Measuring drug performance with a new optimization algorithm

Abstract

Multi stage Monte Carlo optimization (MSMCO) is a new computer mathematics solution technique that shows promise in solving the general nonlinear model. (It also works on many linear systems). This new multi-purpose multivariate solution technique allows one to fit response curves by methods other than least squares. This paper will present several hypothetical multivariate drug interaction studies, where the curve fits (modeling the response function) are done using least absolute deviation and mini max deviation with the MSMCO algorithm. (Mini max curve fitting finds the curve, from the family of functions under consideration, such that the maximum error is minimized). Least absolute deviation and mini max curve fitting help to overcome some of the problems inherent in the least squares technique, namely the squaring of the error term tends to overemphasize outlier points. Also least squares tends to use families of functions such that the resulting normal equations are linear in the betas making the curve fit easier to solve.MSMCO can free one from these problems and allow families of functions for fitting without concern about the form of the normal equations, because the curve fitting is done directly. A sample problem would be a medical condition that has shown signs of remission with six different drugs given to patients separately. Therefore a study is conducted to see if a combination of the six different drugs could provide even better results than just separately. So various amounts of the six drugs (the six independent variables) are given to the patients over time and a response variable (that measures the remission) is recorded. These dependent variable values are then regressed on the six independent variables (using the multivariate nonlinear or linear family of functions selected for the study) with the MSMCO algorithm. Least absolute deviation, mini max (or even least squares) could be used for fitting. The resulting model is then optimized using MSMCO (representing maximum remission) or set equal to a constant, hence an equation, (which is a target goal for remission measurement) and the resulting equation is solved with MSMCO.This technique also allows for multiple health goals for the patient to be pursued simultaneously. This could give rise to systems of nonlinear equations which are also solved with MSMCO.The MSMCO technique will be illustrated and generalized with the sample problems. MSMCO could give the medical researcher one more tool in trying to understand complex multivariate drug interactions.