# Modeling left ventricular dynamics with characteristic deformation modes

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## Abstract

A computationally efficient method is described for simulating the dynamics of the left ventricle (LV) in three dimensions. LV motion is represented as a combination of a limited number of deformation modes, chosen to represent observed cardiac motions while conserving volume in the LV wall. The contribution of each mode to wall motion is determined by a corresponding time-dependent deformation variable. The principle of virtual work is applied to these deformation variables, yielding a system of ordinary differential equations for LV dynamics, including effects of muscle fiber orientations, active and passive stresses, and surface tractions. Passive stress is governed by a transversely isotropic elastic model. Active stress acts in the fiber direction and incorporates length–tension and force–velocity properties of cardiac muscle. Preload and afterload are represented by lumped vascular models. The variational equations and their numerical solutions are verified by comparison to analytic solutions of the strong form equations. Deformation modes are constructed using Fourier series with an arbitrary number of terms. Greater numbers of deformation modes increase deformable model resolution but at increased computational cost. Simulations of normal LV motion throughout the cardiac cycle are presented using models with 8, 23, or 46 deformation modes. Aggregate quantities that describe LV function vary little as the number of deformation modes is increased. Spatial distributions of stress and strain change as more deformation modes are included, but overall patterns are conserved. This approach yields three-dimensional simulations of the cardiac cycle on a clinically relevant time-scale.

## Keywords

Left ventricle Computational modeling Cardiac mechanics 3D simulation Deformation modes## 1 Introduction

The mechanical pumping performance of the left ventricle (LV) depends in a complex way on the ventricular geometry, the passive mechanical properties of the myocardium, the arrangement of cardiac muscle fibers, and the fibers’ contractile force generation. Quantitative understanding of the effects of these characteristics on ventricular function requires the use of theoretical models to simulate the dynamics of the LV. Such models typically employ a continuum mechanics approach in which the active and passive components of stress at each point in the myocardium are expressed as time-dependent functions of local myocardial strain. The equations of equilibrium of mechanical stresses are then solved using approximate methods, subject to boundary conditions that include the external forces acting on the tissue.

Three-dimensional models of the LV are typically developed through the application of the finite element method (FEM) (Nash and Hunter 2000; Costa et al. 2001; Kerckhoffs et al. 2005). In this approach, the integrals of the variational equations (15) are split into integrals over local elements, and the displacement variations \(\delta u\) are given in terms of the local element displacement functions (Zienkiewicz et al. 2005). The resulting large sparse matrix systems may be solved for the local element displacements. FEM simulations of LV dynamics yield detailed descriptions of cardiac deformation and stress. However, such models are computationally demanding due to the many degrees of freedom (DOF) that are needed to describe local element displacements.

While the FEM approach is suitable for many purposes, some problems do not require so many DOF. For example, Nordbø et al. (2014) showed that, despite their use of an elastic FEM model with hundreds of degrees of freedom, the elasticity parameters of a mouse LV were identified with greater certainty using an objective that included only four aggregated quantities: LV long-axis length, short-axis diameter, work, and volume. In such cases, a reduced model of LV kinematics would likely be sufficient and afford better parameter identifiability. Arts et al. (1992) demonstrated that 13 kinematic parameters were sufficient to fit 14 markers recorded in a canine LV model, suggesting that, for studies where only limited data is recorded, a reduced approach is suitable. Simulations of cardiac remodeling are another area where a reduced model would offer an efficient alternative to FEM modeling. The results of a reduced model would likely be similar to those found using an FEM model, as changes to cardiac function based on remodeling are distributed over the myocardium.

The continued prevalence of heart failure (Benjamin et al. 2018; Kapoor et al. 2016; Kovács 2015), among other cardiac pathologies, has lead to increasing interest in improved methods for individualized quantification of cardiac function. Patient-specific computational models of the heart offer information beyond standard clinical indices. While FEM models have been effectively applied to estimate cardiac mechanics for specific geometries (Aguado-Sierra et al. 2011; Krishnamurthy et al. 2013), their theoretical and computational complexity impedes their widespread use. A simplified method that can still represent the 3D geometry and essential deformable characteristics of the LV would provide a more accessible method for use outside the modeling community.

To these ends, we develop a computationally efficient variational method for modeling LV dynamics in Sect. 2. We have previously described this method for an axisymmetric geometry (Moulton et al. 2017) and here extend it to more general geometries and kinematics. The equation of virtual work (15) is used to describe the dynamics of the LV. However, rather than splitting the integrals and displacements into local elements as in the FEM, we describe displacements according to a set of kinematic variables \(\varvec{q}\) that extend over the entire myocardial domain. The displacement variations \(\delta \varvec{u}\) are therefore described in terms of variations of these kinematic variables.

We define a non-axisymmetric unstressed myocardial domain \(\varOmega _0\) in prolate spheroidal coordinates. To simulate LV dynamics, we construct a mechanical model that incorporates muscle fiber orientations, active and passive stresses, and surface tractions. We assume the standard transversely isotropic elastic model. To represent the viscoelastic properties of the myocardium, the passive stress also includes a viscous component. The active stress acts only in the fiber direction and incorporates important fiber properties such as the length–tension and force–velocity relationships. These definitions are introduced into the virtual work equation, yielding a system of ordinary differential equations (ODEs) that characterize the time-dependent deformable mechanics of the LV. The validity of the method is evaluated by comparison with analytical solutions to the strong form equations. Simulations of normal cardiac function are presented and cardiac muscle fiber stress distributions are computed, demonstrating the ability of this approach to describe spatial variations in stress.

## 2 Methods

*x*,

*y*,

*z*) through

*a*defines the focal length of the ellipse. These coordinates are illustrated in Fig. 1. Corresponding coordinates \((x_0,y_0,z_0)\) and \((\mu _0,\nu _0,\phi _0)\) describe the reference configuration.

### 2.1 Myocardial domain

*f*in prolate spheroidal coordinates is described in the supplementary material Section S2. We also define a function \(\nu _\mathrm{up0}(\phi _0)\) that describes the basal boundary. We compute \(\nu _\mathrm{up0}(\phi _0)\) by a one-dimensional periodic spline function. Thus, the reference myocardial domain is

### 2.2 Kinematics

The myocardium may be deformed by an arbitrary displacement from reference coordinates \((\mu _0, \nu _0, \phi _0)\) to deformed coordinates \((\mu , \nu , \phi )\). The deformed myocardial domain \(\varOmega\) is the image of the reference domain \(\varOmega _0\) under such a mapping. In this section, we construct a mapping that describes LV deformations using a limited number of deformation modes. The contribution of these modes to LV displacement is determined by kinematic variables \(q_i\). We evaluate the ability of these modes to represent actual cardiac deformations by analysis of tagged cardiac MRI data in Sect. 2.2.4.

#### 2.2.1 Incompressible framework

#### 2.2.2 Deformation mode definitions

*i*develop a greater degree of local variations in the \(\nu\) coordinate, while increasing values of

*j*and

*k*lead to a higher degree of local variations in the \(\phi\) coordinate. Examples of deformations according to several modes are illustrated in Fig. 3. The total number of terms allowed for each index (

*i*,

*j*,

*k*) determine the number of basis functions in the Fourier series, and therefore the overall deformable freedom for the coordinate under consideration (\(\mu _\mathrm{in}\), \(\nu\), or \(\phi\)).

#### 2.2.3 Strain measures

#### 2.2.4 Evaluation of the characteristic deformation mode kinematic model using cardiac MRI

As described above, we represent the kinematics of the LV with deformation modes that extend over the whole myocardium (Fig. 3). For this approach to be effective, the majority of LV deformation should be representable with relatively few modes. To illustrate that this is possible, we evaluated the ability of such modes to describe the motion of the LV observed with tagged cardiac magnetic resonance imaging (MRI) (see Figure S3). We analyzed two cases: the first from a healthy volunteer and the second from a patient with dilated cardiomyopathy. These data were previously published by Kar et al. (2014) and are used here with permission.

We estimated the deformation of the myocardium from the tagged MRI data using a deformable image registration algorithm. The myocardial walls were outlined manually in each imaging plane before the image registration was performed. The image registration method was described previously (Hong 2018) and is outlined briefly in the supplementary material Section S6. For the purposes of this kinematic evaluation, we assumed the reference configuration to be the end-systolic shape. While this is not the true reference configuration, this choice has little effect on this purely kinematic analysis.

### 2.3 Dynamics

#### 2.3.1 Muscle fiber directions

*s*,

*n*,

*f*) is introduced with base vectors in the fiber direction \(\varvec{e}_f\), the sheet direction \(\varvec{e}_s\)—which lies within the muscle sheet and is perpendicular to the fiber direction, and \(\varvec{e}_n = \varvec{e}_f \times \varvec{e}_s\) (supplementary material Section S7). Quantities that are represented in terms of the local fiber coordinate basis may be transformed to the prolate spheroidal basis using an appropriate unitary rotation matrix

#### 2.3.2 Passive elastic stress

*s*,

*n*,

*f*) representation is

#### 2.3.3 Viscous stress

#### 2.3.4 Active stress

*A*(

*t*), a length–tension relationship \(g(\lambda _{ff})\), and a linear force–velocity relationship:

#### 2.3.5 Surface tractions and chamber volume

For the system to conserve energy, the LV chamber must be closed to define the cavity volume and work done by motion at the base must be incorporated into the virtual work equations. An appropriate closing surface is difficult to define in prolate spheroidal coordinates, and so the surface \(\varGamma\) is defined in cylindrical coordinates (supplementary material Section S9) as shown in Fig. 2.

#### 2.3.6 Virtual work differential equation system

*t*. The total PK2 stress can be expressed as

#### 2.3.7 Numerical solution of the virtual work system

The system (41,42) is an initial value problem for \(\varvec{q}\) and \(P_\mathrm{lv}\). We solve this system by setting initial values of \(\varvec{q}_0\), determining the pressure \(P_\mathrm{lv}\) in that state, and integrating in time using an explicit scheme such as a Runge–Kutta (RK) method. Time is discretized as \(t = m \varDelta t\). When solving for the \(m+1\) time step, the values of the coefficients \(\varvec{\alpha }\), \(\varvec{\kappa }\), and \(\varvec{\eta }\) are computed at time step *m*. The reference myocardial domain \(\varOmega _0\) is discretized into a \(N_\mu \times N_\nu \times N_\phi\) grid to facilitate the numerical evaluation of the coefficients (40). The volume integrals are approximated using a Simpson’s rule that integrates through \(\mu _0\), \(\nu _0\), and \(\phi _0\), in that order (supplementary material Section S11).

### 2.4 Dynamic model solution verification

#### 2.4.1 Comparisons using the strong form of the equilibrium equations

#### 2.4.2 Cardiac cycle simulations

We use simplified lumped-parameter models (Moulton et al. 2017) (supplementary material Section S10) to describe the preload and afterload systems, as illustrated in Fig. 4. The aortic valve (AOV) and mitral valve (MV) are modeled by variable resistances that have a nonlinear dependence on the pressure variable \(P_\mathrm{lv}\) (S67,S66). The full system (41,42) therefore requires a nonlinear solver to evaluate the derivative terms necessary for the time integrator. We use Newton’s method to evaluate the time derivatives (supplementary material Section S11).

## 3 Results

### 3.1 Evaluation of the kinematic model using cardiac MRI

*N*deformation modes \(\varvec{q}\), and \(\varvec{u}_r\) is the displacement data registered from the cardiac MRI. Thus, 6 deformation modes were capable of accounting for 81% of the deformation in the healthy case and 64% in the case of dilated cardiomyopathy. When the number of deformation modes was increased to 19, these percentages increased to 86% and 76%, respectively. We computed the deformation accounted for as the mean error reduction

### 3.2 Comparisons using the strong form of the equilibrium equations

### 3.3 Cardiac cycle simulations

Figure 8 illustrates simulations of the cardiac cycle. The values of material parameters of the LV and the parameters of the lumped circulatory system were chosen to reproduce normal cardiac function (supplementary material Tables S1 and S2). The three simulations illustrate changes to cardiac function and deformation as the number of kinematic parameters is increased.

## 4 Discussion

The development of theoretical models for LV mechanics has been an active area of investigation for decades. A variety of approaches have been used, ranging from varying elastance models, in which the geometry of the LV is not explicitly represented, to finite element models based on detailed descriptions of LV geometry. While this work has provided important insights into cardiac mechanics, all such models have limitations in terms of their applicability. The simpler models have limited ability to make use of information about LV shape and motion, while the more complex models involve challenging issues of parameter specification and impose heavy computational requirements. The goal of the present work is to develop approaches for modeling LV mechanics that can incorporate information about LV shape and deformation, as derived, for example, from echocardiography, and yet are computationally efficient so that they can be used to simulate the cardiac cycle with computational times that would be compatible with clinical applications.

### 4.1 Kinematics

In previous work, we developed models for LV mechanics based on a cylindrical model (Moulton and Secomb 2013) and an axisymmetric spheroidal model (Moulton et al. 2017). In both cases, three deformation modes (radial and axial contraction, and torsion) were used to represent the dominant modes of LV deformation during the cardiac cycle. An important feature of these models was that the assumed deformation modes inherently conserve myocardial volume. Therefore, the resulting systems of equations do not involve stiff constraints resulting from the very high bulk modulus of cardiac tissue, allowing more efficient numerical solution. However, both of these models were restricted to geometries with rotational symmetry about a central axis. In the present model, this constraint is relaxed, and non-axisymmetric reference LV shapes and non-axisymmetric modes of deformation are simulated. The deformation modes retain the property of conserving myocardial volume. The number of deformation modes is increased relative to the earlier models, with an arbitrary number of Fourier-series modes.

We evaluated this kinematic model using two tagged MRI data sets. The results are graphed in Fig. 5. We found that, in a normal volunteer, 5 modes were sufficient to capture 80% of the deformation from end-systole to end-diastole. In a patient with dilated cardiomyopathy, 10 modes were sufficient to capture 70% of the deformation. This illustrates that the majority of LV deformation was accounted for with relatively few deformation modes. This analysis suggests that the characteristic deformation model could be effectively employed to study cardiac function in both normal and pathological cases.

The notion of understanding the heart in terms of a limited number of modes has previously been developed using the Cardiac Atlas Project database, which includes more than 3000 cardiac MRI studies (Fonseca et al. 2011). This database has largely been analyzed using principal component analysis (PCA) of LV shapes at end-systole and end-diastole (Zhang et al. 2014). Farrar et al. (2016) demonstrated that the first 5 modes of the PCA were able to account for 58% of the end-systolic to end-diastolic motion in asymptomatic populations. Their result that 5 modes are sufficient to recapitulate much of the cardiac motion agrees well with our analysis. In this work, we developed deformation modes using a first principles approach by constructing a geometry and deformation modes in prolate coordinates that are naturally suited to the LV shape. While these modes are sufficient to recapitulate much of the cardiac deformation in two tagged MRI data sets, kinematic modes with improved physical relevance could be developed through a statistical analysis of cardiac motion across a large imaging database (such as the PCA used by Farrar et al. 2016).

### 4.2 Dynamics

In order to validate the dynamic model and its numerical solution, we applied three tests. First we compared the model solution to the analytic solution for an incompressible cylinder of a Mooney–Rivlin material under expansion. Figure S5 shows that the model accurately predicts the deviatoric stresses with only five integration points in the \(\mu\) direction. Isotropic stresses contribute no work in incompressible deformations, and are therefore not computed within the model framework. However, the isotropic stress may be computed from the model solution. Figure S5 shows that the isotropic stress computed in the model solution agrees well with the analytic result. Secondly, we constructed an exact solution to the strong form equations using asymmetric deformation modes, and demonstrated that the variational approach converged to the strong form solution as the numerical integration grid was resolved (Fig. 7). Because the strong form solution is independent of the virtual work formulation, these simulations verify the virtual work system (41), as well as the numerical implementation.

The third type of test involved simulations of normal cardiac function (Fig. 8). We computed three model solutions with varying degrees of kinematic freedom. Simulation **a** had \(N_q = 8\) degrees of freedom and required 2.3 seconds per cardiac cycle to compute in parallel on an Intel i9-7980XE workstation with 18 cores clocked at 2.6 GHz. Simulation **b** had \(N_q = 23\) degrees of freedom and required 9.1 s/cycle, while simulation **c** had \(N_q = 46\) and required 27 s/cycle. By comparison, (Kerckhoffs et al. 2007) reported that individual time steps with an FEM model required 2 min to compute. This implies that each cardiac cycle computed with 400 time steps (the temporal resolution used here) would require 13 h. While recent advances in parallel computing would likely improve that estimate, the method presented here still presents an efficient alternative, especially in the cases where 8 or 23 deformation modes are used.

Despite the variation of the degrees of freedom and consequent computation time increases, only minor differences are visible in the cardiac cycle PV-loop between simulations (Fig. 8). End-diastolic and end-systolic shapes are similar in terms of aggregated parameters: volume, long-axis length, and short-axis radius. The work done with 8 deformation modes (1.084 J/cycle) differs from the work done with 46 deformation modes (1.028 J/cycle) by \(5.1\%\). In addition, the volume, long-axis length, and short-axis radius at end-diastole differ by \(0.7\%\), \(0.6\%\), and \(2.6\%\), respectively. These simulations support the use of a restricted set of deformation modes to represent LV dynamics, since the inclusion of additional modes had only small effects on these parameters. This result suggests that, for studies based on aggregated parameters or limited data, the model with 8 modes would be adequate.

Figure 9 shows the effects of varying the number of deformation modes on distributions of stress and strain. Rows (I) and (II) show that all three models have similar mean stretch ratios in the fiber direction. However, as the number of kinematic variables is increased, more localized spatial variations are developed with greater magnitude. These variations result from the nonsymmetric initial shape of the LV. The PK2 fiber stress distributions (III, IV) show similar trends. The stress at end-systole is almost linearly correlated with the fiber stretch \(\lambda _{ff}\) due to the stretch dependence of the active stress generation described in Eq. (32).

### 4.3 Limitations and future development

In the approach presented here, some restrictions are imposed on the allowable deformation modes. The functions \(\phi (\nu _0, \phi _0)\) and \(\nu ( \nu _0, \phi _0 )\) mapping the reference to the deformed configuration are assumed to be independent of \(\mu _0\). This assumption makes possible the integration of (7) with respect to \(\mu _0\), giving an implicit algebraic equation (9) for \(\mu (\mu _0,\nu _0, \phi _0)\). At the same time, this assumption largely restricts shearing motions within the myocardium to the torsional component. The approach presented here could be extended to allow the other two components of shear deformation. In that case, \(\mu\), \(\phi\) and \(\nu\) would all be functions of \(\mu _0\), \(\nu _0\) and \(\phi _0\), and the \(\mu\) mapping would not be expressible in algebraic form.

While providing a natural framework for describing the LV geometry and kinematics, the prolate spheroidal coordinate system suffers from limits to the deformable freedom allowed through the apex due to the singularities at the axis. While the incompressible kinematic model described in Sect. 2.2.1 is dependent on this choice, the variational formulation used to compute LV dynamics in terms of a limited number of kinematic variables is not. It would be possible to describe a geometry and analogous kinematic model in any curvilinear coordinate system, such as spherical or Cartesian coordinates, and the formulation of the dynamic model presented in Sect. 2.3 would apply.

As shown in Fig. 5, a relatively small number of modes accounts for a large proportion of the deformation, as defined in terms of displacement error. The stress and strain depend on derivatives of the displacement field and, consequently, are more strongly affected by inclusion of higher-order displacement modes. A substantially larger number of modes would be required to achieve numerical convergence of the stress and strain fields. The present model is not well suited for computing stress and strain at high spatial resolution, for which a FEM approach would be more suitable.

While we have only described the mechanics of the LV, the framework developed here is naturally extendable to a bi-ventricular geometry. The right ventricle can be directly added to the variational equations (15). Displacement modes can be constructed separately for the RV, although motion continuity would be required at the LV-RV boundary. Further, with appropriate simplified models for the heart valves, this approach could be extended to produce efficient simulations of a four-chamber heart model. The kinematics of each additional chamber would primarily be described by deformation modes that extend only over that region, implying that the increase in computational cost would be generally additive.

A simplified description of myocardial activation and force generation is used. The model for active force takes into account the length–tension and force–velocity characteristics of cardiac muscle. The Frank–Starling effect, in which force generation depends on end-diastolic fiber strain, is not included, although this effect can be represented by a simple modification to (32) (Moulton et al. 2017). The time-dependence of force generation during systole is represented by a spatially independent activation function \(A = A(t)\), and the effects of the time-dependent spatial spread of activation are not included here. Such effects could be introduced in the present approach by using a spatially varying activation function \(A = A(\varvec{x},t)\), provided that the deformation modes were chosen to accommodate the resulting cardiac motions.

### 4.4 Conclusion

A method for simulating LV dynamics using a limited number of deformation modes, which was originally developed for axisymmetric geometries (Moulton et al. 2017), is here extended to general three-dimensional LV shapes and deformations. The method is computationally efficient: simulation of one cardiac cycle using the 23 deformation mode model takes approximately 9 s on a personal computer. Addition of modes beyond this number has only slight effects on overall parameters describing LV function, although local distributions of stress and strain are still affected. The method is suitable for a range of applications in which FEM simulations at high spatial resolution would be computationally impractical. Such applications include systemic exploration of effects of changing parameters describing cardiac mechanics, estimation of such parameters from echocardiographic imaging data, simulations performed over multiple cardiac cycles, and estimation of spatially dependent fiber stresses and strains for use in models of cardiac remodeling.

## Notes

### Acknowledgements

This work was supported by NIH Grant T32 GM084905.

## Supplementary material

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