Cardiovascular regulation in response to multiple hemorrhages: analysis and parameter estimation

Abstract

Mathematical models can provide useful insights explaining behavior observed in experimental data; however, rigorous analysis is needed to select a subset of model parameters that can be informed by available data. Here we present a method to estimate an identifiable set of parameters based on baseline left ventricular pressure and volume time series data. From this identifiable subset, we then select, based on current understanding of cardiovascular control, parameters that vary in time in response to blood withdrawal, and estimate these parameters over a series of blood withdrawals. These time-varying parameters are first estimated using piecewise linear splines minimizing the mean squared error between measured and computed left ventricular pressure and volume data over four consecutive blood withdrawals. As a final step, the trends in these splines are fit with empirical functional expressions selected to describe cardiovascular regulation during blood withdrawal. Our analysis at baseline found parameters representing timing of cardiac contraction, systemic vascular resistance, and cardiac contractility to be identifiable. Of these parameters, vascular resistance and cardiac contractility were varied in time. Data used for this study were measured in a control Sprague-Dawley rat. To our knowledge, this is the first study to analyze the response to multiple blood withdrawals both experimentally and theoretically, as most previous studies focus on analyzing the response to one large blood withdrawal. Results show that during each blood withdrawal both systemic vascular resistance and contractility decrease acutely and partially recover, and they decrease chronically across the series of blood withdrawals.

Appendix: Cardiovascular model equations

We provide here the complete set of equations for the five-compartment cardiovascular model proposed above and summarized in Fig. 2. The evolution of each compartment’s volume is given by:

$$\begin{aligned} \frac{\mathrm{d} V_{\mathrm{ao}}}{\mathrm{d}t}&= q_{\mathrm{av}}-q_\mathrm{a} \\ \frac{\mathrm{d} V_{\mathrm{sa}}}{\mathrm{d}t}&= q_{\mathrm{a}}-q_\mathrm{s} - q_{\mathrm{out}} \\ \frac{\mathrm{d} V_{\mathrm{sv}}}{\mathrm{d}t}&= q_{\mathrm{s}}-q_\mathrm{v} \\ \frac{\mathrm{d} V_{\mathrm{vc}}}{\mathrm{d}t}&= q_{\mathrm{v}}-q_{\mathrm{mv}} \\ \frac{\mathrm{d} V_{\mathrm{lv}}}{\mathrm{d}t}&= q_{\mathrm{mv}}-q_{\mathrm{av}} \end{aligned}$$

Note that this corresponds to the blood withdrawal simulations where the withdrawal rate is $q_{\mathrm{out}}$. The flow between compartments is given by:

$$\begin{aligned} q_{\mathrm{av}}&= \left\{ \begin{array}{ll} \displaystyle \frac{p_{\mathrm{lv}}-p_{\mathrm{ao}}}{R_{\mathrm{av}}}, &{} p_{\mathrm{lv}}> p_{\mathrm{ao}} \ \ \text{( }\text {open valve}) \\ \displaystyle 0, &{} \text{ otherwise } \end{array} \right. \\ q_{\mathrm{s}}&= \frac{p_{\mathrm{sa}}-p_{\mathrm{sv}}}{R_\mathrm{s}}\\ q_{\mathrm{v}}&= \frac{p_{\mathrm{sv}}-p_{\mathrm{vc}}}{R_\mathrm{v}} \\ q_\mathrm{a}&= \frac{p_{\mathrm{ao}}-p_{\mathrm{sa}}}{R_\mathrm{a}} \\ q_{\mathrm{mv}}&= \left\{ \begin{array}{ll} \displaystyle \frac{p_{\mathrm{vc}}-p_{\mathrm{lv}}}{R_{\mathrm{mv}}}, &{} p_{\mathrm{vc}} > p_{\mathrm{lv}} \ \ \text{( }\text {open valve}) \\ \displaystyle 0, &{} \text{ otherwise. } \end{array} \right. \end{aligned}$$

(The expressions for $q_{\mathrm{av}}$ and $q_{\mathrm{mv}}$ are provided again for completeness.)

The pressures in each compartment are given by:

$$\begin{aligned} p_{\mathrm{ao}}&= \frac{V_{\mathrm{ao}}}{C_{\mathrm{ao}}} \\ p_{\mathrm{sa}}&= \frac{V_{\mathrm{sa}}}{C_{\mathrm{sa}}}\\ p_{\mathrm{sv}}&= \frac{V_{\mathrm{sv}}}{C_{\mathrm{sv}}} \\ p_{\mathrm{vc}}&= \frac{V_{\mathrm{vc}}}{C_{\mathrm{vc}}} \\ p_{\mathrm{lv}}&= E_{\mathrm{lv}}(t) (V_{\mathrm{lv}} - V_{\mathrm{lv},0}) \end{aligned}$$

where variables $C_i$ correspond to the compliance in compartment i (see Table 3 for nominal values) and $E_{\mathrm{lv}}(t)$ denotes the time-varying elastance described in Eq. (4).