Blood flows more easily through the vessels when the pulmonary arteries are healthy and flexible. The synergistic effects of pulmonary vascular remodeling, vasoconstriction, and in situ thrombosis lead to an increase in pulmonary vascular resistance (PVR) and result in PHT [15] . However, the elevation of pulmonary artery pressure caused by pulmonary vasoconstriction is reversible in the early stage of PHT. Moreover, the development of stenosis, thickening of the intima and medial membrane, lead to irreversible changes in vascular structure. Furthermore, the compact and rigid walls of the arteries limit blood flow and increase resistance. As the artery narrows further, blood flow is restricted. Pulmonary vascular remodeling is the major pathological change in PHT [15] .

Now, we consider the pulmonary arteries and establish the nonlinear relationship between pressure and volume. Owing to the Poiseuille’s law, the fluid flow rate $Q$ is given by the following relationship:

$Q=\pi \left(\Delta P\right)r^4/8\mu l$ (4)

where $r$ is the radius, $\Delta P$ is the pressure difference, $l$ is the length of the pipe, and $\mu$ is the viscosity of the liquid. We assume that that $l$ and $\mu$ are constants. The quantity $Q$ is proportional to $\Delta P$ and inversely proportional to the blood flow resistance $R$, explained by the Ohm’s law:

$Q=\Delta P/R$ (5)

Then it appears that the resistance to blood flow $R$ is inversely proportional to the fourth power of $r$ i.e.,

$R=8\mu l/\pi r^4$ (6)

In order to simulate the development of pulmonary artery and proximal pulmonary artery narrowing over time, the radius decreases as a function of time according to the relationship [15] :

$r\left(t\right)=r_0{\left(1+g_0\ast t\right)}^{-1/4}$ (7)

in which $r_0$ is the initial radius and $g_0$ is used for the rate of progression ( Table 2 ). Clinical observations showed that resistance develops slowly and its progression may take years [19] . Therefore, it is reasonable to assume that the artery with stenosis does not undergo any short-term changes. In this case, the artery could be in an equilibrium state in a short time and Poiseuille’s law is thus validated. The short time in this study is assumed to correspond to the duration of a single cardiac cycle. Hence, it could be assumed that the artery undergoes no change during a cardiac cycle. This study simulates pulmonary artery stenosis (PAS) with these hypotheses and the relationships between the resistance of pulmonary arteries ( $R_{pa}$), the resistance of peripheral pulmonary ( $R_{pp}$) and $r$ are given as follows:

$R_{pa}\left(r\left(t\right),t\right)={\left(1/r\left(t\right)\right)}^4\ast R_{pa,0}$ (8a)

$R_{pp}\left(r\left(t\right),t\right)={\left(1/r\left(t\right)\right)}^4\ast R_{pp,0}$ (8b)

where $R_{pa,0}$ and $R_{pp,0}$ are the initial values of $R_{pa}$ and $R_{pp}$. Previous studies [20] [21] have shown that the resistance $R$ and the compliance $C$ are inversely related. However, recent evidence suggests that this concept should be revisited [22] , their product decreases as normalized pulmonary vascular stiffness increases. In this study, we accept the new finding that the product of $R$ and $C$, named RC-time, decreases with time:

$R\ast C=\tau \left(t\right)$ (9a)

$\tau \left(t\right)={\tau }_0\ast {\text{e}}^{-\sigma t}$ (9b)

in which ${\tau }_0$ is the initial value of RC-time in the normal heart, and $\sigma$ is a parameter to control the rate of change ( Table 2 ). Hence, the compliance of the pulmonary artery ( $C_{pa}$) and that of the proximal pulmonary artery ( $C_{pp}$) are defined by:

$C_{pa}\left(r\left(t\right),t\right)=\tau \left(t\right)r^4\left(t\right)/R_{pa,0}$ (10a)

$C_{pp}\left(r\left(t\right),t\right)=\tau \left(t\right)r^4\left(t\right)/R_{pp,0}$ (10b)

Based on the pressure-volume relationship, one deduces from Equations (10) the expressions of the pressures of the pulmonary artery and that of the proximal pulmonary artery, respectively:

$P_{pa}=\frac{V_{pa}\ast R_{pa,0}}{\tau \left(t\right)\ast r^4\left(t\right)}$ (11a)

$P_{pp}=\frac{V_{pp}\ast R_{pp,0}}{\tau \left(t\right)\ast r^4\left(t\right)}$ (11b)

in which $V_{pa}$ and $V_{pp}$ are, respectively, the volume of the pulmonary artery and that of the proximal pulmonary artery.

With the progression of PAS, the PVR afterload, and the mean pulmonary artery pressure ( $mPAP$) may progressively increase [23] . In this case, the RV hypertrophy can be reformed by raising the thickness and contractility of the ventricular wall to accommodate the continuous rise in $mPAP$. In this paper, compensation of the RV to accommodate the continuous rise in $mPAP$ is achieved by raising the end-systolic elastance of the right ventricle ( $E_{es-rv}$) according to the relationship:

$E_{es-rv}\left(t\right)=E_{es-rv,0}+k_{10}\ast t$ (12)

where $k_{10}$ is the rate-of-change control parameter and $E_{es-rv,0}$ is the initial value of $E_{es-rv}$ ( Table 2 ).

In accordance with Sagawa et al. [24] , we consider that the isometric pressure-volume relationship is exponential at end-diastole when the ventricle is relaxed and, linear at end-systole when the ventricle is in maximal contraction. The end-diastolic and end-systolic pressure-volume functions of the ventricle are then determined by:

$P_{ES}=E\left(V_{ES}-V_0\right)$ (13)

$P_{ED}=B\exp \lfloor A\left(V_{ED}-V_0\right)\rfloor -B$ (14)

where $E$ represents the end-systolic elastance of the ventricle, $V_0$ is the associated ventricle unconstrained volume, $B$ and $A$ are coefficients characterizing the exponential form of the end-diastolic pressure-volume relationship (EDPVR). The latest are obtained by integrating the time-evolving ventricular activation function $0<\phi \left(t\right)<1$ (with $\phi \left(t\right)=0$ at maximum contraction and $\phi \left(t\right)=1$ at complete relaxation) and admitting that $V_{lv}=V_{ES}=V_{ED}$ at the beginning of the cardiac cycle. Therefore, when the ventricle is maximally contracted, we have:

$P_{max,lv}=\phi \left(t\right)P_{ED}+\left(1-\phi \left(t\right)\right)P_{ED}$ (15)

Hence, the ventricle mechanic model is defined by:

$\begin{array}{c}{P}_{max,lv}=\phi \left(t\right)E_{lv}\left(V_{lv}-V_{lv,0}\right)\\ +\left(1-\phi \left(t\right)\right)\left(B_{lv}\exp \lfloor A_{lv}\left(V_{lv}-V_{lv,0}\right)\rfloor -B_{lv}\right)\end{array}$ (16)

All parameters exploited for simulating the dynamics of the mechanical models of the ventricle are given in Table 3 .

The LV pressure-volume loop is the most direct manifestation of hemodynamic abnormalities. Previous researchers have shown that, the EDPVR moves upward in the LVDD [25] , which is an exponential function controlled by $A_{lv}$ and $B_{lv}$. In order to simulate LVDD pathology, it is necessary to increase the values of $A_{lv}$ and $B_{lv}$ with respect to time to increase the LV diastolic pressure, therefore:

$A_{lv}\left(t\right)=A_{lv,0}+K_{11}\ast t$ (17)

$B_{lv}\left(t\right)=B_{lv,0}+K_{22}\ast t$ (18)

where $K_{11}$ and $K_{22}$ are the coefficients, $A_{lv,0}$ and $B_{lv,0}$ are the initial values of $A_{lv}$ and $B_{lv}$ defined in Table 4 . To simulate the development of PHT in this case, the EDPVR defined by (16) becomes:

$\begin{array}{c}{P}_{max,lv}=\phi \left(t\right)E_{lv}\left(V_{lv}-V_{lv,0}\right)\\ +\left(1-\phi \left(t\right)\right)\left(B_{lv}\left(t\right)\exp \lfloor A_{lv}\left(t\right)\left(V_{lv}-V_{lv,0}\right)\rfloor -B_{lv}\left(t\right)\right)\end{array}$ (19)

This PHT is strongly linked to the RV. The concept of integration of the RV and the pulmonary circulation have been proposed by Noordegraaf et al. [26] . Under normal physiological conditions, the RV is sensitive to elevated pressure. Indeed, it is connected to the pulmonary circulation at low pressure, low resistance, and high compliance. In the early stage of PHT, the RV dynamics will try to compensate the elevated pulmonary artery pressure. With the development of the disease, in order to adapt to the continuous rise in afterload and maintain ejection capacity, the RV enlarges until right heart failure eventually occurs. Acosta et al. [22] attempted unsuccessfully to definitively resolve this RV failure.

With the development of this type of PHT, the RV in the LVDD model overcomes the elevation of afterload by elevating myocardial contractility $E_{es-rv}$ which is given as follows:

$E_{es-rv}\left(t\right)=E_{es-rv,0}+K_{33}\ast t$ (20)

where $K_{33}$ is the rate of change control parameter, $E_{es-rv,0}$ is the initial value of $E_{es-rv}$ ( Table 4 ).

The pathological mechanism in PHT simulation allows for changes in pulmonary artery resistances over time. Figure 4 displays the pressure-volume loops of two heart chambers for PHT due to the PAS. The blue curves describe normal healthy (normal) situation clearly shown in Figure 3(a) while the orange curves deal with pathological conditions (case with PHT). Comparing these results with normal hemodynamic conditions, it becomes relevant to note that the systolic blood pressure of the RV rises over 25 mmHg until it reaches 90 mmHg. Similarly, the rise in pulmonary artery pressure is high enough to shift forward the blood flow in the pulmonary circulation. These results corroborate with those established by Lock et al. [27] .

Now, we check the development of the key pulmonary blood pressures and volume for PHT due to PAS. We focus on temporal evolution of pressure and volume signals in the pulmonary arteries and pulmonary peripheral. The obtained results are plotted in Figure 5 . The graphs of Figure 5(a) exhibit the dynamics of pulmonary artery pressure (Ppa) and peripheral pulmonary pressure (Ppp) showing that increase in pulmonary artery resistance directly causes an increase in pulmonary artery blood pressure. The same observations are made in the peripheral pulmonary artery (orange curves). On the other hand, Figure 5(b) presents the temporal evolution of the pressure and volume signals in the right ventricular. Let us note that during PHT, right ventricular pressure rises leading to a growth of the right ventricular volume (orange).

These results dealing with our model are consistent with previous clinical observations. Indeed, Lock et al. [27] examined children with right and left pulmonary artery stenosis (or hypoplasia) and established that their right ventricular pressure increases to 105.3 ± 37.4 (mean ± SD) mmHg in pre-dilation and

dropped to 83.8 ± 28.6 mmHg in post-dilation. Moreover, Shikata et al. [28] presented a case of PHT pathology and bilateral pulmonary artery stenosis, demonstrating a pulmonary artery pressure of 95/15 (mean 45) mmHg, and a right ventricular pressure of 100/10 (mean 45) mmHg. Furthermore, Tyagi et al. [29] examined the case of an aortoarteritis patient with severe proximal right pulmonary artery stenosis. Hemodynamic measurement showed an elevated main pulmonary artery pressure of 80/24 (52) mmHg. On the same, Baerlocher et al. [30] evaluated the ratio of RV systolic pressure to aortic pressure of patients with branched PAS and obtained that it was 80.6% and 50.6%, respectively, during pre-dilation and post-dilation of primary balloon angioplasty.

In the case of LVDD, the left ventricular end-diastolic pressure is mathematically elevated by linearly raising the parameters of A(t) and B(t). Simulations are carried out with the modified parameters defined in Table 4 . The obtained results are dropped in Figure 6 and Figure 7 .

Figure 6 exhibits the pressure-volume loops of two heart chambers under normal (blue) and pathological (orange) conditions for the left and right ventricles in the LVDD case. Compared with normal hemodynamic conditions, the established results reveal that the LVDD is manifested directly by the growth of the left ventricular end-diastolic pressure ( Figure 6(a) ). Moreover, it appears in Figure 6(b) that event the right ventricular pressure continues to rise until it reaches 70 mmHg. Chatterjee and Massie [31] collected patient data to represent the schematic diagram of P-V relationships in systolic heart failure and in pathologies with diastolic dysfunction. The P-V loop made upward and leftward shifts.

Figure 7(a) presents the shape of the pressure and volume signals in the left atrium. Compared to normal situation, it appears that the amplitude of the pressure or volume signal increases in the pathological case of LVDD (orange curves). Indeed, the left atrium elevates pressure to allow returning of blood to the LV. Long-term blood return is blocked, leading to an accumulation of blood in the left atrium and a consequent increase in its volume. In their clinical work, Kurt et al. [32] noted the same observations where the elevation of left atrial volume in patients with diastolic heart failure compared to the normal control group was represented. Due to the development of PHT, the systolic blood pressure of the left atrium continues to increase. This can be explained by the fact that long-term blood flow obstruction changes the function and structure of the left atrium. Moreover, the clinical data established by Melenovsky et al. [33] lead to similar results showing that the left atrial pressure and volume in heart failure with preserved ejection fraction increased, and the left atrial stiffness also increased compared to the control group. Figure 7(b) displays the decrease to 50mmHg of the symmetric arteries and peripheral pressures, respectively. It can also be seen that growth of systemic artery pressure due to this disease leads directly to the increment of peripheral systemic artery pressure. Figure 7(c) presents the time evolution of pulmonary artery pressure (Ppa) and peripheral pulmonary artery pressure (Ppp) allowing to see that increment up to 80mmHg of Ppa due to this disease induces increment up to 60mmHg of peripheral pulmonary artery pressure (orange). On the other hand, it appears from plots of Figure 7(d) that the right ventricular pressure continues to rise until it reaches 80 mmHg in the case of developing PHT due to LVDD (orange). Hence, the systolic pressure of the RV would rise to overcome the rise in pulmonary artery pressure. In the case of LVDD, the pressure in the left atrium and pulmonary veins rise ( Figure 7(e) ) leading to an increase in pulmonary artery blood pressure ( Figure 7(c) ).