Applied Mathematics for the Analysis of Biomedical Data: Models, Methods, and MATLAB – An interview with author Peter J. Costa

This month Wiley is proud to publish Applied Mathematics for the Analysis of Biomedical Data: Models, Methods, and MATLAB®, which features a practical approach to the analysis of biomedical data via mathematical methods and provides a MATLAB® toolbox for the collection, visualization, and evaluation of experimental and real-life data.

The book presents a practical approach to the task that biological scientists face when analyzing data.The primary focus is on the application of mathematical models and scientific computing methods to provide insight into the behavior of biological systems. The author draws upon his experience in academia, industry, and government–sponsored research as well as his expertise in MATLAB to produce a suite of computer programs with applications in epidemiology, machine learning, and biostatistics. These models are derived from real–world data and concerns. Among the topics included are the spread of infectious disease (HIV/AIDS) through a population, statistical pattern recognition methods to determine the presence of disease in a diagnostic sample, and the fundamentals of hypothesis testing.

In addition, the author uses his professional experiences to present unique case studies whose analyses provide detailed insights into biological systems and the problems inherent in their examination. The book contains a well-developed and tested set of MATLAB functions that act as a general toolbox for practitioners of quantitative biology and biostatistics. This combination of MATLAB functions and practical tips amplifies the book’s technical merit and value to industry professionals.

Through numerous examples and sample code blocks, the book provides readers with illustrations of MATLAB programming. Moreover, the associated toolbox permits readers to engage in the process of data analysis without needing to delve deeply into the mathematical theory. This gives an accessible view of the material for readers with varied backgrounds. As a result, the book provides a streamlined framework for the development of mathematical models, algorithms, and the corresponding computer code.

In addition, the book features:

• Real–world computational procedures that can be readily applied to similar problems without the need for keen mathematical acumen

• Clear delineation of topics to accelerate access to data analysis

• Access to a book companion website containing the MATLAB toolbox created for this book, as well as a Solutions Manual with solutions to selected exercises

Applied Mathematics for the Analysis of Biomedical Data: Models, Methods, and MATLAB® is an excellent textbook for students in mathematics, biostatistics, the life and social sciences, and quantitative, computational, and mathematical biology. This book is also an ideal referencefor industrial scientists, biostatisticians, product development scientists, and practitioners who use mathematical models of biological systems in biomedical research, medical device development, and pharmaceutical submissions.

Statistics Views talks to author Dr Peter J. Costa about the writing process. Dr Costa is a Senior Applied Mathematician at Hologic Incorporated in Marlborough, MA. Dr. Costa is the co-creator of MATLAB’s Symbolic Math Toolbox. He has developed mathematical models for the spread of HIV, the outbreak of AIDS, the transmission of an infectious respiratory disease throughout a population, and the diagnosis of cervical cancer. His research interests include scientific computing and mathematical biology. He received a PhD in Applied Mathematics from the University of Massachusetts at Amherst.


1. Congratulations on the upcoming publication of Applied Mathematics for the Analysis of Biomedical Data. How did the writing process begin?

A friend of mine was diagnosed with stage I prostate cancer. He wanted to know, given his PSA history, how quickly the level might accelerate. I tried putting together a series of mathematical models which provided some indication of how the PSA level might change over a 3 – 6 month period. While not entirely accurate, the models were good enough to permit him to wait for 2 years before undergoing active treatment.

Once I had written down a “report” containing these models, I thought about many of the other ideas I had worked on over the years and began putting the book together.

2. What were your main objectives during the writing process?

My primary objective was to record some of the [what I construe as] valuable techniques I have learned over the years. My first foray into “Mathematical Biology” occurred in the early 1990s (1994 – 95 as I recall) when I was the Director of the Center for Applied Mathematics at the University of St. Thomas. Many of my students were hoping to attend medical or nursing school. They were eager to see whether some of the mathematical techniques (extended Kalman–Bucy filter, linear algebra, differential equations) which I had introduced to them could be applied to biology systems.

Then a local insurance company asked about the spread of HIV/AIDS through a specific population. My students and I combined these interests to construct a more general model for the transmission of HIV through a contained population (such as a college campus). We were able to produce a very realistic model which made sensible predictions.

After receiving tenure, I left my teaching position and chose to pursue an opportunity to apply mathematical techniques to help detect cervical cancer. These two “avenues” of investigation led to two chapters of the book.

3. Please could you explain the main focus of the book and the issues involved?

The primary focus is to demonstrate to the reader the value of “industrial mathematics.” I define this as the application of a mathematical model to clinical data while using scientific computing/software methods to measure effectiveness. Each chapter is rooted in the idea that “data + mathematics + software = good science.” Consequently, each chapter develops the mathematics required to model or interpret data and the corresponding software to incorporate one into the other.

4. If there is one piece of information or advice that you would want your reader to take away and remember after reading your book, what would that be?

It is that mathematical modelling, especially applied to biological systems, is iterative. Try something. See how well it reflects reality (namely, the data). Modify the model and try again. Use scientific computing tools to measure the effectiveness of the model.

5. Who should read the book and why?

Anyone interested in mathematical biology and/or the analysis of biological data. There is a lengthy chapter on hypothesis testing aimed at US review agency submissions. Therefore, statisticians who work for medical device or pharmaceutical companies will find this portion of the book of value. Students keen to know how mathematics is used outside of a university setting should find the contents of the book of interest. Finally, industrial or academic research scientists who need to know how to process or model biological data will find elements in the book to be useful.

6. Why is this book of particular interest now?

I hope it will prove of value to the surge of interest in mathematical biology, hypothesis testing on more sophisticated metrics (such as coefficient of variation), and data analysis. The combination of mathematical models, data, and the software to process and analyze such data should appeal to applied scientists and researchers. As mentioned above, students who seek non–academic careers should find the book a good introduction to industrial work and research.

7. Were there areas of the book that you found more challenging to write, and if so, why?

The section on the use of the extended Kalman–Bucy filter to determine the outbreak of an infectious respiratory disease was particularly challenging. The incorporation of (real) population census data, seroconversion rates, and (simulated) infection data into a series of MATLAB programs (M–file) which correctly implemented the filter required exceptional attention to the details of the model. Also, the development of an M–file which efficiently and properly computed the exact probability density function of the difference in coefficients of variation was difficult due to the changing domain of definition. Finally, the real–time polymerase chain reaction model was tricky to implement because of the appearance of the Lambert–W function. Fortunately, Dr. Cleve Moler (Chief Scientific Officer of the MathWorks and inventor of MATLAB) provided an M–file which calculated values of the Lambert–W function accurately and efficiently.

8. What is it about the areas of applied mathematics and biomedical research that fascinate you?

The interplay of data, mathematics, and software. Why should inherently random phenomenon (as all biological systems are) be so effectively modelled and analyzed via mathematical methods? This continues to surprise and delight me. It is quite rewarding to produce a mathematical model which helps explain (for example) how a disease spreads through a population or how mathematical methods can be used to detect cervical cancer. Mathematics and mathematicians are often utilized for understanding problems arising from applied physics or electrical engineering. While these areas are quite interesting, the ability to bring mathematical methods to help detect or even prevent disease is a deeply enriching personal experience.

9. What will be your next book-length undertaking?

I am not certain I have the stamina to undertake another book! My friend and collaborator Bill Satzer and I have toyed with the idea of writing the next “great partial differential equations” book. This would be an enormous undertaking. Please ask me about this in 2 – 3 years once I have worked through all that might entail.

10. You are currently Senior Applied Mathematician at Hologic Incorporated in Marlborough, MA and the co-creator of MATLAB’s Symbolic Math Toolbox. Please could you tell us more about your educational background and what inspired you to pursue your career in mathematics?

I can only do two things with any facility: Mathematics and playing guitar. Most people who have heard me play guitar tell me that I should stick with mathematics.

I received a BS (bachelor of science) degree in mathematics from SUNY @ Binghamton in 1978. Then I went to graduate school at a large Midwestern (U.S.) university where I was not successful. The faculty there did not think I was a sufficiently strong student to enter their PhD program. They were not necessarily wrong. I did earn a master’s degree at the aforementioned university and I thought it best to study statistics elsewhere. That “elsewhere” turned out to be the University of Massachusetts (Amherst) where, to my great fortune, I met [the late] Professor Melvyn S. Berger. After taking a one–year sequence in applied mathematics from Professor Berger, he asked me to study for the doctorate under his supervision. I explained my previous experience at the Midwestern university and that I likely lacked the intellectual skill to work for such a distinguished mathematician (as was Professor Berger). He would hear none of that and then guided me through a difficult problem (in nonlinear partial differential equations) for my dissertation.

I have never forgotten his kindness and faith in me. Sadly, Professor Berger passed away about 10 years ago. Over the past 7 years, my wife and I have sponsored the Distinguished Lecture in Applied Mathematics in Professor Berger’s memory at the University of Massachusetts. The lecture occurs every autumn and we gladly make the trip back to Amherst to attend the talk. This is a real treat for me as I am able to see what is the current state of academic research. Also, I have asked the coordinators of the lecture to encourage the speaker to address the graduate students in the audience. In this way, they receive the kind of education I did when I studied there many years ago.

11. How did the creation of the Symbolic Math Toolbox come about?

This was mostly Cleve Moler’s idea (see answer 7 above for more information about Dr. Moler). He “glued” Maple into MATLAB so that the latter would have a computer algebra system. I came to the MathWorks as an “expert” in a rival symbolic algebra system and updated the approach and functionality of the Symbolic Math Toolbox.

12. Your work and research has led you to work in a number of different areas of medical research from HIV to cervical cancer. Is there a particular project or work that you are most proud of?

This book is something which gives me great satisfaction and a (I hope contained) sense of pride. The HIV model is something I always look at with a sense of accomplishment; primarily because my students worked so diligently on it. I very much miss interacting with bright young people and hope that my efforts to provide a distinctly different approach to teaching helped them in their careers. The classification methods I have helped develop to detect cervical cancer are still a work in progress. Once we refine them so that the technology becomes less expensive (and more autonomous) then the health of women could be improved world–wide. This would be a significant accomplishment and something I will continue to work towards.