Numerical predictions of shear stress and cyclic stretch in pulmonary hypertension due to left heart failure

Abstract

Isolated post-capillary pulmonary hypertension (Ipc-PH) occurs due to left heart failure, which contributes to 1 out of every 9 deaths in the United States. In some patients, through unknown mechanisms, Ipc-PH transitions to combined pre-/post-capillary PH (Cpc-PH) and is associated with a dramatic increase in mortality. Altered mechanical forces and subsequent biological signaling in the pulmonary vascular bed likely contribute to the transition from Ipc-PH to Cpc-PH. However, even in a healthy pulmonary circulation, the mechanical forces in the smallest vessels (the arterioles, capillary bed, and venules) have not been quantitatively defined. This study is the first to examine this question via a computational fluid dynamics model of the human pulmonary arteries, arterioles, venules, and veins. Using this model, we predict temporal and spatial dynamics of cyclic stretch and wall shear stress with healthy and diseased hemodynamics. In the normotensive case for large vessels, numerical simulations show that large arteries have higher pressure and flow than large veins, as well as more pronounced changes in area throughout the cardiac cycle. In the microvasculature, shear stress increases and cyclic stretch decreases as vessel radius decreases. When we impose an increase in left atrial pressure to simulate Ipc-PH, shear stress decreases and cyclic stretch increases as compared to the healthy case. Overall, this model predicts pressure, flow, shear stress, and cyclic stretch that providing a way to analyze and investigate hypotheses related to disease progression in the pulmonary circulation.

Appendix

As described in detail by Qureshi et al. (2014), for the arterial and venous networks we define a relation between pressure and flow at both ends of each vessel, and derive a boundary condition that matches pressure and flow at the terminal large arteries and veins. To do so, we compute the admittance of each structured tree, relating pressure and flow at the outlet of each large terminal artery to pressure and flow at the inlet to the corresponding large terminal vein. At each junction, we set up an admittance matrix satisfying continuity of pressure and conservation of volume flux. The total admittance is calculated by joining “junction” admittances in series and parallel as illustrated in Fig. 12.

Fig. 12

Relationship between flow and pressure via admittance $\mathbf {Y}$, $\mathbf {Y}^{\Vert }$ and $\mathbf {Y}^{\Leftrightarrow }$ for a single vessel (a), vessels connected in parallel (b), and vessels connected in series (c). Note that $\mathbf {Y}$ is the admittance for a single vessel, $\mathbf {Y}^S$ is the admittance for vessel S, and $\mathbf {Y}^S$ is the admittance for vessel T. Here, the color blue represents deoxygenated arteries, and red represents oxygen-rich veins

Given the proximal and distal flow $Q_{1}=Q(0, \omega )$ and $Q_{2}=Q(L, \omega )$, the admittance matrix is obtained by relating the flow and pressure at the proximal, $x=0,$ and distal, $x=L,$ ends of a vessel of radius $r_0$ yielding

$$\begin{aligned} \left( \begin{array}{l} Q_{1} \\ Q_{2}\end{array}\right) =\frac{i g_{\omega }}{S_{L}} \left( \begin{array}{cc} -C_{L} &{} 1 \\ 1 &{} -C_{L} \end{array}\right) \left( \begin{array}{l} P_{1} \\ P_{2} \end{array}\right) \end{aligned}$$

(6.1)

where $C_{L} \equiv \cos (\omega L / c), S_{L} \equiv \sin (\omega L / c),$ and

$$\begin{aligned} \mathbf {Y}(\omega )=\frac{i g_{\omega }}{S_{L}} \left( \begin{array}{cc} -C_{L} &{} 1 \\ 1 &{} -C_{L} \end{array}\right) \end{aligned}$$

(6.2)

is the admittance matrix for any one artery or vein when $\omega \ne 0.$ For $\omega =0,$ , we have

$$\begin{aligned} \left( \begin{array}{l} Q_{1} \\ Q_{2} \end{array}\right) =\frac{\pi r^{4}}{8 \mu L}\left( \begin{array}{rr} 1 &{} -1 \\ -1 &{} 1 \end{array}\right) \left( \begin{array}{l} P_{1} \\ P_{2} \end{array}\right) . \end{aligned}$$

(6.3)

and therefore

$$\begin{aligned} \mathbf {Y}(0)=\frac{\pi r^{4}}{8 \mu L} \left( \begin{array}{rr} 1 &{} -1 \\ -1 &{} 1 \end{array}\right) . \end{aligned}$$

(6.4)

For two vessels (S and T) in parallel, continuity of pressure between the inlet and outlet and conservation of volume flux across the junction gives

$$\begin{aligned} \left( \begin{array}{l} Q_{1} \\ Q_{2} \end{array}\right) =\mathbf {Y}^{\Vert }\left( \begin{array}{l} P_{1} \\ P_{2} \end{array}\right) , \end{aligned}$$

(6.5)

where $Q_{1}$ and $Q_{2}$ denote the vessel inflow and outflow, $P_1$ and $P_2$ are the corresponding inlet and outlet pressure, and $\mathbf {Y}^{\Vert }=\mathbf {Y}^{S}+\mathbf {Y}^{T}$. We denote $\mathbf {Y}^{S}$ and $\mathbf {Y}^{T}$ as the admittances for vessel S and T, respectively.

Similarly, for two vessels (S and T) connected in series flow and pressure are related by

$$\begin{aligned} Q_{k}^{i}=\sum _{l=1}^{2} Y_{k l}^{i} P_{l}^{i} \end{aligned}$$

(6.6)

where $i=S, T$ and $k=1,2$ with $Y_{k l}$ being the components of the $2 \times 2$ admittance matrix. Assuming that $P=P_{2}^{S}=P_{1}^{T}$ and $Q_{2}^{S}=-Q_{1}^{T}$ at the junction of two vessels in series, the system for the vessels is represented by

$$\begin{aligned} \left( \begin{array}{l} Q_{1}^{S} \\ Q_{2}^{T} \end{array}\right) =\mathbf {Y}^{\Leftrightarrow }\left( \begin{array}{l} P_{1}^{S} \\ P_{2}^{T} \end{array}\right) \end{aligned}$$

(6.7)

where

$$\begin{aligned} \mathbf {Y}^{\Leftrightarrow }&= \frac{1}{Y_{22}^{S}+Y_{11}^{T}} \times \left( \begin{array}{cc} {\text {det}}\left( \mathbf {Y}^{S}\right) +Y_{11}^{S} Y_{11}^{T} &{} -Y_{12}^{S} Y_{12}^{T} \\ -Y_{21}^{S} Y_{21}^{T} &{}{\text {det}}\left( \mathbf {Y}^{T}\right) +Y_{22}^{S} Y_{22}^{T} \end{array}\right) \end{aligned}$$

(6.8)

is the admittance matrix. The symbol $\Leftrightarrow$ denotes that the vessels are joined in series.

To compute pressure and flow in the arterioles and venules, we first prescribe the pressure from the terminal large arteries or veins at the start of the structured tree, i.e. $P^A(0,\omega )$ and $P^V(0,\omega )$. The corresponding flows are calculated using the grand admittance,

$$\begin{aligned} Q^A(0,\omega )&= Y_{11}P^A(0,\omega ) + Y_{12}P^V(0,\omega ), \\ Q^V(0,\omega )&= Y_{21}P^A(0,\omega ) + Y_{22}P^V(0,\omega ). \end{aligned}$$

The small vessel pressure and flow at $x=L$ are determined by

$$\begin{aligned} P^j(L,\omega )&= P^j(0,\omega )C_L - \frac{iS_L}{g_\omega }Q^j(0,\omega ), \end{aligned}$$

(6.9)

$$\begin{aligned} Q^j(L,\omega )&= \frac{i g_\omega }{S_L}\left( P^j(0,\omega ) -C_L P^j(L,\omega )\right) \end{aligned}$$

(6.10)

for $j=A,V$ and $\omega \ne 0$, whereas the zeroth frequency solutions are

$$\begin{aligned} P^j(L,\omega )&=P^j(0,\omega )- \frac{8 \mu L}{\pi r^4}Q^j(0,\omega ), \end{aligned}$$

(6.11)

$$\begin{aligned} Q^j(L,\omega )&=\frac{\pi r^4}{8 \mu L}\left( P^j(0,\omega ) -P^j(L,\omega )\right) . \end{aligned}$$

(6.12)