Introduction

The phrase, mathematical analysis of biomedical data, at first glance seems impossibly ambitious. Does the author assert that inherently irregular biological systems can be described with any consistency via the rigid rules of mathematics? Add to this expression applied mathematics and a great deal of skepticism will likely fill the reader's mind. What is the intention of this work?

The answer is, in part, to provide a record of the author's 30-year career in academics, government, and private industry. Much of that career has involved the analysis of biological systems and data via mathematics. More than this, however, is the desire to provide the reader with a set of tools and examples that can be used as a basis for solving the problems he/she is facing. Some uncommon "tricks of the trade" and methodologies rarely broached by university instruction are provided.

Too often, books are written with only an academic audience in mind. This effort is aimed at working scientists and aspiring apprentices. It can be viewed as a combined textbook, reference work, handbook, and user's guide. The program presented here will be example driven. It would be disingenuous to say that the mathematics will not be emphasized (the author is, after all, a mathematician). Nevertheless, each section will be motivated by the underlying biology. Each example will contain the MATLAB® code required to produce a figure, result, and/or numerical table.

The book is guided by the idea that applied mathematical models are iterative. Develop a set of equations to describe a phenomenon, measure its effectiveness against data collected to measure the phenomenon, and then modify the model to improve its accuracy. The focus is on solving real examples by way of a mathematical method. Sophistication is not the primary goal. A symbiosis between the rigors of mathematical techniques and the unpredictable nature of biological systems is the point of emphasis.

The book reflects the formula that "mathematics + data + scientific computing = genuine insight into biological systems." The computing software of choice in this work is MATLAB. The reader can think of MATLAB as another important mathematical tool, akin to the Fourier transform. It (that is, MATLAB) helps transform data into mathematical forms and vice versa.

The presentation of concepts is as follows.

This introduction gives an overview of the book and ends with a representative example of the "mathematics + data + software" paradigm. The first chapter lists a set of guidelines and methods for obtaining, filtering, deciphering, and ultimately analyzing data. These techniques include data visualization, data transformations, data filtering/smoothing, data clustering (i.e., splitting one collection of samples into two or more subclasses), and data quality/data cleaning. In each case, a topic is introduced along with a data set. Mathematical methods used to examine the data are explained. Specific MATLAB programs, developed for use in an industrial setting, are applied to the data. The underlying assumption of this book is that, unlike most academic texts, data must be examined, verified, and/or filtered before a model is applied.

Following the discussion of data, the second chapter provides a view of the utility of differential equations as a modeling method on three distinct medical issues. The interaction of glucose and insulin levels within a human body is described by way of an elementary interaction model. This same approach is applied to the transition of HIV to AIDS within a patient. The HIV/AIDS example portends the susceptible-exposed-infected-recovered/removed models detailed in Chapter 3. The renowned polymerase chain reaction is presented as a coupled set of differential equations. In all of the cases above, the models are either applied to real clinical data or tested for their predictive value. MATLAB functions and code segments are included.

Chapter 3 focuses on mathematical epidemiology. The approach here is decidedly more involved than the examples provided in Chapter 2. The first section concerns a model, built on reported clinical data, that governs the transmission of HIV/AIDS through a population. It is, to the author's knowledge, the only such unified approach to the spread of a contagious disease. The second example within Chapter 3 concerns a mathematical method developed to predict the outbreak of a contagious disease based on simulated data (that mimic clinical data) of respiratory infections recorded at Boston Children's Hospital (BCH). Due to HIPAA (Health Insurance Portability and Accountability Act) laws, the author was unable to include the actual BCH data in the analysis. The simulated data, however, very closely resemble the clinical data. In each case, the need to estimate certain parameters crucial to the overall model reflecting real measurements and recorded populations is emphasized.

The fourth chapter concerns statistical pattern recognition methods used in the classification of human specimen samples for disease identification. Again, the application of mathematics to clinical data is the central focus of the exposition. All of the methods described are applied to data. Numerous figures and MATLAB code segments are included to aid the reader. The chapter ends with a presentation of support vector machines and their applications to the classification problem. Special software, developed at the University of Newcastle (Australia), is used with the permission of the design team to calculate the support vector boundaries. This cooperation itself is an example of how things are done in industry: collaborate with other experts; do not reinvent the wheel.

Chapter 5 is dedicated to a key component of biostatistics: hypothesis testing. The mathematical infrastructure is developed to produce the calculations (sample size, test statistic, hypothesis test, p-values) required by review agencies for submissions. This is an encyclopedic chapter listing the most important hypothesis tests and their variations including equivalence, non-inferiority, and superiority tests. As a point of reference, note that the author is on the organizing committee for the annual statistical issues workshop attended by industrial and review agency scientists, statisticians, and policy analysts. Thus, some of the key statistical matters as presented in these workshops are included in the chapter.

The final chapter examines clustered (that is, multi-reader/multi-category) data and the mathematical methods developed to render scientifically justified conclusions. The techniques include hypothesis testing and analysis of variance on clustered data.

0.1 How to Use This Book

Throughout these chapters, every effort has been made to present the material in as direct and clear a manner as possible. It is assumed that the reader is familiar with elementary differential equations, linear algebra, and statistics. An appendix includes a brief review of the mathematical underpinnings of these subjects. Further, it is hoped that the reader has some familiarity with MATLAB. In order to use the M-files written for this text, the reader must have access to MATLAB and the MATLAB Statistics Toolbox. A summary of most of the pertinent M-files and MAT-files which contain the data sets are provided in the Glossary of MATLAB Functions located at the end of the book. Also within this section is a recommendation for setting up a workspace to access the M- and MAT-files associated with this text. The reader is strongly urged to follow the recommendations contained therein. To gain access to the quadratic programming solver implemented in Chapter 4, the reader must contact Professor Brett Ninness of the University of Newcastle in New South Wales, Australia (http://sigpromu.org/quadprog/index.html). Expertise in any of the aforementioned areas, however, is not crucial to the use and understanding of this work.

The chapters are, by design, independent units. While methods developed in each chapter can be applied throughout the book (especially the chapter on data), each topic can be read without reliance on its predecessor. Whenever possible, it is recommended that the reader have MATLAB at the avail so that the examples can be traced along with the provided code.

0.2 Data and Solutions

With these ideas in mind, a few words about the source of all modeling efforts are presented: data. How does an industrial scientist deal with data? As a theoretical physicist once noted, if the data do not fit your model, change the data. Naturally, this comment was made with tongue firmly implanted in cheek. Here are some guidelines.

1. Data Normalization. When dealing with time-dependent data, it is often advisable to "center time" based upon the given interval. That is, if t ranges over the discrete values {t0, t1, ., tn}, then make calculations on times which start at 0 by subtracting off the initial time t0. More precisely, map t into t´ = t - t0: tk ? t´k = tk - t0. Similarly, some large measurements (say, population) can be given as 6,123,000, 6,730,000, etc. Rather than reporting such large "raw" numbers (which can cause overflow errors in...