The Application of Adaptive Step-Size Control in the Numerical Simulation of Calcium Release in Vascular Smooth Muscle

Title of Thesis : The Application of Adaptive Step-Size Control in the Numerical Simulation of Calcium Release in Vascular Smooth Muscle. Michael J. Sydor, Master of Science in Biomedical Engineering, 1991 Thesis directed by : Dr. F. P. J. Diecke, Chairman of Physiology An algorithm for the adaptive control of numerical integration step-size is developed and implemented for the simulation of a three compartment model for Vascular Smooth Muscle. The three compartment model accounts for the simultaneous diffusion of Ca, 45Ca, EGTA, CaEGTA, and 45Ca-EGTA, and is an extension of a two compartment model by Diecke for the simultaneous diffusion of Ca, EGTA, and Ca-EGTA. The addition of the third compartment is to account for the presence of the Sarcoplasmic Reticulum which stores the calcium needed for contraction and is the primary regulator of calcium in the VSM cell. The SR has been implicated as the slow component in calcium release, as measured by Stout and Diecke in saponin skinned VSM. The compartmental model is developed from mass-balance equations and is solved numerically with a Runge-Kutta-Gill algorithm. Step-size is controlled with an adaptive algorithm which adjusts the integration interval (step-size) for the transient and steady-state phases of the simulation. The implementation of the various programs is designed to accommodate automated execution and analysis of a high volume of simulation runs. Some introduction into the methodology of modeling and simulation, as well as the complex physiology of the SR is discussed and the complete process of modeling, simulation, and analysis is illustrated for a simple model of a two-compartment leaky tank. A more comprehensive introduction into numerical integration is included to provide sufficient background for the development of the adaptive algorithm. The adaptive algorithm allows a complex simulation to be executed in one-fifth the time required for constant step (non-adaptive) numerical integration without incurring significant error. This reduction in the amount of computer time required permits more aggressive protocols for the determination of parameters and model responses by allowing more simulation runs to be processed in the course of a study. The simulation of calcium release from VSM revealed that the response of the system is not a multiple of the increase in rate but is instead related via a linear function representative of the buffering capacity of the model. Results from a similar two-compartment model suggest that the EGTA buffer system has a significant impact of the perceived rates of release. This Thesis is dedicated to the memory of my friend Richard "Butter" Waterworth.