Application of parametric sensitivity analysis to calcium handling in cardiac myocytes

Heart failure disease (HF) kills 220,000 people in America every year. HF is characterized by the ineffective handling of calcium ions by the mechanisms present in cardiac myocytes resulting in decreased contractile force in the heart. This inefficiency leads to necrosis of the myocardium, cardiac hypertrophy, and eventual death of the patient. The mechanisms affected include various ion channels and active transport mechanisms. Numerous technologies and techniques have been developed recently to apply genetic and pharmacological therapies to these cells. Unfortunately, there are a large number of target parameters that can be manipulated in the myocytes. Mathematical models have been developed that accurately reproduce the calcium mechanisms. These models are reproducible and rapidly evaluated on desktop computers using current mathematical software.

Sensitivity analysis is useful in determining the best targets for manipulation by external sources in the hopes of restoring the proper calcium ion handling by ion channels and calcium transporters. Sensitivity analysis uses matrix algebra and calculus to determine the normalized response of a control variable to a change in a parameter. More specifically, this technique calculates the responses of all control variables to changes in all of the parameters individually. This generates a matrix of values that can easily be analyzed. This method will save valuable time and resources that would have been spent on performing multiple experiments or simulations to determine the best targets for manipulation. The cytosolic calcium concentration is targeted as it is directly related to the contractile force of the heart. Reduction of complex models is also possible using the results of sensitivity analysis. Rota et al. describe a method of normalizing fluxes and sensitivities against their greatest magnitude as a characteristic for suggesting unimportant fluxes and parameters.

Two models are examined: the Tang-Othmer (T-0) model and the Winslow, Rice, and Jafri (WRJ) model. Both of these models seek to describe the handling of calcium ions within cardiac myocytes; however the level of their complexities differs greatly. The T-O model contains much fewer state variables, parameters, and fluxes than the WRJ model. The greater incorporation of states and the related parameters and fluxes leads the WRJ model to yield a much greater complexity than that presented with the T-O model. This complexity is magnified upon formation of the sensitivity matrices for these models, where the T-O model yields a fifty-five element matrix and the WRJ model yields 1,881 elements in its sensitivity matrix. The T-O model is used to develop the application method whereas the WRJ model is used to determine the best targets for manipulation and to illustrate the efficacy of model reduction.

Sensitivity analysis upon the T-O model yields the sodium-calcium exchanger parameters and the sarcolemmal leak coefficient as the optimal targets for manipulation to restore proper cytosolic calcium concentration. The best targets for genetic or pharmacological manipulation according to the WRJ model are the Na+ and Ca2+ background currents, the maximum current for the sodium-potassium pump, and the half saturation constants for the sodium-potassium pump. These parameters are not present in the T-O model and have greater sensitivity magnitudes than those that carry over into the WRJ model.

Model reduction by the method of Rota et al. reveals that the Tang-Othmer model is irreducible in its present state. The WRJ model was found to have no reducibility with regards to the number of fluxes in the model. However the integration of some sensitivities was unattainable, and some of these parameters may be found to be removable upon further analysis of the model once these sensitivities are obtained. Further integration methods will be attempted, such as the use of a hard-coded implicit integrator.

Sensitivity analysis also revealed a crossover phenomenon in both models. This phenomenon describes the change in sign of a sensitivity of a state variable to a parameter during the course of a heartbeat. When this occurs the desired effect of manipulating the parameter yields the opposite effect upon the state variable. This phenomenon may generate interesting side effects that require further study.

The results of sensitivity analysis provide future direction for physical experiments. These experiments will both confirm the calculated sensitivities, and investigate their application to failing myocytes. The crossover phenomenon will provide interesting avenues of research into the side-effects of parameter manipulation based on the magnitude and location in the cycle of the crossover. Model reduction may play a key role in simplifying models for easier computation and analysis.