in the so-called Runge-Kutta methods, which simultaneously include implicit and explicit integration schemes. A detailed overview of these finite difference schemes can be found in Hairer and Wanner [64] and references therein. However, the drawback of such a stepwise time integration on a spatially fixed FE discretisation is that the system of DAE turns into a system of stiff differential equations. Thus, an explicit time integration scheme will result in unstable numerical results unless the step size is chosen unreasonably small. Regarding the soft biological tissues under study, the deformation processes of interest are on a much larger time scale and thus, explicit time integration schemes are excluded in favour of the implicit ones. Following this, the so-called stiffly accurate s-stage diagonally implicit Runge-Kutta (DIRK) method has been proven to provide a suitable numerical integration scheme for the underlying problem. For more information about the DIRK method with regard to the computability of large DAE systems, the stability of the solution, as well as the possibility for an error controlled time increment control, the reader is referred to the works of Diebels et al. [28], Ehlers and Ellsiepen [41] or Ellsiepen [52].
5.3 Solution of the Resulting Nonlinear System
Following this, it is devastating to solve the nonlinear function vector R_{n}(Δy_{n}) with the Jacobian (252), because the partially sparse structure would be completely destroyed. Hence, a suitable blockwise solution is needed in order to retain the sparse structure of the time- and space-discrete FE system G_{n} and to solve the equations L_{n} at the integration points in a decoupled fashion on element level.
5.3.1 Multilevel Newton Method
In order to exploit the block-structured nature of (252), a generalisation of the Block Gauß-Seidel-Newton method is applied, which is known as multilevel or two-stage Newton procedure. Herein, the implicit dependence of the local L-equations on the linearisation of the global G-equations is considered. This method is strongly related to the term “algorithmically consistent linearisation”, which was established by Simo and Taylor [145] in the context of computational finite elastoplasticity. For a more detailed introduction into the solution procedure, the reader is referred to Diebels et al. [28], Ehlers and Ellsiepen [41], Ellsiepen [52] or Ellsiepen and Hartmann [53].
Note in passing that the linear system (256) belongs to the class of saddle point problems which are characterised by their indefiniteness and frequently occurring poor spectral properties. In this regard, special care must be taken regarding the choice of a suitable solver. A good overview on this topic can be found in Benzi et al. [12] and references therein. In general, the so-called Krylov subspace methods provide stable solutions. For the purpose of this contribution, the generalised minimal residual method (GMRES) of Saad and Schultz [129] is applied, which is extensively discussed in Wieners et al. [182] in the context of an inelastic multi-phasic model.
- 1.
Compute the defect \(\boldsymbol {G}_{n}^{k}(\boldsymbol {u}_{n}^{k},\boldsymbol {q}_{n}^{k})\) of the new time step including the Neumann boundary conditions and proceed to the next time step, if the user-defined criterion for \(\|\boldsymbol {G}_{n}^{k}\|\) is met. If \(\|\boldsymbol {G}_{n}^{k}\|>\mathit{TOL}_{\boldsymbol {G}_{n}}\), start the global Newton iteration and proceed to step two. Note that the defect is computed element wise via a quadrature rule. In order to do so, the stress needs to be computed at every integration point which requires the solution of the local evolution equations (253)_{1}. Thus, the evaluation of the defect \(\boldsymbol {G}_{n}^{k}(\boldsymbol {u}_{n}^{k},\boldsymbol {q}_{n}^{k})\) automatically leads to the determination of the current vector of the internal variables \(\boldsymbol {q}_{n}^{k}(\boldsymbol {u}_{n}^{k}):=\boldsymbol {q}_{n}^{k,m+1}=\boldsymbol {q}_{n}^{k,m}+\Delta \boldsymbol {q}_{n}^{k,m}(\Delta \boldsymbol {u}_{n}^{k})\) in m local Newton iterations, respectively.
- 2.
Compute the consistent Jacobian matrix \(\boldsymbol {J}_{\scriptsize \boldsymbol {G}_{n}}^{k}\) as is given in (255).
- 3.
Solve the sparse linear system (256) for \(\Delta \boldsymbol {u}_{n}^{k}\).
- 4.
Update the global variables \(\boldsymbol {u}_{n}^{k+1} = \boldsymbol {u}_{n}^{k} + \Delta \boldsymbol {u}_{n}^{k}\).
- 5.
Compute the new defect \(\boldsymbol {G}_{n}^{k+1}(\boldsymbol {u}_{n}^{k+1},\boldsymbol {q}_{n}^{k+1})\) and check for convergence. If \(\|\boldsymbol {G}_{n}^{k+1}\|>\mathit{TOL}_{\boldsymbol {G}_{n}}\), repeat from step two, whereas if \(\|\boldsymbol {G}_{n}^{k+1}\|<\mathit{TOL}_{\boldsymbol {G}_{n}}\), proceed to next time increment and start with step one again.
5.3.2 Solid Stress Computation
During every iteration step of the multilevel Newton method, a finite element code requires an algorithm which computes the solid extra stress tensor \(\mathbf {T}^{S}_{\mathrm{E}}\) and the consistent material tangent \((\pmb {\mathcal {C}}^{S})^{\underline{4}}\) at every integration point of the numerical quadrature. In particular, the solid stress is needed for the evaluation of the defect, while the computation of the consistent Jacobian matrix (255) demands for the stress and the material tangent.
Note that the determination of eigenvalues and eigenvectors for the coefficient matrices of the left deformation tensors B_{S} and B_{Se} is a very sensitive issue in computational mechanics. Following this, a robust algorithm is needed regarding the treatment of identical eigenvalues, like they naturally occur in the undeformed referential frame. According to Markert [100], one of the most efficient techniques for finding eigenvalues and eigenvectors of real symmetric matrices is the combination of the Householder reduction followed by a QL decomposition, cf. Press et al. [123].
5.3.3 Admissible Initial Conditions
Moreover, if the initial osmotic pressure is always added onto the mechanical solid extra stress, then the corresponding material tangent \((\pmb {\mathcal {C}}^{S}_{\mathrm{OSM}})^{\underline{4}}\) needs to be modified as well.
5.3.4 Linearisation of the Balance Equations
5.4 Parallelisation of the Finite Element Simulation
The underlying model as well as the numerical solution scheme were implemented into the sequential research code PANDAS^{4} which traces back to Ehlers and Ellsiepen [40, 41] and Ellsiepen [52]. However, as already mentioned before, the numerical solvability (computability) of the presented nonlinear coupled differential equations is reached relatively soon on single CPU machines. This is mainly due to the involved complex geometries in computational biomechanics, like in the case of the spine, for instance. Herein, a large number of elements is needed to approximate the geometry, which in turn leads to a large number of unknowns and thus, to large linear systems to be solved in (256). Especially the quadratic approximation of the displacements in the Taylor-Hood elements leads to a quickly rising number of the DOF in 3D. Moreover, 3-d adaptive strategies with remeshing are rather difficult to achieve due to the irregular geometries and lie beyond the scope of this contribution. Alternatively, the described solution procedure can be carried out in parallel using several PCU simultaneously. Since the development of a new parallel code lies also beyond the focus of this work, the research program PANDAS is coupled with the parallel solver M++^{5} by extending an already existing interface. In this regard, the interface was originally designed by Ammann [2] and Wieners et al. [184] for the computation of large geotechnical BVP, where it is possible to set up a coarsely discretised BVP in PANDAS and almost directly solve finer discretised grids in parallel using M++.
The purpose of this section is to provide a brief overview on the capabilities of the two codes as well as to give an introduction to the basic idea of the interface.
5.4.1 Capabilities of PANDAS and M++
PANDAS is a very sophisticated FE code which was originally developed for research purposes and is continuously enhanced at the Institute of Applied Mechanics (Continuum Mechanics) at the Universität Stuttgart. Its structure is designed to allow for a very efficient treatment of mixed finite elements and is held general enough to enable the implementation of almost every possible coupled problem. Even the treatment of general inelastic material behaviour is structured in such a way that it is possible to implement the respective theories without going into the details of the whole FE code. Herein, the user is provided with several interfaces to the PANDAS core, which allow for a convenient handling of the material parameters, the history variables, the element properties, the numerical integration of the governing equations and the consistent linearisation on element level, as well as the solution of the local DAE system among many other features. Especially useful is a library containing all the routines needed for tensor calculus, i.e., it is possible to directly compute the respective tensor products without the need to program them using loops. Moreover, it contains space (only in 2D) and time adaptive strategies as well as a direct and several iterative solvers for linear systems of equations. Another really big benefit is the way the boundary conditions can be incorporated. Herein, one has almost any possibility to set initial conditions as well as Neumann and Dirichlet conditions on external boundaries or even at distinct nodes in the inside of the discretised domain.
On the other hand, there is the program M++ which is designed to pursue a different goal. Following the name Mesh, Multigrid and More (M++), it becomes obvious that the main interest is not on the physical side but on its parallel algorithms for the numerical solution of the arising of strongly coupled problems. In this regard, M++ was built to perform on MIMD (multiple instruction stream, multiple data stream) architectures and is based on the parallel programming model MPI (Message Passing Interface) of Walker and Dongarra [173]. Note that an MIMD architecture is basically a cluster of several individual workstations which are connected by a network. The workstations themselves are mostly classical personal computers (PC) of the von Neumann type [60] which have a single instruction and a single data stream (SISD). This requires the need for a parallel data structure which uniquely defines the elements or cells of a finite element mesh on all the computing nodes. In M++ this is realised using the so-called Distributed Point Objects (DPO) as is described in Wieners et al. [184]. Regarding the solution of the nonlinear and time-dependent simulation, the most sensible part of the parallelisation is the solution of the linear problems (256) within every Newton step. Herein, M++ uses a parallel GMRES method, as is described in Wieners et al. [182], together with a domain decomposition preconditioner.
From numerical experiments it is well-known that overlapping domain decomposition preconditioners with coarse grid correction applied to the Navier-Stokes system using Taylor-Hood elements are very efficient, see Klawonn and Pavarino [82, 83]. Hence, this type of preconditioning is a good choice for the discussed application to complex bio-mechanical structures which leads to a similar system. Although, two major modifications are required due to the following reasons: Only a moderate number of processors is used so that the subdomain problems are too large for exact solving. Moreover, the underlying geometry is too complex to allow for a small coarse mesh. Thus, an inexact sub-domain solver is used (a multilevel incomplete LU factorisation (ILU) with pivoting and dropping strategy by Mayer [104, 105]) and the coarse problem is constructed on an independent overlay mesh. Last but not least, M++ provides the possibility to uniformly refine a given finite element mesh which allows for convergence studies regarding the mesh size.
Thus, in order to simultaneously use the conveniences of PANDAS during the design of a user element as well as the power of the parallel solver M++, an interface needs to be defined which will be discussed in the following subsection.
5.4.2 Interface M++/PANDAS
Following the idea of keeping PANDAS for the definition of the physics-specific routines as well as for the definition of the BC, the simplest possibility is to incorporate the interface on element level. The benefit is that PANDAS does not need to be modified at all and a BVP set up in PANDAS can be computed directly in parallel using M++ as a black box solver. In this regard, M++ manages only the global FE mesh with its unknowns as well as the corresponding element specific data (history variables, inhomogeneous material parameters). PANDAS is then initiated in the background and called E times during the global assembly of the defect \(\boldsymbol {G}_{n}^{k}(\boldsymbol {u}_{n}^{k},\boldsymbol {q}_{n}^{k})\) and the Jacobian matrix \(\boldsymbol {J}_{\scriptsize \boldsymbol {G}_{n}}^{k}\). Thus, these routines have a key function, as they are called fairly often within a Newton step. Moreover, due to the parallel data structure of M++, these routines can be executed simultaneously on several processors without any network communication. The difficult task of solving the global system on several processors is then performed by M++ needing network communication.
Pandas_Init: Initiates PANDAS with the required types of finite elements. Herein, the PANDAS data structure is constructed and the element is linked with the desired physics routine containing the weak forms of the global system, the evolution equations as well as the corresponding material parameters.
InitInhomo: Initially computes the inhomogeneous distribution of the fibre vectors and the involved material parameters as is described in Sect. 6.1.1. This has to be carried out within the interface, as the respective algorithms for the computation of the inhomogeneous distributions depend on data of the overall mesh, which is never fully available in PANDAS.
Pandas_Dirichlet: Evaluates the Dirichlet boundary conditions.
Pandas_Defect: Evaluates the Neumann boundary conditions and computes the defect (residual) on element level. Herein, the solution of the local system on quadrature point level is performed by PANDAS. Thus, besides the element residual vector, the newly obtained history variables are also returned to M++ in this routine.
Pandas_Tangent: Computes the algorithmically consistent tangent on element level.
Pandas_Update: Updates the history variables in PANDAS.
InitElem: Converts the data format of M++ into the data structure of PANDAS, every time before one of the four routines above is called.
6 Application to the Intervertebral Disc
Concerning the simulation of soft biological tissues in general and the IVD or spine in particular, the creation of the underlying finite element mesh is not straightforward, as the involved geometries are rather complex. Thus, an extra section about this topic will be presented, wherein the creation of a realistic geometry model as well as the inclusion of inhomogeneously distributed quantities like the fibre vectors is addressed.
In a next step, the theoretically introduced parameters need to be identified. As the author does not have the possibility to perform experiments on living tissues, the identification is carried out with experimental results obtained from literature. Subsequently, a numerical sensitivity analysis is carried out in order to judge the influence of the respective parameters on a compression-bending experiment of a motion segment as a representative deformation mode of the spine. Herein, the model can be understood as a numerical laboratory, where the influence of certain parameters or effects can be obtained nicely.
Finally, the determined parameters will be used for a computation of the lumbar spine consisting of four motion segments. Due to the enormous amount of finite elements needed to resolve the geometry, these computations are carried out in parallel on multiple CPU using the interface M++/ PANDAS.
6.1 Modelling the Intervertebral Disc
Modelling the Intervertebral Disc This section offers a brief introduction into the arising problems, when the previously developed governing equations are applied to numerically simulate the behaviour of the IVD. Herein, the 3-d geometry needs to be acquired and spatially discretised using arbitrary amounts of finite elements. Moreover, as the IVD exhibits a very inhomogeneous material behaviour, the respective theoretical material parameters, which characterise the mechanical behaviour, need to be inhomogeneously distributed in the discretised domain. In particular, this addresses the varying fibre orientations and the material parameters affecting the anisotropic stress response in the AF, as well as the inhomogeneous distribution of the fixed negative charges throughout the IVD.
6.1.1 Modelling Inhomogeneities
As is described in Sect. 2.1, the inhomogeneities of the IVD mainly occur in the AF. Following this, an algorithm needs to be defined which allows for a radial and tangential variation of the mechanical properties. In the context of the FEM, the inhomogeneous material behaviour is captured via location-dependent material parameters inside the IVD and thus, extra storage needs to be provided on quadrature (Gauß) point \((\mathcal {G}\mathcal {P})\) level, just like it is the case for the internal (history) variables. Hence, additional storage is needed for scalar quantities which, in contrast to the history variables, stay constant over the computation. In particular, these are the initial molar concentration \(c^{fc}_{0S}\) of the fixed negative charges, the parameters characterising anisotropic stress response, \(\tilde{\mu}^{S}_{m}\) and \(\tilde{\gamma}^{S}_{m}\), as well as the fibre angle \(\phi^{S}_{0}\). Moreover, the varying fibre alignment is captured by the referential fibre vectors \(\mathbf {a}^{S}_{0}\) and \(\mathbf {b}^{S}_{0}\), which leads to six additional scalar values to be stored.
Moreover, in order to allow for a tangential distribution of the material parameters, the polar angle δ is introduced in the axial symmetry plane in order to define the tangential position of a \(\mathcal {G}\mathcal {P}\), cf. Fig. 14(d). Concerning the radial distribution in the AF, the non-circular cross section of the IVD impedes the usage of the simple distance between the COG and the \(\mathcal {G}\mathcal {P}\). Hence, the distances d_{1} and d_{2} are introduced instead, which result from the projection of the \(\mathcal {G}\mathcal {P}\) onto the surrounding inner and outer side surfaces of the AF, respectively. It is now possible to define interpolating functions in tangential and radial direction of the AF. For the purpose of this contribution, the normalised polar angle δ^{∗}=δ/180^{∘} as well as the normalised distance d^{∗}=d_{1}/(d_{1}+d_{2}) are introduced, thereby allowing for a convenient linear interpolation in the tangential and radial direction, respectively, provided that extremal values are given at the respective start and end points.
6.2 Parameter Identification
In general, it is extremely difficult to realise suitable experiments on living soft biological tissues, which is mainly due to the following problems. Because it is virtually impossible to perform in vivo experiments, i.e., while still in the living organism, an ex vivo testing setup is preferable. In this regard it is extremely difficult to obtain “fresh” specimens within a reasonable time frame, which is usually only 24 hours post decease before degeneration effects start to evolve. Besides the difficulty of obtaining these fresh tissue samples, the experimental testing is also not a trivial task. Firstly, the tissue exhibits a strongly coupled behaviour in time and thus, it is almost impossible to distinguish between dissipative effects resulting from the viscoelastic solid skeleton or the viscous fluid flow. Moreover, concerning the swelling-active material behaviour, the surrounding fluid in the test chamber is also an important issue. In this regard, one can either perform tests using a physiological solution surrounding the tissue (in order to prevent swelling or shrinking processes) or a humidity chamber, which prevents the specimens from drying out. If these difficulties are overcome, one still has the problem that the acquired data is extremely “patient specific” and may vary if a person of different age is considered.
Due to the extreme lack of representative experimental data in this field, the present contribution is making use of real experiments only when they meet the above described restrictions. To the knowledge of the author, there are only two data sources available in literature which are directly suitable for the identification of the involved material parameters in the presented TPM model. These are the very detailed data on the behaviour of the collagen fibres in the AF acquired by Holzapfel et al. [71] and the torsional shear experiments on NP specimens performed by Iatridis et al. [76]. Moreover, an extensive study of the related literature has been carried out in order to obtain mean values for the remaining parameters.
6.2.1 Structural Collagen in the Anulus Fibrosus
As is already described in Sect. 2.1, the collagen fibres in the AF exhibit not only structural inhomogeneities but also a location-dependent mechanical behaviour. Both will be addressed in the following.
Structural Observations
Mechanical Behaviour
6.2.2 Isotropic Viscoelasticity of the Nucleus Pulposus
Herein, the lines indicate the computed curves using the presented model, whereas the circles display the measured values of Iatridis et al. [76]. The material parameters were fit manually to the curve corresponding to an angular displacement of φ=0.036. The other two curves, i.e., for an angular displacement of φ=0.018 and φ=0.054, were computed thereafter keeping the identified parameters constant.
Material parameters of the biphasic model for the vertebrae, nucleus pulposus and anulus fibrosus, respectively, whereas the isotropic contributions are always described using the neo-Hookean model
Vertebrae: Treated with no distinction between cortical shell and spongiosa | |||||||
---|---|---|---|---|---|---|---|
\(K^{S}_{0S}=2.7\cdot10^{-5}\) | [mm^{2}] | \(n^{S}_{0S}=0.2\) | [–] | \(\mu^{S}_{0}=96.0\) | [MPa] | κ=0.0 | [–] |
μ^{FR}=3.8⋅10^{−8} | [MPa⋅s] | \(c^{fc}_{0S}=0.0\) | [mol/l] | \(\varLambda ^{S}_{0}=112.7\) | [MPa] | \(\gamma^{S}_{0}=1.0\) | [–] |
Nucleus pulposus: Treated as isotropic, viscoelastic and charged material | |||||||
---|---|---|---|---|---|---|---|
\(K^{S}_{0S}=3.5\cdot10^{-12}\) | [mm^{2}] | \(n^{S}_{0S}=0.2\) | [–] | \(\mu^{S}_{0}=0.5\) | [kPa] | κ=0.0 | [–] |
μ^{FR}=6.9⋅10^{−10} | [MPa⋅s] | \(c^{fc}_{0S}=0.3\) | [mol/l] | \(\varLambda ^{S}_{0}=0.3\) | [kPa] | \(\gamma^{S}_{0}=50.0\) | [–] |
1st Maxwell element: | \(\zeta^{S}_{1}=0.37\) | [kPa⋅s] | \(\mu^{S}_{1}=2.8\) | [kPa] | [–] | ||
\(\eta^{S}_{1}=0.37\) | [kPa⋅s] | \(\varLambda ^{S}_{1}=1.9\) | [kPa] | \(\gamma^{S}_{1}=12.0\) | [–] | ||
2nd Maxwell element: | \(\zeta^{S}_{2}=10.0\) | [kPa⋅s] | \(\mu^{S}_{2}=0.85\) | [kPa] | [–] | ||
\(\eta^{S}_{2}=10.0\) | [kPa⋅s] | \(\varLambda ^{S}_{2}=0.57\) | [kPa] | \(\gamma^{S}_{2}=12.0\) | [–] |
Anulus fibrosus : Treated as inhomogeneous anisotropic charged material | |||||||
---|---|---|---|---|---|---|---|
\(K^{S}_{0S}=6.2\cdot10^{-12}\) | [mm^{2}] | \(n^{S}_{0S}=0.35\) | [–] | \(\mu^{S}_{0}=0.95\) | [MPa] | κ=0.0 | [–] |
μ^{FR}=6.9⋅10^{−10} | [MPa⋅s] | \(c^{fc}_{0S}=0.1\) | [mol/l] | \(\varLambda ^{S}_{0}=2.2\) | [MPa] | \(\gamma^{S}_{0}=1.0\) | [–] |
VLE | VLI | DE | DI | |
---|---|---|---|---|
\(\tilde{\mu}_{1}^{S}\) [kPa] | 40.20 | 34.33 | 50.79 | 5.973 |
\(\tilde{\gamma}_{1}^{S}\) [–] | 148.1 | 44.05 | 54.24 | 30.46 |
Finally note that the main deformation mode of the NP under bending is volumetric compression and thus, the information gathered by Iatridis et al. [76] is not sufficient to reliable determine this behaviour. Moreover, a volumetric compression is also coupled to the dissipative effects resulting from the viscous fluid flow inside the NP. Regarding the time scale on which the intrinsic dissipative effects of the solid skeleton occur, one can assume that the intrinsic viscoelasticity plays a negligible role in the context of long term analyses. This is because interstitial fluid flow takes place over a much greater period of time compared to the few seconds of stress-relaxation in the torsion experiment.
6.2.3 Inhomogeneous Distribution of the Fixed Negative Charges
6.2.4 Parameters Obtained from the Literature
The remaining parameters were determined by an extensive study of the related literature. In this context, it was not always possible to directly find exactly the same parameters for the respective nonlinear material laws of this contribution, because most of the times, other authors use linear material models. As a consequence, these parameters are adopted, thereby knowing well that they only apply to the small strain domain. As a consequence of the lack of reliable data, only the neo-Hookean approach is used for the solid skeleton, while the more sophisticated Mooney-Rivlin or Ogden approaches are left as a possibility for further studies, when more detailed data is available.
Moreover, because this work is concerned with the behaviour of the IVD, the vertebrae are only modelled as a smeared porous continua, thereby neglecting the proper composition of a dense cortical shell and a porous spongy bone. Herein, the respective quantities found in literature were averaged according to the volume fractions of the cortical shell and the spongy bone in the overall vertebra.
The parameters \(\gamma^{S}_{j}\) for j=0,1,2 were chosen to limit the disc bulge in the swelling experiment of the IVD, as seen in Fig. 24(c), whereas the remaining isotropic parameters for the AF as well as for the vertebrae were taken in accordance to averaged values given in Argoubi and Shirazi-Adl [5], Eberlein et al. [31], Iatridis et al. [75, 76], Lee et al. [91], Ochia and Ching [119], Shirazi-Adl et al. [143], Wu and Chen [189] and references therein. Note that according to Gu et al. [62], the young AF has an anisotropic permeability which becomes isotropic with age or degeneration. Due to the almost impermeable character of the AF and the fact that the directional variation of the permeability is “only” of factor two, while the values given in the related literature span over decades, the anisotropic permeability of the AF is neglected.
6.2.5 Swelling Behaviour of the Nucleus Pulposus
Comparison of the swelling mechanism with more sophisticated models:
In order to generally validate the capabilities of the presented model with respect to the swelling mechanisms in soft biological tissues, a 1-d swelling experiment is carried out, thereby comparing the presented biphasic approach with a triphasic approach of Acartürk [1], Ehlers and Acartürk [39] and Lai et al. [87]. Herein, only the results of the respective computations will be presented without going into the details of the more sophisticated triphasic model.
Swelling Experiment of a Sagittally Cut NP
The second example concerns the swelling capability of the presented model, when it is applied to the IVD. In this context, a swelling phenomenon is computed, which occurred while Holzapfel et al. [71] performed an experiment on a sagittally cut motion segment. Right after the specimen was cut in half and placed on the laboratory table, the NP started to swell out of the IVD and reached a maximum bulge of 4.9 mm, cf. Fig. 15.
The simulation is carried out on a sagittally cut geometry of an L4–L5 motion segment, which is discretised using 1898 20-noded Taylor-Hood elements yielding a total of 28832 DOF, cf. Fig. 24(a), where the vertebrae, the NP, and the AF are highlighted in blue, red and yellow, respectively. The corresponding material parameters are listed in Table 3, whereas the inhomogeneities are modelled as is described in the preceding subsections. Following the swelling experiment, only essential boundary conditions are applied, which do not lead to a mechanical loading of the tissue. In this regard, the top and bottom surfaces of the vertebrae are fixed in space and a drainage () is ensured on all free surfaces. Furthermore, the concentration of the external solution is lowered from \(\bar{c}_{m}=0.15\) to 0.0 mol/l within 50 s and is then held constant at zero for another 5000 s. Again, a tissue temperature of 25 ^{∘}C was assumed yielding RΘ=2477.6 J/mol.
Due to the almost impermeable characteristics of the IVD, the inflow is constrained, and hence, the volume dilatation inside the IVD happens really slow. Thus, the excess osmotic pressure difference is firstly carried via a suction power of the hydraulic pressure (part I in Fig. 25(a)) and is then gradually released into tension carried by the dilated solid skeleton (part II). The duration of this process is strongly dependent upon the filter velocity of the fluid that gets sucked inside, which is finally a function of the gradient of the negative hydraulic pressure and the resistance (permeability) of the tissue. In principle, this behaviour is reverse to the consolidation problem from geomechanics, where a load on the top surface is firstly carried by the pore fluid alone. Then, some of the incompressible pore fluid is gradually expelled due to the arising hydraulic pressure gradient, thereby causing a volumetric compression of the aggregate. Finally, when the consolidation process is finished, the load is carried by the volumetric solid extra stress alone. Note that if the medium was permeable without any resistance, the consolidation or the swelling process would be finished just after applying the load or concentration drop. The endpoint of the present swelling process is reached, when the negative hydraulic pressure inside the tissue is discharged. According to Fig. 25, this is the case at about t≈3000 s. The resulting bulge of the NP can be seen in Fig. 24(c) with the maximum bulge of 5.7 mm, whereas the development of the bulge is depicted in Fig. 25(b).
6.3 Compression-Bending Experiment of an L4–L5 Motion Segment
As seen in the previous section, the identification of the involved material parameters is not straightforward, because many effects occur in a coupled fashion. As a consequence, these parameters may more conveniently be determined via inverse computations. Following this, the influence (sensitivity) of the involved parameters on dominant deformation modes of the IVD needs to be computed, in order to conclude a statement about their determinability. Here, a compression-bending experiment of an L4–L5 motion segment with dissected processes and ligaments will be used, where the material parameters of Table 3 are taken as a reference. Several computations will then be carried out, thereby varying only one parameter at a time.
Starting with Fig. 27(a), an initial osmotic pressure Δπ_{0S}=0.3 MPa is shown in the NP which is fully compensated by the pre-stressed solid skeleton without any initial deformations, cf. Sect. 5.3.3. Subsequently, Figs. 27(b)–(d) show the gradually evolving total pore pressure p, which is mainly dominated by the hydraulic pressure contribution as a result of the volumetric deformation of the solid skeleton. Since a bending moment is applied, the stiffer vertebrae undergo a tilting movement in the axial plane, thereby squeezing the IVD in the ventral area, while expanding the AF in the dorsal region. Thus, the pore pressure p is highest, where the stiff vertebrae intrudes the anterior AF, whereas a suction power can be observed at the posterior AF.
6.3.1 Survey on the Influence of the Involved Parameters
Computed absolute arithmetic mean values of the sensitivities (275) for all parameters on the measured responses. “Sensitive” values are highlighted in black
Variation | Location | h_{d} | h_{v} | d_{d} | d_{v} | p_{d} | p_{v} | Δϕ |
---|---|---|---|---|---|---|---|---|
\({\tilde{\mu}}^{S}_{1},\,{\tilde{\gamma}}^{S}_{1}\) | AF | 1.48 | 0.57 | 0.67 | 0.38 | 5.00 | 0.26 | 0.79 |
\(\mu^{S}_{0}\) | AF | 0.04 | 0.29 | 0.05 | 0.19 | 1.84 | 0.21 | 0.12 |
\(\mu^{S}_{0}\) | NP | 0.01 | 0.08 | 0.03 | 0.02 | 0.88 | 0.32 | 0.04 |
\(c^{f\!c}_{0S}\) | NP + AF | 0.00 | 0.01 | 0.00 | 0.00 | 1.65 | 0.36 | 0.01 |
\(K^{S}_{0S}\) | NP + AF | 0.00 | 0.01 | 0.00 | 0.00 | 0.01 | 0.00 | 0.00 |
\(n^{S}_{0S}\) | NP + AF | 0.00 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.01 |
\(\gamma^{S}_{0}\) | NP + AF | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.01 | 0.00 |
Computed sensitivities (275) of the parameter variations for \(\tilde{\mu}^{S}_{1}\) and \(\tilde{\gamma}^{S}_{1}\) (fibre stiffness in the AF) on the measured responses during the L4–L5 compression-bending experiment
Variation | h_{d} | h_{v} | d_{d} | d_{v} | p_{d} | p_{v} | Δϕ |
---|---|---|---|---|---|---|---|
130 % | −0.79 | −0.43 | −0.20 | −0.30 | 3.69 | 0.20 | −0.47 |
70 % | −1.38 | −0.55 | −0.56 | −0.44 | 5.12 | 0.25 | −0.76 |
40 % | −2.28 | −0.72 | −1.24 | −0.40 | 6.19 | 0.34 | −1.15 |
Absolute mean | 1.48 | 0.57 | 0.67 | 0.38 | 5.00 | 0.26 | 0.79 |
Influence of the additional Mooney-Rivlin parameters on the L4–L5 compression-bending experiment. The values denote absolute percentage deviations from the reference computation using the neo-Hookean model and the corresponding set of reference parameters
Case | Location | h_{d} | h_{v} | d_{d} | d_{v} | p_{d} | p_{v} | Δϕ |
---|---|---|---|---|---|---|---|---|
(a) | NP | 0.00 | 0.00 | 0.00 | 0.00 | −0.04 | −0.01 | 0.00 |
(a) | AF | 0.00 | 0.00 | 0.01 | 0.00 | −0.04 | −0.01 | 0.00 |
(b) | NP | 0.00 | 0.00 | 0.00 | 0.00 | −0.07 | −0.02 | 0.00 |
(b) | AF | 0.00 | 0.00 | 0.01 | 0.00 | −0.07 | −0.03 | −0.01 |
(c) | NP | 0.00 | 0.00 | 0.00 | 0.00 | −0.11 | −0.03 | 0.00 |
(c) | AF | 0.00 | 0.01 | 0.00 | −0.01 | −0.11 | −0.04 | −0.01 |
(d) | NP | 0.00 | 0.00 | 0.00 | 0.00 | −0.14 | −0.04 | 0.00 |
(d) | AF | 0.00 | 0.01 | 0.01 | −0.01 | −0.14 | −0.05 | −0.01 |
(e) | NP | 0.00 | 0.00 | 0.00 | 0.00 | −0.1 | −0.04 | 0.00 |
(e) | AF | 0.00 | 0.01 | 0.01 | −0.01 | −0.18 | −0.06 | −0.01 |
Influence of the Lamé constants of the NP and AF on the L4–L5 compression-bending experiment using the Mooney-Rivlin model. The values denote the mean average sensitivities according to (275) and “sensitive” values are highlighted in black
Case | Location | h_{d} | h_{v} | d_{d} | d_{v} | p_{d} | p_{v} | Δϕ |
---|---|---|---|---|---|---|---|---|
(a) | AF | 0.07 | 0.45 | 0.09 | 0.31 | 3.23 | 0.24 | 0.23 |
(b) | AF | 0.07 | 0.40 | 0.09 | 0.32 | 2.93 | 0.39 | 0.27 |
(c) | AF | 0.07 | 0.48 | 0.08 | 0.31 | 3.31 | 0.41 | 0.23 |
(d) | AF | 0.06 | 0.47 | 0.09 | 0.30 | 3.54 | 0.36 | 0.23 |
(e) | AF | 0.06 | 0.48 | 0.09 | 0.33 | 3.38 | 0.40 | 0.23 |
(a) | NP | 0.01 | 0.07 | 0.03 | 0.02 | 0.92 | 0.32 | 0.04 |
(b) | NP | 0.01 | 0.07 | 0.03 | 0.02 | 0.97 | 0.32 | 0.04 |
(c) | NP | 0.01 | 0.07 | 0.03 | 0.02 | 1.05 | 0.32 | 0.05 |
(d) | NP | 0.01 | 0.07 | 0.03 | 0.02 | 1.11 | 0.32 | 0.05 |
(e) | NP | 0.01 | 0.08 | 0.03 | 0.02 | 1.13 | 0.32 | 0.05 |
6.3.2 Interpretation of the Results
After having performed several computations with varying material parameters, it is now possible to conclude and interpret the results of the last subsection. Proceeding from the idea of using the L4–L5 compression bending experiment to either determine or validate the involved material parameters, it is obvious that this cannot be achieved for parameters which do not have an influence on the measurable response. Moreover, a strong and coupled influence of all responses is also undesirable for such an approach. In this case, the material parameter plays a major role and many measurements of the response have to be recorded and controlled simultaneously. Hence, these parameters should, if possible, rather be determined directly using independent experiments instead. In contrast, a material parameter with a medium influence can conveniently be determined using inverse computations by measuring the response at the most important points in the IVD.
From Table 6, it is straightforward to conclude that a striking dominance can be adjudicated to the AF, where particularly the type-I collagen fibres have by far the greatest influence. In conclusion, it is preferable to determine the corresponding material parameters \(\tilde{\mu}{}^{S}_{1}\) and \(\tilde{\gamma}{}^{S}_{1}\) via independent fibre elongation tests, as they were performed by Holzapfel et al. [71], for instance. Moreover, the dominant behaviour also reveals the need to incorporate the inhomogeneous distribution and alignment of the structural collagen fibres as well as their location-dependent mechanical properties. However, in order to describe the distribution of the involved material parameters more accurately, it would be of great benefit to perform lamellae tension tests at even more distinct points in the AF.
Moreover, also the Lamé constants \(\mu^{S}_{0}\) and \(\varLambda ^{S}_{0}\) of the AF, which characterise the isotropic elasticity, seem to play an important role. However, since they are not as dominant as the collagen fibres, they may be determined via inverse computations using the compression bending experiment. This is actually of great benefit, because it is not possible to obtain an AF specimen that is decoupled from the influence of its type-I collagen fibres. In particular, the difficulty arises from the ingrown (embedded) characteristic of the structural collagen in the PG network. Hence, it is not possible to determine the isotropic characteristic of the AF via an independent experiment. Similar to the Lamé constants of the NP which can also be obtained by inverse computations of compression bending experiment, but in contrast to the AF, it is also possible to obtain a NP specimen for separate and independent testing. However, due to the almost impermeable nature of NP tissue, it is rather difficult to measure the influence of the solid skeleton alone, as deformations are coupled to viscous interstitial fluid flow.
A possibility to overcome this problem is to apply the so-called porous indenter test, where the indenter is locally driven into the tissue sample. But again, due to the impermeable characteristic of the NP, such an experiment takes place over hours until the specimen is fully drained and decoupled from fluid flow. Another possibility is the torsional shear test which is free of any volumetric deformations within the small strain domain and thus, in theory independent of interstitial fluid flow. Such an experiment was carried out by Iatridis et al. [76] and is used for the identification of the viscous parameters in Sect. 6.2.2. However, the drawback of this experiment is that the NP specimen has to be pre-compressed in order to obtain a sufficient grip between the testing device and the specimen, which again leads to undesired volumetric deformations. Thus, the method of inverse computations is convenient, even if only used to verify parameters obtained from other experiments.
Regarding the measured response of the IVD, the Lamé constants of the NP seem to play a minor role in comparison to the influence of the structural collagen as well as the isotropic part of the solid skeleton in the AF. Moreover, the second study of the effect of the isotropic mechanical behaviour of the NP and AF on the deformation behaviour of the IVD reveals that the influence of the five different possibilities for the Mooney-Rivlin approach are not relevant in comparison to the neo-Hookean approach. In particular, there is a slight noticeable increase in the influence of the dorsal pore pressure, as the parameters \(\mu^{\ast}_{0(1)}\) and \(\mu ^{\ast}_{0(2)}\) are varied from case (a) to (e) in (276). However, a variation of the respective Lamé constants for the combinations (a) through (e) does not append new sensitive responses to the ones already given in Table 6. It is therefore not suggested to model the EQ part of the AF and NP using a more sophisticated approach than the elastic neo-Hookean model. Following this, it is also not recommended to account for the numerically expensive viscoelastic model of the NP during the simulation of the compression-bending experiment, because the gain in information is in no relation to the quickly rising numerical effort. However, the intrinsic viscoelasticity may play a role, when focusing on the damping characteristic of the IVD or other short-term load cases.
Naturally, the initial molar concentration \(c^{fc}_{0S}\) of the fixed negative charges has only an influence on the internal pore pressure. This is somewhat obvious, as the initial osmotic pressure Δπ_{0S} is always subtracted from the solid extra stress in order to obtain a stress-free reference configuration without any initial volumetric deformations. Hence, this parameter will only have an influence, if the molar concentration \(\bar{c}_{m}\) of the solutes in the surrounding solution changes. As a consequence, this parameter cannot be determined using the compression-bending experiment but must be identified separately as is done in Urban and Maroudas [167].
For various reasons, the remaining parameters of the study have no influence at all. In particular, the permeability \(K^{S}_{0S}\) of the solid skeleton has no effect on the deformation behaviour of the IVD, because the time frame, during which the compression-bending experiment takes place, is too short to uncover consolidation effects. Hence, as long as the permeability is not greater than a certain upper bound, the loading time of one second is too short for a reduction of the pore pressure inside the IVD. Note that this effect also explains why the isotropic elasticity of the NP is playing a minor role in the deformation behaviour of the short-term compression-bending experiment. Because the tissue has not enough time for consolidation, there is not much volumetric compression noticeable which in turn leads to a negligible contribution of the (compressive) solid extra stress. As a consequence, the solidity \(n^{S}_{0S}\) has also no influence, because the compaction point is rarely reached due to the almost impermeable characteristic of the IVD. Hence, the overall aggregate of the IVD behaves almost incompressible during short term loadings. The same holds for the parameter \(\gamma^{S}_{0}\) which activates the volumetric extension term in (211) by governing its non-linearity. Thus, this parameter will only influence great swelling mechanisms which are accompanied by large volumetric dilatations, cf. Fig. 10, as well as long-term deformations of the IVD, where the compaction point is reached. Following this, all three parameters need to be determined independently or can be taken from the related literature. However, in the proceeding subsection, long-term loading conditions are observed with respect to the influence of the permeability \(K^{S}_{0S}\) of the solid skeleton.
6.4 Long-Term Compression of a Motion Segment
After the performance investigation of the proposed model during short-term loading conditions like the compression-bending experiment, this section addresses the effects resulting from an axial compression of the IVD over a much longer period of time. Following this, the present study is motivated by the diurnal variation of the body height which results from a long loading period during the daytime and an “unloaded” resting period thereafter. In particular, the loads during every day activities cause the pore fluid to be expelled from the IVD which in turn leads to a higher concentration of the fixed negative charges inside the disc as well as an increase of the solid extra stress. As a consequence, the excess of \(c^{fc}_{m}\) triggers the osmotic effect which causes a diffusion process of the surrounding fluid into the IVD. Whenever the disc is unloaded during rest at night, the solid extra stress is released and the original disc height will be obtained in the next morning. In this regard, the development of the disc height over time is strongly influenced by the permeability of the tissue. Hence, this study is concerned with different values for \(K^{S}_{0S}\) as well as for κ, which triggers the deformation dependency of the intrinsic permeability.
For convenience, the discretised geometry of the L4–L5 motion segment of the preceding section is used again for the numerical simulations in the present long-term compression study. The same holds for the material parameters given in Table 3. Moreover, as the influence of the intrinsic viscoelasticity of the NP is only noticeable during short-term loadings, it is neglected and relaxed material parameters (\(\mu^{S}_{0}=0.5~\mbox{MPa}\) and \(\varLambda ^{S}_{0}=0.75~\mbox{MPa}\)) are chosen according to the previous study.
For the intention of this contribution, however, it is sufficiently proved that the presented model is capable of reproducing the effect of the diurnal variation of the body height. Note that a more accurate description using the present approximation with absence of processes as well as ligaments and spine muscles is not possible in the first place, as it is not clear how big the actual loads on the IVD are. Moreover, the applied load of 400 N is probably an underestimation of the actual loads applied during normal day activities, which may also include dynamic load cases. Thus, in order to exploit this kind of experiment for parameter identification, the complete spine should be modelled and representative loads need to be determined.
6.5 Bending of the Lumbar Spine (L1–L5)
Finally, the applicability of the proposed model is demonstrated by numerical simulations of large-scale problems which address the deformation behaviour of the lumbar spine with removed processes. This leads to the challenge that even on up-to-date PC systems, the numerical solution of an initial BVP involving several motion segments cannot be computed efficiently on a single-CPU machine anymore. In this regard, computation time and memory utilisation, needed to solve the resulting large linearised systems of equations, increase quickly to multiple days and several GB of Random Access Memory (RAM), respectively. Hence, a parallel solution strategy will be applied using the interface M++/PANDAS as is described in Sect. 5.4. Following this, two computations are carried out. One consisting of intact discs throughout the lumbar spine and one having a stiffened L4–L5 motion segment, where the parameters for the lowest IVD are set such that they reproduce the mechanical behaviour of the surrounding stiff vertebrae.
In particular, the simulation of the healthy section of the lumbar spine is presented in the following subsection, thereby investigating the efficiency of the numerical implementation of the parallel solution strategy. Thereafter, the deformation behaviour of the degenerated lumbar spine is computed in order to analyse the influence of a stiffened L4–L5 motion segment in comparison to the results obtained for the healthy state.
6.5.1 Deformation Behaviour of the Healthy State
In the present study, the lumbar spine is modelled without processes and is represented by the geometry shown in Fig. 14(a). The 3-d geometry is discretised using 72320 quadratic 20-noded Taylor-Hood elements yielding a total of 982044 DOF. In order to induce a bending movement on the lumber spine, boundary conditions are set such that the lower surface of the L5 vertebra is totally fixed in space (\(\mathbf {u}_{S}=\mbox {$\mathbf {0}$}\)), while the top surface of the L1 vertebra is loaded with the traction vector \(\bar{\mathbf {t}}\). Herein, the Neumann boundary condition yields a horizontal (F_{H}) and vertical (F_{V}) force when integrated over the top-surface. For convenience, the traction vector is linearly increased over time until the resulting tip-loads of F_{H}=31.7 N and F_{V}=126.7 N are achieved at t=0.3 s, respectively. Note that all free side surfaces are always drained () and the molar concentration of the surrounding fluid is held constant at \(\bar{c}_{m}=0.15~\mbox{mol/l}\). The computation is then carried out in parallel on the Beowulf Linux cluster of the Institute of Applied Mechanics (Chair of Continuum Mechanics) at the Universität Stuttgart using 84 CPU simultaneously, cf. footnote on page 66. During the simulation, a constant time increment of Δt_{n}=0.003 s is used for the backward Euler time integration. Taking into account three smaller time steps in the beginning of the simulation leads to a total of 102 time (Euler) steps.
Important indicators needed to obtain an impression on the convergence behaviour of the parallel solver M++ during the solution of the nonlinear problem
Comp. time | Newton iter. | GMRES iter. | ||
---|---|---|---|---|
Average | 5:09 [min:s] | 1.6 | 146 | per Euler step |
Maximum | 15:19 [min:s] | 2 | 355 | |
Minimum | 0:23 [min:s] | 1 | 40 | |
Average | 3:10 [min:s] | – | 91 | per Newton step |
Maximum | 8:38 [min:s] | – | 270 | |
Minimum | 1:35 [min:s] | – | 40 | |
Average | 2:45 [min:s] | – | – | of the linear solver |
Maximum | 8:20 [min:s] | – | – | |
Minimum | 1:11 [min:s] | – | – |
6.5.2 Healthy Versus Stiffened Motion Segment
As expected, the force which is required to accomplish the same deformation with the stiffened lumbar spine is remarkably higher than the force needed for the healthy state of the lumbar spine. Moreover, the maximum total intradiscal pressure p in the L3–L4 disc is also raised from 0.7 MPa to 1.0 MPa comparing the healthy with the stiffened state. Note that this is even a higher value than the 0.95 MPa of total disc-pressure reached in the healthy L4–L5 disc. Hence, it is obvious that the stiffening of a motion segment causes a surplus load in the adjacent IVD, when deformations are kept constant. These studies are of particular interest, when a totally degenerated IVD has to be replaced. As flexible implants are still a challenging field for research, a simple cage implant is frequently used instead to fully demobilise the corresponding motion segment.
7 Summary and Outlook
7.1 Summary
The goal of this contribution was to present a model which is capable of describing the electro-chemically coupled deformation behaviour of saturated biological tissues in general and the intervertebral disc (IVD) in particular. In order to understand the necessary fundamentals from a biological point of view, a brief excursus into basic anatomy was presented firstly, thereby embracing the disc’s structural and bio-chemical composition.
Moreover, a detailed continuum-mechanical modelling approach was followed which is based on the thermodynamically consistent Theory of Porous Media (TPM). Herein, the general idea of the TPM was presented starting from the general case of a saturated soft biological tissue, where its pore-fluid is actually a mixture of an incompressible liquid and dissolved ions. This general approach was conveniently reduced to the presented extended biphasic model using Lanir’s assumption, which is the simplest possibility to still include electro-chemical effects like osmosis driven swelling phenomena. The missing constitutive relations that characterise the actual mechanical behaviour of the tissue were introduced in a modular manner. Herein, the focus was laid on the thermodynamical consistency of the proposed relations as well as on their admissibility in the finite deformation regime. In this regard, the isotropic part of the solid skeleton was modelled to behave viscoelastic using a generalised Maxwell model, where a very general Ogden law was used for the respective strain energies. In order to incorporate the purely elastic type-I collagen fibres, a polynomial strain energy was developed in accordance to the existing Ogden law, where it is possible to reproduce almost any nonlinear mechanical stress-strain behaviour. Moreover, the osmotic pressure contribution as well as the deformation-dependent Darcy filter law were included into the model such that they do not violate any thermodynamical restrictions.
As a next step, the obtained governing equations were rewritten in weak form which allowed for a convenient numerical treatment of the coupled system of partial differential equations using the Finite Element Method (FEM). In this regard, the spatial as well as the temporal discretisation was carried out using quadratic Taylor-Hood elements and the backward Euler time integration scheme, respectively. This finally led to a discrete system of coupled nonlinear partial differential equations. As the presented model also includes inelastic material behaviour of the isotropic solid skeleton, the global system of equations contains several local systems of evolution equations, which have to be solved before the global system can be solved. Following this, a multi-level Newton method was presented which allows for the efficient numerical treatment of both systems simultaneously. Finally, some concrete ideas were presented to couple the existing FE-code PANDAS to the parallel solver M++ or to any other commercial FE package.
The last part of this contribution is concerned with numerical examples which clearly embrace the capabilities of the presented time- and space-discrete set of governing equations. As a prerequisite, an algorithm was defined which is capable of computing the occurring inherent inhomogeneities with respect to fibre alignment and their corresponding location-dependent mechanical behaviour as well as the unequal distribution of the fixed negative charges. In a next step, the theoretically introduced material parameters were identified using experiments from the related literature. In this context, it was not always possible to directly determine all parameters using real-life experiments. Hence, a vast literature study was performed and the corresponding mean values for the respective parameters were presented.
Furthermore, in order to obtain an impression on the influence of the involved parameters on representative deformation modes of the spine, two parameter studies were performed on an L4–L5 motion segment in short-term bending and long-term compression. In this regard, it turned out that for short-term loading conditions, the IVD is mainly influenced by parameters characterising the solid skeleton, but especially the type-I collagen fibres were of prime importance. Hence, it can be summarised that special care should be taken regarding their inhomogeneous distribution as well as their location-specific mechanical behaviour, no matter if a single- or multiphasic model is used for simulation. A different result was obtained for the case of long-term loading conditions, where the permeability of the tissue plays also a key role.
Finally, a benchmark computation was presented addressing the deformation behaviour of the overall lumbar spine, thereby proving the efficiency of the underlying parallel solution strategy. Moreover, the same computation was performed on a degenerated lumbar spine having its L4–L5 motion segment stiffened. This led to a load increase in the neighbouring discs compared to the results of a similar deformation state of the healthy state.
7.2 Outlook
The present contribution successfully proved that the proposed extended biphasic model outlines a reasonable approach for the numerical simulation of the IVD. Following this, it is now possible to describe the healthy state of an IVD and the model can function as a numerical laboratory. This feature can significantly support research concerning the influence of the involved structural elements as well as the occurring inhomogeneities during other representative deformation modes, like pure torsion or torsional bending for instance. In this regard, it would be interesting to understand the reason for the inhomogeneously distributed constituents and what would happen, if the inhomogeneities were not included. Moreover, the influence of the loss of fixed negative charges with age on the fluid content of the IVD during long-term loading conditions can be studied straightforwardly. In this context, it is of general interest to extend the model towards degeneration effects resulting from regular ageing, mechanical damage or combinations of both. In order to do so, evolution equations need to be developed which describe the effects like the calcification process, the loss of fixed negative charges, etc. over time or during mechanical stimulus. However, the corresponding mechanisms, which are responsible for ageing or degeneration, are still not fully investigated and understood, as they are often coupled to each other. Thus, a numerical model could substantially help to fully reveal these complicated processes.
Furthermore, the described model should be implemented into a commercial FE code, which is capable of capturing the contact forces in the facet joints of the articular processes. In this regard, the geometry model should be advanced to capture all attached processes as well as the surrounding ligaments. At the same time, material parameters, which could not be clearly determined in the present study, should be identified from additional experiments and most important, the found parameters need to be validated thereafter. For the passive state, i.e., having no active muscle contractions, this can be accomplished using ex-vivo experiments. This state will allow to completely simulate the healthy state of a spine segment, which may remarkably aid the design of new implants. However, the respective boundary conditions for the active case, i.e., where muscle contractions are also responsible for the applied loads, are not fully understood yet. In this regard it is suggested to firstly develop a coarser and numerically cheaper model, for instance, a model based on multi-body dynamics. Herein it is possible to conveniently describe the spine as a multi-body system having active muscles as well as “dead loads” (resulting from the surrounding tissue) attached, while the presented IVD model can be included in a homogenised integral sense, cf. Karajan et al. [81]. With such a discrete multi-body system it will be possible to obtain reasonable magnitudes of the loads during everyday activities or even during traumatic situations, which can be applied as boundary conditions for simulations using the presented more detailed model.
Nuclei pulposi from young subjects absorb fluid equal to 95 % of their original weight when soaked in saline, while those from adult and aged subjects absorbed the equivalent of 55 % and 25 % of their weight, respectively, cf. Naylor [116] and qualitatively in [157].
Clayey silt has a hydraulic conductivity of about 10^{−8} m/s, while IVD tissue reaches down to a value of about 10^{−11} m/s.
Porous media Adaptive Nonlinear finite element solver based on Differential Algebraic Systems (http://www.get-pandas.com).
Calculations were executed on the Beowulf Linux cluster of the Institute of Applied Mechanics (Chair of Continuum Mechanics) at the Universität Stuttgart. It consists of 98 Opteron CPU (2.2 GHz, 47 dual boards and one quad board), 104 GB RAM (at least 1 GB/CPU) and two Gigabit networks, which share the load stemming from the system administration and the data transfer of the parallel computations.
Acknowledgements
I thank the German Research Foundation (DFG) for funding the research project “Diffusions- und Strömungsprozesse in der anisotropen menschlichen Bandscheibe” under grant number ‘Eh 107/16’.