This work will extend two mathematical models for the dissolution and release process occurring in sustained release matrix tablets. The first of these models uses random walks on the weighted contact graph of a random dense packing of spheres of multiple diameters. The second model consists of a system of partial differential equations of reaction-diffusion type for the solved and unsolved excipient and drug, respectively. We will develop new computational tools, borrowing from theories of random walks on graphs, probability density estimation, and numerical partial differential equations. The theoretical models will be implemented in fast, robust code, and model parameters will be calibrated to actual data in close collaboration with researchers in the pharmaceutical sciences.
The team of investigators will develop mathematical and computational methods to predict the release kinetics of sustained release matrix tablets. These tablets are used to deliver an active drug and to gradually release it over an extended period of time. Sustained release tablets offer considerable advantages over immediate release tablets, namely maintaining more constant drug levels in the patient's body while minimizing the number of tablets that need to be taken each day. In previous work, jointly with collaborators in New Zealand (initiated by NSF grant DMS-0737537), we have proposed mathematical models for the dissolution and release process, predicting qualitatively excellent release curves for different compositions of the powder mixture. This research will streamline the design process of new and better pharmaceutical delivery devices and will introduce computational methods into a new field of science. All programs will be written in open source software and will be freely available online to interested researchers.
The overarching theme of this project has been to apply mathematical and computational methods to problems arising from pharmaceutical science. Clinicians prescribing a particular therapy need to answr the following key dosing questions: How much? How often? Where? Part of a successful therapy is a directed delivery of the drug to the site of its action, without causing toxicity elsewhere in the body. Pharmaceutical scientists have developed many delivery devices such as matrix tablets, liposomes and drug-loaded colloidal particles. Animal experiments to determine the drug release behavior of these devices are often difficult and fraught with ethical issues. Mathematical modeling has great potential to help understand the underlying delivery processes, to streamline treatment developments and testing, and to reduce the expenses involved in translating a treatment from "bench to bedside". Matrix tablets release a water-soluble drug over an extended period of time. We have proposed a flexible cellular automaton model for the dissolution and diffusion of a water-soluble drug from a water-insoluble polymer matrix. The paper is accompanied by a freely-available software tool that we hope will be used liberally by the pharmaceutical science community. A sample simulation is shown in the attached Figure. In contrast to other regions of the body, brain capillaries are impenetrable to many macromolecular chemicals, making them a part of the important blood-brain barrier. The low permeability poses a major challenge to the delivery of drugs, in particular those that are used to treat dementia, movement disorders and brain cancers. Use of colloidal carrier particles has opened promising new avenues for the treatment of diseases of the brain. The particles are loaded with drug and injected into the bloodstream where they freely circulate without releasing their contents. Triggered release of the drug is achieved by application of focused ultrasound. We have propose a differential equation model for delivery of a drug to the brain from a colloidal carrier. There is a remarkable amount of complexity due to the interactions of the free drug with proteins in the plasma and the brain tissue and nonlinear active transport mechanisms. Our allows optimization of the ultrasound-mediated drug delivery to avoid for example an overheating of the sensitive brain tissue. Over the course of this project we have also pursued mathematical topics, such as the existence and uniqueness of steady states for random dynamical systems, the search for jammed configurations of disks and spheres and several theoretical works in the area of structured population dynamics. We have also initiated other projects in bio-medical modeling, including an analysis of anti-coagulant treatments and protein folding and lipid interfaces. Many projects that were started during the grant are continuing. During the work we have advised one graduate student, multiple undergraduate students and participated in the supervision of graduate students at other institutions. These students are co-authors of some of the dozen peer-reviewed publications that have benefited from the support of this grant. The work has been highly collaborative and interdisciplinary, with contributions from pharmaceutical scientists, physicists, mathematicians and biologists in Canada, New Zealand, the United Kingdom and the United States.
