Hemodynamic assessment of pulmonary hypertension in mice: a model-based analysis of the disease mechanism

Abstract

This study uses a one-dimensional fluid dynamics arterial network model to infer changes in hemodynamic quantities associated with pulmonary hypertension in mice. Data for this study include blood flow and pressure measurements from the main pulmonary artery for 7 control mice with normal pulmonary function and 5 mice with hypoxia-induced pulmonary hypertension. Arterial dimensions for a 21-vessel network are extracted from micro-CT images of lungs from a representative control and hypertensive mouse. Each vessel is represented by its length and radius. Fluid dynamic computations are done assuming that the flow is Newtonian, viscous, laminar, and has no swirl. The system of equations is closed by a constitutive equation relating pressure and area, using a linear model derived from stress–strain deformation in the circumferential direction assuming that the arterial walls are thin, and also an empirical nonlinear model. For each dataset, an inflow waveform is extracted from the data, and nominal parameters specifying the outflow boundary conditions are computed from mean values and characteristic timescales extracted from the data. The model is calibrated for each mouse by estimating parameters that minimize the least squares error between measured and computed waveforms. Optimized parameters are compared across the control and the hypertensive groups to characterize vascular remodeling with disease. Results show that pulmonary hypertension is associated with stiffer and less compliant proximal and distal vasculature with augmented wave reflections, and that elastic nonlinearities are insignificant in the hypertensive animal.

Appendices

Appendix

A Vascular compliance

The volumetric compliance, defined as $C_\mathrm{v}= \hbox {d}V/\hbox {d}p$ (ml/mmHg) for a cylindrical vessel with volume V, is computed from the linear ($C_\text {lin}$) and nonlinear ($C_\text {nlin}$) models. For a longitudinally tethered vessel i in the network

$$\begin{aligned} C_\mathrm{v} = \frac{\hbox {d}V}{\hbox {d}p} \equiv L\frac{\hbox {d}A}{\hbox {d}p}, \end{aligned}$$

(28)

where L is the fixed length of the vessel and $\hbox {d}A/\hbox {d}p$ is computed from Eqs. (3) and (4), giving

$$\begin{aligned} C_\text {lin}= C_{0.\text {lin}}\left( \frac{p}{\beta }+1\right) ,\qquad \text {and}\qquad C_\text {nlin}= C_{0.\text {nlin}}\left( \frac{p_1^2}{p^2 + p_1^2}\right) , \end{aligned}$$

(29)

where $C_0$ denotes the reference compliance at $p=0$, given by

$$\begin{aligned} C_{0.\text {lin}} = \frac{2A_0L }{\beta },\qquad \text {and}\qquad C_{0.\text {nlin}}=\frac{\gamma A_0 L}{\pi p_1}. \end{aligned}$$

(30)

B Pulse wave velocity

The pulse wave velocity (PWV), c (cm/s), is computed from the eigenvalues of the hyperbolic system of Eq. (2), from $\lambda _{1,2} = q/A \pm c$ where

$$\begin{aligned} c=\sqrt{\frac{A}{\rho }\frac{\hbox {d} p}{\hbox {d}A}} = \sqrt{\frac{AL}{\rho \, C_\mathrm{v}}}. \end{aligned}$$

(31)

Setting $C_\mathrm{v} = C_\text {lin}$ and $C_\text {nlin}$ in Eq. (31) gives the squared PWV computed for the linear and nonlinear wall models, respectively

$$\begin{aligned} c^2_\text {lin}= & {} c^2_{0.\text {lin}}+\frac{p}{2\rho },\nonumber \\ c^2_\text {nlin}= & {} c^2_{0.\text {nlin}}\left( \frac{\gamma }{\pi }\tan ^{-1}(p/p_1)+1\right) \left( 1+(p/p_1)^2\right) , \end{aligned}$$

(32)

where $c_0^2$ is the square of the reference PWV at $p=0$, given by

$$\begin{aligned} c^2_{0.\text {lin}}=\frac{\beta }{2\rho },\qquad \text {and}\qquad c^2_{0.\text {nlin}}=\frac{\pi p_1}{\gamma \rho }. \end{aligned}$$

(33)

We use PWV in wave intensity analysis, described in Sect. 2.7, for separating the incident and reflected waves.

C Nominal parameter values

Table 4 Nominal values for wall parameters and compliance for individual mice in each group

As described in Sect. 2.5, nominal values for the nonlinear model are set as $p_1=\beta /\pi$ and $\gamma = 2$ for all cases. Moreover, nominal values for $R_{\mathrm{T},j}$ are computed from $R_\mathrm{T}$ reported in the table above using methods described in Sect. 2.5 and the network dimensions stated in Table 2. For all cases, the resistance ratio $a\equiv R_1/R_\mathrm{T} = 0.2$.

D Optimized parameter values

For all cases, we optimized $\beta , \gamma ,$ and $p_1$ for the wall models, and the global scaling parameters $r_1, r_2, c_1$ for the Windkessel model, such that

$$\begin{aligned} {R}_{1,j} = r_1R_{10,j}, \quad {R}_{2,j} = r_2R_{20,j},\quad {C}_{\mathrm{p},j} = c_1C_{p0,j} \end{aligned}$$

where 0 indicate the nominal quantity. Upper and lower bounds for the optimization intervals are given in Table 5.

Table 5 Bounds for optimization

Table 6 Optimized values for individual mice in each group

E Convergence of optimization algorithm

To test the convergence of our optimization algorithm, we carried out repeated optimizations in four- and five-dimensional parameter spaces for the linear and nonlinear wall models, respectively. This was done only for the representative control mouse. Optimizations were initialized from 20 initial values, drawn using the Sobol sequence to uniformly cover the entire domain (i.e. predefined interval). Regardless of the starting value, the algorithm converged to the same values for a given parameter. Also, Fig. 13 shows the convergence to a unique minimum of the objective function regardless of the starting point. For the sake of computational efficiency, parallel optimization was conducted starting from only four initial values for the remaining mice. Figures 11 and 12 show the optimization history (starting from 20 initial points) for the linear and nonlinear cases from the control mouse.

Fig. 11

Time history of optimization algorithm for the case of linear wall model. As a test case, 20 starting points were sampled from the parameter interval (vertical axis). Each color represents an iteration chain associated with a given starting point. All of them converged to the same final value. On average, it took 28 iterations to converge to an optimal value using the linear wall model

Fig. 12

Time history of optimization algorithm for the case of nonlinear wall model. As a test case, 20 starting points were sampled from the parameter interval (vertical axis). Each color represents an iteration chain associated with a given starting point. All of them converged to the same final value. On average, it took 52 iterations to converge to an optimal value using the nonlinear wall model

Fig. 13

Time history of objective function’s values during the optimization process starting from 20 starting points. Each color represents an iteration chain associated with a given starting point. The plots are shown on a linear-log scale ($\log (S)$) both using the linear and the nonlinear wall models for the representative control mouse. Optimization converges to a minimum of objective function irrespective of the starting point