An in silico study of the effects of left ventricular assist device on right ventricular function and inter-ventricular interaction

Abstract

Background:

Left ventricular assist device (LVAD) is associated with a high incidence of right ventricular (RV) failure, which is hypothesized to be caused by the occurring inter-ventricular interactions when the LV is unloaded. Factors contributing to these interactions are unknown.

Methods:

We used computer modeling to investigate the impact of the HeartMate 3 LVAD on RV functions. The model was first calibrated against pressure–volume (PV) loops associated with a heart failure (HF) patient and validated against measurements of inter-ventricular interactions in animal experiments. The model was then applied to investigate the effects of LVAD on (1) RV chamber contractility indexed by $V_{60}$ derived from its end-systolic PV relationship, and (2) RV diastolic function indexed by $V_{20}$ derived from its end-diastolic PV relationship. We also investigated how septal wall thickness and regional contractility affect the impact of LVAD on RV function.

Results:

The impact of LVAD on RV chamber contractility is small at a pump speed lower than 4k rpm. At a higher pump speed between 4k and 9k rpm, however, RV chamber contractility is reduced (by ~3% at 6k rpm and ~10% at 9k rpm). The reduction of RV chamber contractility is greater with a thinner septal wall or with a lower myocardial contractility at the LV free wall, septum, or RV free wall.

Conclusion:

RV chamber contractility is reduced at a pump speed higher than 4k rpm, and this reduction is greater with a thinner septal wall or lower regional myocardial contractility. Findings here may have clinical implications in identifying LVAD patients who may suffer from RV failure.

1 |. INTRODUCTION

Left ventricular assist devices (LVADs) are used as a bridge to transplantation or as long-term indefinite therapy in patients who have advanced heart failure (HF). This treatment is, however, associated with right ventricular (RV) failure that is found in 13% to 44% of LVAD recipients.¹ The occurrence of RV failure in LVAD patients has been attributed to the changes in RV function resulting from a change in the loading condition of the LV during systole and diastole.² In particular, it has been hypothesized that the unloading of LV during LVAD operation causes the inter-ventricular septum to shift into the LV and decreases LV pressure generation, both of which contribute to reduced RV contractility.

The basis of this explanation rests on the so-called “ventricular interdependence,” where changes in the loading condition of one ventricle affect the diastolic and systolic functions of the other ventricle.^3,4 Based on animal experiments using isolated hearts, it was established that the LV chamber contractility (indexed by a change in the end-systolic pressure–volume relationship, ESPVR, as well as the peak LV pressure) is augmented with an increase in RV pressure and vice versa,³ indicating systolic ventricular interdependence. On the other hand, it has also been shown that the LV volume is increased at a given filling pressure when the RV chamber is empty (i.e., unloaded),⁴ indicating diastolic ventricular interdependence. As such, the unloading of LV, where its chamber pressure and volume are both reduced by LVAD, may affect RV function via ventricular interdependence through the septum wall.

Clinical observations, however, have found that the effects of LVAD on RV function via inter-ventricular interactions are not consistent and are heterogeneous in patients. For example in one recent study,⁵ invasive pressure measurements made in patients revealed a wide range of effects of LVAD on the RV function, with some showing no effects and some showing effects on the systolic and diastolic RV functions. It is not clear what gives rise to the heterogeneous effects of LVAD on RV function. Understanding the mechanistic effects of LVAD on RV function is significant in that such knowledge can be used to predict patients who may suffer from RV failure post-LVAD implantation. This is especially so as previous studies have found that left and right filling pressures, mean pulmonary artery pressure, heart rate, cardiac index, RV ejection fraction (EF), and RV end-diastolic volume (EDV) and end-systolic volume (ESV) cannot predict which patients require RV support after LVAD implantation (non-negative preoperative predictors).²

Factors contributing to the impact of LVAD on RV function via inter-ventricular interactions, however, are difficult to discern and isolate from pure clinical studies. To address this issue, we apply a computational (finite element, FE) model developed based on our previous work^6–9 to investigate factors affecting the effects of LVAD on RV systolic and diastolic functions.

2 |. METHODS

2.1 |. Biventricular computational framework with an implanted LVAD

Based on previous work,^8,9 the computational modeling framework consists of a biventricular FE model and an LVAD connected to a closed-loop lumped representation of the systemic and pulmonary circulations (Figure 1A). The computational model is formulated based on mass conservation and equilibrium of active and passive stress in the myocardium with respect to its loading conditions. The detailed formulation for the closed-loop lump parameter model is given in Appendix A.

FIGURE 1.

(A) Schematic of the closed-loop lumped parameter framework that couples the left ventricular assist device (LVAD) and finite element (FE) biventricular model; (B) Pump characteristics curve used to model the LVAD, where dP is the pressure difference between the inflow and outflow cavities ($P_{sa}\left(t\right)$ and $P_{LV}\left(t\right)$), Q is the flow rate in LVAD, each curve represents a different pump speed with unit (k rpm); (C) The dimension of the biventricular geometry and material regions associated with the right ventricular free wall (RVFW), septum and left ventricular free wall (LVFW), respectively. [Color figure can be viewed at wileyonlinelibrary.com]

Flow through the LVAD was sourced from the LV and ejected into the systemic arterial system. The LVAD was described based on the measured pressure gradient (dP) – flow characteristics of a commercially available system (HeartMate 3, Abbot Laboratories, Abbott Park, IL) with an operating pump speed $N$ between 4k and 9k rpm^5,10 (Figure 1B). The LVAD flow rate is dependent on (1) the pressure difference (dP) between the inflow and outflow cavities (i.e., $\text{dP}=P_{sa}\left(t\right)-P_{LV}\left(t\right)$) and (2) the operating pump speed. In the computational framework, LVAD flow rate, $q_{\mathit{LVAD}}\left(t\right)$, was linearly interpolated between discrete values from a continuous $\text{dP}-Q-N$ relationship. The biventricular geometry is idealized as the fusion of two truncated ellipsoids representing the LV and RV. The dimensions of the biventricular geometry are prescribed based on those used to describe a failing heart in a previous study (Figure 1C),¹¹ where the resultant LV EDV is comparable to heart failure patients with dilated cardiomyopathy.¹²

2.2 |. Finite element formulation of the biventricular model

The relationships between pressures and volumes in the LV and RV are described based on the biventricular FE model as

$$P_{LV,RV}\left(t\right)=f^{BiV}\left(V_{LV,RV}\right(t)).$$ (1)

Finite element formulation of the biventricular can be generalized by minimizing the Lagrangian function

$$\mathcal{L}\left(u,\hspace{0.33em}p,\hspace{0.33em}{P}_{LV},\hspace{0.33em}{P}_{RV},\hspace{0.33em}{c}_1,\hspace{0.33em}{c}_2\right)={\int }_{{\text{Ω}}_0}W\left(u\right)dV-{\int }_{{\text{Ω}}_0}p(J-1)dV-P_{LV}\left(V_{LV}\left(u\right)-V_{LV,p}\right)-P_{RV}\left(V_{RV}\left(u\right)-V_{RV,p}\right)-c_1•{\int }_{{\text{Ω}}_0}u\hspace{0.17em}dV-c_2•{\int }_{{\text{Ω}}_0}X\times u\hspace{0.17em}dV.$$ (2)

In Equation (2), ${\text{Ω}}_0$ is the reference configuration of the biventricular unit, $u$ is the displacement field, $p$ is a Lagrange multiplier to enforce incompressibility of the tissue (i.e., Jacobian of the deformation gradient tensor $J=1$), $P_{LV}$ and $P_{RV}$ are the Lagrange multipliers to constraint the cavity volumes $V_{LV}\left(u\right)$ and $V_{RV}\left(u\right)$ to the prescribed values $V_{LV,p}$ and $V_{RV,p}$, respectively,¹³ and both $c_1$ and $c_2$ are Lagrange multipliers to constrain rigid body translation (i.e., zero mean translation) and rotation (i.e., zero mean rotation),¹⁴ $X$ is the coordinate of a material point and $W$ is the strain energy function of the myocardial tissue. The LV and RV cavity volume can be described by

$$V_{LV,RV}\left(u\right)=\underset{{\text{Ω}}_{inner,(LV,RV)}}{\int }d{v}_{LV,RV}=-\frac{1}{3}\underset{{\text{Γ}}_{inner\text{,(}LV,RV)}}{\int }x.n\hspace{0.33em}d{a}_{LV,RV},$$ (3)

where ${\text{Ω}}_{inner\text{,(}LV,RV)}$ denotes the LV and RV volumes enclosed by their inner surface ${\text{Γ}}_{inner\text{,(}LV,RV)}$, and $n$ denotes the outward unit normal vector of the surfaces. The geometry of the biventricular unit and constitutive law of biventricular are given in Appendix B.

2.3 |. Quantification of systolic and diastolic inter-ventricular interactions

Based on a previous experimental study on systolic interdependence between the LV and RV,³ we define (left to right) systolic inter-ventricular interaction as the changes in the RV ESPVR slope $E_{es}$ (i.e., $\Delta E_{es}$) and volume intercept $V_0$ i. e. , $\Delta V_0$ due to a change in the LV pressure arising from varying the contractility of the left ventricular free wall (LVFW) region in the computational model (Figure 1C). Because $E_{es}$ and $V_0$ varies in the opposite direction with a change in the chamber contractility, an index that integrates both of these quantities for the RV, namely, $V_{60}=60\text{/}{E}_{es}+V_0$ or the RV volume at an end-systolic pressure of 60 mm Hg is proposed and evaluated here. The choice of using an end-systolic pressure of 60 mm Hg is based on the simulation results (see Section 3), where the peak RV pressure varies between 45 and 55 mm Hg. We note here that a lower end-systolic pressure of 20 mm Hg was used in another study⁵ as the range of peak RV pressure there is lower (20–45 mm Hg). A similar index $V_{120}=120\text{/}{E}_{es}+V_0$, which reflects the higher chamber pressure in the LV (~ 120 mm Hg) was used to describe the chamber contractility of LV as with previous studies^15,16 although we note that $V_{100}$ is also used in some studies.^17,18 Here, we used $V_{120}$ and $V_{60}$ to index LV and RV chamber contractility, respectively. Higher values of $V_{120}$ and $V_{60}$ are associated with decreased LV and RV chamber contractility, respectively. Similarly, (left to right) diastolic inter-ventricular interaction is defined as the change in the volume intercept of the end-diastolic pressure–volume relationships (EDPVR) at a specified filling pressure.¹⁹ Here, we chose the filling pressure for evaluating the volume intercept to be 20 mm Hg for the index $V_{20}$, which is adapted from $V_{30}$ used for assessing the LV.^5,15 As before, this choice is guided by the range of end-diastolic pressure found in the simulations (15–25 mm Hg). The relative changes in systolic and diastolic inter-ventricular interactions induced by LVAD operation that affect the RV are quantified by the ratio of differences of these indices ($V_{60}$ and $V_{20}$) at different pump speeds with respect to their value at zero pump speed $(N=0\hspace{0.33em}\text{rpm)}$. Specifically, the relative changes in systolic and diastolic inter-ventricular interactions at LVAD pump speed $(N=\alpha \hspace{0.33em}\text{rpm)}$ are quantified by $\frac{(V_{60,N=\alpha }-V_{60,N=0})}{V_{60,N=0}}$ and $\frac{(V_{20,N=\alpha }-V_{20,N=0})}{V_{20,N=0}}$, respectively. Right ventricular ESPVR and EDPVR are determined, respectively, from the end-systolic and end-diastolic points in PV loops with different RV preload (by varying $V_{sv0}$ in the model).

2.4 |. Simulation cases

The model was first calibrated to match the clinically measured pressure and volume waveforms of the LV and RV from a HF patient without LVAD $(N=0\text{k}\hspace{0.33em}\text{rpm)}$.⁵ Then, the calibrated model was applied to simulate the effects of LVAD operating at different pump speeds typically found in the clinic $(N=4\text{k,}\hspace{0.33em}5\text{k,}\hspace{0.33em}6\text{k}\hspace{0.33em}\text{rpm)}$.^5,10 We also simulated the effects of the LVAD operating at high pump speeds between 6k and 9k rpm on RV function to compare with animal experiments.²⁰ To investigate the effects of biventricular geometry and mechanics on the inter-ventricular interactions arising from LVAD operation, we simulated the effects of (1) varying septal wall thickness from 0.6 to 3.0 cm while the wall thickness in the LVFW and right ventricular free wall (RVFW) is kept the same as that in baseline; (2) varying myocardial contractility in either the LVFW, septum, RVFW or LVFW + septum from 40% to 100% of its original value. Relative changes in systolic and diastolic interventricular interactions due to LVAD operation were then calculated as described above.

3 |. RESULTS

3.1 |. Heart failure case with LVAD implantation

The computational model was first calibrated to represent the heart function of a patient with end-stage HF in the baseline case,⁵ in which the model predicted PV loops agree with the measurements (Figure 2). Inter-ventricular interaction in the baseline case was quantified by changing the myocardial contractility $T_{max}$ of one chamber's free wall from 0 to its baseline value and assessing the ratio of pressure increase in the two chambers. The model-predicted peak LV and RV pressure, $V_0$ and $E_{es}$ in the ESPVRs are summarized in Table 1. The ratio of an increase in the peak RV pressure to the increase in the peak LV pressure resulting from an increase in the LVFW's myocardial contractility $T_{max}$ from 0 to its baseline value (LV → RV) was found to be 0.14 in the model (Figure 2A). Similarly, when RVFW's myocardial contractility $T_{max}$ was increased from 0 to its baseline value, the volume intercept of LV ESPVR (i.e., $V_0$) was reduced by 3.5 mL, whereas its slope was unchanged (Figure 2B). The ratio of the increase in peak LV pressure to the increase in peak RV pressure (RV → LV) was found here to be 0.67. These results are consistent with experimental measurements.³ In addition, $V_{60}$ associated with the RV ESPVR was reduced from 195 to 164 mL when LVFW's myocardial contractility was increased from 0 and to its baseline value, whereas $V_{120}$ associated with the LV ESPVR was reduced from 184 to 181 mL when RVFW's myocardial contractility was increased from 0 to its baseline value. This result indicates the contractility of each chamber is derived partly from that of the other chamber. Consistent with experiments,³ the model also predicts that the effect of LV function on RV function is more significant compared to the effect of RV function on LV function (Figure 2A,B).

FIGURE 2.

(A) Right ventricular (RV) pressure–volume (PV) loops and end-systolic pressure–volume relationships (ESPVRs) with left ventricular free wall (LVFW) $T_{max}$ as 0 and its calibrated value, denoted by “LV0” and “LV1,” respectively; (B) Left ventricular (LV) PV loops and ESPVRs with right ventricular free wall (RVFW) $T_{max}$ as 0 and its calibrated value, denoted by “RV0” and “RV1,” respectively; (C) Deformation patterns of the biventricular in heart failure (HF) case and HF case with left ventricular assist device (LVAD) at 6k rpm; (D) LV PV loops; (E) LV pressure waveforms; (F) LV volume waveforms; (G) RV PV loops; (H) RV pressure waveforms; (I) RV volume waveforms, for HF case with different LVAD pump speed; Scattered points show the clinical measurements from a HF patient. [Color figure can be viewed at wileyonlinelibrary.com]

TABLE 1.

A summary of the model predicted peak LV and RV pressures, $V_0$ and $E_{es}$ in the ESPVRs. “LV → RV” denotes the effects of LVFW contractility on the RV and vice versa for “RV → LV.”

Conditions	Regions	$T_{max}$ (100%)	Peak LVP (mm Hg)	Peak RVP (mm Hg)	$V_0$ (mm (mL)	$E_{es}$ Hg/mL)
LV → RV	LVFW	0	60	41	41	0.39
		1	102	47	57	0.56
RV → LV	RVFW	0	90	31	136	2.47
		1	102	49	133	2.49

Next, the model was then applied to simulate the effects of the LVAD operating at normal pump speeds between 4k and 6k rpm, and an extreme pump speed of 9k rpm (Figure 2C–I). Operation of LVAD causes the septum to shift into the LV, resulting in an increased RV cavity volume (Figure 2C). The model predicted LV EF was increased from 16% to 22%, LVEDV was reduced from 203 to 187 mL and LVEDP was reduced from 30 to 17 mm Hg when pump speed was increased to 6k rpm (Figure 2E,F). The unloading of LV with increased pump speed was accompanied by a rightward shift of the RV PV loop toward higher volumes and pressures. The increase in LVAD pump speed to 6k rpm produced a small increase in RVEDP (17 to 21 mm Hg) as well as a moderate increase in both RVEDV (160 to 179 mL) and RVEF (19% to 25%) when compared to the baseline HF case (Figure 2G–I).

3.2 |. Effects of septal wall thickness

The effects of septal wall thickness changing from 0.6 cm to 3.0 cm on the RV ESPVR and EDPVR at different pump speeds (0k, 4k-9k rpm) were predicted from the model (Figure 3A–D). At fixed septal wall thickness, $V_0$ and $E_{es}$ from the RV ESPVR, and $V_{20}$ from the RV EDPVR were increased with increasing pump speeds from 0k to 9k rpm (Figure 3A–D). Based on $V_{60}$ that integrates both $V_0$ and $E_{es}$, an increase in pump speed increased $V_{60}$ for each value of septal wall thickness, indicating a reduction in RV contractility in these cases (Figure 3A–D). Without LVAD (baseline HF case), an increase in septal wall thickness reduced $V_{60}$, indicating an increase in RV contractility with increasing septal wall thickness (Figure 3A–D). With LVAD operating at 9k rpm, an increase in septal wall thickness led to a decrease in $V_{60}$ with respect to the baseline HF case (i.e., 18% with thickness of 0.6 cm but 3% with thickness of 3.0 cm) (Figure 3E), indicating that the reduction of RV chamber contractility by LVAD was reduced with increasing septal wall thickness.

FIGURE 3.

End-systolic pressure–volume relationship (ESPVR) and end-diastolic pressure–volume relationship (EDPVR) with left ventricular assist device (LVAD) speed from 0 to 9k rpm at swt of (A) swt0; (B) swt1; (C) swt2; (D) swt3. Effects of swt on (E) $V_{60}$ and (F) $V_{20}$. “swt” denotes septal wall thickness. “ES” and “ED” denote end-systolic pressure–volume relationship and end-diastolic pressure–volume relationship, respectively. [Color figure can be viewed at wileyonlinelibrary.com]

For diastolic interaction, the relative change of $V_{20}$ between HF and HF + LVAD cases at 9k rpm was reduced from 26% to 8% when septal wall thickness was increased from 0.6 to 3.0 cm (Figure 3F). This result indicates that the diastolic interaction was reduced with increasing septal wall thickness.

3.3 |. Effects of regional contractility

The effects of regional myocardial contractility $T_{max}$ (in LVFW, septum, RVFW or LVFW + septum) reducing from 100% to 40% of its value in the baseline case on the RV systolic and diastolic interactions with LVAD were predicted from the model (Figure 4). The RV contractility index $V_{60}$ was 160 mL in the baseline HF case (without LVAD) When $T_{max}$ was reduced to 40% of its value in the LVFW, septum and RVFW of the biventricular unit, this value was increased to 169 mL, 164 mL, and 200 mL, respectively, indicating a reduction in RV chamber contractility (Figures C1–C4).

FIGURE 4.

Effects of left ventricular free wall (LVFW) contractility on (A) $V_{60}$ and (B) $V_{20}$. Effects of septal contractility on (C) $V_{60}$ and (D) $V_{20}$. Effects of right ventricular free wall (RVFW) contractility on (E) $V_{60}$ and (F) $V_{20}$. Effects of LVFW and septal contractility on (G) $V_{60}$ and (H) $V_{20}$. [Color figure can be viewed at wileyonlinelibrary.com]

The reduction in RV chamber contractility (indexed by an increasing $V_{60}$) arising from systolic interaction at different pump speeds was greater at smaller values of $T_{max}$ in the myocardium (Figure 4A,C,E,G). The degree of reduction, however, was different depending on the pump speed and the region at which $T_{max}$ was reduced. For pump speed between 0k and 4k rpm, $V_{60}$ was not affected at the baseline value of $T_{max}$, but the value was increased with increasing pump speed at smaller value of $T_{max}$ in different regions of the myocardium. At a pump speed of 4k rpm, the greatest increase in $V_{60}$ (~4%) is found when $T_{max}$ was decreased (to 40% of its value) at the LVFW + septum. We note that this increase in $V_{60}$ is more than the sum of the increase of ~2% and ~ 1% when $T_{max}$ was decreased only in the LVFW and septum, respectively. At higher pump speed, $V_{60}$ was increased with increasing pump speed and the increase was greater with smaller value of $T_{max}$. At the highest pump speed of 9k rpm, the increase in $V_{60}$ becomes lesser with the reduction of $T_{max}$ at the LVFW and RVFW (Figure 4A,E), but remains substantial when $T_{max}$ at the septum was reduced (Figure 4C,G). Specifically at a pump speed of 9k rpm, $V_{60}$ was increased by ~2% and ~ 1% when $T_{max}$ at the LVFW and RVFW was reduced to 40% of its value (cf. ~ 8% when $T_{max}$ at the septum was reduced by the same amount).

For diastolic interaction, $V_{20}$ increased at a lower rate up for pump speed between 0k and 4k rpm, and increased at a higher rate for pump speed greater than 9k rpm. The behavior is the same for different values of $T_{max}$ (at different regions). The largest increase in $V_{20}$ (~5%) occurred at 9k rpm when $T_{max}$ was reduced from 100% to 40% of its baseline value in the LVFW, septum, and LVFW + septum. In contrast, the smallest increase in $V_{20}$ (~2%) occurred when $T_{max}$ in the RVFW was reduced by 60% from its baseline value.

4 |. DISCUSSION

The main finding of this study is that RV chamber contractility (indexed by $V_{60}$) is reduced when the LVAD is operating at pump speeds higher than 4k rpm. The degree of reduction depends on several factors. Specifically, the reduction is larger with (1) a thinner septal wall and (2) lower myocardial contractility in the LVFW, septum or RVFW (Table 2). The computer model predicts the RV chamber's passive compliance (indexed by $V_{20}$) is increased at higher pump speed; the increase is larger with a thinner septal wall and lower myocardial contractility in the LVFW, septum, or RVFW.

TABLE 2.

Key findings of impact of left ventricular assist device (LVAD) operation on right ventricular (RV) systolic function.

Impact of LVAD	Pump speed, $N$ (k rpm)	Confounding factors	RV chamber contractility
HF + LVAD	$0\le N<4$	–	→
	$4<N\le 9$	–	↓
Impact of factors on RV function with LVAD operation	Pump speed, $N$ (k rpm)	Confounding factors	RV chamber contractility
HF + LVAD + varying factors	$0\le N\le 4$	LVFW contractility ↓	↓ ↓
		Septum contractility ↓	↓ ↓
		RVFW contractility ↓	↓ ↓
		LVFW + Septum contractility ↓	↓ ↓
		Septal wall thickness ↓	↓ ↓

We note here that the model can also reproduce the experimental findings associated with inter-ventricular interactions, where the change in systolic function of one ventricle affects that of the other ventricle.^2,3 The ability to consider and reproduce inter-ventricular interactions between the LV and RV found in the experiments represents a significant improvement over previous modeling frameworks,^21–23 and enables one to investigate how LVAD affects RV function via inter-ventricular interactions.

4.1 |. Factors affecting inter-ventricular interactions and impact of LVAD on RV systolic function

The unloading of LV by LVAD support is accompanied by a relatively unchanged RV chamber contractility as indexed by a constant $V_{60}$ up to a pump speed of 4k rpm. With increasing pump speed from 4k to 6k rpm that is within the typical operating range, $V_{60}$ is increased slightly (by ~5%). When pump speed is further increased from 6k to 9k rpm, $V_{60}$ became ~10% higher than that without LV support, implying that excessive increase in the LVAD operating speed can reduce RV chamber contractility (Figure 3E). Correspondingly, this finding suggests that LVAD support within the normal range of operating pump speed of 4k-6k rpm does not affect RV chamber contractility in the HF patient significantly (less than 5%). Beyond that range, however, our finding suggests that LVAD operation impairs RV chamber contractility more.

Clinical and experimental studies on the effects of LVAD on RV function are conflicting. In a clinical study involving HeartMate 2,²⁴ implantation of the LVAD led to an improvement in RV contractility that was measured qualitative based on echocardiographic appearance. Another two clinical studies also showed that HeartMate II LVAD implantation led to an improved RV function as indexed by right atrial pressure, RV stroke work, and mean pulmonary artery pressure.^25,26 These studies, however, investigated the (chronic) effects of LVAD after a period of time. However, an early experimental acute study on dogs showed that LVAD reduces the slope of RV ESPVR, $E_{es}$, from 2.4 to 1.7 mm Hg/mL but did not change the volume intercept $V_0$ (41 mL). Correspondingly, $V_{60}$ is increased, implying that RV contractility is reduced with LVAD²⁰, which is consistent with our findings.

We investigated several factors that may affect how LVAD impacts the RV function. Chosen to reflect the conditions and features present in some heart diseases, these factors were analyzed in isolation to identify the important and significant ones. Among these factors, we found that a thinner septal wall as well as a lower myocardial contractility in either the LVFW, septum, RVFW or LVFW + septum regions produce a greater reduction in RV chamber contractility during LVAD support (Figures 3 and 4). The finding that the septal wall thickness is a key factor is not too surprising as the inter-ventricular septum is a wall that separates the LV and RV and acts as a conduit for the transmission of forces across these two chambers. This finding is particularly relevant for HF patients suffering from regional ischemia or had septal remodeling as it suggests that patients having substantially lower myocardial contractile force or a thinner septum (i.e., due to occlusion of the left anterior descending artery) may be more susceptible to the adverse effects of LVAD on RV chamber contractility. Conversely, the finding suggests that patients having a thicker septal wall, such as those with hypertrophic cardiomyopathy (HCM),^27–30 may be less susceptible to the adverse effects of LVAD on RV chamber contractility. It must be noted that although the inter-ventricular septal wall shows a more significant thickening in HCM,²⁸ the wall thickness in other regions is also increased, which we do not consider here. This finding is also relevant to the findings in one animal study³¹ that myocardial perfusion is affected by LVAD operation, which may indirectly affect regional myocardial contractility.

4.2 |. Effects of LVAD operation on RV diastolic function

In terms of diastolic function, the model predicts that an increase in pump speed led to an increase in RV preload as indexed by the right atrium pressure and an increase in RVEDV (Figure 2A,D). This result is consistent with early studies showing that a decrease in the volume of one chamber causes the other to become more distensible (i.e., increasing $V_{20}$).⁴ Here, the model predicts that the increase in LVAD pump speed reduces LV volume and causes the RV to become more distensible. As such, $V_{20}$ increases monotonically with increasing pump speed from 0k to 9k rpm. The impact of LVAD operation on RV diastolic function is increased with a decrease in septal wall thickness and regional contractility of the biventricular unit.

4.3 |. Clinical implications

The insights developed in this computational study on the factors affecting the inter-ventricular interactions and RV functions during LVAD operation (i.e., septal wall thickness and regional contractility) may have clinical implications. Specifically, factors causing RV failure post-LVAD implantation have not been established³² and the performance of existing risk models for predicting a patienťs postoperative risk of RV failure after LVAD is poor.³³ The use of these risk models in the clinic is also limited.³³ As such, the findings here may help identify patients who may be at a risk of RV failure following LVAD implantation. Moreover, the computational model can also be extended in future studies for patient-specific modeling that considers individual geometry and heart function, which in turn, can be applied for individual risk assessment of RV failure after LVAD implantation.

4.4 |. Limitations

There are limitations associated with this study. First, besides changes in septal wall thickness, we did not investigate other changes in size and geometry of the biventricular unit that may affect the results. Future studies can include patient-specific biventricular geometry to increase its realism in terms of the geometrical changes found in HF. Second, the position and direction of the LVAD inflow cannula tip, which may affect septum shifting and immobilize the ventricular apex, are not considered in this model. To overcome this limitation, a three-dimensional LVAD model can be incorporated into the modeling framework to investigate the effects of device positioning in future studies.

5 |. CONCLUSION

We applied a novel computational model that can reproduce experimental findings on inter-ventricular dependence to investigate factors affecting the impact of LVAD on RV function. The model predicts RV chamber contractility is reduced at pump speed higher than 4k rpm, and this reduction is greater with a thinner septal wall and when myocardial contractility in the LVFW, septum, RVFW or LVFW + septum regions is decreased.

APPENDIX A

CLOSED-LOOP LUMPED PARAMETER MODEL

The governing system of equations of the closed-loop lumped parameter representation^34,35 is given as follows:

$$\frac{d{V}_{LA}\left(t\right)}{dt}=q_{pv}\left(t\right)-q_{mv}(t\text{),}$$ (A1)

$$\frac{d{V}_{LV}\left(t\right)}{dt}=q_{mv}\left(t\right)-q_{av}\left(t\right)-q_{\mathit{LVAD}}(t\text{),}$$ (A2)

$$\frac{d{V}_{sa}\left(t\right)}{dt}=q_{av}\left(t\right)-q_{sa}\left(t\right)+q_{\mathit{LVAD}}(t\text{),}$$ (A3)

$$\frac{d{V}_{sv}\left(t\right)}{dt}=q_{sa}\left(t\right)-q_{sv}(t\text{),}$$ (A4)

$$\frac{d{V}_{RA}\left(t\right)}{dt}=q_{sv}\left(t\right)-q_{tv}(t\text{),}$$ (A5)

$$\frac{d{V}_{RV}\left(t\right)}{dt}=q_{tv}\left(t\right)-q_{pvv}(t\text{),}$$ (A6)

$$\frac{d{V}_{pa}\left(t\right)}{dt}=q_{pvv}\left(t\right)-q_{pa}(t\text{),}$$ (A7)

$$\frac{d{V}_{pv}\left(t\right)}{dt}=q_{pa}\left(t\right)-q_{pv}\left(t\right).$$ (A8)

In Equation (A1)–(A8), $V_{LA}$, $V_{LV}$, $V_{sa}$, $V_{sv}$, $V_{RA}$, $V_{RV}$, $V_{pa}$, and $V_{pv}$ are the volumes of the eight compartments with the subscripts denoting the left atrium (LA), LV, systemic arteries (sa), systemic veins (sv), right atrium (RA), RV, pulmonary arteries (pa), and pulmonary veins (pv), respectively. Flow rates in the vascular components are denoted by $q_{pv}$, $q_{mv}$, $q_{av}$, $q_{sa}$, $q_{sv}$, $q_{tv}$, $q_{pvv}$, and $q_{pa}$, whereas the flow rate of the LVAD is denoted by $q_{\mathit{LVAD}}$.

Blood flow rates in the arteries and veins are determined by each segmenťs resistance and the pressure difference between the two connecting storage compartments as

$$q_{mv}\left(t\right)=\left\{\begin{array}{cc}\frac{P_{LA}\left(t\right)-P_{LV}\left(t\right)}{R_{mv}}& P_{LA}\left(t\right)\ge P_{LV}\left(t\right)\\ 0& P_{LA}\left(t\right)<P_{LV}\left(t\right)\end{array}\right.,$$ (A9)

$$q_{av}\left(t\right)=\left\{\begin{array}{cc}\frac{P_{LV}\left(t\right)-P_{sa}\left(t\right)}{R_{av}}& P_{LV}\left(t\right)\ge P_{sa}\left(t\right)\\ 0& P_{LV}\text{(t)}<P_{sa}\left(t\right)\end{array}\right.,$$ (A10)

$$q_{sa}\left(t\right)=\frac{P_{sa}\left(t\right)-P_{sv}\left(t\right)}{R_{sa}},$$ (A11)

$$q_{sv}\left(t\right)=\frac{P_{sv}\left(t\right)-P_{RA}\left(t\right)}{R_{sv}},$$ (A12)

$$q_{tv}\left(t\right)=\left\{\begin{array}{cc}\frac{P_{RA}\left(t\right)-P_{RV}\left(t\right)}{R_{tv}}& P_{RA}\left(t\right)\ge P_{RV}\left(t\right)\\ 0& P_{RA}\left(t\right)<P_{RV}\left(t\right)\end{array}\right.,$$ (A13)

$$q_{pvv}\left(t\right)=\left\{\begin{array}{cc}\frac{P_{RV}\left(t\right)-P_{pa}\left(t\right)}{R_{pvv}}& P_{RV}\left(t\right)\ge P_{pa}\left(t\right)\\ 0& P_{RV}\left(t\right)<P_{pa}\left(t\right)\end{array}\right.,$$ (A14)

$$q_{pa}\left(t\right)=\frac{P_{pa}\left(t\right)-P_{pv}\left(t\right)}{R_{pa}},$$ (A15)

$$q_{pv}\left(t\right)=\frac{P_{pv}\left(t\right)-P_{LA}\left(t\right)}{R_{pv}}.$$ (A16)

In Equations (A9)–(A16), $P_{LA}$, $P_{sa}$, $P_{sv}$, $P_{RA}$, $P_{RV}$, $P_{pa}$, and $P_{pv}$ are the pressures at the LA, systemic arteries, systemic veins, RA, RV, pulmonary arteries, and pulmonary veins, respectively, and $R_{mv}$, $R_{av}$, $R_{sa}$, $R_{sv}$, $R_{tv}$, $P_{pvv}$, $P_{pa}$, and $R_{pv}$ are the resistances associated with the mitral valve, aortic valve, systemic arteries, tricuspid valve, pulmonary veins, pulmonary arteries, and pulmonary valves, respectively. We note that the flow across the heart valves is unidirectional (assuming no valvular regurgitation).⁸ Pressures in the storage compartments representing the peripheral (proximal and distal) arterial and venous networks are given as

$$P_{sa}=\frac{V_{sa}\left(t\right)-V_{sa0}}{C_{sa}},$$ (A17)

$$P_{sv}=\frac{V_{sv}\left(t\right)-V_{sv0}}{C_{sv}},$$ (A18)

$$P_{pa}=\frac{V_{pa}\left(t\right)-V_{pa0}}{C_{pa}},$$ (A19)

$$P_{pv}=\frac{V_{pv}\left(t\right)-V_{pv0}}{C_{pv}},$$ (A20)

where $\left(V_{sa0},\hspace{0.33em}{C}_{sa}\right)$, $\left(V_{sv0},\hspace{0.33em}{C}_{sv}\right)$, $\left(V_{pa0},\hspace{0.33em}{C}_{pa}\right)$, and $\left(V_{pv0},\hspace{0.33em}{C}_{pv}\right)$ are the prescribed (resting volume, compliance) pair associated with the systemic arteries, systemic veins, pulmonary arteries, and pulmonary veins, respectively.

The pumping characteristics of the RA and LA are represented by a time-varying elastance model. The instantaneous pressure is related to the instantaneous volume by a time-varying elastance function, $e_i\left(t\right)$ given as

$$P_i\left(V_i(t\text{),}\hspace{0.17em}t\right)=e_i\left(t\right)P_{es}\left(V_i\left(t\right)\right)+\left(1-e_i\left(t\right)\right)P_{ed}\left(V_i\left(t\right)\right),$$ (A21)

where

$$P_{es}\left(V_i\left(t\right)\right)=E_{es,i}\left(V_i\left(t\right)-V_{i0}\right),$$ (A22)

$$P_{ed}\left(V_i\left(t\right)\right)=A_i\left\{exp\left[B_i\left(V_i\left(t\right)-V_{i0}\right)\right]-1\right\},$$ (A23)

$$e_i\left(t\right)=\left\{\begin{array}{l}\frac{1}{2}\left[\left(sin\frac{\pi }{t_{max,i}}t-\frac{\pi }{2}\right)+1\right]t<\frac{3}{2}{T}_{max,i}\hfill \\ \frac{1}{2}exp\left[-\left(t-\frac{3t_{max,i}}{2}\frac{1}{{\tau }_i}\right)\right]t\ge \frac{3}{2}{t}_{max,i}\hfill \end{array}\right..$$ (A24)

In Equation (A21), the subscript $i$ denotes either LA or RA. $P_{es}$ and $P_{ed}$ are the end-systolic and end-diastolic pressures, respectively. $E_{es,i}$ is the maximal chamber elastance, $V_{i0}$ is the volume with zero end-systolic pressure, and both $A_i$ and $B_i$ are parameters defining the end-diastolic pressure volume relationship. In Equation (A24), $t_{max,i}$ is the time to end systole and ${\tau }_i$ is the relaxation time constant.

APPENDIX B

PRESSURE–VOLUME RELATIONSHIPS IN FINITE ELEMENT METHODS

Pressure–volume relationships of the LV and RV required in the lumped circulatory model are defined by the solution obtained from minimization of the functional. Taking the first variation of the functional in Equation (2) leads to the following expression as

$$\begin{array}{l}\delta \mathcal{L}\left(u,\hspace{0.17em}p,\hspace{0.17em}{P}_{LV},\hspace{0.17em}{P}_{RV},\hspace{0.17em}{c}_1,\hspace{0.17em}{c}_2\right)={\int }_{{\text{Ω}}_0}\left(P-p{F}^{-T}\right):\hspace{0.33em}\nabla \delta u\hspace{0.33em}dV-{\int }_{{\text{Ω}}_0}\delta p(J-1)dV\\ \hspace{1em}\hspace{1em}-\left(\left(P_{LV}+P_{RV}\right){\int }_{{\text{Ω}}_0}cof(F\text{):}\hspace{0.33em}\nabla \delta u\hspace{0.33em}dV-\delta P_{LV}\left(V_{LV}\left(u\right)-V_{LV}\right)-\delta P_{RV}\left(V_{RV}\left(u\right)-V_{RV}\right)\right.\\ \hspace{1em}\hspace{1em}-\delta c_1•{\int }_{{\text{Ω}}_0}u\hspace{0.33em}dV-\delta c_2•{\int }_{{\text{Ω}}_0}X\times u\hspace{0.33em}dV-c_1•{\int }_{{\text{Ω}}_0}u\hspace{0.33em}dV-c_2•{\int }_{{\text{Ω}}_0}X\times \delta u\hspace{0.33em}dV,\end{array}$$ (B1)

where $P$ is the first Piola Kirckhoff stress tensor, $F$ is the deformation gradient tensor, $\delta u$, $\left(\delta p,\delta P_{LV,RV}\right)$, $\delta c_1$ and $\delta c_2$ are the variation of the displacement field, Lagrange multipliers for enforcing incompressibility and volume constraints, zero mean translation and rotation, respectively. The Euler–Lagrange problem then becomes finding $u\in H^1\left({\text{Ω}}_0\right)$, $p\in L^2\left({\text{Ω}}_0\right)$, $P_{LV,RV}\in ℝ$, $c_1\in {ℝ}^3$, $c_2\in {ℝ}^3$ that satisfies

$$\delta \mathcal{L}\left(u,\hspace{0.33em}p,\hspace{0.33em}{P}_{LV},\hspace{0.33em}{P}_{RV},\hspace{0.33em}{c}_1,\hspace{0.33em}{c}_2\right)=0$$ (B2)

and $u(x,\hspace{0.17em}y,\hspace{0.17em}0).n{\text{|}}_{base}=0$ (for constraining the basal deformation to be in-plane) $\forall \delta u\in H^1\left({\text{Ω}}_0\right)$, $\delta p\in L^2\left({\text{Ω}}_0\right)$, $\delta P_{LV,RV}\in ℝ$, $\delta c_1\in {ℝ}^3$, $\delta c_2\in {ℝ}^3$. In the implementation, the displacement field, $u$, is discretized by quadratic elements and the Lagrange multiplier, $p$, is discretized by linear elements.

Geometry of the biventricular unit

The biventricular geometry is idealized as the fusion of two truncated ellipsoids representing the LV and RV. The dimension of the biventricular geometry is prescribed based on a failing heart (Figure 1C).¹¹ The geometry is discretized using tetrahedral elements. Helix angle associated with the myofiber direction ${\mathrm{e}}_{f_0}$ is prescribed to vary transmurally from 60° at the endocardium to − 60° at the epicardium throughout the biventricular wall based on previous experimental measurements.³⁶

Constitutive law of the biventricular

An active stress formulation was used to describe the mechanical behavior of the biventricular unit in the cardiac cycle. In this formulation, the stress tensor $P$ can be decomposed additively into a passive component $P_p$ and an active component $P_a$ (i.e., $P=P_p+P_a$). The passive stress tensor is defined by $P_p=dW/dF$, where $F$ is the deformation gradient tensor and $W$ is a strain energy function of a Fung-type transversely-isotropic hyperelastic material³⁷ given by

$$W=\frac{1}{2}C\left(e^Q-1\right).$$ (B3)

In Equation (B3),

$$Q=b_{ff}{E}_{ff}^2+b_{xx}\left(E_{ss}^2+E_{nn}^2+E_{sn}^2+E_{ns}^2\right)+b_{fx}\left(E_{fn}^2+E_{nf}^2+E_{fs}^2+E_{sf}^2\right),$$

where $E_{ij}$ with $(i,\hspace{0.17em}j)\in (f,\hspace{0.17em}s,\hspace{0.17em}n)$ are components of the Green-Lagrange strain tensor $E$ with $f$, $s$, $n$ denoting the myocardial fiber, sheet, and sheet normal directions, respectively. Material parameters of the passive constitutive model are denoted by $C$, $b_{ff}$, $b_{xx}$, and $b_{fx}$. The active stress $P_a$ was calculated along the local fiber direction using the active constitutive relationship³⁸ as

$$P_a=T_{max}\frac{C_{a0}^2}{1+E{C}_{a50}^2\left(E_{ff}\right)}\frac{1-\text{cos}\left(\omega \left(t,\hspace{0.33em}{t}_{init},\hspace{0.33em}{E}_{ff}\right)\right)}{2}{\mathrm{e}}_f\otimes {\mathrm{e}}_{f_0},$$ (B5)

where $C_{a0}$ is the peak intracellular calcium concentration, $T_{max}$ is myocardial contractility, and $E{C}_{a50}$ is the length-dependent calcium sensitivity given by

$$E{C}_{a50}=\frac{{\left(C_{a0}\right)}_{max}}{\sqrt{\text{exp}\left(B\left(l-l_0\right)\right)-1}}.$$ (B6)

In Equation (B6), $B$ is a material constant, ${\left(C_{a0}\right)}_{max}$ is a prescribed maximum peak intracellular calcium concentration, $l$ is instantaneous sarcomere length based on prescribed initial length of sarcomere $l_{s0}$ and calculated by $l=l_{s0}\sqrt{f_0•C•f_0}$, and $l_0$ is sarcomere length at which no active tension develops.

APPENDIX C

FIGURE C1.

Right ventricular (RV) end-diastolic pressure–volume relationships (EDPVRs) and end-systolic pressure–volume relationships (ESPVRs) with different pump speed at left ventricular free wall (LVFW) contractility (A) 100%; (B) 80%; (C) 60%; (D) 40% of its original value. [Color figure can be viewed at wileyonlinelibrary.com]

FIGURE C2.

Right ventricular (RV) end-diastolic pressure–volume relationships (EDPVRs) and end-systolic pressure–volume relationships (ESPVRs) with different pump speed at septum contractility (A) 100%; (B) 80%; (C) 60%; (D) 40% of its original value. [Color figure can be viewed at wileyonlinelibrary.com]

FIGURE C3.

Right ventricular (RV) end-diastolic pressure–volume relationships (EDPVRs) and end-systolic pressure–volume relationships (ESPVRs) with different pump speed at right ventricular free wall (RVFW) contractility (A) 100%; (B) 80%; (C) 60%; (D) 40% of its original value. [Color figure can be viewed at wileyonlinelibrary.com]

FIGURE C4.

Right ventricular (RV) end-diastolic pressure–volume relationships (EDPVRs) and end-systolic pressure–volume relationships (ESPVRs) with different pump speed at left ventricular free wall (LVFW) + septum contractility (A) 100%; (B) 80%; (C) 60%; (D) 40% of its original value. [Color figure can be viewed at wileyonlinelibrary.com]