In-silico assessment of the effects of right ventricular assist device on pulmonary arterial hypertension using an image based biventricular modeling framework

Abstract

Pulmonary arterial hypertension (PAH) is a heart disease that is characterized by an abnormally high pressure in the pulmonary artery (PA). While right ventricular assist device (RVAD) has been considered recently as a treatment option for the end-stage PAH patients, its effects on biventricular mechanics are, however, largely unknown. To address this issue, we developed an image-based modeling framework consisting of a biventricular finite element (FE) model that is coupled to a lumped model describing the pulmonary and systemic circulations in a closed-loop system. The biventricular geometry was reconstructed from the magnetic resonance images of two PAH patients showing different degree of RV remodeling and a normal subject. The framework was calibrated to match patient-specific measurements of the left ventricular (LV) and RV volume and pressure waveforms. An RVAD model was incorporated into the calibrated framework and simulations were performed with different pump speeds. Results showed that RVAD unloads the RV, improves cardiac output and increases septum curvature, which are more pronounced in the PAH patient with severe RV remodeling. These improvements, however, are also accompanied by an adverse increase in the PA pressure. These results suggest that the RVAD implantation may need to be optimized depending on disease progression.

Graphical Abstract

1. Introduction

Pulmonary arterial hypertension (PAH) is a cardio-pulmonary disease that is characterized by an abnormally elevated pulmonary artery (PA) pressure (> 25 mmHg), which can be due to idiopathic reasons or caused by other conditions (e.g., presence of ventricular septal defect). Without treatment, PAH progresses rapidly and adversely affects the right ventricular (RV) function, eventually leading to right heart failure and death [1]. There are currently no effective treatments for PAH, and existing therapies for this disease have been mostly palliative [2]. Given the success of left ventricular assist device (LVAD) as a treatment for left heart failure, right ventricular assist device (RVAD) is recently proposed as a therapeutic option for PAH patients, especially when the disease is refractory to vasodilator therapy [3]. Unlike its LVAD counterpart, however, the effects of this device on RV mechanics are not well understood. Moreover, our understanding on the effects of PAH on RV mechanics in humans is also lacking as most patient studies of this disease are based largely on measuring regional myocardial deformation or kinematics from clinical images [4].

Computational models are increasingly developed to improve our understanding on the effects of PAH on RV mechanics, although the number of models is significantly lesser compared to those developed to study left ventricular (LV) mechanics [5–8]. While able to produce insights of PAH, existing computational heart models developed to investigate this disease currently suffer from one or more of the following limitations: (a) not calibrated based on human data [9,10], (b) focusing only on RV passive mechanics and ignore active mechanics [11], and (c) do not couple both systemic and pulmonary circulatory systems [9,10,12]. Specifically, the latter limitation places a restriction on our ability to fully assess how alterations in the pulmonary circulation consisting of the RV, such as by RVAD implantation, can impact the systemic circulation (including LV mechanics), and vice versa. On the other hand, while it is possible to use a simplified lumped circulatory modeling framework [3] to simulate the effects and interactions of RVAD with the systemic and pulmonary circulations in PAH, this approach cannot be used to quantify the effects of RVAD and its different operating configurations on important physiological quantities such as septal curvature and RV myofiber stress, which is directly related to oxygen consumption [13].

To address these limitations, we present a computational framework consisting of a patient-specific biventricular finite element (FE) model that is coupled to a lumped parameter model describing both the pulmonary and systemic circulations in a closed-loop system. The biventricular FE model was reconstructed from magnetic resonance (MR) images and the computational framework was calibrated against measurements of the pressure and volume waveforms acquired from two PAH patients with different stages of remodeling as well as a normal subject. An RVAD model based on a realistic pump characteristic was also incorporated into the calibrated computational framework to investigate its effects on ventricular stresses and deformation in the patients. The framework described here lays the foundation for subsequent development of patient-specific computational heart models to elucidate the complex ventricular interdependence [14] and pulmonic-systemic interactions associated with PAH and its treatments.

2. Methods

2.1. Image and data acquisition

Cine MR images from one normal human subject and two PAH patients were acquired using a 3T Philips scanner. The characteristics of these 3 patients are given in Table 1. Among the two PAH patients, the RV of one patient underwent severe remodeling, which was evident from the large RV end-diastolic volume (EDV) to LV EDV ratio [15]. We denote this patient as PAHR in this study. The second PAH patient had RV chamber size and RV EDV to LV EDV ratio close to that found in the normal subject. We denote this patient as PAHN. Right heart catheterization (RHC) was performed on the PAH patients to acquire the left and right ventricular, atrial and arterial pressure. All enrolled participants gave their written consent and all data were acquired in the National Heart Center of Singapore.

Table 1.

Characteristics of the normal and PAH patients

	Normal	PAHR	PAHN
HR, bpm	75	58	76
LV EDV, ml	83	84	84
LV ESV, ml	27	31	32
LV EF, %	67	63	62
MAP, mmHg	-	100	99.7
RV EDV, ml	102	147	79
RV ESV, ml	42	92	39
RV EF, %	59	37	51
RV EDV/LVEDV	1.2	1.75	0.95
mPAP, mmHg	-	47.7	32.3

^aMAP (mean arterial pressure) and mPAP (mean pulmonary artery pressure) were calculated by the formula (systolic pressure + 2*diastolic pressure)/3.

^bHR: heart rate; LV: left ventricle; RV right ventricle; EDV: end-diastolic volume; ESV: end-systolic volume; EF: ejection fraction; PAHR: PAH with remodeling (RVEDV/LVEDV >1.5); PAHN: PAH with normal RVEDV/LVEDV

We segment the LV endocardial, RV endocardial and epicardial surfaces at different time points in the cardiac cycle from the short and long-axis views of the cine MR images of the PAH patients and normal subject using the medical image analysis software MeVisLab (http://www.mevislab.de) [12]. Patient-specific LV and RV cavity volume waveforms over a cardiac cycle were obtained from these surfaces. As in a previous study, these volume vs. time curves were synchronized with the pressures measured from RHC in the PAH patients to obtain the LV and RV pressure-volume (PV) loops. Because invasive RHC was not performed on the normal subject, a normal RV pressure waveform and a scaled LV pressure waveform (with end-systolic pressure equal to 0.9 of the measured cuff pressure) were used as surrogate for the normal subject and paired with the measured cavity volumes to obtain the LV and RV PV loops [12].

2.2. Biventricular geometry and microstructure

Finite element meshes associated with the biventricular geometry for the three cases (Normal, PAHR and PAHN) were generated from the volume enclosed by the segmented epicardial, LV endocardial and RV endocardial surfaces at the cardiac time point corresponding to the lowest pressure using GMSH [16]. These meshes served as the unloaded configuration of the biventricular unit of the normal subject and PAH patients (Fig. 1). Approximately 5800 quadratic tetrahedral elements and 18000 nodes were used in the FE meshes to discretize the biventricular geometries. Using a Laplace-Dirichlet Rule-Based algorithm [17], the myofiber helix angle was prescribed to vary linearly in the transmural direction across the LV wall from 60° at the endocardium to −60° at the epicardium based on previous experimental measurements [18]. The RV free wall myofiber orientation was assumed to be same due to the lack of experimental measurements in humans.

Fig. 1:

Left: Biventricular FE geometry reconstructed from MR images with LVFW (red), septum (green), and RVFW (blue) material regions. Right: Unloaded FE geometry of normal, PAHN and PAHR cases respectively.

2.3. Closed loop circulatory system

The biventricular FE models were coupled to a closed loop lumped parameter circulatory model that describes both systemic and pulmonary circulations (Fig. 2a). The modeling framework consists of eight compartments with four cardiovascular components (ventricle, atrium, artery and vein) each in the systemic and pulmonary circulation. Conservation of total mass of blood in the circulatory model requires the net change of inflow and outflow rates of each compartment be related to the rate of change of the volume by the following relation

$$\frac{d{V}_{LV}(t)}{dt}=q_{mv}(t)-q_{av}(t),$$ (1a)

$$\frac{d{V}_{sa}(t)}{dt}=q_{av}(t)-q_{sa}(t),$$ (1b)

$$\frac{d{V}_{sv}(t)}{dt}=q_{sa}(t)-q_{sv}(t),$$ (1c)

$$\frac{d{V}_{RA}(t)}{dt}=q_{sv}(t)-q_{tv}(t),$$ (1d)

$$\frac{d{V}_{RV}(t)}{dt}=q_{tv}(t)-q_{pvv}(t)-q_{RVAD}(t),$$ (1e)

$$\frac{d{V}_{pa}(t)}{dt}=q_{pvv}(t)-q_{pa}(t)+q_{RVAD}(t)$$ (1f)

$$\frac{d{V}_{pv}(t)}{dt}=q_{pa}(t)-q_{pv}(t),$$ (1g)

$$\frac{d{V}_{LA}(t)}{dt}=q_{pv}(t)-q_{mv}(t).$$ (1h)

In Eqn. (1), V_LV, V_sa, V_sv, V_RA, V_RV, V_pa, V_pv and V_LA are the volumes of the eight compartments with the subscripts denoting the left ventricle (LV), systemic artery (sa), systemic vein (sv), right atrium (RA), right ventricle (RV), pulmonary artery (pa), pulmonary vein (pv), and left atrium (LA), respectively. Flow rates at different segments of the circulatory model are denoted by q_mv, q_av, q_sa, q_sv, q_tv, q_pvv, q_pa and q_pv (Fig. 2a), whereas flow rate of the RVAD is denoted by q_RVAD.

Fig. 2:

(a) Schematic of the coupled biventricular FE - closed loop modeling framework, (b) pump characteristics curve used to model the RVAD, (c) schematic showing the contours used to calculate the septum and LVFW local curvature in the LV endocardium in Cartesian coordinates.

To model the RVAD, we used the pressure gradient – flow characteristics of the Synergy^™ continuous flow miniature pump (CircuLite Inc, Saddle Brook, NJ) with operating speeds between 20 to 28krpm (Fig. 2b). This pump has the characteristics that are well suited for RVAD application [19] and was applied in a previous study [3]. Flow through the RVAD was sourced from the RV and ejected to the pulmonary artery. For a particular operating speed, the RVAD flow rate was determined from the characteristics curve based on the pressure gradient across the pulmonary artery and RV i.e.,

$$q_{RVAD}(t)=\frac{-K_1\left[P_{pa}(t)-P_{RV}(t)\right]+K_2}{R_{can}}.$$ (2)

In Eqn. (2), K₁ and K₂ are constants determined from the RVAD pump characteristics at a prescribed operating speed, P_pa is the pulmonary artery pressure, P_RV is the RV pressure, and R_can is the total resistance of the inlet and outlet cannula connecting the pump to the RV and pulmonary artery, respectively. On the other hand, the systemic and pulmonary arteries and veins were modelled using their electrical analogs based on Ohm’s law. At each segment, the flowrate depends on the pressure gradient and resistance to the flow as described in the following equation

$$q_{mv}(t)=\{\begin{array}{cc}\frac{P_{LA}(t)-P_{LV}(t)}{R_{mv}}& when,P_{LA}(t)\ge P_{LV}(t)\\ 0& when\text{, }{P}_{LA}(t)<P_{LV}(t)\end{array},$$ (3a)

$$q_{av}(t)=\{\begin{array}{cc}\frac{P_{LV}(t)-P_{sa}(t)}{R_{av}}& when,P_{LV}(t)\ge P_{sa}(t)\\ 0& when,P_{LV}(t)<P_{sa}(t)\end{array},$$ (3b)

$$q_{sa}(t)=\frac{P_{sa}(t)-P_{sv}(t)}{R_{sa}},$$ (3c)

$$q_{sv}(t)=\frac{P_{sv}(t)-P_{RA}(t)}{R_{sv}},$$ (3d)

$$q_{tv}(t)=\{\begin{array}{cc}\frac{P_{RA}(t)-P_{RV}(t)}{R_{tv}}& when,P_{RA}(t)\ge P_{RV}(t)\\ 0& when,P_{RA}(t)<P_{RV}(t)\end{array},$$ (3e)

$$q_{pvv}(t)=\{\begin{array}{cc}\frac{P_{RV}(t)-P_{pa}(t)}{R_{pvv}}& when,P_{RV}(t)\ge P_{pa}(t)\\ 0& when,P_{RV}(t)<P_{pa}(t)\end{array},$$ (3f)

$$q_{pa}(t)=\frac{P_{pa}(t)-P_{pv}(t)}{R_{pa}},$$ (3g)

$$q_{pv}(t)=\frac{P_{pv}(t)-P_{LA}(t)}{R_{pv}}.$$ (3h)

In Eqn. (3), R_mv, R_av, R_tv and R_pvv are the resistances associated with the mitral, aortic, tricuspid and pulmonary valves, respectively. On the other hand, the vessel resistances are denoted by R_sa, R_sv, R_pa and R_pv, respectively. To describe the compliance of the systemic and pulmonary vessels, we used the following pressure-volume relationships

$$P_{sa}(t)=\frac{V_{sa}(t)-V_{sa,0}}{C_{sa}},$$ (4a)

$$P_{sv}(t)=\frac{V_{sv}(t)-V_{sv,0}}{C_{sv}},$$ (4b)

$$P_{pa}(t)=\frac{v_{pa}(t)-V_{pa,0}}{c_{pa}},$$ (4c)

$$P_{pv}(t)=\frac{V_{pv}(t)-V_{pv,0}}{c_{pv}},$$ (4d)

where V_sa,0, V_sv,0, V_pa,0, and V_pv,0 are the resting volumes of the systemic and pulmonary vessels, and C_sa, C_sv, C_pa, and C_pv are their corresponding total compliance.

Contraction of the LA and RA was modeled using a time varying elastance function that is given by the following pressure – volume relations

$$P_k(t)=e(t)P_{es,k}\left(V_k(\text{t})\right)+(1-e(t))P_{ed,k}\left(V_{\text{k}}(\text{t})\right),$$ (5a)

$$P_{es,k}\left(V_k(\text{t})\right)=E_{es,k}\left(V_k(\text{t})-V_{0,k}\right),$$ (5b)

$$P_{ed,k}\left(V_k(\text{t})\right)=A_k\left(e^{B_k\left(V_k(\text{t})-V_{0,k}\right)}-1\right).$$ (5c)

In Eqn. (5), the subscript k denotes either LA or RA. The volume, end-systolic elastance, and volume-intercept of the end-systolic pressure-volume relationship (ESPVR) of the corresponding atrium are denoted by V_k, E_es,k and V_0,k, respectively. The parameters A_k and B_k define the atrium curvilinear end-diastolic pressure volume relationship (EDPVR) and the driving function is defined as

$$e(t)=\{\begin{array}{cc}\frac{1}{2}\left(\text{sin}\left[\left(\frac{\pi }{t_{max}}\right)t-\frac{\pi }{2}\right]+1\right);& 0<t\le 3/2t_{max}\\ \frac{1}{2}{e}^{-\left(t-3/2t_{max}\right)/\tau };& t>3/2t_{max},\end{array},$$ (6)

where t_max is the point of maximal chamber elastance and τ is the time constant of relaxation.

Last, the relationships between pressure and volume of the LV and RV were computed from the biventricular FE model (see next section), which can be expressed as a non-closed form function i.e.,

$$P_{RV}(t),P_{LV}(t)=f^{BiV}\left(V_{LV}(t),V_{RV}(t)\right).$$ (7)

2.4. Finite element formulation

The weak form associated with the biventricular FE model was derived based on minimization of the following Lagrangian functional

$$\mathcal{L}\left(u,p,P_{LV},P_{RV},c_1,c_2\right)={\int }_{{\mathrm{\Omega }}_0}W(u)dV-{\int }_{{\mathrm{\Omega }}_0}p(J-1)dV-P_{LV}\left(V_{LV,cav}(u)-V_{LV}\right)+P_{RV}\left(V_{RV,cav}(u)-V_{RV}\right)-c_1\cdot {\int }_{{\mathrm{\Omega }}_0}udV-c_2\cdot {\int }_{{\mathrm{\Omega }}_0}X\times udV,$$ (8)

where Ω₀ is the reference configuration of the biventricular unit, u is the displacement field, P_LV and P_RV are, respectively, the Lagrange multipliers that constrain the LV cavity volume V_LV,cav(u) to a prescribed value V_LV and the RV cavity volume V_RV,cav(u) to a prescribed value V_RV [20]. We note that V_LV and V_RV are prescribed from the closed-loop circulatory model in Eqn. (7). The Lagrange multiplier p was used to enforce incompressibility of the tissue (i.e., Jacobian of the deformation gradient tensor J = 1). The vectors c₁ and c₂ are Lagrange multipliers applied to constrain, respectively, the rigid body translation (i.e., zero mean translation) and rotation (i.e., zero mean rotation) [21]. In Eqn. (8), X denotes a material point in Ω₀ and W is the strain energy function of the myocardial tissue. The cavity volume of the LV and RV were obtained from the displacement field by using the following functional relationship (k = LV or RV)

$$V_{k,\text{cav}}\left(u\right)={\int }_{{\mathrm{\Omega }}_{inner\text{, }k}}d{v}_k=-\frac{1}{3}{\int }_{{\mathrm{\Gamma }}_{inner,k}}x.nd{a}_k,$$ (9)

where Ω_inner,k is the volume enclosed by the inner surface Γ_inner,k of the LV or RV, and n denotes the outward unit normal vector of those surfaces. Taking the first variation of the Lagrangian functional given in Eqn. (8) leads to

$$\delta \mathcal{L}\left(u,p,P_{LV},P_{RV},c_1,c_2\right)={\int }_{{\mathrm{\Omega }}_0}\left(P-p{F}^{-T}\right):\nabla \delta udV-{\int }_{{\mathrm{\Omega }}_0}\delta p(J-1)dV-\left(P_{LV}+P_{RV}\right){\int }_{{\mathrm{\Omega }}_0}cof(F):\nabla \delta udV-\delta P_{LV}\left(V_{LV}(u)-V_{LV}\right)-\delta P_{RV}\left(V_{RV}(u)-V_{RV}\right)-\delta c_1\cdot {\int }_{{\mathrm{\Omega }}_0}udV-\delta c_2\cdot {\int }_{{\mathrm{\Omega }}_0}X\times udV-c_1\cdot {\int }_{{\mathrm{\Omega }}_0}\delta udV-c_2\cdot {\int }_{{\mathrm{\Omega }}_0}X\times \delta udV.$$ (10)

In Eqn. (10), P is the first Piola Kirchhoff stress tensor and F is the deformation gradient tensor. The variations of the displacement field, Lagrange multiplier for enforcing incompressibility and volume constraint, zero mean translation and rotation are denoted by δu, δp, δP_LV, δP_RV, δc₁, and δc₂, respectively. Together with the constraint that the basal deformation at z = 0 is in-plane in the biventricular unit, the solution of the Euler-Lagrange problem was obtained by finding $u\in H^1\left({\mathrm{\Omega }}_0\right),p\in L^2\left({\mathrm{\Omega }}_0\right),P_{LV}\in ℝ,P_{RV}\in ℝ,c_1\in {ℝ}^3,c_2\in {ℝ}^3$ that satisfies

$$\delta \mathcal{L}\left(u,p,P_{LV},P_{RV,},c_1,c_2\right)=0,$$ (11a)

$$u(x,y,0),{n|}_{base}=0,$$ (11b)

for all $\delta u\in H^1\left({\mathrm{\Omega }}_0\right),\delta p\in L^2\left({\mathrm{\Omega }}_0\right),\delta P_{LV}\in ℝ,\delta P_{RV}\in ℝ,\delta c_1\in {ℝ}^3,\delta c_2\in {ℝ}^3$.

2.5. Mechanical behavior of the cardiac tissue

Mechanical behavior of the myocardial tissue was described by an active stress formulation in which the first Piola stress tensor P in Eqn. (10) was additively decomposed into a passive and an active component, i.e.

$$P=P_p+P_a{e}_f\otimes e_{f_0}.$$

In Eqn. (12), P_p is the passive stress tensor, P_a is the magnitude of the active stress, whereas e_f and e_f0 are the local basis vectors that define the cardiac muscle fiber directions in the current and reference configuration, respectively. The passive stress tensor P_p is related to the strain energy function W_p and deformation gradient tensor F by

$$P_p=\frac{d{W}_p}{dF}.$$ (13)

A Fung-type transversely-isotropic hyperelastic strain energy function [22]

$$W_p=\frac{1}{2}C\left(e^Q-1\right),$$ (14a)

with

$$Q=b_{ff}{E}_{ff}^2+b_{xx}\left(E_{ss}^2+E_{nn}^2+E_{sn}^2+E_{ns}^2\right)+b_{fx}(E_{fn}^2+E_{nf}^2+E_{fs}^2+E_{sf}^2).$$ (14b)

was prescribed. In Eqn. (14b), E_ij with (i, j) ∈ (f, s, n) denote the components of the Green-Lagrange strain tensor $E=\frac{1}{2}\left(F^{T}F-I\right)$ with f, s, n denoting the myofiber, sheet and sheet normal directions, respectively. Material parameters of the Fung-type constitutive model are C, b_ff, b_xx and b_fx.

To describe the active stress behavior, a previously developed active contraction model [23] was used. The magnitude of the active stress P_a was described by

$$P_a=\frac{l_s}{l_{s0}}{f}^{iso}\left(l_c\right)f^{twitch}\left(t,l_s\right)\left(l_s-l_c\right)E_a,$$ (15)

where l_s is the sarcomere length, l_c is the length of the contractile element, l_s0 is the sarcomere length in a prescribed reference state (relaxed sarcomere length), and E_a is the stiffness of the serial elastic element. The function f^iso(l_c) denotes the dependency of the isometrically developed active stress on l_c and is given by

$$f^{iso}\left(l_c\right)=\{\begin{array}{cc}{T}_0{\text{tanh}}^2\left[a_6\left(l_c-a_7\right)\right]& \text{when},l_c<a_7\\ 0& \text{ when},l_c>a_7\end{array},$$ (16)

where T₀ is a model parameter that scales the active tension. Both a₆ and a₇ are model parameters. The time course of the active tension development is controlled by

$$f^{twitch}\left(t,l_s\right)=\{\begin{array}{cc}0& \text{ when},t<0\hfill \\ {\text{tanh}}^2\left(\frac{t}{t_r}\right){\text{tanh}}^2\left(\frac{t_{max}-t}{t_d}\right)& \text{ when},0<t<t_{max}\hfill \\ 0& \text{ when},t>0,\hfill \end{array}$$ (17a)

$$t_{max}=b\left(l_s-l_d\right).$$ (17b)

In Eqn. (17), t_r is the activation rise time constant, t_d is the activation decay time constant, b relates activation duration t_max to the sarcomere length l_s, l_d and is the sarcomere length at the start of the activation time, i.e., when t_max = 0. The time course of the contractile element l_c was expressed by an ordinary differential equation

$$\frac{\partial l_c}{\partial t}=\left[E_a\left(l_s-l_c\right)-1\right]v_0,$$ (18)

where ν₀ is the unloaded shortening velocity. The sarcomere length l_s was calculated from the myofiber stretch λ and the relaxed sarcomere length l_s0 by

$$\lambda =\sqrt{e_{f_0}\cdot F^{T}F{e}_{f_0}},$$ (19a)

$$l_s=\lambda l_{s0}.$$ (19b)

2.6. Model parameterization and simulation of the cardiac cycles

In the 3 cases, the biventricular FE models were divided into three material regions, namely the LV free wall (LVFW), the septum and the RV free wall (RVFW). Similar to a previous study [24], passive stiffness C and contractility T₀ were prescribed to be the same in the LVFW and septum (denoted as C_LV and T_0,LV) but had different values in the RVFW (denoted as C_RV and T_0,RV). For each FE models, the parameters were adjusted to fit the experimentally measured LV and RV PV loops, volume and pressure waveforms throughout the cardiac cycle. Specifically, the LV and RV end diastolic pressures were matched by adjusting the passive parameters C_LV and C_RV. The regional contractility parameters (T_0,LV, T_0,RV) were adjusted to match the LV and RV systolic pressures and stroke volumes. On the other hand, the contraction model parameters t_r, t_d and b were adjusted to match the time course of the volume and pressure waveforms measured in the LV and RV. Circulatory model parameters (resistances and compliances) were also adjusted to match the systolic pressure (afterload), preload and systemic and pulmonary vein pressures. All the model parameters are listed in Table 2. The model is implemented using FEnICS [25] and the code is publicly available (https://bitbucket.org/shaviksh/biv_rvad/src/master/).

Table 2.

Model parameters for Normal and PAH cases.

	Normal	PAHR	PAHN
Passive material model	CLV = 0.1Pa, CRV = 1.0Pa	CLV = 4.0Pa, CRV = 30Pa	CLV = 7Pa, CRV = 57Pa
Active contraction model	T0,LV = 800kPa, T0,RV = 195kPa, tr= 310ms, td = 150ms, b = 0.22ms.μm−1	T0,LV = 500kPa, T0,RV = 220kPa, tr = 320ms, td = 100ms, b = 0.24ms.μm−1	T0,LV = 650kPa, T0,RV = 430kPa, tr = 320ms, td = 100ms, b = 0.24ms.μm−1
Circulatory model	Csa = 0.005Pa.ml, Cpa = 0.2Pa.ml, Csv = 0.3Pa.ml, Cpv = 0.09Pa.ml, Rsa = 132kPa.ms.ml−1, Rpa = 7kPa.ms.ml−1, Rsv = Rpv =2kPa.ms.ml−1, Rav = 3kPa.ms.ml−1, = 0.9kPa.ms.ml−1, Rtv = 0.4kPa.ms.ml−1, Rpvv = 2kPa.ms.ml−1, Vsa,0 = 610ml, Vpa,0 = 50ml, Vsv,0 = 3315ml, Vpv,0 = 400ml	Csa = 0.0055Pa.ml, Cpa = 0.006Pa.ml, Csv = 0.3Pa.ml, Cpv = 0.09Pa.ml, Rsa = 265kPa.ms.ml−1, Rpa = 115kPa.ms.ml−1, = Rpv =2kPa.ms.ml−1, Rav = 3kPa.ms.ml−1, Rmv = 0.9kPa.ms.ml−1, Rtv = 0.4kPa.ms.ml−1, Rpvv = 2kPa.ms.ml−1, Vsa,0 = 610ml, Vpa,0 = 400ml, Vsv,0 = 3335ml, Vpv,0 = 415ml	Csa = 0.0053Pa.ml, Cpa = 0.0055Pa.ml, Csv = 0.3Pa.ml, Cpv = 0.09Pa.ml, Rsa = 206kPa.ms.ml−1, Rpa = 82kPa.ms.ml−1, Rsv = Rpv =2kPa.ms.ml−1, Rav = 3kPa.ms.ml−1, Rmv = 0.9kPa.ms.ml−1, Rtv = 0.4kPa.ms.ml−1, Rpvv = 2kPa.ms.ml−1,Vsa,0 = 590ml, Vpa,0 = 425ml, Vsv,0 = 3550ml, Vpv,0 = 280ml

Steady-state pressure-volume loop was established by running the simulation over several cardiac cycles. The cardiac cycle time was prescribed based on the measured heart rate in each of the cases. Specifically, a cycle time of 800ms (equivalent to 75 bpm), 1030ms (equivalent to 58 bpm) and 790ms (equivalent to 76 bpm) was prescribed in the simulations for the normal subject, PAHR and PAHN, respectively.

2.7. Post processing of septum curvature

Regional curvature κ of the LVFW and septum was computed at each time point of the cardiac cycle by first fitting a curve to the 2D slice of the LV endocardial surface taken at the mid-ventricular level (halfway between the apex and the base) of the deformed mesh (Fig. 2c). Based on the fitted plane curve y = f(x) in the Cartesian coordinate system, the local curvature was then computed by

$$\kappa =\frac{\left|y^{\prime \prime }\right|}{{\left(1+y^{\prime 2}\right)}^{3/2}}$$ (20)

where y′ and y″ are the 1^st and 2^nd derivative of the function y = f(x).

3. Results

3.1. Comparison between simulations and measurements

The PV loops, volume and pressure waveforms of the LV and RV predicted by the model were closely matched with the measured data for the normal subject as well as the PAH patients (Fig. 3). Specifically, the coefficient of determination R² associated with the fitting of volume and pressure at all cardiac time points are 0.983 and 0.978, respectively (Fig. 3a). Compared to the normal subject, the peak RV pressure was 2.5 times higher in the PAHR case (76 mmHg vs. 30 mmHg) and 1.8 times higher in the PAHN case (55 mmHg vs. 30 mmHg). The peak LV pressure was also increased slightly in the PAH patients compared to the normal (133 mmHg in PAHR and 137 mmHg in PAHN vs. 124 mmHg in normal). The LV end-diastolic and end-systolic volumes (EDV and ESV) were comparable between the normal and PAH cases, resulting in a large ejection fraction (EF) for all the cases (63% in PAHR and 62% in PAHN vs. 67% in normal). However, the PAHR case had significantly larger RV EDV (147 ml vs. 102 ml) and RV ESV (92 ml vs. 42 ml) compared to normal (and PAHN), which resulted in a substantially lower RV EF (37% vs. 59%). On the other hand, PAHN had slightly lower RV EDV (79 ml vs. 102 ml) and slightly larger RV ESV (39 ml vs. 42 ml) than the normal, which also resulted in a lower RV EF (51% vs. 59%).

Fig. 3:

(a) Scatter plot of the simulated vs. measured volume (left) and pressure (right) at all cardiac time points for the three cases. A y = x line is also plotted to show the zero-error reference. (b) Measurements and model predictions of LV and RV PV loops (first row), volume waveforms (2^nd row) and pressure waveforms (3^rd row) for the normal, PAH patient with severe RV remodeling (PAHR) and PAH patient with normal RV (PAHN).

3.2. Effect of RVAD on hemodynamics

Implantation of RVAD in the PAH patients produced a triangular shaped RV PV loop without any isovolumic (contraction and relaxation) phases as well as a lower RV EDV and RV ESV compared to the baseline (no RVAD) (Fig. 4a). With increasing RVAD speed, both RV EDV and RV ESV decreased progressively, with the PAHR case undergoing a larger decrease compared to the PAHN case. At the highest RVAD speed of 28 krpm, RV EDV was reduced by 18% and 11%, respectively, in the PAHR case (121 ml vs. 147 ml at baseline) and the PAHN case (70 ml vs. 79 ml at baseline). In the PAHR case, the RV peak pressure decreased moderately with increasing RVAD speed whereas it increased slightly in the PAHN case when RVAD speed was increased. Both LV EDV and peak pressure (Fig. 4b) increased with RVAD speed for both PAH cases, and the increase was more evident in the PAHR case compared to the PAHN case. Cardiac output (CO) steadily improved for PAHR with RVAD flow reaching to 3.33 L/min. at 28krpm compared to 3.09 L/min. at baseline (no RVAD). In contrast, CO remained fairly constant at different RVAD operating speed up to 28 krpm in the PAHN case.

Fig. 4.

(a) RV, (b) LV PV loops. (c) RV, PA and RA pressure waveforms. (d) LV, SA and LA pressure waveforms for the PAHR and PAHN cases with different RVAD speed. Scattered points show the measurements.

In terms of arterial hemodynamics, both PA diastolic and mean pressures (Fig. 4c) increased with increasing RVAD speed. Specifically, at an RVAD speed of 28 krpm, the PA diastolic pressure increased by 69% (58.4 mmHg vs. 34.5 mmHg) and the mean PA pressure increased by 32% (63.3 mmHg vs. 48 mmHg) compared to baseline (no RVAD) in the PAHR case. The increase is more in the PAHN case at the same operating speed, where the PA diastolic pressure increased by 88% (41.4 mmHg vs. 22 mmHg) and mean PA pressure increased by 42% (46.6 mmHg vs. 32.8 mmHg). In addition, the RA pressure decreased by 40% (4.8 mmHg vs. 8 mmHg) for PAHR compared to baseline (no RVAD) at 28krpm, whereas, for PAHN case it decreased by only 20% (5.6 mmHg vs. 7mmHg) at the same RVAD speed. Compared to PA pressure, the change in aortic pressure was very small in both cases (Fig. 4d).

3.3. Effect of RVAD on septum curvature

The septum has substantially lower curvature κ in the PAH cases compared to that in the normal case (Fig. 5a). Septum curvature in the PAH patients was also lower compared to the LVFW curvature, which is opposite of what is found in the normal subject. The PAHR case with severely dilated RV had the lowest LVFW and septum curvatures. To eliminate the size effect, we also computed the normalized septum curvature by dividing the septum curvature with that average curvature over the entire LV (Fig. 5b). We found that the PAH patients had lower normalized septum curvature over all hand, increased the septum curvature for both PAH patients, cardiac time points. Implantation of the RVAD, on the other particularly during the filling (diastolic) phase (Fig 5b). The overall increase in the septum curvature is evident in the mid-ventricular short-axis slices taken from the PAHR and PAHN cases (Fig. 5c), revealing that the septum moved towards the RV when RVAD was operated at 28krpm.

Fig. 5.

(a) LVFW, septum and LV curvature for normal, PAHR and PAHN cases, (b) comparison of the normalized septum curvature between normal, PAHR and PAHN cases (left), normalized septum curvature for PAHR (middle) and PAHN (right) cases with different RVAD speeds, (c) mid-ventricular short-axis slice of the PAHR (left) and PAHN (right) cases showing the motion of septal wall with RVAD over a cardiac cycle.

3.4. Effect of RVAD on myofiber stress

Average myofiber stresses in the LVFW, septum and RVFW were higher in both PAH cases compared to the normal (Fig. 6). Peak fiber stresses (over a cardiac cycle) were highest in the PAHR case in all 3 regions (values given in the figure insets). Implantation of RVAD reduced only the myofiber stress at the RVFW in the PAHR case. At an RVAD speed of 28krpm, the peak myofiber stress was reduced by 12% in the RVFW (86.3 vs 98 kPa in the baseline). In the PAHN case, the average LVFW, septum and RVFW myofiber stresses all remained relatively unchanged with RVAD.

Fig. 6.

Average LVFW (left), Septum (middle) and RVFW (right) fiber stress (a) in the normal, PAHR and PAHN. (b) PAHR case with different RVAD speed. (c) PAHN case with different RVAD speed. (d) Myofiber stress shown by color map in long-axis slices at mid-ventricular level in Normal, PAHR and PAHN cases at end-systole.

4. Discussion

We have developed a computational framework that couples an image-based biventricular FE model to a lumped parameter representation of the pulmonary and systemic circulations in a closed-loop system. Based on biventricular geometries that were reconstructed from the MR images of two PAH patients and a normal subject, the computational framework was calibrated and matched well with the corresponding patient-specific measurements of 1) PV loops, 2) LV and RV volume waveforms and, 3) LV and RV pressure waveforms. The calibrated computational framework was used to simulate the effects of RVAD on the hemodynamics and ventricular mechanics of the two PAH patients, with different degree of RV remodeling. The major findings of this study suggest that 1) the effects of RVAD depend on the degree of RV remodeling in PAH and 2) the improvement in RV mechanics and septal curvature by RVAD are accompanied by alterations in arterial hemodynamics that may be detrimental.

The modeling results show that the spatially-averaged stresses in the LVFW, septum and RVFW are all increased in both PAH cases compared to normal subject, with the largest increase of ~190% found in the RVFW of the PAHR case (i.e., PAH patient with severe RV remodeling). Similar results showing that LVFW stress is increased in PAH are also found in a previous study[12]. Abnormalities in the septal wall motion was also found in the PAH cases, which are manifested as a bulging movement of the septum wall towards the LV (i.e., left ventricular septal bow) that is a well-known feature of this disease [26]. As a result, septum curvature computed at the mid-ventricular level are substantially lower in the PAH cases compared to the normal case, indicative that the septum in these cases are more “flattened”.

The effects of RVAD on the two PAH cases can be different or similar depending on the quantity of interest. In terms of similar features, the models show that RVAD 1) reduces both RV EDV and RV ESV, 2) improves CO, 3) decreases RA pressure and 4) increases the septum curvature during the filling phase of the cardiac cycle in both PAHN and PAHR cases, all with greater effects found in the latter case. In terms of difference, we found that RVAD produces 1) a slight increase in the RV peak pressure in the PAHN case but a small reduction in that in the PAHR case, and 2) only a reduction in the RVFW myofiber stress in the PAHR case by about 12% at an operating speed of 28 krpm and no change in the myofiber stresses in the PAHN case at different operating speeds. We note that even with the 12% reduction in RVFW myofiber stress in the PAHR case after RVAD implantation, the resultant myofiber stress is still about 153% higher than that found in the normal case. Nevertheless, these findings suggest that RVAD may produce more benefits when implanted in PAH patients exhibiting severe RV remodeling.

The positive effects of RVAD as described above, however, are compromised by an increase in the mean and diastolic PA pressure as well as LV EDV, which are all present in the two PAH cases. The increase in PA pressure (mean and diastolic) is, however, more prominent in the PAHN case. Specifically, the increase in PA pressure can severely damage the pulmonary vasculature, which can produce pulmonary hemorrhage or pulmonary edema [27]. These findings are consistent with previous case reports [28,29] and a study based on lumped parameter circulatory model [3], which all reported an increase in the mean and diastolic PA pressures with RVAD implantation in PAH patients.

Taken together, our finding that the effects of RVAD on hemodynamics and ventricular mechanics are not the same in PAH patients with different degree of remodeling suggests that the decision concerning the implantation of this device and its operation may need to be determined and optimized individually for each patient depending on disease progression. Moreover, the beneficial effects of RVAD (e.g., reduction of RV myofiber stress in PAH patients with severe RV remodeling) may also need to be balanced against its adverse effects on arterial hemodynamics. Indeed, while RVAD has been implanted in patients who have right heart failure caused by RV infarction or LV assist device (LVAD) implantation [30–34], this device has not been used in PAH patients due to the high risk of pulmonary hemorrhage resulting from high RVAD flow. This is especially because the pump used in RVAD is generally designed to support the LV with higher flow even though newer pump design with lower flow rate [19] and partial-assist pumps [35] have been recently proposed to assist the RV in providing sufficient flow for circulation without damaging the pulmonary vasculature.

As shown in this study, the image-based computational framework can help evaluate patient-specific effects of PAH and RVAD implantation not only on ventricular hemodynamics, deformation and myofiber stresses but also on arterial hemodynamics and wall stresses when coupled with a FE model of the vasculature as we have done previously [36]. Previous computational heart models [9,10,12] developed to investigate PAH in humans do not consider the bi-directional coupling between the heart and both the pulmonary and systemic circulation. On the other hand, a computational study that investigate the effects of RVAD in PAH patients [3] is based entirely on a lumped parameter modeling framework, which cannot be used to evaluate regional ventricular stresses and deformation (e.g., septal curvature). The image-based computational framework presented here, which couples patient-specific biventricular FE model with a closed-loop pulmonary and systemic circulatory model, overcomes all these limitations and enables us to assess the effects of remodeling in PAH and different RVAD operating speed on regional myofiber stresses, biventricular deformation and hemodynamics. The computational framework can be extended in future to include a FE model of the pulmonary vasculature to develop more insights in PAH, particularly, in understanding the complex ventricular interdependence and ventricular-vascular coupling associated with this disease.

5. Model limitations

There are some limitations associated with this study. First, the zero-stress unloaded biventricular geometry was reconstructed from the MR images corresponding to the lowest pressure in diastole. While previous studies have either assumed the unloaded geometry to correspond to that obtained at end systole [6] or mid systole [37], or have computed it from the end diastolic geometry using unloading algorithms based on some prescribed material properties [24,38], there are, unfortunately, no general consensus in the literature on what is the best approach to approximate the unloaded geometry. Second, we have used a “rule-based” method to prescribe the local myofiber direction, which varies transmurally from at 60° the endocardium to −60° at the epicardium in the biventricular FE for both the normal subject as well as the PAH patients. Thus, we did not take into account possible changes in the myofiber direction during RV remodeling [39] that may occur in humans. Nevertheless, as we have focused on comparing the effects of RVAD on the overall heart function in PAH patients, we do not expect these limitations and assumptions to highly impact the findings of our study.

Highlights.

• An image-based computational modeling framework consisting of a biventricular finite element model that is coupled to the pulmonary and systemic circulations was developed and calibrated with the clinical measurements from pulmonary arterial hypertension (PAH) patients.

• The effects of right ventricular assist device (RVAD) on the biventricular mechanics were investigated using the framework in PAH patients exhibiting different degrees of right ventricular (RV) remodeling.

• The model results show that RVAD can improve the RV mechanics and septum curvature but these positive effects are accompanied by detrimental effects on the pulmonary vasculature.

• The model results show that the effects of RVAD depends on the degree of RV remodeling in the PAH patients.