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Algebraic Solutions for Homeostatic Mechanisms in a Minimal Closed‐Loop Cardiovascular System Model

To characterize the interaction of the heart with the pulmonary and systemic circulations, engineers developed a minimal closed‐loop cardiovascular model, predicting cardiac output and pressures from 18 critical parameters. Despite the need for numerical solutions, it has potential to be used by clinical investigators to study the mechanical basis of altered hemodynamic status and by physiologists to study homeostatic mechanisms. To capture the complex interplay between hemodynamics and structural adaptation, we modified the standard minimal model to make structural parameters adapt to local hemodynamics. First, we linearized all pressure‐volume and pressure‐flow relationships. Then we made the simplifying assumption that end‐systolic pressure approximates mean arterial pressure. Finally, we assumed that each parameter adapts as a linear function of regional pressure. Yielding cardiac output and mean pressures, our model reproduces basic adaptive behaviors such as cardiomegaly in response to decreases in contractility and diastolic dysfunction in response to decreases in sensitivity of the adaptive process. More importantly, by judicious use of simplifying assumptions, all solutions can be expressed algebraically, which makes this fundamental model accessible to physiologists, clinical investigators, and students.