Mathematical Modeling

Much of the investigation at Ordway Research Institute (ORI) examines intervention in pathologic processes with small molecules. Part of the activity is in finding novel targets to allow new drugs to be found. Another area of activity is in screening of diverse, drug-like libraries of compounds for these targets with the end of identifying lead compounds for further optimization. Yet another area of interest is in identifying doses and schedules of drugs or drug combinations that will achieve the desired goal of generating therapeutic effectiveness while engendering only a small probability of producing a drug concentration-related toxicity.

It is in this latter area that the Mathematical Modeling Core relates to many of the activities within ORI. The core spans the activities related to anti-infective therapeutics, oncolytic therapy and central nervous system intervention.

In order to fashion exposure-response relationships, it is necessary to relate some measure of drug exposure to the effect that a drug produces. The effect of the drug can be measured in an in vitro setting (Hollow Fiber Core), in an animal system (see, for example, Emerging Infections and Pharmacodynamics [PD] Laboratory), or in humans.

Mathematical Algorithms and Software: One strength of our mathematical modeling core is that our group has extensive experience with essentially all relevant nonlinear mixed-effects modeling algorithms, including parametric and nonparametric algorithms. Our most commonly applied software packages for estimation are NPAG and NPOD, S-ADAPT, ADAPT V, NONMEM®, and WinBUGS. For several years this core has implemented parallelized versions of NPOD/NPAG and S-ADAPT and greatly benefits from the computation power that almost linearly increases with the number of nodes on our 128-node computer cluster. Our core applies various software packages for simulation and population (stochastic) optimal design, including Berkeley Madonna, POPT / WinPOPT, ADAPT, and others.

Themes: This laboratory has been a leader in the development and validation of optimal sampling theory, so that system information-rich sampling times are identified for study. Once the data are obtained, population pharmacokinetic modeling techniques are applied in each of these settings, with Maximal A-posteriori Probability (MAP) Bayesian estimation employed to estimate drug exposure in specific instances. The estimated drug exposures are then linked to the effect by one of a number of different techniques. In some instances (e.g., prevention of resistance) the drug exposures and the effect are all estimated simultaneously by virtue of large models (e.g., 3-6 parallel inhomogeneous differential equations of 13-29 parameters) in a population analysis. In later projects, such models involved up to 49 differential equations with typically 10 to 30 model parameters that are estimated based on multiple dependent variables in a nonlinear mixed-effects modeling framework. In other instances, the Bayesian estimates are used as an independent variable in logistic regression analyses when the endpoint is dichotomous or in a Cox Proportional Hazards model when the endpoint is the time to an event. Other approaches are also possible.

Optimal Dosing of Patients: Once an exposure-response relationship is identified and a target is decided upon (e.g., a large enough drug exposure to suppress resistance), it is important to ascertain whether a specific drug dose will reliably obtain the desired goal when a large population of patients is considered. This laboratory pioneered the use of Monte Carlo simulation to evaluate drug doses for robustness of target attainment. This new technique has been used for dose assessment and has been adopted by the Clinical and Laboratory Standards Institute (CLSI; formerly called National Committee for Clinical Laboratory Standards - NCCLS) as the method of choice for determining susceptibility breakpoints.

Mechanism-based Modeling: One focus of our laboratory is to identify and quantify the effects of one or several biochemical pathways on the observed pharmacological and physiological changes during and after drug therapy. The mechanism-based models developed under this objective aim at 1) quantifying the time course of phenotypic and genotypic responses to anti-infective, antineoplastic, and other compounds, 2) identify and predict optimal combination therapy strategies and optimize the experimental design of prospective validation studies, and 3) providing a rational basis for the translation from in vitro and animal data to optimal dosing strategies in humans.

Translational Pharmacology: To support optimal translation from in vitro studies, for example from the hollow-fiber system, to animal and human clinical trials, our group relies on both 2-stage and 3-stage hierarchical nonlinear mixed-effects modeling techniques. While the latter are computationally more intensive, the 3-stage hierarchical approach allows one to borrow information from preliminary experiments with uncertainty. This concept is extremely useful for the transition from information rich in vitro studies to animal infection models and human clinical trials for which often only one observation per patient can be obtained.

Disease Progression Modeling: Another aim of our core is to dissect the time course of natural disease progression from the effects of drug therapy. For this task, simultaneously modeling the disease progression in a no-treatment control group, a placebo group, and the treated groups is very important. Nonlinear mixed effects modeling provides the most rational and powerful approach to this data analytical problem. This part of the mathematical modeling core also seeks to identify and quantify the time course of the effects of the immune system and its potential interactions with drugs, for example in anti-infective and antineoplastic therapy.

In these ways, the Mathematical Modeling Core takes the data emanating from the laboratories in ORI and optimizes the utility of this data to provide insight into the best use of drugs to help protect our patients.