Efficient implementation of non-linear valve law and ventricular interaction dynamics in the minimal cardiac model

A minimal model of the cardio-vascular system (CVS) with ventricular interaction and inertial effects that accurately captures the physiological trends of a variety of disease states has been developed. However, the physiologically accurate open on pressure, close on flow valve law is computationally heavy to implement, reducing the model's potential clinical benefit. A significantly simpler representation of the valve law using Heaviside functions is derived and the ventricular interaction equations are reformulated to obtain a unique closed form analytical solution. The new formulation is tested and compared with the previous formulation for a healthy human and four clinically significant disease states: mitral and aortic stenosis, pulmonary embolism and septic shock. The new model formulation matches the previous model definition, differing by a mean model response error of no more than 0.2%. Computationally, it is 24 x faster than the previous method. More specifically, a short 20-beat simulation that took 102 s now requires 4.3 s, significantly improving the model's potential for practical use in a diagnostic and/or decision support role in the intensive care unit.