Document Type


Date of Award

Fall 1-31-2013

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)


Mathematical Sciences

First Advisor

Linda Jane Cummings

Second Advisor

Lou Kondic

Third Advisor

Michael Siegel

Fourth Advisor

Shahriar Afkhami

Fifth Advisor

Treena Livingston Arinzeh


In this dissertation we develop a comprehensive model to simulate a tissue engineering experiment. The experiment takes place in a bioreactor in which a cell seeded porous scaffold is placed, and the scaffold experiences a perfused flow of a nutrient-rich culture medium. The goal of the model is to assist experimentalists in evaluation of different parameter scenarios as the time needed to simulate an experiment is significantly less than the time needed for the experiment itself. We provide the full two-dimensional model development, as well as investigation into possible variations of specific model choices, and we demonstrate the robustness and versatility of the model.

Simulation results are presented with different initial cell seeding scenarios which increase in complexity with each simulation. We next model the effect of printing a growth factor onto the scaffold in an attempt to direct cell motility and enhance proliferation via a process known as haptotaxis. While a quantitative representation of these phenomena requires more experimental data than are yet available, qualitative agreement with preliminary experimental studies is obtained, and appears promising. The ultimate goal of such modeling is to ascertain initial conditions (growth factor distribution, initial cell seeding, etc.) that will lead to a final desired outcome.

A simplified 2D mathematical model for tissue growth within a cyclically-loaded tissue engineering scaffold is then analyzed. Such cyclic loading has the potential to improve yield and functionality of tissue such as bone and cartilage when grown on a scaffold within a perfusion bioreactor. The cyclic compression affects the flow of the perfused nutrient, leading to flow properties that are inherently unsteady, though periodic, on a timescale short compared with that of tissue proliferation. A two-timescale analysis based on these two well-separated timescales is exploited to derive a closed model for the tissue growth on the long timescale. Some sample numerical results are given for the final model, and the comparison with the unloaded case is discussed.

Finally, we simulate to hypothetical extensions to the basic model. We first test the hypothesis of a death rate which varies as a function of the local fluid flow and compare the results to the original model. The second test is the introduction of a channel through the center of the porous scaffold thought to aid in nutrient delivery to the cells in the interior of the scaffold. The last two simulations are presented to illustrate the ability that the model has to incorporate many different supplemental experimental situations, whether they have yet been experimentally considered or not.

Included in

Mathematics Commons



To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.