Biomechanics of Hydrocephalus

Abstract

In this chapter, a state of the art of the biomechanics of hydrocephalus is carried out. Firstly, the history of the mathematical modeling of the condition is presented, with the contribution from every researcher analyzed. With the first works on the field dating back from the 1970s, numerous approaches have been used, be it of thermodynamical, mechanical, or electrical nature. The improvements in computational power also have allowed to make extensive use of the numerical tool: this has led to more realistic geometries (notably with the help of developments in medical imaging) and much more accurate assessments of the relevant physiological parameters. Secondly, an in-depth investigation of the most popular models aims to reveal their strengths and insufficiencies. This part also shows how to build a rigorous model for ventricular dynamics, along with its coupling to surrounding compartments. The way hydrocephalus manifests itself in the mathematical equations is evidenced, and an application to pediatric hydrocephalus is presented.

Keywords

Introduction

A recent editorial of the prestigious journal Physical Review Letters of the American Physical Society (Beggs 2015) raises the important matter of an emerging “brain physics.” Far beyond the development of new investigation tools such as TEP, MEG, or phase-contrast MRI which reveal functional features of the brain, this questioning regards the ability of physics to deal with biological complexity regarded as the ultimate form of organization raising insurmountable difficulties. Understanding the emergence of consciousness from the brain dynamics and simply clarifying cognitive processes are typical goals of such initiatives. But the brain is just one piece of the wider intracranial system, composed of many coupled compartments endowed with their own individual dynamics. The notion of complexity we referred to can be thought of as the coordination of the individual dynamics through appropriate couplings between the components of the system and the intricacy of space and time characteristic scales. It is clear that this coordination process encodes the architecture of the whole system. Dynamical and structural complexities are thus tightly connected. These are the main obstacles toward a deeper understanding of the dynamics of such complex systems. On the clinical side, the most invaluable benefits of a relevant physical approach to the complexity of biological systems regard the possibility to predict and classify the pathological (or oppositely the healthy) conditions of such systems in keeping with specific dynamical behaviors. Though not fully realized yet, this research program can result in new curative strategies through a better understanding of the processes involved in any disease. Mixing the structural complexity of the intracranial system and alterations of brain dynamics due to a change in the mechanical state of the brain, hydrocephalus illustrates the previous considerations. Hydrocephalus is a typical complex condition of the intracranial system exhibiting features allowing its association with pathological dynamics. Because of the large number of organs, tissues, and fluids it affects, hydrocephalus is considered a very complex condition to model mathematically. The main obstacle to a rigorous description of this affection is the complexity of the intracranial system, inherent first to its anatomical structure involving numerous liquid compartments such as arteriovenous blood and cerebrospinal fluid or CSF , coupled by fluid exchange and mechanical deformations but also and mainly to all the processes of biological regulation acting within the intracranial system. In order to write the equations of evolution of the system, of which the rigorous form is most certainly complicated, it is very usual within the community researching in this field to proceed to simplifications, of very different natures. At the geometrical level (representation of the brain, the spinal cord, etc.), for example, the complex shape of liquid compartments can be simplified, and it is naturally possible to reduce the number of compartments by retaining only those playing a dominant role in the encountered problem. The simplification of the dynamics may also be performed through a careful linearization of these dynamics after having identified the steady (out of equilibrium) states around which the system is maintained. But this linearization scheme of the rigorous dynamics is generally not applied in the literature relative to this field, and this is even more understandable as the accurate dynamics is typically unknown, owing to the biological complexity of the systems.

Nevertheless, the search for linearized dynamics can be fulfilled in a phenomenological way. In this way, one of the most common approaches is the construction of electrical analogies of intracranial biomechanics. Behind the choice of the electrical modeling lie two things. On one hand, there is the strong idea that the dynamics are linear (presented in the literature as a work hypothesis) and on the other hand, lies the intrinsic simplicity of the laws of electricity. This approach can be justified by the fact that within the physical laws governing the intracranial processes prevail hydrodynamics which display strong analogies with the laws of electricity. For example, we may cite the parallel between Poiseuille’s law and Ohm’s law.

The ad hoc nature of these models arises from the fact that they are not derived from the exact dynamics. As such, several parameters of the models suffer from a defect of physical interpretation, because they are not linked to biological or physiological parameters of the system. The possibilities of measurements of those parameters are therefore very limited. One of the main features of such electrical approaches is their holistic essence; they are mostly global models of the intracranial system, i.e., imply the whole set of intracranial compartments. These models accurately reflect the anatomical structure through a subdivision into several compartments: arterial and venous blood, ventricular and subarachnoid CSF, and brain. Each compartment is characterized by its compliance, which depends on both its geometry and its elastic properties. From the electrical point of view, this compliance is the equivalent of a capacitance. The compartments are connected by hydrodynamical resistances to the flow which mimic the dissipative properties of both the exchanged fluids and the biological tissues. These resistances and capacitances represent the fundamental parameters of a minimal linear model of the system. When the structure of the electrical analogue is chosen, the determination of the components is done by performing a comparison with the measured data (e.g., evaluated using flow MRI). Unfortunately, available data generally are not enough to determine all parameters of the global model: an infinity of solutions are accessible, each of which corresponding to a possible model subject. To a first level of arbitrary displaying our lack of a perfect knowledge of the intracranial system succeeds a second level of arbitrary associated with the shortage of experimental data (that is, the overabundance of the global model parameters). Regarding electrical analogies, while the main aspects will be presented in a later section, one may refer to Ursino (1988), Ursino and Lodi (1997), Takemae et al. (1987), Linninger et al. (2005), and Ambarki et al.’s works (2007) for the main references.

Other approaches to the mathematical modeling of hydrocephalus include pressure-volume models and purely biomechanical or biophysical models. The former ones are linear models (often of the first order) relating the CSF pressure variations of one or several compartments to its corresponding volume variations. We may cite the articles of Guinane (1972), Marmarou et al. (1975), Sklar and Elashvili (1977), Rekate et al. (1988), and Drake et al. (1994) for deeper understanding. This class of models is abundant in the literature but has lately mostly been succeeded by the latter type of models. Those are very refined models making use of the laws of physics, generally designed by physicists, mechanicians, or mathematicians interested in the modeling of such abnormalities, such as Nagashima et al. (1987), Kaczmarek et al. (1997), Bouzerar et al. (2005), and Tenti et al. (1999, 2008). The numerical tool (in the form of finite element and/or finite volume analysis) is also extensively used, as the equations governing the dynamics are coupled and may exhibit highly nonlinear behaviors (Tekaya 2012).

Beginning with expressing basic geometrical and physical laws, the details of previously cited models will be exposed, in order to exhibit their principles and limitations. Extensions of such models to pediatric cases will also be shown, evidencing the very complex nature of intracranial dynamics.

From the Monro-Kellie Hypothesis to the Electrical Analogies

Numerous authors operating electrical analogies use the Monro-Kellie hypothesis as their basis (Ambarki 2006; Tekaya 2012). This doctrine, enunciated during the eighteenth century, depicts the constraint imposed by the cranial rigidity to all the matter that the skull encloses, that is, the liquids and other intracranial matter (blood, CSF, brain matter).

In a publication edited in 1783, Alexander Monro, a Scottish anatomist, applied certain physical principles to the intracranial compartment. He formulated four “hypotheses” expressing well-established empirical facts:

• The brain is confined in a nonexpendable case of bone.

• The brain matter is almost incompressible.

• The volume of intracranial blood is nearly constant.

• A continuous outflow of venous blood out of the cranial compartment is required to allow for arterial blood to continuously flow within the skull.

His pupil, George Kellie, then achieved several experimental analyses on both the human and animal brains and concluded that the hypotheses emitted by Monro were right (Kellie 1824). Later on, the Monro-Kellie doctrine was validated by other observations, such as the exsanguination experiments carried on animals by Abercrombie in 1828.

Monro, Kellie, and Abercrombie ignored the presence of CSF in the craniospinal enclosure. In fact, the first complete description of the CSF dates back to 1842. It arises from the work of François Magendie (1842), a famous French physiologist who was the first to analyze the production, absorption, and flow of CSF. He described the existence of a communication between the fourth ventricle and the subarachnoid spaces. The work of Magendie showed that the CSF is a crucial element of the craniospinal system. This important discovery implied deep consequences on the understanding of the repartition of the intracranial liquids. Indeed, quickly later (in 1848), a reconsideration of the Monro-Kellie hypothesis was proposed by George Burrows. This English physician repeated Kellie’s experiments while taking into account the presence of this precious liquid. He characterized the compensation relationships between the intracranial volumes of CSF and blood: their respective variations were opposed. In 1926, Harvey Cushing made the Monro-Kellie relationship even more accurate by stating that, for a rigid skull, the sum of CSF, brain substance, and blood volumes is always constant. In other words, the increase of one of these volumes leads to a decrease of one or both other volumes. From a more formal point of view, this Monro-Kellie relationship expresses a purely geometrical constraint imposed to the volumes of all intracranial substances by the rigidity of the skull. In the adult subject, the intracranial content is distributed between the brain volume V_M (80%), the blood volume V_B (10%), and the CSF volume V_CSF (10%) (Grévy et al. 1998). These facts allow to write the law of conservation of the intracranial volume V_IC,

$$ {V}_{IC}={V}_M+{V}_B+{V}_{\mathrm{CSF}}=\mathrm{constant} $$

(1)

This law is of fundamental importance in the understanding of the dynamics of the intracranial system; it enables a better understanding of how these volumes change during a cardiac cycle and which parameters are the most determinant for their variations (Figs. 1 and 2).

Fig. 1

Schematic illustration of the Monro-Kellie-Cushing hypothesis (generalized Monro-Kellie hypothesis). An increase in any of these compartments will lead to a decrease in one or more of the remaining compartments

Fig. 2

Relationship between the CSF and blood flow rates. ${\dot{V}}_A(t)$: arterial blood flow rate entering the skull. ${\dot{V}}_V(t)$: venous blood flow rate exiting the skull. ${\dot{V}}_{\mathrm{CSF}}(t)$: CSF flow rate oscillating between the skull and the spinal cord

Within the skull, the CSF is distributed between the ventricles and the subarachnoid spaces (SAS), which allows discerning between the “ventricular volume” and the “SAS volume.”

Also, the blood present in the arterial and venous network will be characterized by their volume and designated, respectively, by “arterial volume” and “venous volume.”

Most authors who published in the field of the central nervous system hydrodynamics considered the CSF, blood, and brain tissue to be incompressible. Few other authors insist on the compressibility of the brain parenchyma.

The cerebrospinal fluid and the brain tissue are subject to oscillatory motions (Enzmann and Pelc 1992, 1993) due to cardiac contractions that generate pulses transmitted by the blood flow to the brain parenchyma. During the systole, a massive amount of arterial blood flows into the head. Because the bones that make up the skull are rigid, this blood intake causes a transient volume increase of the intracranial arteries and small craniocaudal displacements of the cerebral structures ranging from about 0.1–0.5 mm (Enzmann and Pelc 1993). These movements are spread to the intracranial CSF that in turn causes a blood expulsion through the veins (Greitz et al. 1994). Another part of the CSF is moved from the intracranial SAS to the spinal cord.

During the diastole, the moved CSF comes back to its initial place in the skull. The volume variation induced by the blood intake within the skull is distributed in two components. The largest part regards the blood expulsion in the internal jugular veins, and the other component deals with the movement of part of the CSF toward the spinal cord.

This volume repartition can be mathematically written in the form of the following equilibrium equation,

$$ {\dot{V}}_B(t)+{\dot{V}}_{\mathrm{CSF}}(t)=0 $$

(2)

where ${\dot{V}}_B(t)$ and ${\dot{V}}_{\mathrm{CSF}}(t)$, respectively, denote the volumetric variations (time derivatives of volumes) of both blood and intracranial CSF through time, during a cardiac cycle. The volume variations of intracranial blood during the cardiac cycle ${\dot{V}}_B(t)$ are given by the instantaneous difference between the incoming arterial flow and the outgoing venous flow, that is,

$$ {\dot{V}}_B(t)={\dot{V}}_A(t)-{\dot{V}}_V(t) $$

(3)

Here, ${\dot{V}}_A(t)$ and ${\dot{V}}_V(t)$ are, respectively, the instantaneous arterial and venous blood flow rates. ${\dot{V}}_B(t)$ represents the “vascular flow rate” or “arteriovenous flow rate.”

Classification of the Different Models of Intracranial Dynamics

The Pressure-Volume Models

Such models require a preliminary description of the pressure-volume (p − V) relationship inside the skull. This relationship is generally unknown because of the difficulty to measure it, as the necessary insertion of probes into the intracranial compartments is a delicate task. The relationship between these quantities takes the form of a functional that in the linear case writes,

$$ V\left(\overline{p}+\delta p\right)=V\left(\overline{p}\right)+\underset{-\infty }{\overset{+\infty }{\int }}\delta p\left(t-{t}^{\prime}\right)h\left({t}^{\prime}\right){dt}^{\prime } $$

(4)

In this expression, V denotes the volume of the liquid compartment considered, $\overline{p}$ is the steady pressure or possibly subject to slow variations, and the function h(t) characterizes the physical properties of the compartment (such as its ability to deform). The component δp reflects the contribution of the quick processes to the pressure variations (the response in pressure to abrupt excitations). It is worth noting that it is possible to obtain similar equations describing possible additional compartments coupled to the first one, thus generating a differential system depicting the whole system. In this approach, the dynamics of the compartment may be written,

$$ \frac{dV}{dt}=\dot{V}=S-R+{\dot{V}}_e+{\dot{V}}_{exc} $$

(5)

The quantities S and R describe the CSF secretion and resorption rates (that can possibly be zero) in the compartment. The difference between these terms controls the slow variations of the pressure p. The two other terms, respectively, denote the flow rate of exchanged fluid with the other compartments (${\dot{V}}_e$) and the excitation (${\dot{V}}_{exc}$), describing the volume variations induced by arterial or venous pulses. The most frequent form (but also the simplest) of the exchange term is that given by Poiseuille’s law, where the flow rate is linearly proportional to the pressure difference between the compartments involved. The previous relationship simply depicts the law of conservation of mass. A development of the convolution immediately leads to an equation that takes the form,

$$ S-R+{\dot{V}}_{exc}- G\delta p\approx \frac{d\delta p}{d t}\underset{-\infty }{\overset{+\infty }{\int }}h\left({t}^{\prime}\right)d{t}^{\prime }-\frac{d^2\delta p}{d{t}^2}\underset{-\infty }{\overset{+\infty }{\int }}{t}^{\prime }h\left({t}^{\prime}\right)d{t}^{\prime }+\dots $$

(6)

Here, G represents the hydrodynamic conductance of the paths of fluid flow connecting the various coupled compartments. The missing terms that depend upon the steady pressure $\overline{p}$ have been absorbed in the excitation term. Then, depending on the degree of biophysical realism desired and the relevance in the considered phenomena, the authors retain more or less terms. In the literature relative to the topic, only the first derivative of the pressure modulations is kept, and the following quantity is introduced,

$$ C=\underset{-\infty }{\overset{+\infty }{\int }}h\left({t}^{\prime}\right){dt}^{\prime } $$

(7)

thus defining the static compliance of the compartment. It can also be expressed in terms of the Fourier transform H(ω) of the function h(t) by C = H(ω = 0), that is, the zero-frequency component of H. This compliance evidently depends upon the steady pressure of the compartment, its geometry, and its elastic properties. The various subclasses of the pressure-volume models are easily deduced from Eq. (5) by eliminating certain terms in favor of the processes sought to be isolated. The following table presents these sub-models encountered in the literature (Table 1).

Table 1 Classification of the various types of p − V models in the literature with the objectives sought

The authors familiar with this type of approach often focus their efforts on the dependence of the compliance upon the pressure. The C(p) relationships invoked are often empirical relationships arising from experimentations on animals: one of the most frequent of them is the one stated by Marmarou et al. (1975), very used despite the system instability it generates. The experimental process followed by Marmarou’s team and repeated by other authors consists in the injection (or withdrawal) of a certain quantity of CSF in intracranial cavities, closed beforehand, and simultaneously measures the corresponding pressure variation. The relationship proposed by Marmarou et al. derives from a fit of the curve obtained: he suggests an exponential evolution of the pressure with the injected volume. Similar studies (Lofgren and Zwetnow 1973) carried out after an injection/withdrawal of CSF in the spinal cord showed that the exponential relationship advanced by Marmarou and colleagues also fitted their data, even though the curves measured were different at several levels from Marmarou’s ones. The distinct geometry and possibly the different mechanical properties may partially justify these differences. Yet, the disparity of the experimental setups used could have a non-negligible role in these differences. It indeed seems that the pistons within the tubes that release the physiological saline influence this relationship. A deeper examination of these experiments would therefore be interesting. Additionally, the relationship obtained experimentally is also influenced by the properties of the fluid (equation of state) itself. The compliance deduced thus does not characterize the elastic properties alone of the compartments considered. This problem is never evoked by the authors, most likely due to its complexity.

The Electrical Models

Modeling a dynamical system by using electrical analogies relies on the similarity of the laws (or at least, in certain dynamical regimes of the system) governing the system at the linear approximation and the laws of electricity. Indeed, the linearized system can always be considered as a filter, once the excitation and response are identified (or chosen). The electrical analogue may then appear as a physical realization of this filter. Another justification of the analogical approach of intracranial dynamics is that the flows of physiological fluids are ruled by the laws of hydrodynamics which display a striking similitude with the laws of electrodynamics. However, this similarity is often made inaccurate by the limitation of some authors to restrain their description of the fluid flows to Poiseuille type flows, ruled by a kind of “Ohm’s law,” of which the effective realization within the intracranial case is not fulfilled. This forms one of the main limitations of this analogical method. The first way, based on the notion of linear filter, is superior but limited by the fact that the filter is not physically achievable. At this first problem is added that of the stability of the filter (in the sense of the formalism of signal processing), that is rarely, if not never, looked upon by the scientists. Yet this question is fundamental: representing a healthy subject by using an unstable filter has no physiological meaning.

It is actually the complexity of these questions that forces the “ad hoc” character of the electrical analogy. The linearity imposed to the model is done without justification, and the nature of the components (often passive) that compose the electrical circuit often lacks justification. Nevertheless, the most prevalent minimal electrical model in the literature systematically includes resistors and capacitors. This kind of structure is relevant from a physical point of view because it accounts for two fundamental characteristics of the studied systems: the capacitors mimic the compliant property, that is, the elasticity of the liquid compartments (able to deform, that is, undergo a volume variation under the effects of a pressure variation), and the resistors describe the dissipative character of dynamics. This minimal structure is present in all electrical analogues, independently if the author’s motivation. In this manner, in Alperin et al.’s model (1996), the intracranial and spinal contents constitute a sole body characterized by a compliance C in parallel with a conductance G. He thus associates a first-order linear filter to the system of which the excitation arises from heartbeats. More precisely, this excitation is transmitted by the arteriovenous flows, considered an input variable, and the corresponding responses (output variables) lie in the spinal and intracranial CSF flows. From a quantitative point of view, Alperin and associates calculated the corresponding transfer functions under the form of a low-pass filter of which the cutoff frequency yields the time constant of the system examined. Therefore, the identification of these transfer functions allowed them to establish an electrical analogue of the intracranial system. Its global feature is very clear as it compacts in only one circuit the whole intracranial structure along with the spinal cord. This simplification illustrates the necessary reduction of the complexity of the intracranial space in order to obtain a model of it. However, a frequently encountered problem is the spatial location of the distributed properties, such as the elastic or dissipative properties. This approach from Alperin’s team still holds an undeniable quality: the inputs or outputs of the model are variables that are measurable, using phase-contrast MRI, for example.

In 1969, Agarwal et al. suggested a new multicompartmental model based on the Monro-Kellie doctrine, which represents distinctly the arterial, venous, jugular, and capillary compartments and the spaces occupied by CSF . Their model, presented in hydrodynamical terms (since the heart is considered a hydraulic pump causing the blood to flow) and shown on Fig. 3, also relies on the electrical analogy. It is worthwhile noting that in this model, the skull and brain parenchyma are not represented. The blood and CSF are mechanically coupled through compliances C₁, C₂, C₃, C₄, materializing the deformability of the vessels. In addition, Agarwal et al. took into account CSF secretion/resorption around the capillaries with resistances R₂, R₄ describing the CSF resorption within the venous sinuses. The other parameters are the vascular resistances (R₁, R₃, R₅, R₆).

Fig. 3

Hydrodynamical model of the intracranial system, as proposed by Agarwal et al. (1969)

This model, although quite complete and close to the anatomo-physiological reality, still holds the problem of the spatial location of the distributed biophysical properties. A deformable blood vessel is certainly not equivalent to a series RC circuit (or parallel). Besides, this model could not be used practically because of the difficulty to measure pressures within the skull.

With the same idea, Takemae et al. (1987), inspired by Agarwal et al.’s work, proposed a simplified electrical model to explain the transmission of pulsatile flows to the cerebral circulation (Fig. 4). This model made up of four compartments does not include the CSF secretion and resorption processes (indeed negligible during one or some cardiac cycles). It is supplied by a voltage source mimicking the physiological arterial pressure curve. The elements (R_a, C_a), (R_k, C_k), and (R_v, C_v), respectively, denote the resistances and compliances of the intracranial arterial, capillary, and venous compartments. C_b is the compliance of the whole intracranial content, and ${R}_{v_0}$ is the resistance of the extracranial veins. According to Takemae et al., the intracranial pressure (ICP) modulations are reproduced by the voltage across the compliance C_b.

Fig. 4

Electrical model of the intracranial biomechanics as proposed by Takemae et al. (1987)

The model, developed by Ursino (1988a, b), completes that of Takemae and colleagues by taking into account the CSF secretion/resorption processes but also by describing the self-regulation of blood flows through an adjustment of the vascular resistances. This last process is rarely described by authors because of its extreme complexity. Its structure is presented on Fig. 5. The author focuses on the morphology and dynamics of the intracranial pressure by investigating the influence of the parameters of his model on this physiological signal. These parameters were estimated from anatomical and physiological data of the human intracranial system. This model comprises two sources of excitation: the electric voltage P_a simulates the arterial pressure, and the current I_i mimics the injection of small CSF volumes within the skull. From a normal arterial pressure curve, the author proceeds to simulations of the vascular and intracranial pressures and studies the relationships between the ICP and the pressures within the arteries, capillaries, and veins.

Fig. 5

Electrical model of the intracranial system suggested by Ursino (1988a). C_vi, C_ve, and C_tiss are the compliances of the intracranial arteries and veins, the extracranial veins, and the brain parenchyma (or CSF in this model). R_ai, R_pv, R_vs, and R_O, respectively, denote the resistances of the intracranial arteries, the proximal veins, the sinuses, and the CSF resorption

The Biomechanical Models

As stated earlier, the very first models for hydrocephalus in the 1970s (p − V models) basically did not take into account any spatial variations of these physical quantities. As such, the pressure and volume distributions were uniforms. A refinement was necessary in order to describe correctly the development of abnormalities such as hydrocephalus. This was the beginning of the biomechanical approach to these models. Hakim et al. (1976), Hakim and Hakim (1984) were among the first to take into consideration the porosity of the brain and its reaction to an enlargement of brain ventricles. As the brain is a porous medium, it would be unnatural to think that CSF might not flow through it. This class of models is known as the poroelastic models, which are approaches that take into account both the elasticity and the porosity of the media. Then, Nagashima et al. (1987) extended these studies by including the effects of the viscosity of CSF, that is, proposing a model relying on Biot’s theory of consolidation (1941). This theory, extensively used in soil mechanics, helps in describing the mechanical behavior of the interaction of a viscous fluid flowing through a porous medium. Applying the finite element method on a two-dimensional system made up of ventricles surrounded by a porous brain, they managed to compute the pressure and strain distribution within the brain as a result of a ventricular dilation. Nagashima and associates did however the choice of overly simplified boundary conditions and did not mention what led them to choose them, which is debatable. As all problems in physics and mechanics, the choice of appropriate boundary conditions is crucial and strongly determines the profile of solutions found. This work yet had the merit to exploit real geometries of the spaces in study, by exploiting computed tomography scans of brains. In addition, the main problem encountered, also met by many authors in the domain, is the lack of accurate values of the relevant biophysical parameters. In the case of Nagashima and colleagues, the Poisson ratio (a mechanical measure of compressibility) was the most problematic parameter. This usually leads to results outside of physiological ranges and incoherent consequences. But in this study, the researchers found that the zones of largest stresses and strains were the anterolateral angle of the frontal horn and the occipital and temporal horns. That is in agreement with observations in clinical cases of acute obstructive hydrocephalus. They also showed that the interstitial pressure increased around the periventricular area as time went on. These regions, in clinical situations, are those of fluid accumulations and edema. While these studies were early, they clearly showed the feasibility and predictive potential of numerical models. Following Nagashima’s way in the use of the numerical tool, Tada et al. (1990) and Peña et al. (1999) carried out numerical simulations on the subject as well, much like Taylor and Miller (2004) who performed a 2D finite element simulation to better assess the Young’s modulus of the brain parenchyma, using commonly accepted data of brain strain rates under hydrocephalic conditions. They although noted a difficulty in the fact that as the brain is modeled as a biphasic substance, the Young’s modulus will change as load is applied: as the brain is filled with CSF, an initial load can be tolerated, but as the load goes up, CSF will be squeezed out, and the parenchyma will solely support the load and will undergo deformation. The numerical results show that the Young’s modulus can therefore decrease by one order of magnitude, which is substantial. In Peña’s work, which basically refined Nagashima’s linear poroelastic model, a time-dependent evolution of ventricular expansion was presented. Their results indicated that increases in ventricular pressure led to dilation of the frontal and occipital horns and lateral flattening of the thalami (Clarke and Meyer 2007). We shall also make mention of extensions to porous viscoelastic models that were primarily done by Tenti et al. (1999, 2000) and Sivaloganathan (2005a, b). Unlike most researchers who made use of simulations, they employed simplified geometries to find analytical solutions of the pressure and deformation.

An interesting work is that of Wirth (2005) who accurately refined existing models by using a fully nonlinear and strain-dependent permeability and integrating a CSF source or sink in the brain parenchyma. Lacking the constitutive laws to model nonlinear elasticity, he used linearized elasticity but paved the way for the inclusion of nonlinear elasticity in future models. His study could, for example, account for CSF flow through the brain tissue as a result of a blockage of the aqueduct, by taking into account the flow diffusion process through the ependyma. The symptoms of hydrocephalus such as ventricular dilation, edema, and increased ICP could be interpreted by mechanical principles, and assessments of the distribution of pressure, stresses, and other important biophysical quantities within the brain matter could be performed. He could also determine the change in brain compliance as a result of a variation of the biomechanical parameters (especially the brain poroelasticity) (Fig. 6).

Fig. 6

Distribution of pressure (upper row) and fluid content (lower row) for two bent ventricles, as found by Wirth (2005). The graphs are cross sections at three different heights. In the lower row, the ventricles are shaded to make the increased fluid content around them more evident. Light areas correspond to higher pressure or fluid content values

Interestingly, he considered the case of infantile hydrocephalus by introducing a crude model of the cranium which captures the main features, although lacking precious data about the geometry and mechanical properties of the child’s cranial bones. Replacing the standard boundary conditions he used for an adult with new ones taking into account the nonrigid properties of the infant’s skull, he obtained solutions presented on Fig. 7.

Fig. 7

Simulation results for infantile hydrocephalus with head enlargement for a blocked aqueduct (dotted line) and for an impaired subarachnoid absorption (dash-dot line), as exposed by Wirth (2005). The solid and the dashed line represent the same simulations; only head enlargement is prevented, e.g., by compressive head wrapping. Top row, from left to right: CSF pressure, CSF velocity, and increment of CSF content. Bottom row, from left to right: brain displacement and brain stress tensor normal components. All quantities are plotted as functions of the ventricular radius

It is worth noting that most authors to model the CSF flow within the aqueduct of Sylvius used Poiseuille’s law, but this assumption is severely undermined by the fact that the flow is pulsatile in nature. Indeed, due to the heartbeats, corresponding pressure waves flow along the arteriovenous network, making the fluid flows within the body oscillatory. Womersley (1955), while studying the arterial flows, showed that a viscous fluid moving in a circular tube under a pulsatile pressure gradient had an oscillatory motion as well, with a phase lag between the pressure gradient and the flow rate increasing with a dimensionless number, dubbed the Womersley number. Sivaloganathan et al. (2005a, b) were among the first to include this important fact to further improve the depiction of the CSF flow. Numerous authors, seeking to model the CSF flows within the ventricles, subarachnoid spaces, and spinal cord, resort to a Womersley representation of the flow. We may cite the works of Loth et al. (2001), Peng et al. (2003), Linninger et al. (2005, 2007), Gupta et al. (2010), Sweetman (2011), Bouzerar et al. (2012), and Tekaya (2012).

While the two previous parts were devoted to a survey and a tentative classification of the biophysical approaches to hydrocephalus, in the next part, we intend to point out and discuss the unavoidable weaknesses of these models and especially those arising from the problematical handling of the anatomical and functional complexity of the intracranial system. Possible and workable solutions to these difficulties are proposed which lead to a unified approach to different pathologies of the system.

Main Interests and Insufficiencies of the Usual Approaches to Hydrocephalus

Within the large variety of mathematical attempts to capture the main features of hydrocephalus, the modeling strategies differ primarily by the way the complexity of the brain/CSF system is handled. The survey of the available literature evidenced an irreducible anatomical ingredient common to all types of models, the intracranial system subdivision into well-defined compartments. This subdivision comprises the ventricular space, the parenchyma irrigated with the complex arterial blood network, and the subarachnoid spaces. Though its architecture is clearly identified, the difficulties in modeling the system arise from the physical couplings between the compartments, may it be direct, through matter transfer, for instance (CSF flow through the aqueduct of Sylvius), or indirect through the brain strain modulations (brain volume variations under blood flow excitation). Whatever the model, the mathematical representation of these couplings determines its nature, abstract, or intuitive. However, dealing with an abstract model, though quantitatively relevant, can lead to strong difficulties in its handling and could not be useful in diagnosis assistance. The study of these models and their formal structure suggest further criticism. These approaches to hydrocephalus can be divided into three classes, according to their main focus and the incorporation of the complexity of the system:

• Thermodynamical models (as pressure-volume models) based on a CSF state equation

• Biomechanical models (e.g., porous viscoelastic brain models)

• Electrical analogies of the whole intracranial system

The transition from one category to the next goes clearly with increasing structural complexity. Thermodynamics-based approaches can indeed be applied to individual compartments (e.g., ventricular CSF), but global electrical analogues provide a holistic approach. As can be clearly seen from the literature survey, descriptions of the “pressure-volume” type are linear or most rarely nonlinear models of zeroth order (static model) or first order (dynamical model), relying on a p − V relationship either hypothetical or extracted from any empirical pressure dependence C(p) of the compliance. The same notion of compartment compliance is of course one of the main components of any electrical analogue where it is interpreted as a capacitance. The residual nonlinearity (reflecting the most general C(p) relationships) and the multiplication of the compartments and their mutual couplings should be interpreted as a tentative account for the complexity of the intracranial system.

The complexity account is not the only realistic feature of such models. Indeed, some of the adjustable parameters of any model can be set to a realistic value reflecting the physiological reality through a comparison to observational data. When the complexity is too high (large number of free parameters), the set of observational data being incomplete, the missing information appears as a residual arbitrariness of the model (undetermined parameters). This is the reason why these models can so hardly differentiate healthy dynamical behavior from pathological ones. Most models try to account for the physiological consequences of hydrocephalus, but these are often ad hoc models, that is, they ignore the underlying mechanisms and fail deriving pathological behaviors as a consequence of the approach. The very origin of this prediction failure arises from oversimplifications and difficulties in the assessment of some parameters making unfeasible the comparison to available medical data. Ideally, a relevant physical description of the mechanisms underlying intracranial processes is thus needed for a realistic and complete modeling which should also predict abnormal behaviors, likely in keeping with existing pathologies. This requirement of predictive capabilities is certainly a laudable aim but difficult to reach. Nevertheless, the gain on the clinical side and especially the applications to diagnosis assistance would be invaluable. Obviously, the connection of these models with clinical facts is an important feature. At large, the consideration of hydrocephalus varies from one category to the next though it consists primarily of the many possible causes of that trouble hypothesized by physicians. Accordingly, we may distinguish as major causes the blocking of the aqueduct of Sylvius or any obstacle in the CSF pathways within ventricles and the anomalous CSF secretion or resorption within the ventricles or within the parenchyma: these causes are the main focus of many models of hydrocephalus.

The specific difficulties to face are well identified for each category. The exact CSF state equation needed in any thermodynamical model depends crucially on the deformability of the shell demarcating the corresponding compartment. The elasticity of the shell should be accounted for properly. The porous viscoelastic model of the brain captures CSF absorption, but the CSF hydrodynamics through a porous random network coupled to the brain deformations remains problematic. The third category is of peculiar importance since it groups most of the models encountered in the literature. The use of electrical analogies relies partly on the similarity between the linear sector of the intracranial dynamics (assumed to exist even if it is not derived from the rigorous dynamical laws, far beyond our practical knowledge of the intracranial system) and the well-known laws of electricity. In fact, electrical analogy is also made natural by the presence of physiological fluid flows governed by the laws of hydrodynamics, very similar to the laws of electrodynamics. In the practice of electrical analogy, the compartments are reduced to one or a few electrical components as capacitances and resistances which mimic their elastic and dissipative properties, but this is achieved through a strong simplification, the localization of the elastic properties of the tissues (shells, arteries), as discussed in the first part of the chapter. Electrical analogies are very interesting and useful because they provide remarkably simple and workable models, but this approach raises unavoidable issues. The lumped elasticity of the tissues demarcating a liquid compartment is a strong approximation since each part of any elastic envelope (blood vessels, ependyma) manifests that elastic behavior. More acutely, that approximation warps the actual dynamics of the compartment, skewing the conclusions drawn from such models. This can be easily understood. Though the actual dynamics of an extended elastic shell incorporates the mechanical coupling between distant parts of it, this property is lost in the lumping limit as well as the information about the whole compartment. Due to the compartments being devoid of any structure (extreme complexity reduction), the description can be made unrealistic. These conclusions regard elastic properties of the tissues as well as their dissipative properties. Indeed, elastic membranes obey a dissipative proper dynamics expressed by each portion of the membrane. Misleading conclusions due to the lumping approximation applied to dissipative elastic tissues can be illustrated in a simple situation.

Let’s consider an elastic vessel transporting any fluid (e.g., blood) as schematized on Fig. 8 below.

Fig. 8

Schematic view of an elastic vessel deformed by the fluid flow pressure inside (field p(x, t)) and associated electrical analogue. Each portion of the vessel is depicted as a simple RC circuit with a voltage across it given by the pressure

The n-th elementary portion of the vessel of length dx is viewed as a simple RC circuit fed with a “voltage” p_n(t) (fluid pressure within this portion). The identity of the R and C parameters for each portion manifests the homogeneity of the mechanical and physical properties of the vessel tissues. In this discrete scheme, the actual pressure field p(x, t) is approximated to p_n(t), and the number of portions should be as great as possible. With the current flowing through the n-th portion being denoted i_n(t), it is easy to establish the following equations relating the currents and voltages (Kirchhoff’s laws),

$$ \left\{\begin{array}{c}{p}_n={p}_{n+1}-R{i}_n\\ {}{i}_{n+1}={i}_n+C\frac{d{p}_{n+1}}{dt}\end{array}\right. $$

(8)

We are thus led to the discrete assessment of the pressure and current field space derivatives (pressure and current gradients within the portion n),

$$ \left\{\begin{array}{c}{\left(\frac{\partial p}{\partial x}\right)}_n=\frac{p_{n+1}-{p}_n}{\delta x}=\frac{R}{\delta x}{i}_n\\ {}{\left(\frac{\partial i}{\partial x}\right)}_n=\frac{C}{\delta x}\frac{\partial {p}_n}{\partial t}=\frac{\delta x}{R}\frac{{\left(\frac{\partial p}{\partial x}\right)}_{n+1}-{\left(\frac{\partial p}{\partial x}\right)}_n}{\delta x}\approx \frac{\delta x}{R}{\left(\frac{\partial^{{}^2}p}{\partial {x}^{{}^2}}\right)}_n\end{array}\right. $$

(9)

To recover the actual vessel, the number of portions should be infinite or equivalently the length δx → 0 (continuum limit of the discrete model), but the ratio δx²/RC remains finite (condition for a nontrivial continuum limit) resulting in a diffusion equation for the pressure fluid,

$$ \frac{\partial p}{\partial t}\left(x,t\right)=D\frac{\partial^{{}^2}p}{\partial {x}^{{}^2}} $$

(10)

with a diffusion constant D = δx²/RC. This result means that the characteristic time RC of any elementary portion of the vessel scales as the squared length of that portion, provided this length is small enough. Equivalently, RC corresponds to the transit time through diffusion of a portion of length δx. The consequences of the simple example treated here can be summarized in three points:

• It clearly shows how the local properties attached to an elementary portion of the vessel define a global parameter, the diffusion constant.

• This brief study evidences the close connection between the choice of the electrical analogue and the physical behavior of the system: in our example, a series of RC elements leads unavoidably to a diffusion process. Changing the structure of the modeling circuit would lead to different physical processes.

• The parameters of the vessel lumping manifesting as a large collection of elementary circuits are not equivalent to a unique simple circuit associated with the whole vessel: it shows the necessity for considering an infinite collection of elements.

In fact, the preliminary knowledge of the dynamical regimes of the pressure within the vessel along with the viscoelastic behavior of the vessel determines unambiguously the local electrical structure of the analogue. The lumping approximation applied to the global properties of the vessel does not necessarily have a macroscopic sense and can deeply modify the nature of the dynamical regimes actually present within the system.

To these sized difficulties must be added another problem rarely addressed. In practice, the electrical analogue is a linear filter associated with a predefined couple of input-output signals. A fundamental question is therefore that of the stability conditions of the filter. This study is necessary to avoid associating unstable filters with healthy subjects since this situation would have no physiological sense.

Oppositely, it opens the way to the interpretation of some pathological states as unstable conditions. As a consequence of the complexity reduction implemented in electrical analogue studies, the numerous parameters of the circuit are adjustable ones though such a holistic model requires a great amount of experimental data which is often unavailable. This lack of experimental knowledge can be compensated only through a prediction of some of the constitutive parameters, that is, their connection to a more fundamental description (biophysical features of the system). This lack of predictability is problematic. Elaborated in very different backgrounds, these attempts suffer a lack of unity, but they share common features: they are based on two key ingredients, the notion of compliance and the strong assumption of a linear dissipative dynamics. In the next section, we discuss the importance of these ingredients and describe the general structure of a possible unifying scheme providing a possible way out of the difficulties raised by the current approaches.

Toward a Unified Scheme of Intracranial Dynamics

Two sides of the intracranial dynamics are of peculiar importance in our present study: healthy and pathological behaviors. Prior to any modeling attempt, a question comes to mind: What is a disease from a physical point of view? This question is a quite delicate one, first because, up to now, the known physics deals with a complexity level far below biological complexity as was emphasized in the previous sections. Though it sounds naive, this question should be given an answer in order to connect clinical facts to a predictive physical scheme. In the available literature devoted to tentative modeling of intracranial dynamics, the explicit or implicit answer to that question varies notably, resulting in different strategies to approach the problem, among which those depicted in the previous section.

Structural Complexity and Approximate Dynamics

Any system, be it biological, ecological, economical, or any other type, is fundamentally a dynamical object whose evolution and responses to some stimulations are governed by specific dynamical laws. The identification and study of these dynamical laws are precisely the main goal of physics. But this step is certainly a difficult task, darkened by the system’s degree of complexity. This situation holds especially for the intracranial system. Fortunately, a separation of the space and time characteristic scales occurs, due to their very different orders of magnitudes. This separation results in a dynamical hierarchical organization: at the fundamental level, the molecular one holds the fastest dynamical processes, while the intermediate mesoscopic level is dominated by neurodynamics, the last level, the macroscopic one, being ruled by the slowest dynamical degrees of freedom. Though the complete and rigorous dynamical laws are not known, fast processes can be eliminated through any appropriate time averaging as usually carried out in statistical physics to let survive the only slow dynamics. Oppositely, when focusing on the fast sector of the dynamics, the slow degrees of freedom appear as frozen, that is, as a slowly varying environment. Within this first and necessary step of complexity reduction through the approximate decoupling of the different dynamical sectors, physics, or at least the appropriate field, can be implemented to explore the intracranial dynamics at two relevant scales, the meso- and the macroscopic scales. The complexity reduction described before doesn’t dismiss the interplay between the two levels but simplifies the description by substituting effective dynamical laws for the exact ones. This interplay can particularly manifest itself in abnormal behaviors of the intracranial system such as hydrocephalus. From a clinical point of view, this fact is clearly pointed out in the set of symptoms reported in Hakim’s triad (Adams et al. 1965) where a macroscopic change in the mechanical state of the brain results in cognitive damage. Our goal is now made clear: bridging the gap between clinical facts about intracranial pathologies and the (bio-) physical modeling of intracranial dynamics with the hope to get predictions about the occurrence of abnormal behaviors. Though a little more modest, our situation resembles proportionately to that encountered in brain physics (Beggs 2015): bypassing its structural and dynamical complexity, we aim to understand the emergence of pathological behaviors of the intracranial system on physical grounds. But how can physics allow a deeper understanding of the intracranial system permeated with a so high degree of complexity?

General Structure of a Dynamical Model of the Intracranial System

As can be clearly noticed from the survey of the most relevant hydrocephalus models, the typical mathematical structure common to these models can be easily identified. Most models of hydrocephalus can indeed be encoded by a graph structure underlying the linear dynamical behavior of the system around a well-defined steady configuration. The corresponding linear approximation to the dynamics can be expressed by a system of differential equations of the general form,

$$ \sum \limits_j\left({C}_{ii}{\delta}_{ij}+{C}_{ij}\right)\frac{d{p}_j}{dt}=-\sum \limits_j{\gamma}_{ij}\left({p}_i-{p}_j\right)+{S}_{iexc},i\ \epsilon\ \left\{1,2,\dots, N\right\} $$

(11)

It governs the pressure variation in any compartment indexed by the integer number i and connected to the others through fluid exchange or any other way (e.g., mechanical deformations). The parameters of the left side member consist mainly of the compliances of the (CSF) compartments, divided into two types: the self-compliances C_ii of individual compartments that are disconnected from the surrounding ones and the inter-compliances C_ij (nonzero for different compartments only) capturing the potential effects of interactions between compartments. The minimal approach requires the only self-compliances, neglecting the inter-compliances because these terms physically less intuitive are much more difficult to compute. The right side exhibits the hydrodynamic conductances γ_ij between two different compartments exchanging some fluid. In Eq. 11 we have set to zero the γ_ii. The last terms S_iexc sum up the relevant excitation sources accounting for the modulating effect of cardiac pulses as well as CSF secretion or resorption processes. Though we restrict ourselves to deterministic models, the right side can host an additional stochastic term accounting for the likely influence of the physiological noise or any other random excitation source. It can thus be treated as a simple generalization of the excitation term and subsequently does not spoil the physical conclusions drawn from the study of Eq. 11. Though rarely addressed, the consequences of a noise term, whatever its origin, could be interesting in pathological situations.

Restricting to a first-order dynamical equation (that is, assuming a static compliance only) is a rather good approximation for slowly varying excitation terms or for slow variations of the environment of the system. This approximation amounts to assume excitation terms with a Fourier spectrum dominated by frequencies much lower than the relaxation frequency (cutoff) of the system. Of course, additional corrections to the first-order approximation admit second-order (and higher) pressure derivative terms. In fact, the actual dynamics is of infinite order, as can be deduced from the more general linear Eq. 5 discussed in the literature review section. It is the expansion (Eq. 6) of Eq. 5, truncated to the relevant order, which allows to generate the successive approximations of the pressure dynamics. This truncation allows handling simpler electrical analogues of the pressure dynamics. It is worth noticing that the linearity assumption and the order of the dynamical pressures are two different approximations of the actual dynamical scheme with different physical interpretations. As the situation is often met in the literature, we will restrict now our discussion to the first-order approximation of the dynamics, that is, we consider only static compliances.

The linear dynamics ruled by Eq. 11 can be given a more intuitive representation schematized on Fig. 9 below.

Fig. 9

Typical graph structure associated with a collection of interacting liquid compartments ruled by a linear dynamical law. The set of parameters C_ij and γ_ij form two matrices determining completely the dynamical equations

This graph structure expresses the linearity of the dynamics and accounts for the thermodynamical constraints on the intracranial system: for this reason, the graph structure is a fundamental prerequisite of any model of the intracranial system. From a thermodynamical point of view, the linear structure of the dynamical laws is allowed through the assumption of a steady (out of equilibrium) configuration of the intracranial system. This fundamental assumption and its consequences are rarely addressed in the literature though they establish the nonarbitrariness of the graph structure expressed by the matrices C_ij and γ_ij. This way of thinking about the intracranial dynamics and more especially hydrocephalus leads to a predictive approach of the disease: the actual nonlinear dynamics is linearized around any relevant steady state, and its pathologies can be identified to unstable steady configurations predicted through linear stability analysis of that state. It is then clear that the graph structure underlying relevant electrical analogues of the system is associated with the linearized dynamics: the components of the electrical circuit (built from the compliances and conductances) depend on the biophysical features of a peculiar steady state. Moreover, dissipation within the system is required not only to ensure the existence of steady configurations of the system but also to trap (regulation processes) the system around the appropriate steady state. In other words, the graph structure given by the compliances and conductances network expresses thermodynamical features of the steady states of the system.

From a more practical point of view, the graph interpretation of any dynamical system is a very intuitive representation summarizing the architecture of the system and allowing easier modifications of the dynamics through further approximations (e.g., suppression of connections between compartments) and providing natural (re-) interpretation of the dynamical equations as electrical circuits. The construction of electrical analogues is indeed systematic if we associate to Eq. 11 the previous graph with a discrete function on the graph associating to each summit (the compartments) a pressure p_i and a flow on that graph associating to each pair of summits a conductance γ_ij. Equation 11 constrains severely the nature of the circuit: whatever the electrical interpretation of the graph, this circuit should be topologically equivalent to that graph. Equivalent graphs define the same dynamical equations: a linear map applies the pressure values in one model onto the pressure values in the other and so do the conductances and compliances. An infinite set of such mappings exist. Nevertheless, from a practical point of view, the physical representation of the system (which should be regarded as the natural one) is set by the choice of any experimental configuration to assess pressures, conductances, and compliances. Topological equivalence implies the existence of an “irreducible” graph, that is, a common structure to all equivalent graphs. This common structure can be reached through a mathematical procedure we now describe.

The main physical information contained in the associated graph reduces to the characteristic relaxation times of the system. Introducing the pressure state (column) vector [p] with components of the individual pressures p_i, a little algebra allows giving Eq. 11 the equivalent form,

$$ \left(\forall i\right)\ \frac{d\left[p\right]}{dt}=-\widehat{\Phi}.\left[p\right]+\left[{S}_{exc}^{\prime}\right] $$

(12)

where the matrix $\widehat{\Phi}={\widehat{C}}^{-1}.\widehat{G}$ is built up from the matrix G and the inverse compliance matrix, having coefficients,

$$ {G}_{ij}=\left(\sum \limits_l{\gamma}_{il}\right){\delta}_{ij}-{\gamma}_{ij},{\widehat{C}}^{-1}={\left({C}_{ii}{\delta}_{ij}+{C}_{ij}\right)}^{-1} $$

(13)

while the excitation column vector is turned into $\left[{S}_{exc}^{\prime}\right]={\widehat{C}}^{-1}\left[{S}_{exc}\right]$. The last formulation of the pressure dynamics defines the irreducible graph we referred to previously. This is the simplest form of the dynamical equations. In some sense, this equation describes a RC circuit with a “characteristic time” associated with the inverted matrix ${\widehat{\Phi}}^{-1}$. Indeed, the matrix $\widehat{\Phi}$ has coefficients with inverse time units. These coefficients appear as very complex functions of the compliances and conductances. Only the eigenvalues of the matrix $\widehat{\Phi}$, which are also complex functions of the structural parameters of the system, have an intrinsic significance: if nonzero and positive, their reciprocal values give the characteristic relaxation times of the system (or equivalently the graph), that is, the typical lifetimes of pressure fluctuations (around their steady values). Negative eigenvalues result in exponentially divergent pressure fluctuations within one or more compartments: these compartments will consequently either expand or collapse according to the initial pressures within. This behavior matches the usual manifestation of hydrocephalus. This is an exciting proposal suggesting a unified view of hydrocephalus and SVS syndrome regarded as unstable conditions of the intracranial dynamics. We can thus conclude that the main information contained in the graph consists mainly of eigenvalue spectrum of matrix $\widehat{\Phi}$ and their sign. This view highlights the way the spectrum of the matrix $\widehat{\Phi}$ defines the structure common to all equivalent graphs. Nevertheless, as discussed previously, the available representation is usually set by any experimental protocol, but it doesn’t allow accessing the actual dynamics: we deal only with an approximate and incomplete graph as those met in the available literature. Unluckily, usual electrical analogues, made of classical conductances and compliances which are always positive, are always stable as can be proved from the general class of models depicted in Eq. 11. The stability analysis has then no interest on the clinical side since the question of the nature of the intracranial pathologies remains unsolved. To match the instability idea, some of the conductances or compliances should be negative. This is not a shocking assumption if we remember that it is already known in physics. For instance, negative resistance is a concept used to model tunneling diodes which are inserted in any circuit leading to instabilities (Esaki 1958), Esaki and Tsu (1970). Negative compliances have also their counterpart in peculiar ferroelectrics behaving as negative capacitances (Khan et al. 2011; Salvatore et al. 2012). We thus see that the corresponding situations should apply to hydrocephalus modeling through effective parameters, that is, an appropriate combination of classical parameters driving instabilities. But such a combination reflects the features of the actual dynamics.

The usual situation and its problematic issues hold because of the lack of knowledge of the actual dynamics from which the linear graph should be deduced. Because of this lack of information, the irreducible graph is not known: we can only try to guess its features. A deeper understanding of the actual nonlinear dynamics is thus required.

Theory of the Instabilities of the Intracranial System

The previous description of the dynamics is not complete: the dynamical system (Eq. 11) is deduced from a more fundamental equation expressing mass conservation or, as we are dealing with incompressible fluid flows, the fluid/compartment volume variations. We have then to supplement Eq. 11 with the mechanical equations governing the deformation dynamics of the involved compartments. The deformation dynamics regards mainly the shape changes of the shells demarcating the fluid compartments as well their effects on the fluid transport to adjacent compartments. This brief description of the structure of the actual dynamics gives a better idea of the complexity of these systems. The set of Eq. 11 appears as an effective equation combining the mass conservation equation coupled to the deformation dynamics and the fluid transport equations. The compliances concentrate the information about the compartments’ geometry and their mechanical properties. The conductances arise from the fluid transport properties as well as the dissipation within the shell tissues. To understand how to implement the unifying scheme and generate electrical analogues or build the relevant graph, we have to highlight the tight connection between these parameters and the actual dynamics.

As a building block of instability-based approach to intracranial dynamics, a general theory of compliance has been outlined in previous studies (Tekaya 2011, 2012). To avoid unnecessary technical difficulties, we explore the basis of a general theory of compliance through a simple toy model. This will make it easier to understand the physical content of the compliance. This simple mechanical model is schematized in the next figure (Fig. 10).

Fig. 10

Structure of a simple mechanical model substituted to a system of two fluid compartments exchanging fluid. The fluid space is demarcated by a rigid wall and a mobile plate hanged up to a spring mimicking the viscoelastic properties of the true tissues

To simplify this overview, we consider only two compartments exchanging some fluid with well-identified properties. The elastic shell enveloping the liquid space is modeled as a cylindrical liquid space (e.g., the ventricular space) demarcated by a rigid wall and bounded by a mobile plate (e.g., the ependyma) hanged up to a spring with stiffness k and a damper (damping coefficient per unit mass η) accounting for the dissipative motion along the cylindrical axis (one degree of freedom). The spring models the elastic properties of the actual shell tissues. The fluid can be transported to an adjacent compartment (not depicted on the figure) through any aperture with a known hydrodynamical conductance γ (e.g., the aqueduct of Sylvius) concentrating the main physical information about the flow regime through the aperture. The steady position (equilibrium position) of the plate corresponds to a height h of the liquid space determined by the steady pressures within the cylinder and in the surrounding compartments. The pressure above the plate is assumed to be constant (${\overline{p}}^{\prime }$), and we denote p_e(t) the pressure variations in the second compartment. Let x(t) be the vertical displacement of the plate around the equilibrium position (deformation of the shell) and p(t) the corresponding pressure variations within the cylinder. The set of equations governing the motion of the plate (mass M) and the mass conservation equation of the incompressible fluid is given by,

$$ \left\{\begin{array}{c}M\frac{d^{{}^2}x}{dt^{{}^2}}+ kx+\eta M\frac{dx}{dt}= Ap\\ {}A\frac{dx}{dt}=-\gamma \left(p-{p}_e\right)\end{array}\right. $$

(14)

The steady configuration of the system is determined by the secretion and resorption rate of the liquid considered constant in this simplified model,

$$ \left\{\begin{array}{c} kh=A\left(\overline{p}-{\overline{p}}^{\prime}\right)\\ {}-\gamma \left(\overline{p}-{\overline{p}}_e\right)+S-R=0\end{array}\right. $$

(15)

Introducing the Fourier transform of the displacement $\overset{\sim }{x}\left(\omega \right)$ and of the pressure $\overset{\sim }{p}\left(\omega \right)$ in the motion equation yields,

$$ \left\{\begin{array}{c}A\overset{\sim }{x}=C\left(\omega \right)\overset{\sim }{p}\\ {}C\left(\omega \right)=\frac{A\overset{\sim }{x}}{\overset{\sim }{p}}=\frac{A^{{}^2}}{-M{\omega}^2+k+ i\eta M\omega}\end{array}\right. $$

(16)

The dynamical compliance C(ω) has thus been deduced from the knowledge of the laws governing the motion of the system. It admits as a zero-frequency limit the static compliance C = A²/k. Combining these results with the mass conservation equation leads simply to the pressure/pressure transfer function between the coupled compartments or equivalently the Fourier transform of the pressure gradient along the aperture. The compliance of our system appears as a generalized susceptibility connecting the fluid volume variations and its pressure variations: in the time domain, the fluid volume variations appear as a convolution product of the pressure variations and a response function of which the Fourier spectrum is precisely the dynamical compliance. But more fundamentally, it connects the compliance to the elastic properties of the shell and shows that the (static) compliance scales as the squared area of the shell. This result is very close to the exact one since the compliance of a shell of width l and elastic moduli (Young’s and Poisson’s) E and ν can be shown to be $C={k}_G\frac{A^2}{El}\left(1-{\nu}^{{}^2}\right)$ where the prefactor k_G depends only on the shape of the shell (e.g., for a spherical shell, k_G = 1/8π). This general result indicates that the compliance of an elastic compartment increases with its size (area of the shell) and that an alteration of the elastic properties of the shell results in a change of its compliance. This simple approach leads to additional fundamental results. Indeed, it describes steady states of the whole system as critical points (we deal with only one degree of freedom) of the motion equation. Critical points are in fact non-varying states (steady): in our simplified model, we get only one such state x = 0, dx/dt = 0 corresponding to mechanical equilibrium, and this state is stable (k > 0). Within the approximation of a static compliance and a first-order dynamics, the mass conservation equation is turned into,

$$ \frac{A^{{}^2}}{k}\frac{dp}{dt}+\gamma p\approx {\gamma p}_e $$

(17)

evidencing the pressure relaxation time τ = A²/kγ due to fluid transport. As expected, within such approximations, the pressure dynamics can be interpreted as a simple RC circuit. Usual approaches ignoring the deformation dynamics of the shell miss an important part of the information about the system. Indeed, the deformation dynamics of a shell with tissue density ρ and thickness l is controlled by a relaxation time τ_d = ηρAl/k, to be compared to the pressure relaxation time. The most obvious difference between these characteristic times is clear: the pressure relaxation time scales as A²/El, whereas the deformation relaxes as A/E. The sensitivities of the pressure dynamics and the deformation dynamics to the geometry and size of the compartment are thus very different.

The previous model, though extremely simple, captures the essential features of the mechanical behavior of a deformable liquid compartment (through the compliance) and its thermodynamical features (through the pressure relaxation dynamics). Only a predictive approach can lead to such results.

This conclusion along with the trends reported about the characteristic parameters of a deformable liquid compartment and especially their sensitivity to the shape and size of the compartment is confirmed by the treatment of actual compartments. Due to the calculations being much more complex, we will only indicate the main results and compare them to the predictions of our simplified mechanical model.

Contrary to our one-degree-of-freedom simplified mechanical system, an actual liquid compartment demarcated by an elastic shell is a continuum system with infinitely many degrees of freedom encoded by a displacement field accounting for the deformations of the shell around its stationary configuration (linearized dynamics). A stationary state of a real liquid compartment corresponds to the geometrical shape of its shell realized for constant fluid pressures (and stress field) within that compartment and in the surrounding ones. The shell is indeed shaped by the mechanical action of the fluid or matter in adjacent compartments, the constant values of the pressure being set by the physiological conditions imposed to the system. From a geometrical point of view, the stationary shape corresponds to a surface (Σ) of ordinary Euclidean space. Provided with these mathematical notions, the general expression for the static compliance extracted for the general form of the linear dynamics of a shell reads (Tekaya 2010, 2012),

$$ {C}_{\Sigma}=\frac{1}{k}\underset{\Sigma}{\iint}\left[-{\Delta}_{\Sigma}+{H}^2-2{k}_G\right]\delta \left(r-{r}^{\prime}\right) dSd{S}^{\prime } $$

(18)

where δ denotes the Dirac function on the surface (Σ) and the couple of integrals indicate two successive integrations over the surface. The integrated term can thus be regarded as the Green function of the continuum operator to be inverted. This mathematically complicated expression involves characteristic features of the surface (Σ) such as the attached Laplace operator Δ_Σ as well as the mean curvature H and its Gauss curvature k_G (Jost 2002). These two last parameters quantify the shape of the considered surface. The relevant dependence of the compliance on the shape of the shell is thus complicated and corresponds to a functional dependence. Nevertheless, this result is the relevant extension of the simple result derived from our toy model since the right area dependence is recovered for simple surfaces such as a sphere (H and k_G constant) or a cylinder (H constant and k_G = 0). This allows understanding the way the shape of a liquid compartment influences the pressure dynamics through its compliance. Referring to the simplest result and emphasizing the influence of the shape, the compliance can be rewritten as ${C}_{\Sigma}=\frac{A^2}{\kappa_{\Sigma}k}$ to evidence a shape factor,

$$ {\kappa}_{\Sigma}={\left(\frac{1}{A^2}\underset{\Sigma}{\int}\underset{\Sigma}{\int }{\left[-{\Delta}_{\Sigma}+{H}^2-2{k}_G\right]}^{-1}\delta \left(r-{r}^{\prime}\right) dSd{S}^{\prime}\right)}^{-1} $$

(19)

to be interpreted as a double averaging of the Green function over the surface (Σ). As a stable shape is thermodynamically constrained, we understand the deep significance of the compliance as a thermodynamical parameter: the role of thermodynamics is thus fundamental in these approaches to intracranial dynamics based on a realistic description of the relevant geometry of the liquid compartments and their deformation dynamics. We notice also that the compliance in Eq. 17 is positive because of the positive elastic modulus k (the continuum operator within the square brackets is itself a positive operator): no instability can occur in a purely elastic model. Additional features are required. This issue will be addressed in the next section.

Effective Parameters and Instabilities of Intracranial Dynamics

In view of an appropriate depiction of its dynamical behavior, a clear account of the complexity of any biological system is needed. Among the attempts to obtain such a description, models focusing on the many couplings manifesting that complexity are the most important. Their efficiency relies on the substitution of renormalized parameters to the actual parameters of the system. This is precisely what should be meant by effective parameters (or more exactly effective theories): these new parameters incorporate the effect of the couplings, leading to a simplification of the models depicting these complex systems.

As stated previously, the compliance of any liquid compartment with a known stationary geometry bears information about its mechanical behavior but seems independent from the liquid features contained within. In fact, due to blood transport, pressure variations of the fluid act as an excitation source of the shell deformations: the corresponding deformation dynamics and the fluid pressure dynamics are strongly coupled. As noticed in our simplified mechanical model of the compliance, the fluid pressure variations are involved in the equations ruling the fluid transport to neighboring compartments (mass conservation). To account for CSF secretion through the choroid plexus, the model should be completed with a specific source term leading to the complete mass conservation equation relating the volume variation to the CSF fluid pressure variation,

$$ \delta \dot{V}+\left({\gamma}_{aq}+\frac{DA}{l}\right)\delta p\approx \delta S+{\gamma}_{aq}\delta {p}_e+\frac{DA}{l}\delta {p}_b $$

(20)

This equation accounts for the various CSF pathways within the cranium space. It can flow either to external compartments (subarachnoid space pressure δp_e) through the aqueduct of Sylvius or to the brain (pressure δp_b) through the ependyma. Accordingly, the last equation incorporates a transependymal CSF diffusion term acting as a resorption process due to the permeability of the ependyma to CSF (Bouzerar et al. 2005) and proportional to the ependymal area A. This process is associated with an additional conductance $\frac{DA}{l}$, l being the thickness of the ependyma (about 500 μm), which completes the conductance γ_aq of the aqueduct of Sylvius. The right-hand side describes the pressure variations within the subarachnoid space, the brain, and the secretion term as excitation terms. These terms are of course modulated at a frequency induced by heartbeats. Introducing a “secretion” conductance γ_C so that δS ≈ γ_Cδp (CSF secretion increases pressure within the ventricles), the last equation leads after linearization to an effective characteristic relaxation time τ, which reads,

$$ \tau =\frac{C_{\Sigma}}{\gamma_{aq}+\frac{DA}{l}-{\gamma}_C}=\frac{1}{k\left(\gamma -{\gamma}_C\right)}\underset{\Sigma}{\int}\underset{\Sigma}{\int }{\left[-{\Delta}_{\Sigma}+{H}^2-2{k}_G\right]}^{-1}\delta \left(r-{r}^{\prime}\right) dSd{S}^{\prime } $$

(21)

and where we have introduced the overall CSF conductance associated with its many pathways $\gamma ={\gamma}_{aq}+\frac{DA}{l}$. It is worth noticing the opposite effects of the conductance of the aqueduct (and diffusion through the ependyma) and the secretion conductance: the first one controls the ability of the aqueduct to evacuate excess CSF from the ventricles (to lower the pressure within), while the second ensures its renewal. The competition between these processes leads to a non-necessary positive relaxation time, justifying their effective nature. When positive (γ > γ_C), the CSF pressure is regulated around its well-defined stationary value. But, if γ < γ_C, any pressure variation increases exponentially: the pressure regulation ability of the aqueduct is lost. Such instabilities or more precisely bifurcations, which we will refer to as transport instabilities, will certainly be associated with pathological states of the intracranial system. This raises an interesting issue: Does hydrocephalus consist of these instabilities? This hypothesis has been partially confirmed by numerical simulations of some models of the deformation dynamics of the ventricles (Tekaya 2012; Bouzerar et al. 2012). The main clinical manifestations of hydrocephalus as ventricle inflation and stationary pressure increase are accounted for by these numerical simulations, reinforcing this hypothesis. Another quite different and more subtle instability has been evidenced in some theoretical studies (Bouzerar 2005; Tekaya 2012). These instabilities involve the elasticity of the ependyma (stiffness k of the simplified model presented in this paper) and the CSF wetting properties (surface tension). In the situation of a failure of the ependymal elasticity, the stationary state of the ventricular space is suppressed, resulting again in uncontrolled pressure variations. Due to their tight connection with the complexity of the ventricular space, these elastic instabilities are more difficult to simulate, and no definite conclusion can be drawn from these numerical simulations. Nevertheless, we can legitimately wonder if all hydrocephalus could be embraced by these instabilities. The likely connection between instabilities of the intracranial dynamics and hydrocephalus will be discussed further in the next section.

Applications to Hydrocephalus

There are many ways to tackle the consequences of hydrocephalus, but the issues related to their mechanisms are rarely addressed due to the complexity of the disease. The various and many attempts to their mathematical description reinforce this conclusion. The identification of causes is always a complex issue in medicine and biology. Nevertheless, the set of (effective) parameters playing an obvious role in these pathologies suggests a common origin to different types of hydrocephalus. With the compliance of the intracranial system or of any compartment within (e.g., ventricles), the conductance of the cerebral aqueduct and other foramina and quantities associated with the coupling between the blood system and CSF flow are such parameters which capture the main features of the structural and dynamical complexity of the intracranial system. These parameters play a fundamental role in the effective theory of intracranial dynamics. These are not purely phenomenological parameters but can be assessed from appropriate measurements, opening the way to an experimental validation of the models. A study carried out by Tekaya (2012) and Tekaya et al. (2012) provides a method for extracting the conductance of the aqueduct from flow MRI data. This study accounts for the realistic CSF flow transported by the aqueduct through a Womersley flow excited by heartbeats and uses an accurate Fourier analysis of the CSF pressure gradient and flow rate deduced from MRI data to derive the dynamical conductance. A typical example of the result is presented in the figure below (Fig. 11).

Fig. 11

Example of numerical results deduced from phase MRI data carried out on a healthy subject (Tekaya 2012). Top: hydrodynamic conductance (m³.Pa⁻¹.s⁻¹) of the cerebral aqueduct as a function of the frequency of the CSF pressure modulations (Hz). Bottom: CSF flow rate (left, mm³/s) and pressure gradient (right, Pa/cm) through the aqueduct with respect to time (s) during a cardiac cycle

Apart from the difficulty raised by the low resolution of the measurements, the main weakness of that method arises from the necessary interpolation of the sampled velocity field to produce a solution to the Navier-Stokes equations from which the pressure gradient can be assessed. This is a very delicate procedure which deserves a complete study since it makes possible a classification of the pathological subjects and can lead to a prediction of pathological conditions.

Disease Kinetics

The potential clinical applications of such a method are rather obvious. The study of the variability of the parameters among a large population of subjects will certainly help highlighting the disease mechanisms. As hydrocephalus involves progressive processes, the development of the disease with time and more especially the manifestation of the slow dynamics are a key issue. The slow dynamics corresponds to the disease kinetics, that is, the long-term evolution of the state of the intracranial system in the presence of hydrocephalus. The disease kinetics proceeds directly from the dynamical laws governing the system. The first experimental report of kinetic aspects of hydrocephalus dates back to a study carried out by Milhorat et al. in 1970 and presents the evolution in the ventricular size of a population of chimpanzees. The incorporation of kinetics to any model of hydrocephalus is a very difficult task. The usual approaches to hydrocephalus, such as these reviewed in the first section of this chapter, do not deal with kinetic aspects. For instance, electrical analogue-based models rather focus on the response to cardiac excitation. To tackle kinetic processes, the electrical analogue should evolve with time through, for example, a very slow change of its parameters (structure of the circuit). The bifurcation approach allows naturally the exploration of the disease kinetics. This is illustrated in Fig. 12 below, in the peculiar situation of the constriction of an aqueduct due to a tumor growth described by the Gompertz growth function (Laird 1964). This constriction effect leads to a transport instability (which can be easily understood) as can be noticed from the figure.

Fig. 12

Simulation of the effects of a tumor growth blocking the aqueduct of Sylvius over time. Top: decrease in hydrodynamic conductance (m³.Pa⁻¹.s⁻¹) with time (weeks). Bottom: corresponding increase in CSF ventricular pressure (Pa) as the tumor grows (Tekaya 2012)

These pathological conditions can be realized in many ways: aqueduct constriction due to tumor growth, for instance, changes in the CSF flow regime through it (vortex excitations) due to anomalous curvature of the aqueduct.

Adult and Infantile Hydrocephalus: A Physical Basis

Many prevalence studies for infantile hydrocephalus lead to frequencies lying between 1 and 30 per 10,000 births (Tully and Dobyns 2014), these estimates depending on the definition retained for hydrocephalus as well as the statistical population. Clinicians involved in the field of infantile hydrocephalus usually distinguish acquired hydrocephalus from developmental hydrocephalus. The first category groups pathological conditions due to extrinsic causes such as hemorrhage or any infection, often met in premature infants. The second one refers to genetic mutations. Be it intrinsic or extrinsic, the evoked causes will be referred to as “primitive causes” since these are of a biological nature. The statistical study of large populations of subjects will certainly mix these primitive causes proportionately to their occurrence probability. The assessment of these probabilities is a very delicate but important issue because they are associated with fundamental biological events (e.g., gene mutation), clearly in keeping with microscopic or at least mesoscopic scales. Oppositely, the pathological conditions identified by clinicians belong to the macroscopic scale. Though the pathological conditions associated with hydrocephalus arise from primitive causes, while not clearly connected, we will treat these macroscopic conditions as a set of secondary causes. The intricacy between primitive and secondary causes has darkened the clinical picture and has probably led to the actual clinical positions about hydrocephalus, regarded as multifactorial.

In the absence of any clear connection between the microscopic and macroscopic scales, only the secondary causes can be investigated in physical approaches to hydrocephalus (which are clearly macroscopic descriptions of the intracranial system). Accordingly, infantile hydrocephalus differs from adult cases mainly by some mechanical differences in the skull and CSF features. For very young children, the cranial bones are separated into different pieces which will fuse only after the third year of life. Before that fusion, the dependence upon pressure of the compliance of infant cranium exhibits two regimes controlled by the facilitated skull deformation, while in an adult’s skull, only one regime is expected. These regimes are defined by the comparison of the CSF volume to the cranium volume, namely, the ratio $\eta =\frac{\mathrm{CSF}\ \mathrm{volume}}{\mathrm{cranium}\ \mathrm{volume}}$. For adults, when, after enlargement, the ventricular volume reaches its maximum value (η = 1), that is, the volume of the rigid cranium, the compliance practically vanishes because of the extreme rigidity of the skull bones. In that case we observe an abrupt decrease of the compliance at high CSF pressure. For very young children with unfused skull bones, the ratio saturates at η = 1 but with a variable cranium volume due to its deformability. This skull-dominated regime is a striking feature of infantile hydrocephalus associated with a nonvanishing compliance $C\sim \frac{A_{\mathrm{skull}}^2}{k_{\mathrm{skull}}}$ (see previous section) where A_skull is the skull variable area (depending on CSF pressure) and the stiffness k_skull contains the elastic modulus of the skull. To highlight the differences between adult and infant hydrocephalus, we can compare the corresponding electrical analogues. Handling electrical analogues is a simple and intuitive way to represent the CSF pressure dynamics. Nevertheless, the treatment of hydrocephalus as an instability of the intracranial dynamics makes that discussion more delicate. To simplify that discussion and bypass the evoked difficulties, we assume that once enlarged, the ventricles recover their ability to regulate the CSF pressure (but with very different characteristic times).

Remembering from the previous sections that the electrical analogues proceed from the Fourier analysis of the linearized (around the stationary state) dynamical equations (grouping the ventricle deformation dynamics and CSF mass conservation equation), we are led to CSF pressure dynamics approximated to a low-pass filter (simplest model). We can then discriminate adult/infant situations through a comparison of the corresponding structures of the low-pass filters and cutoff frequencies. For adults undergoing extreme enlargement, the associated filter is a second-order one (because of the vanishing compliance) combining an inductance L and a resistance R, the current being an image of the CSF flow through the aqueduct. For infants, because of the unfused skull bones, the associated filter is a first-order RC series circuit, C being the residual skull-dominated compliance. The respective cutoff frequencies R/L and 1/RC have different orders of magnitude: with the inductance being small, R/L is high, while 1/RC is rather small (the skull compliance proportional to its squared area is high). This means that high-frequency pressure variations (acoustic waves) can propagate through the ventricles of hydrocephalic adults, while these high frequencies are suppressed for hydrocephalic infants. These different mechanical responses should be given more attention because of their potential clinical importance.

Another important difference between adult and infant hydrocephalus arises from the difference in disease kinetics. The disease kinetics can be derived from the description of hydrocephalus as an instability of the intracranial system. The longtime development of the ventricular expansion depends on the mechanical features of the intracranial system which are clearly very different for young children (deformable skull) and adults (rigid skull). Such a comparison can be achieved only through complex numerical simulations. These studies can be also of a great clinical interest.

The instability approach to hydrocephalus leads to other interesting conclusions: the notion of virtual stationary states. A virtual stationary state corresponds to ventricular sizes greater than the cranium radius. This new concept explains the large expansion amplitude observed in young subjects and extreme mechanical compression of brain against the rigid cranium for adults. In the last situation, the ventricles try to reach the new stable stationary state located outside the cranium!

Bridging the Gap Between Mechanical Features and Cognitive Abilities of the Brain

The connection with brain matter regards the consequences of the hydrocephalic state on the infant brain development due to severe mechanical conditions. Alteration of the cognitive development (Mataro et al. 2001), e.g., learning disabilities, can be approached through a physical model describing the effect of stresses on the neuronal network. Such models do not exist in the available literature. Yet, such models would appreciably complete the usual “mechanical” descriptions and help understanding the clinical picture. Such a simplified model of the brain treated as a neuronal network is presented below, emphasizing its sensitivity to mechanical stresses.

The simplest model should treat neurons as binary automatons (McCulloch and Pitts 1943) with two individual states available (active or inactive neurons) and summation capabilities of signals received from other neurons. The connection between different neurons, as schematized on the figure below, is controlled by synapses whose electric currents are modeled by the so-called synaptic coefficients w_ij (connecting neurons i and j) (Fig. 13).

Fig. 13

McCulloch and Pitts’ formal neuron. The signals (S_j) from surrounding neurons are weighted by the synaptic coefficients (w_ij) to generate an input signal (ϵ_i) compared to the threshold S. The output signal (S_i) is then either zero (inactive neuron) or 1 (active neuron)

We need in fact a model of the brain as simple as possible to highlight the consequence of a hydrocephalic brain on its cognitive functioning, not a realistic description of the individual neuron dynamics: this is why we can adopt an oversimplified neuron. To tackle the delicate problem of the alteration of the cognitive abilities of the brain, we only have to emphasize the change in the synaptic coefficients due to the unusual stress field associated with the hydrocephalic state. The model discussed here relies on a simple assumption: synaptic coefficients depend on the mechanical state of the brain, that is, on the stress field ${\sigma}_{ij}\left(\overrightarrow{r},t\right)$ imposed to the brain. In our case the stress field is generated by the pressure of CSF within the ventricles and propagating throughout the brain. We then understand how a pathology affecting the mechanical state of the brain can result in malfunctions of the neural network. The dynamical laws governing that network have a simple form,

$$ \left\{\begin{array}{c}{S}_i(t)=\theta \left({\epsilon}_i(t)-S\right)\\ {}\frac{d{\epsilon}_i}{dt}=-\frac{1}{\tau}\left({\epsilon}_i(t)-{\sum}_i{w}_{ij}\left({\sigma}_{ab}\right){S}_j(t)\right)\end{array}\right. $$

(22)

where θ is the usual Heaviside function (decision function of the formal neuron) and the second equation controls the time evolution of the neuron individual state ϵ(t). It tells us that within a short time lapse τ (response time of the neuron), its state evolves to the appropriate weighted combination of the input signals. We have dismissed useless nonlinear corrections, which are not essential to achieve our goal. The synaptic coefficients depend on the CSF pressure within the ventricles and the distance of the neurons to the ependyma. This simple model combines a fast dynamics (reflecting the usual cognitive activity of a brain) and a slow dynamics associated with a possible drift of the pressure reflecting the disease kinetics. It is clear that the elastic moduli of brain tissues/ependyma and skull bones are encoded in the stress field: it should be different for infant and adult hydrocephalic subjects. Introducing the vector of the individual states of the neurons, the solution to the previous equations reads,

$$ \overrightarrow{\epsilon}(t)\approx \widehat{W}(p){\int}_{\mathbb{R}}h\left(t-{t}^{\prime}\right)\overrightarrow{S}\left({t}^{\prime}\right)d{t}^{\prime } $$

(23)

where $\overrightarrow{S}(t)$ is the vector of the neuron binary output signals, h(t) the response function of the neuron treated as linear filter (its Fourier transform is the corresponding transfer function) and $\widehat{W}\left(p(t)\right)$ the synaptic coefficients matrix. This solution is approximated to the relevant situation of a disease kinetics (time variation of the CSF pressure p(t)) much slower than the neuron characteristic time τ. In the presence of abnormal stresses, the synaptic coefficients can change abruptly. Locally, if the stress exceeds a critical threshold, the synaptic coefficient vanishes. This corresponds to the breakdown of synapses. The topology of the neuronal network depends crucially on the mechanical state of the brain: it can contain an increasing number of broken connections. The topological order of the assembly of neurons can thus be broken. From a dynamical point of view, this situation can be represented by the condition accounting for the synaptic local changes,

$$ \frac{d\widehat{W}}{dt}={\sum}_{a,b}\frac{\partial \widehat{W}\left({\sigma}_{ij}\right)}{\partial {\sigma}_{ab}}{\dot{\sigma}}_{ab}+{\left.\frac{\partial \widehat{W}}{\partial t}\right|}_{\sigma_{ij};\mathrm{Hebb}} $$

(24)

It exhibits two contributions: the first one is due to the stress field induced by the hydrocephalic state (or any other mechanical pathology affecting the brain) and the second one, the usual contribution reflecting the normal behavior of the synapses, described, for instance, by the well-known Hebbian rule (Hebb 1949), accounting for the brain plasticity. When the stationary mechanical state of the brain is reached, only the second term persists so that Hebb’s usual rule can be applied but with the stationary stress field built in the brain. The coupling to the evolving mechanical state of the brain is thus clearly stated by this equation which shows how the pathological state can affect the normal brain behavior. The symbols $\frac{\partial {w}_{ij}}{\partial {\sigma}_{ab}}$ (elements of the matrix $\frac{\partial \widehat{W}\left({\sigma}_{ij}\right)}{\partial {\sigma}_{ab}}$) can be interpreted as the sensitivities of synapses to mechanical stresses.

Defining naively the cognitive loss (efficiency loss) of a neuronal assembly as the proportion of broken links, we get an index to measure the influence of the pathological mechanical state of the brain on its functioning. For a connection density almost constant over the brain (initial total number of connections N_C), that efficiency is given, at any time t, by

$$ \eta =\frac{N(t)}{N_C}=\frac{1}{V_B^0}{\int}_BP\left(w\left(\overrightarrow{r},t|{\sigma}_{ab}\right)=0\right){d}^3\overrightarrow{r} $$

(25)

where N(t) is the number of broken connections, ${V}_B^0$ is the initial brain volume, and the integration is performed over the domain (B) filled with brain matter. The quantity $P\left(w\left(\overrightarrow{r},t|{\sigma}_{ab}\right)=0\right)$ is the probability for the breakdown of the connection (synapsis) between neurons with relative position $\overrightarrow{r}$ in the presence of a stress field σ_ab. The knowledge of these probabilities is a key ingredient of any model of the brain cognitive feature degradation. As we need only to assess the influence of any abnormal stress field on the functioning of the brain, we will not address this issue here and assume that the dependence upon the stress field is known. The sensitivity of the cognitive loss index to the CSF pressure p can be approximated to,

$$ \frac{\partial \eta }{\partial p}\approx \overline{\frac{\partial P\left(w\left(\overrightarrow{r},t|{\sigma}_{ab}\right)=0\right)}{\partial p}} $$

(26)

where the overbar indicates an averaging over the domain engulfing the brain matter. This last quantity controls the time variation of the number of lost connections $\dot{\eta}\approx \dot{p}\frac{\partial \eta }{\partial p}$. In fact, these quantities are equivalent to the synaptic sensitivity mentioned previously. Severe hydrocephalus is associated with high “$\frac{\partial \eta }{\partial p}$” conversely suggesting the clinical relevance of such a coefficient. The CSF pressure sensitivity of the cognitive loss index allows discriminating between infant and adult cases since it depends on the stress field generated within the brain: the boundary conditions on the skull are very different in these cases since the deformation is zero for adult subjects (rigid skull) but doesn’t vanish for infantile situations. Abnormally high values of these indexes indicate anomalous sensitivity to CSF pressure and related malfunctions (e.g., instabilities). Subsequent alterations of the brain plasticity are expected. These indexes deserve a deeper study because they provide interesting diagnosis assistance tools for cognitive disorders induced by hydrocephalus.

Conclusion

While the first examinations of the ventricular compartments are quite old (they date back from the eighteenth to nineteenth century with the works of Abercrombie, Magendie, Monro, and Kellie), the advances in physics, mechanics, and mathematics at that time did not allow for breakthroughs in the understanding of hydrocephalus. Therefore, it is clear that the history of the modeling of hydrocephalus is quite recent. It began in the early 1970s with works from physicians and researchers such as Hakim and Marmarou and has ever since brought a lot of insight as to how hydrocephalus arises and develops. Three distinct approaches are usually used, and each of them offers promising results. Modern researches also have brought a clearer view of kinetic aspects of the condition. These are essential as they may lead to a way to predict the evolution of the syndrome and to perfect the designs of shunt treatments in particular conditions.

With the late advances during the last decades in the field of medical imaging and the increase in computational power, technology has become crucial in this field. We may cite the use of simulation tools based on finite element models that allow for accurate mechanical calculations but also the use of phase-contrast MRI to assess the flow of the CSF through the cerebral aqueduct, for example.

Infantile hydrocephalus remains tough to model as the circumstances with adult hydrocephalus are quite distinct. Indeed, aside from the fact that the mechanical situations singularly differ (the adult cranium is rigid and thus its compliance is much lower than that of the infant cranium), this condition in the case of children may either have a congenital origin or develop as a result of an external cause. It is therefore much more difficult to design a biophysical model for infantile hydrocephalus than for adult hydrocephalus. Thus, a lot of work has to be done on that matter.

Research dealing with cognitive impairments as a result of acute hydrocephalus is not yet matured, but this area of research is very encouraging. It is a very complex field as it requires good knowledge of neurons, their coupling, and the geometry of neural networks. Preliminary studies show how altered mechanical states of the brain induce broken links in the neural network, but how cognitive losses are engendered remains to be investigated.