|Authors||L. Nikitushkina, S. W. Funke, H. Finsberg, L. C. Lee and S. Wall|
|Title||Sensitivity Analysis of Cardiac Growth Models|
|Project(s)||OptCutCell: Simulation-based optimisation with dynamic domains, Center for Biomedical Computing (SFF)|
|Year of Publication||2017|
|Place Published||FEniCS 2017 conference, Luxembourg|
Introduction The heart is a dynamic organ capable of changing its shape in response to the body’s demands. For example, the human heart continuously adapts in size and geometry to meet greater blood flow needs of the growing body during normal development. In this case, a gradually imposed volume overload leads to progressive chamber enlargement. Another example of a normal physiological growth can be found in the athlete’s heart, where a sustained elevated chamber pressure results in the chamber wall thickening and an overall increase in cardiac mass. Growth processes, however, can also be maladaptive as found in many cardiovascular diseases where structural changes in the heart progressively decompensate cardiac function.
In order to better understand this balance between adaptive and maladaptive cardiac growth, we examine the effect of known growth stimuli using a mechanical model of the heart. We perform a sensitivity analysis of existing growth models in order to assess the relative importance of model parameters and respective mechanisms. This work can eventually lead to simplifications in the model systems for prediction of growth, or help in localizing shortcomings that need to be addressed in the existing modeling frameworks.
Methods In order to simulate the motion of the heart throughout the cardiac cycle, we use a nonlinear finite element (FE) model of a realistic left ventricle (LV) coupled to a lumped-parameter model of the systemic circulation. Under the quasi-static assumption, this problem is reduced to finding the displacement u, hydrostatic pressure p and LV pressure p LV that minimize the incompressible strain energy functional Π parameterised by the LV volume V LV : The muscular tissue of the heart is modeled as a transversely isotropic hyperelastic material via the strain energy density Ψ :
where m = (a, b, a f , b f ) is a set of passive material parameters, C e is the elastic right Cauchy-Green. By introducing an activation parameter γ representing the active tensor, I1 = trC e and I shortening in the fiber direction f 0 at zero-load, the model incorporates muscle contraction using the active strain approach, which is based on a multiplicative decomposision of the deformation gradient F = I + Grad(u) into an elastic and an active parts F = F e F a with F a defined as:
The dynamic changes in the ventricular blood pressure and volume over the entire cardiac cycle are modeled by a three-element Windkessel model described by a system of ordinary differential equations . At each time step, the coupling between the FE model and the circulatory model is achieved through an additional Lagrange multiplier p LV which represents the LV cavity pressure. The problem (1) is solved such that the simulated LV cavity volume V (u) matches the target volume value V LV obtained from the circulatory model.
Growth process in the heart wall is modeled by deforming the reference unloaded geometry to a new grown configuration, again through a multiplicative decomposition of F into an elastic and, this time, a growth part, where F = F e F g . The constitutive laws for finite growth can be expressed using a generic format for the growth tensor F g = θ f f 0 ⊗ f 0 + θ s s 0 ⊗ s 0 + θ n n 0 ⊗ n 0. The evolution of the local tissue growth parameter θ g = [θ f , θ s , θ n ] T can be defined in terms of a growth activation function φ (F e ) and a growth rate function k(θ θ g ) which specifies the speed and nonlinearity of the growth process .
Using the above described model it is possible to simulate various physiological conditions together with the associated structural adaptation of the heart walls in response to change in loadings. In each case, a growth model can be chosen depending on the nature of the considered physiology. The sensitivity of the system’s grown state to the prescribed growth model can then be estimated. For this, we define an objective functional J(u), the model output of interest, which is to serve as a qualitative and/or quantitative measure of how well the growth model reproduces the expected behavior. More specifically, if we are to compare the model response to a real measurement, then the objective functional can be defined as the mismatch between the simulated u and the measured u exp grown states at a reference time tref :
Finally, the sensitivities of J to θ , where θ is a set of growth parameters specific to a given model, are defined as dJ(u)dθ.
The solver has been developed within the FEniCS  framework and the functional gradients are computed by solving an automatically derived adjoint equations .
Results We first focus on implementing and testing a strain-driven growth law to simulate a volume overload state of the left ventricle. To achieve this, the initial physiological equilibrium of the heart is altered by increasing a diastolic filling pressure. This initiates cardiac growth that continues until a new equilibrium state is reached. The model is able to reproduce qualitatively experimental observations reported in the literature, such as LV cavity dilation due to fiber over-stretching and a gradual increase in myocardium volume. At the next step, we perform a sensitivity analysis of the model with respect to the growth model parameters and evaluate its performance in reproducing the expected growth behavior.