1. What Is Mathematical Biology and How Useful Is It? Avner Friedman Tiffany Nguyen and Dr. Dana Clahane

2. Recent Progress in the Bioscience That Led to the Necessity of Biological Mathematics Completion of the Human Genome Project —Identify all the approximately 20,000-25,000 genes in human DNA, —Determine the sequences of the 3 billion chemical base pairs that make up human DNA Discovery of non-coding genes Advancement of technology: medical imaging, nanoscale bioengineering, gene expression arrays — Collect data from usage of these technologies — Find trends in the data and analyse them to discover how things work Completion of the Human Genome Project —Identify all the approximately 20,000-25,000 genes in human DNA, —Determine the sequences of the 3 billion chemical base pairs that make up human DNA Discovery of non-coding genes Advancement of technology: medical imaging, nanoscale bioengineering, gene expression arrays — Collect data from usage of these technologies — Find trends in the data and analyse them to discover how things work Genes: segments of deoxyribonucleic acid (DNA) that carry instructions for how to construct other cells and contain traits from the parent Four base pairs of DNA can combine in an infinite amount of orders (factorials can be applied) to determine the coding of proteins Non-coding genes: genes that do not code for proteins Medical imaging: non-invasive techniques using x-rays, radiowaves, soundwaves, or magnetic fields to record images of internal organs Nanoscale bioengineering: manufacturing artificial tissues, organs, etc. to replace damaged parts - on a nanoscale Gene expression arrays: the process and order through which genes convert information to carry out the construction of proteins

3. What is Mathematical Biology? Proposing a mechanism derived from collected data for cellular processes Determining the relationship among the variables of a process Developing a model (equation) that correlates with the experimental data Using the trends in the model to predict future outcomes Proposing a mechanism derived from collected data for cellular processes Determining the relationship among the variables of a process Developing a model (equation) that correlates with the experimental data Using the trends in the model to predict future outcomes

4. Applications of Mathematical Biology Ischemic Wounds (wounds with a shortage of blood flow, which impairs healing) Mathematical biology is used to find ways to heal ischemic wounds — Find relationships between variables of the healing process: types of cells and tissues involved, chemicals secreted that facilitate closure of the wound, density of the tissue Use of partial differential equations where the open wound’s surface is the unknown free boundary Ischemic Wounds (wounds with a shortage of blood flow, which impairs healing) Mathematical biology is used to find ways to heal ischemic wounds — Find relationships between variables of the healing process: types of cells and tissues involved, chemicals secreted that facilitate closure of the wound, density of the tissue Use of partial differential equations where the open wound’s surface is the unknown free boundary Free boundary problem: solving for an unknown function u and an unknown domain Ω in a partial differential equation Partial differential equation: equation involving the derivative of more than one variable, with respect to one variable

5. Mathematics of Ischemic Wounds {0 ≤ r ≤ R(t)} is the open wound region {R(t) ≤ r ≤ R(0)} is the partially healed region {R(0) ≤ r ≤ L} is the normal healthy tissue Small incisions of size  are made at r=L, separated by distances of   is a measure of ischemia where  near 1 means extreme ischemia u is a boundary condition, u s is a solution of ∆u=f u=u s before the incisions changed into a boundary condition {0 ≤ r ≤ R(t)} is the open wound region {R(t) ≤ r ≤ R(0)} is the partially healed region {R(0) ≤ r ≤ L} is the normal healthy tissue Small incisions of size  are made at r=L, separated by distances of   is a measure of ischemia where  near 1 means extreme ischemia u is a boundary condition, u s is a solution of ∆u=f u=u s before the incisions changed into a boundary condition (1-  )(u-u s )+  (  u/  r)=0 at r=L

6. Mathematics of Surgical Tissue Transfer Model consists of: Variables of venuous blood cell volume fractions  c (x),  b (x), and  a (x) Oxygen concentration transport/diffusion equations Partial differential equations describing the transport and diffusion (high to low concentration) of heat, mass, momentum, etc. Arterial and venuous blood flow conservation laws Laws that govern the mass of blood-flow such that the mass remains constant Model consists of: Variables of venuous blood cell volume fractions  c (x),  b (x), and  a (x) Oxygen concentration transport/diffusion equations Partial differential equations describing the transport and diffusion (high to low concentration) of heat, mass, momentum, etc. Arterial and venuous blood flow conservation laws Laws that govern the mass of blood-flow such that the mass remains constant Determine the success rate of tissue transfers depending on diameter of perforating vessel (vessel at the operating site that delivers blood to the transferred tissue)