Coupling of a 3D Finite Element Model of Cardiac Ventricular Mechanics to Lumped Systems Models of the Systemic and Pulmonic Circulation

Abstract

In this study we present a novel, robust method to couple finite element (FE) models of cardiac mechanics to systems models of the circulation (CIRC), independent of cardiac phase. For each time step through a cardiac cycle, left and right ventricular pressures were calculated using ventricular compliances from the FE and CIRC models. These pressures served as boundary conditions in the FE and CIRC models. In succeeding steps, pressures were updated to minimize cavity volume error (FE minus CIRC volume) using Newton iterations. Coupling was achieved when a predefined criterion for the volume error was satisfied. Initial conditions for the multi-scale model were obtained by replacing the FE model with a varying elastance model, which takes into account direct ventricular interactions. Applying the coupling, a novel multi-scale model of the canine cardiovascular system was developed. Global hemodynamics and regional mechanics were calculated for multiple beats in two separate simulations with a left ventricular ischemic region and pulmonary artery constriction, respectively. After the interventions, global hemodynamics changed due to direct and indirect ventricular interactions, in agreement with previously published experimental results. The coupling method allows for simulations of multiple cardiac cycles for normal and pathophysiology, encompassing levels from cell to system.

Appendices

Appendix A: time-varying elastance model for ventricles that includes direct ventricular interaction

In the heart, the relation between ventricular volumes and pressures is written as:

$$ \Delta \vec {V}=\underline{C} \cdot \vec {p}=\left[ {{\begin{array}{l} {\Delta V_{\rm L} }\\ {\Delta V_{\rm R}}\\ \end{array}}} \right]=\left[ {{\begin{array}{ll} {C_{\rm LL} }& {C_{\rm LR} (p_{\rm L} )}\\ {C_{\rm RL} (p_{\rm R} )}& {C_{\rm RR} }\\ \end{array} }} \right]\left[ {{\begin{array}{l} {p_{\rm L} }\\ {p_{\rm R} }\\ \end{array} }} \right]=\left[ {{\begin{array}{ll} {\frac{dV_{\rm L}}{dp_{\rm L}}}&{\frac{dV_{\rm L}}{dp_{\rm R}}}\\ {\frac{dV_{\rm R} }{dp_{\rm L} }}& {\frac{dV_{\rm R} }{dp_{\rm R} }}\\ \end{array}}}\right]\left[{{\begin{array}{l} {p_{\rm L} }\\ {p_{\rm R} }\\ \end{array} }}\right] $$

(A1)

In which $\Delta\vec{V}$ is a vector with LV and RV instantaneous ventricular volumes minus the rest volumes:

$$ {\vec{V}}_{\rm rest}=(1-y_{\rm v}) \left[{{\begin{array}{ll} V_{\rm L,rest,d}&-V_{\rm L,rest,s}\\ V_{\rm R,rest,d}&-V_{\rm R,rest,s}\\ \end{array}}}\right]+ \left[{{\begin{array}{l} V_{\rm L,rest,s}\\ V_{\rm R,rest,s}\\ \end{array}}}\right] $$

(A2)

where y _v is a ventricular activation function:

$$ y_{\rm v}=\left\{{{\begin{array}{ll} -\frac{1}{2}\cos(2\pi t_{\rm ventricle}/t_{\rm twitch})+0.5 &t_{\rm ventricle} < t_{\rm twitch}\\ 0&t_{\rm ventricle}\geq t_{\rm twitch}\\ \end{array}}}\right. $$

(A3)

with

$$ t_{\rm ventricle}=\left\{ {{\begin{array}{ll} {\bmod(t-t_{\rm active} ,t_{\rm cycle} )}& {t\geq t_{\rm active}}\\ 0& {t < t_{\rm active} }\\ \end{array}}}\right. $$

(A4)

where V _x,rest,d and V _x,rest,s are diastolic and systolic unloaded volumes.

Matrix $\underline{C}$ is the ventricular time-varying compliance matrix:

$$ \underline{C}=y_{\rm v}(\underline{C}_{\rm max }-\underline{C}_{\rm min} )+\underline{C}_{\rm min} $$

(A5)

$\underline{C}_{\rm max}$ and $\underline{C}_{\rm min}$ are compliances for the fully active and passive state, respectively.

From the pressure and volume curves in Fig. 2 it can be seen that ventricular co-compliances are pressure-dependent: i.e. the RV volume change for a LV pressure change is different at a constant low and high RV pressure. Hence C _LR and C _RL in Eq. A1 are written as a function of pressure (see also Table A1).

The same procedure is performed for maximally activated ventricles (Fig. 2b).

Using the time-varying elastance model in the more common way (with input volume and output pressure), Eq. A1 is rewritten as:

$$ \vec {p}={\underline C}^{-1}(t)\cdot (\vec {V}-\vec {V}_{\rm rest}) $$

(A6)

Using this equation, the co-compliances need to be written as a function of volume:

$$ C_{\rm min,LR}=C_{\rm min,LR,slope} V_{\rm L} +C_{\rm min,LR,intercept} $$

(A7)

$$ C_{\rm min,RL} =C_{\rm min,RL,slope} V_{\rm R} +C_{\rm min,RL,intercept} $$

(A8)

$$ C_{\rm max,LR}=C_{\rm max,LR,slope} V_{\rm L} +C_{\rm max,LR,intercept} $$

(A9)

$$ C_{\rm max,RL}=C_{\rm max,RL,slope} V_{\rm R} +C_{\rm max,RL,intercept} $$

(A10)

Parameter values and results of the time-varying elastance model are shown in Table A1 and Fig. 2, respectively, for passive and fully activated myocardium.

TABLE A1. Parameters of the time-varying elastance model.

Appendix B: circulatory model

Time-Varying Elastances for Atria

The atrial elastances are driven by an activation function

$$ y_{\rm a}=\left\{{{\begin{array}{ll} {-\tfrac{1}{2}\cos(2\pi t_{\rm atrium}/t_{\rm twitch} )+0.5}& {t_{\rm atrium} < t_{\rm twitch}}\\ 0& {t_{\rm atrium} \geq t_{\rm twitch} }\\ \end{array}}}\right. $$

(B1)

where

$$ t_{\rm atrium} =\left\{ {{\begin{array}{ll} {\bmod (t-t_{\rm active} +\Delta t_{\rm PR} ,t_{\rm cycle} )}& {t\geq t_{\rm active} -\Delta t_{\rm PR} }\\ 0& {t < t_{\rm active} -\Delta t_{\rm PR} }\\ \end{array}}}\right. $$

(B2)

Left atrial pressure is given by

$$ p_{\rm LA} =E_{\rm LA}\times (V_{\rm LA} -V_{\rm LA,rest}) $$

(B3)

where LA elastance and rest volume (volume at zero pressure) are given by

$$ E_{\rm LA} =(E_{\rm LA,max } -E_{\rm LA,min})\times y_{\rm a} +E_{\rm LA,min} $$

(B4)

and

$$ V_{\rm LA,rest} =(1-y_{\rm a} )(V_{\rm LA,rd} -V_{\rm LA,rs} )+V_{\rm LA,rs} $$

(B5)

Right atrial pressure, elastance, and rest volume are given by

$$ p_{\rm RA} =E_{\rm RA} \times(V_{\rm RA} -V_{\rm RA,rest}) $$

(B6)

$$ E_{\rm RA}=(E_{\rm RA,max}-E_{\rm RA,min} )\times y_{\rm a}+E_{\rm RA,min} $$

(B7)

$$ V_{\rm RA,rest} =(1-y_{\rm a} )(V_{\rm RA,rd}-V_{\rm RA,rs} )+V_{\rm RA,rs} $$

(B8)

Systemic Circulation

$$ p_{\rm as} =V_{\rm as} /C_{\rm as} $$

(B9)

$$ p_{\rm vs} =V_{\rm vs} /C_{\rm vs} $$

(B10)

$$ Q_{\rm ao} =\left\{ {{\begin{array}{ll} {(p_{\rm LV,estim} -p_{\rm as} )/R_{\rm ao} }& {p_{\rm LV,estim} \geq p_{\rm as} }\\ 0& {p_{\rm LV,estim} < p_{\rm as} }\\ \end{array} }}\right. $$

(B11)

$$ Q_{\rm as} =(p_{\rm as} -p_{\rm vs} )/R_{\rm as} $$

(B12)

$$ Q_{\rm vs} =(p_{\rm vs} -p_{\rm RA} )/R_{\rm vs} $$

(B13)

$$ Q_{\rm mitral} =\left\{ {{\begin{array}{ll} {(p_{\rm LA} -p_{\rm LV,estim})/R_{\rm mitral} }&{p_{\rm LA}\geq p_{\rm LV,estim} }\\ 0& {p_{\rm LA} < p_{\rm LV,estim} }\\ \end{array} }} \right. $$

(B14)

$$ \frac{dV_{\rm LA} }{dt}=Q_{\rm vp} -Q_{\rm mitral} $$

(B15)

$$ \frac{dV_{\rm LV,circ} }{dt}=Q_{\rm mitral} -Q_{\rm ao} $$

(B16)

$$ \frac{dV_{\rm as} }{dt}=Q_{\rm ao} -Q_{\rm as} $$

(B17)

$$ \frac{dV_{\rm vs} }{dt}=Q_{\rm as} -Q_{\rm vs} $$

(B18)

Pulmonary Circulation

$$ p_{\rm ap} =V_{\rm ap} /C_{\rm ap} $$

(B19)

$$ p_{\rm vp} =V_{\rm vp} /C_{\rm vp} $$

(B20)

$$ Q_{\rm pa} =\left\{ {{\begin{array}{ll} {(p_{\rm RV,estim} -p_{\rm ap} )/R_{\rm pa} }& {p_{\rm RV,estim} \geq p_{\rm ap} }\\ 0& {p_{\rm RV,estim} < p_{\rm ap} }\\ \end{array}}}\right. $$

(B21)

$$ Q_{\rm ap} =(p_{\rm ap} -p_{\rm vp} )/R_{\rm ap} $$

(B22)

$$ Q_{\rm vp} =(p_{\rm vp} -p_{\rm LA} )/R_{\rm vp} $$

(B23)

$$ Q_{\rm tricus} =\left\{ {{\begin{array}{ll} {(p_{\rm RA} -p_{\rm RV,estim} )/R_{\rm tricus} }& {p_{\rm RA} \geq p_{\rm RV,estim} }\\ 0& {p_{\rm RA} < p_{\rm RV,estim} }\\ \end{array}}}\right. $$

(B24)

$$ \frac{dV_{\rm RA} }{dt}=Q_{\rm vs} -Q_{\rm tricus} $$

(B25)

$$ \frac{dV_{\rm RV,circ} }{dt}=Q_{\rm tricus} -Q_{\rm pa} $$

(B26)

$$ \frac{dV_{\rm ap} }{dt}=Q_{\rm pa} -Q_{\rm ap} $$

(B27)

$$ \frac{dV_{\rm vp} }{dt}=Q_{\rm ap} -Q_{\rm vp} $$

(B28)

Notice that ventricular volume changes (Eqs. B16 and B26) are purely determined by ventricular in- and outflows. In the case of coupling between the FE and circulatory model, cavity pressures come from the update algorithm. In the case of the procedure for determining the initial conditions, cavity pressures are calculated by the ventricular time-varying elastance model.

See Table B1 for a description of state variables and their initial conditions. Tables B2 and B3 contain descriptions of circulatory variables and parameters, respectively.

TABLE B1. State variables of the circulatory model.

TABLE B2. Variables of the circulatory model.

TABLE B3. Parameters of the circulatory model.