# Mathematical model of the pulmonary circulation: Effect of lung inflation and cardiac output

#### Abstract

A hemodynamic model of the pulmonary circulation was developed to integrate extant geometric, biomechanics, and rheologic data, and new experimental data on the distribution of the pulmonary arterial: capillary: venous resistance. In the model construct, this information was utilized in functional relationships between the independent variables intravascular pressure, P, pleural pressure, P$\rm\sb{pl}$, alveolar pressure, P$\rm\sb{A}$, blood flow, Q, and feed hematocrit, Hct$\rm\sb{f}$, and the dependent variables of arterial and venous dimensions in each order, capillary sheet dimensions, lung volume, V, and perivascular pressures. The resulting implicit steady-state model equations were solved iteratively given the input values for the venous pressure, P$\rm\sb{v}$, Hct$\rm\sb{f}$, Q, P$\rm\sb{A}$, and P$\rm\sb{pl}$. For the dynamic form of the model, each order and the capillary sheet were represented by a system of first order differential equations, the coefficients of which are the hemodynamic resistance, compliance, and inertance values calculated from the steady-state model. The dynamic model allows for simulations of the pulmonary vascular responses to the pulsatile cardiac output, respiratory movement, and transient changes in blood flow and pressure. In this study, the steady-state model was used to explore the impact of lung volume on the sum of the vascular resistances of the extra-alveolar vessels, which distend with lung volume, and the alveolar vessels, which narrow with lung volume. Experimentally, this sum (the total vascular resistance) versus V is U-shaped, initially decreasing as V increases and then increasing with further increases in V. In the model simulations, the impact of the vessel geometry on the hematocrit dependence of blood viscosity dominates the variation of vascular resistance with V through its impact on the alveolar vessel resistance. These simulations suggest ways of examining this potential importance of geometry dependence of viscosity which has not previously been considered in explanations of the impact of V on pulmonary vascular resistance. The responses of the dynamic form of the model to oscillatory flow and transient flow maneuvers were carried out to determine how well the model predicts the previously published experimental pressure responses. These examples suggest that the mathematical model provides a useful tool for understanding the complex hemodynamic interactions within the pulmonary vasculature and for guiding future experimental research.

#### Recommended Citation

Haworth, Steven Thomas, "Mathematical model of the pulmonary circulation: Effect of lung inflation and cardiac output" (1996). *Dissertations (1962 - 2010) Access via Proquest Digital Dissertations*. AAI9634266.

https://epublications.marquette.edu/dissertations/AAI9634266