Biomedical engineering programs have exploded in popularity and number over the past 20 years. In many programs, the fundamentals of engineering science are taught from textbooks borrowed from other, more traditional, engineering fields: statics, transport phenomena, circuits. Other courses in the biomedical engineering curriculum are so multidisciplinary (think of tissue engineering, Introduction to BME) that this approach does not apply; fortunately, excellent new textbooks have recently emerged on these topics. On the surface, numerical and statistical methods would seem to fall into this first category, and likely explains why biomedical engineers have not yet contributed textbooks on this subject. I mean … math is math, right? Well, not exactly.

There exist some unique aspects of biomedical engineering relevant to numerical analysis. Graduate research in biomedical engineering is more often hypothesis driven, compared to research in other engineering disciplines. Similarly, biomedical engineers in industry design, test, and produce medical devices, instruments, and drugs, and thus must concern themselves with human clinical trials and gaining approval from regulatory agencies such as the US Food & Drug Administration. As a result, statistics and hypothesis testing play a bigger role in biomedical engineering and must be taught at the curricular level. This increased emphasis on statistical analysis is reflected in special “program criteria” established for biomedical engineering degree programs by the Accreditation Board for Engineering and Technology (ABET) in the USA.

Introduction

The job of a biomedical engineer often involves the task of formulating and solving mathematical equations that define, for example, the design criteria of biomedical equipment or a prosthetic organ or physiological/pathological processes occurring in the human body. Mathematics and engineering are inextricably linked. The types of equations that one may come across in various fields of engineering vary widely, but can be broadly categorized as: linear equations in one variable, linear equations with respect to multiple variables, nonlinear equations in one or more variables, linear and nonlinear ordinary differential equations, higher order differential equations of nth order, and integral equations. Not all mathematical equations are amenable to an analytical solution, i.e. a solution that gives an exact answer either as a number or as some function of the variables that define the problem. For example, the analytical solution for

1. (1) x2 + 2x + 1 = 0 is x = ±1, and

2. (2) dy/dx + 3x = 5, with initial conditions x = 0, y = 0, is y = 5x − 3x2/2.

Sometimes the analytical solution to a system of equations may be exceedingly difficult and time-consuming to obtain, or once obtained may be too complicated to provide insight.

Introduction

The primary goal of the field of bioinformatics is to examine the biologically important information that is stored, used, and transferred by living things, and how this information acts to control the chemical environment within living organisms. This work has led to technological successes such as the rapid development of potent HIV-1 proteinase inhibitors, the development of new hybrid seeds or genetic variations for improved agriculture, and even to new understanding of pre-historical patterns of human migration. Before discussing useful algorithms for bioinformatic analysis, we must first cover some preliminary concepts that will allow us to speak a common language.

Deoxyribonucleic acid (DNA) is the genetic material that is passed down from parent to offspring. DNA is stored in the nuclei of cells and forms a complete set of instructions for the growth, development, and functioning of a living organism. As a macromolecule, DNA can contain a vast amount of information through a specific sequence of bases: guanine (G), adenine (A), thymine (T), and cytosine (C). Each base is attached to a phosphate group and a deoxyribose sugar to form a nucleotide unit. The four different nucleotides are then strung into long polynucleotide chains, which comprise genes that are thousands of bases long, and ultimately into chromosomes.

MATLAB is a technical computing software that consists of programming commands, functions, an editor/debugger, and plotting tools for performing math, modeling, and data analysis. Toolboxes that provide access to a specialized set of MATLAB functions are available as add-on applications to the MATLAB software. A Toolbox addresses a particular field of applied science or engineering, such as optimization, statistics, signal processing, neural networks, image processing, simulation, bioinformatics, parallel computing, partial differential equations, and several others.

The MATLAB environment includes customizable desktop tools for managing files and programming variables. You can minimize, rearrange, and resize the windows that appear on the MATLAB desktop. The Command Window is where you can enter commands sequentially at the MATLAB prompt “≫.” After each command is entered, MATLAB processes the instructions and displays the results immediately following the executed statement. A semicolon placed at the end of the command will suppress MATLAB-generated text output (but not graphical output).

The Command History window is a desktop tool that lists previously run commands. If you double click on any of these statements, it will be displayed in the Command Window and executed. The Current Directory browser is another desktop tool that allows you to view directories, open MATLAB files, and perform basic file operations. The Workspace browser displays the name, data type, and size of all variables that are defined in the base workspace.

Introduction

As previously discussed in Chapter 2, a system of linear equations can be expressed as Ax = b, where A and b are not functions of x. On the other hand, a system of equations written as Ax = b, in which A and/or bis a function of x, contains one or more nonlinear equations. A polynomial in x of order ≥ 2 has nonlinear dependency on the variable x and is an example of a nonlinear equation. Equations involving trigonometric and/or transcedental functions such as sin(x), cos(x), log(x), ex, and coth(x) are also nonlinear in x. The solution x of a nonlinear equation f(x) = 0 is termed a root of the equation or a zero of the function f(x). Application of numerical techniques to obtain the roots of nonlinear equations is the focus of this chapter.

Nonlinear equations that are not amenable to an analytical solution may be solved using iterative numerical schemes. A system of consistent linear equations will always yield a solution. In fact, linear equations that are consistent but not underdetermined will always have a unique solution, regardless of the technique used to solve the system of linear equations. On the other hand, a numerical method used to solve a nonlinear equation may not necessarily converge to a solution. Often, certain conditions must be met before a particular numerical technique can successfully produce a solution.

Introduction

Mathematical modeling is used to analyze and quantify physical, biological, and technological processes. You may ask, “What is modeling and what purpose does a model serve in scientific research or in industrial design?” Experimental investigation generates a set of data that describes the nature of a process. The quantitative trend in the data, which describes system behavior as a function of one or more variables, can be fitted to a mathematical function that is believed to represent adequately the relationship between the observations and the independent variables. Fitting a model to data involves determining the “best-fit” values of the parameters of the model.

A mathematical model can be of two types.

1. (1) A model formulated from physical laws is called a mechanistic model. A model of this type, if formulated correctly, provides insight into the nature of the process. The model parameters can represent physical, chemical, biological, or economic properties of the process.

2. (2) An empirical model, on the other hand, is not derived from natural laws. It is constructed to match the shape or form of the data, but it does not explain the origin of the trend. Therefore, the parameters of an empirical model do not usually correspond to physical properties of the system.

Mechanistic models of scientific processes generate new insights into the mechanisms by which the processes occur and proceed in nature or in a man-made system. A mathematical model is almost always an incomplete or imperfect description of the system.

Introduction

Variability abounds in our environment, such as in daily weather patterns, the number of accidents occurring on a certain highway per week, the number of patients in a hospital, the response of a group of patients to a particular type of drug, or the birth/death rate in any town in a given year. The “observed” occurrences in nature or in our man-made environment can be quantitatively described using “statistics” such as mean value, standard deviation, range, percentiles, and median. In the fields of biology and medicine, it is very difficult, and often impossible, to obtain data from every individual that possesses the characteristic of interest that we wish to observe. Say, for instance, we want to assess the success rate of a coronary drug-eluting stent. To do this, we must define our population of interest – patients who underwent coronary stent implantation. It is an almost impossible task to track down and obtain the medical records of every individual in the world who has been treated with a drug-eluting stent, not to mention the challenges faced in obtaining consent from patients for collection and use of their personal data. Therefore we narrow our search method and choose only a part of the population of interest (called a sample), such as all Medicare patients of age >65 that received a coronary stent from May 2008 to April 2009.

Introduction

Often, a researcher, engineer, or clinician will formulate a statement or a belief about a certain method, treatment, or behavior with respect to one or more populations of people, animals, or objects. For example, a bioengineer might be interested in determining if a new batch of drug-coated stents has a drug coat thickness that is different from a previous batch of stents. The engineer makes several measurements on a small number of stents chosen at random. A substantial deviation in drug coating thickness from the standard range of acceptable values indicates that the process conditions have unexpectedly changed. The stents from the new batch may have either a thicker or a thinner drug coating compared to those of the previous batches, depending on the direction of deviation in coat thickness. Or, it is possible that the drug coat thickness widely fluctuates in the new batches of stents beyond the range specified by design criteria. The engineer formulates a hypothesis based on her preliminary survey. She will need to perform statistical tests to confirm her hypothesis.

Suppose a biomaterials researcher is testing a group of new promising materials that enhance the durability or lifetime of a contact lens. The researcher poses a number of hypotheses regarding the performance qualities of different lens materials. She must use statistical techniques to test her hypotheses and draw appropriate conclusions.