Sensitivity Analysis and Model Assessment: Mathematical Models for Arterial Blood Flow and Blood Pressure

Abstract

The complexity of mathematical models describing the cardiovascular system has grown in recent years to more accurately account for physiological dynamics. To aid in model validation and design, classical deterministic sensitivity analysis is performed on the cardiovascular model first presented by Olufsen, Tran, Ottesen, Ellwein, Lipsitz and Novak (J Appl Physiol 99(4):1523–1537, 2005). This model uses 11 differential state equations with 52 parameters to predict arterial blood flow and blood pressure. The relative sensitivity solutions of the model state equations with respect to each of the parameters is calculated and a sensitivity ranking is created for each parameter. Parameters are separated into two groups: sensitive and insensitive parameters. Small changes in sensitive parameters have a large effect on the model solution while changes in insensitive parameters have a negligible effect. This analysis was successfully used to reduce the effective parameter space by more than half and the computation time by two thirds. Additionally, a simpler model was designed that retained the necessary features of the original model but with two-thirds of the state equations and half of the model parameters.

Appendix

Equations, Full Model

• (1) Compartment ODE’s:

  $$ \begin{aligned} \frac{dp_{\rm a}}{dt} &= \left(\frac{p_{\rm lv}(t)-p_{\rm a}(t)}{R_{\rm av}}-\frac{p_{\rm a}(t)-p_{\rm au}(t)}{R_{\rm au}}- \frac{p_{\rm a}(t)-p_{\rm ac}(t)}{R_{\rm ac}}-\frac{p_{\rm a}(t)-p_{\rm af}(t)}{R_{\rm af}}\right)/C_{\rm a}\\ \frac{dp_{\rm au}}{dt}&= \left(\frac{p_{\rm a}(t)-p_{\rm au}(t)} {R_{\rm au}}-\frac{p_{\rm au}(t)-p_{\rm al}(t)}{R_{\rm al}}- \frac{p_{\rm au}(t)-p_{\rm vu}(t)}{R_{\rm aup}}\right)/C_{\rm au} \\ \frac{dp_{\rm al}}{dt}&= \left(\frac{p_{\rm au}(t)-p_{\rm al}(t)}{R_{\rm al}} -\frac{p_{\rm al}(t)-p_{\rm vl}(t)}{R_{\rm alp}}\right)/C_{\rm al} \\ \frac{dp_{\rm af}}{dt}&= \left(\frac{p_{\rm a}(t)-p_{\rm af}(t)}{R_{\rm af}} - \frac{p_{\rm af}(t)-p_{\rm v}(t)}{R_{\rm afp}}\right)/C_{\rm af} \\ \frac{dp_{\rm ac}}{dt}&= \left(\frac{p_{\rm a}(t)-p_{\rm ac}(t)}{R_{\rm ac}} -\frac{p_{\rm ac}(t)-p_{\rm vc}(t)}{R_{\rm acp}}\right)/C_{\rm ac} \\ \frac{dp_{\rm v}}{dt} &= \left(\frac{p_{\rm vu}(t)-p_{\rm v}(t)} {R_{\rm vu}} +\frac{p_{\rm vc}(t)-p_{\rm v}(t)}{R_{\rm vc}}+ \frac{p_{\rm af}(t)-p_{\rm v}(t)}{R_{\rm afp}} -\frac{p_{\rm v}(t)-p_{\rm la}(t)}{R_{\rm v}}\right)/C_{\rm v} \\ \frac{dp_{\rm vu}}{dt}&=\left(\frac{p_{\rm vl}(t)-p_{\rm vu}(t)}{R_{\rm vl}} +\frac{p_{\rm au}(t)-p_{\rm vu}(t)}{R_{\rm aup}} - \frac{p_{\rm vu}(t)-p_{\rm v}(t)}{R_{\rm vu}}\right)/C_{\rm vu} \\ \frac{dp_{\rm vl}}{dt}&= \left(\frac{p_{\rm al}(t)-p_{\rm vl}(t)} {R_{\rm alp}}-\frac{p_{\rm vl}(t)-p_{\rm vu}(t)}{R_{\rm vl}}\right)/C_{\rm vl} \\ \frac{dp_{\rm vc}}{dt} &=\left(\frac{p_{\rm ac}(t)-p_{\rm vc}(t)}{R_{\rm acp}} -\frac{p_{\rm vc}(t)-p_{\rm v}(t)}{R_{\rm vc}}\right)/C_{\rm vc} \\ \frac{dV_{\rm lv}}{dt}&= \frac{p_{\rm la}(t)-p_{\rm lv}(t)} {R_{\rm mv}(t)} -\frac{p_{\rm lv}(t)-p_{\rm a}(t)}{R_{\rm av}(t)} \\ \frac{dV_{\rm la}}{dt}&= \frac{p_{\rm v}(t)-p_{\rm la}(t)} {R_{\rm v}}-\frac{p_{\rm la}(t)-p_{\rm lv}(t)}{R_{\rm mv}(t)} \end{aligned} $$

  Parameters include R _i (mmHg s/cm³), C _i (cm³/mmHg).

• 2) Valve equations R _av, R _mv, R _vu:

  $$ \begin{aligned} R_{\rm av}&=\min(R_{\rm av,open} + e^{(-2(p_{\rm lv}-p_{\rm a}))},20)\\ R_{\rm mv}&=\min(R_{\rm mv,open} + e^{(-2(p_{\rm la}-p_{\rm lv}))},20)\\ R_{\rm vu}&=\min(R_{\rm vu,open} + e^{(-2(p_{\rm vu}-p_{\rm v}))},20) \end{aligned} $$

• 3) Left atrium and ventricle:

  $$ \begin{aligned} p_j(t) &= a_j(V_j(t)-b_j)^2+(c_j V_j(t)-d_j)f_j(t), \quad j = \hbox{lv, la} \\ f_j(t) &= \left\{ \begin{array}{ll} \tilde{p}_j(H)\frac{\tilde{t}^{n_j}(\beta(H)-\tilde{t})^m_j} {n_j^{n_j}m_j^{m_j}[(\beta_j(H))/(m_j+n_j)]^{m_j+n_j}} & 0 \leq \tilde{t} \leq \beta_j(H)\\ 0 & \beta_j(H) < \tilde{t} < T \end{array}\right.\\ \tilde{t} &= \hbox{mod}(t,T) \\ t_j &= (t_{diff,j}) \frac{\theta_j^{\nu_j}} {H^{\nu_j}+\theta_j^{\nu_j}}+t_{{\rm min},j}\\ \tilde{p}_j &= (p_{diff,j}) \frac{H^{\eta_j}} {H^{\eta_j}+\phi_j^{\eta_j}}+p_{{\rm min},j}\\ \beta_j(H)&= \frac{n_j+m_j}{n_j}t_j(H) \\ \end{aligned} $$

Equations, Reduced Model

• (1) Compartment ODE’s:

  $$ \begin{aligned} \frac{dp_{\rm a}}{dt}&= \left(\frac{p_{\rm lv}-p_{\rm a}}{R_{\rm av}}-\frac{p_{\rm a}-p_{\rm as}} {R_{\rm as}}- \frac{p_{\rm a}-p_{\rm ac}}{R_{\rm ac}}\right)/C_{\rm a}\\ \frac{dp_{\rm v}}{dt}&=\left(\frac{p_{\rm vs}-p_{\rm v}}{R_{\rm vs}}+\frac{p_{\rm vc}-p_{\rm v}}{R_{\rm vc}} -\frac{p_{\rm v}-p_{\rm lv}}{R_{\rm mv}}\right)/C_{\rm v}\\ \frac{dp_{\rm as}}{dt}&= \left(\frac{p_{\rm a}-p_{\rm as}}{R_{\rm as}} -\frac{p_{\rm as}-p_{\rm vs}}{R_{\rm asp}}\right)/C_{\rm as} \\ \frac{dp_{\rm vs}}{dt}&= \left(\frac{p_{\rm as}-p_{\rm vs}}{R_{\rm asp}} -\frac{p_{\rm vs}-p_{\rm v}}{R_{\rm vs}}\right)/C_{\rm vs} \\ \frac{dp_{\rm ac}}{dt}&= \left(\frac{p_{\rm a}-p_{\rm ac}}{R_{\rm ac}} -\frac{p_{\rm ac}-p_{\rm vc}}{R_{\rm acp}}\right)/C_{\rm ac} \\ \frac{dp_{\rm vc}}{dt}&= \left(\frac{p_{\rm ac}-p_{\rm vc}}{R_{\rm acp}} -\frac{p_{\rm vc}-p_{\rm v}}{R_{\rm vc}}\right)/C_{\rm vc} \\ \frac{dV_{\rm lv}}{dt}&= \frac{p_{\rm v}-p_{\rm lv}}{R_{\rm mv}} -\frac{p_{\rm lv}-p_{\rm a}}{R_{\rm av}} \end{aligned} $$

• (2) Valve equations, R _av, and R _mv:

  $$ \begin{aligned} R_{\rm mv}&=\hbox{min}(R_{\rm mv,open}+e^{(-2(p_{\rm v}-p_{\rm lv}))},20)\\ R_{\rm av}&=\hbox{min}(R_{\rm av,open}+e^{(-2(p_{\rm lv}-p_{\rm a}))},20) \end{aligned} $$

• (3) Left ventricle: Equations for the left ventricle are identical to those displayed above.