1
Pressure Waves for Diagnostics and Therapy

1.1 Introduction

To understand the role of mathematics in an engineering discipline, we may talk about the capability of an engineer. Inspired by the laws of physics, an engineer uses mathematics as a tool to convert nature's resources to a product. This definition may be considered out of date now. A better definition may take the form: inspired by the laws of physics, using mathematics as a translator, to convert nature's resources to a product or to study a phenomenon (or phenomena) or a criterion (criteria). This latter definition takes engineers to go far into space and deep into the human body microstructure in order to investigate, analyze, and apply this knowledge to innovations for the betterment of the humankind. The process modeling could be done in the form of mathematical modeling or/and computer simulation.

1.2 Significance of Biological System Modeling

Various physical and biological systems can be modeled in the form of mathematical equations. The question often raised is why we put so much emphasis on mathematical modeling? The answer to this is the fact that mathematical modeling has so many advantages including but are not limited to:

1. It is a tool that helps to convert basic laws of nature (physics, chemistry, biology, etc.) to industrial application.
2. It converts real systems of interest into models which can be analyzed and tested on a piece of paper or a screen.
3. Nowadays, computers and new technologies have made math the most valuable tool to convert complicated real-life system into a virtual environment on the screen.

Of course, talking about math here implies all types of mathematical approaches including but not limited to differential mathematics, computational, statistical, and others. Statistical models applied to experimental data, as an example, may help in understanding the results for the purpose of assurance or development of a useful future formulation.

The two "Engineering" definitions stated earlier may sound the same; however, the second one is more general, and it represents the current methodology for engineering research. No more the engineer only designs and develops. Engineers go beyond the scope of developing a product to help humankind, by studying and investigating various systems and behaviors whether natural or synthetic. A biomedical engineer is not only responsible to build a medical device, a biomedical tool, or an artificial body part. Many biomedical engineers focus on investigating biological behaviors or responses in an optimum goal of serving the human body needs. Thus, this aligns with the second definition of "Engineers" stated earlier. The question which is normally raised is "what do we do with this modeling?" The response to this is summarized in Figure 1.1. While changing a realistic system, whether extremely large or at a nanoscale, to a system which can be represented virtually on a screen and reducing the process of experimental trial and error to a minimum time and cost without jeopardizing integrity are typical processes of mathematical modeling. Modeling could be used to simulate a system or a process to understand and assess, to change system parameters, and to investigate performance before building the device while reducing the time and costs of experimental investigations of the complicated very large or very small systems to investigate them. With this in mind, the main outcomes of the modeling process are

Figure 1.1 Advantages of mathematical modeling.

1. To turn unreachable systems to accessible ones.
2. To allow studying the real system without affecting or damaging it.
3. To keep a system model ready on the screen to optimize, design, analyze, etc.
4. To change the parameters and see how the system behaves.
5. To create specifications for the system to be ready for manufacturing.

Accessing unreachable systems and converting them to reachable has a very wide range of applications, particularly in the biomedical field. A study of an organ or a cell behavior is difficult if not impossible within the living human body. Thus, having a model on the screen helps to introduce many variations without affecting the human body as a living system itself. Further, if a model for a device to support the human body such as an artificial kidney is developed and needs optimization. A mathematical model for this device is extremely helpful in the process of designing and optimizing without jeopardizing the health of the person. Multiple variations of parameters and its effects could be studied affecting the human body itself. Obviously, out of the last design iteration and optimization process, new specifications could be generated for optimal operation and performance. However, the question which is always raised is how reliable modeling is? The answer is summarized by the following facts:

• Pace of technological change drives industries to implement modeling.
• Aircraft, underwater, space, and other industries have produced good and reliable models for the development of their products.
• Most research groups have developed reliable models in their fields of research and have worked with confidence to move toward innovation.
• Experience plays an important role in reliability.
• If a model is not perfect, its value at least is to show the trend of the behavior or variations.

Over the years and with the development of the computational capabilities, mathematical modeling has demonstrated advancement and success in all modern technologies, from space engineering to nano cells within the human body. All systems generated from modeling have worked reliably and with improved performance. Of course, like any other discipline, experience plays an important role in developing the best model for a particular device or a process. In general, if the model is not perfect, it will show the general trend of variation for the process or operation.

Without going into details of modeling, in this chapter, we will focus on the development of a single equation, as an example of modeling of one of the biological systems such as the respiratory or cardiovascular system and for the purpose of developing a medical diagnostic tool or therapy device/method or to investigate the behavior and measure performance. This simple equation is the wave equation. The nature of several systems within the human body conveys fluid for the purpose of living, particularly, the cardiovascular and the respiratory systems. In spite of the large differences between the functions of these two systems, wavepropagation through them can be tuned into a diagnostic or a therapy approach. The significance of this equation will be clarified in Chapters 2, 3 and 5, but at this stage we will derive the equation and outline its significance for biomedical diagnostics and therapies.

1.3 Wave Equation

In this book, we focus on how mathematics can be applied as a modeling tool to convert basic laws of physics, biology, and chemistry sciences to a process, protocol, method, or/and a medical device, which can deliver an outcome for diagnostics or treatment for some critical lung, heart, or arterial diseases. Our focus will be on the simple wave equation, its basic principle, application, and how it can be used to investigate various biological processes and how this powerful equation could be used to develop a medical device for diagnostic purposes as well as various tools for treatments. There are many ways of deriving this equation from a simple string to a rod (bar) and to the flow in pipes. In this chapter, we focus on equations defining the flow in the pipes as it resembles an artery or an airway passage in the context of the human body.

Various transmission lines in the respiratory and cardiovascular systems may be treated as branching compliant tubes conveying fluid. Blood flow in the arterial system and airflow in the respiratory system induce forces and stresses in the arterial and respiratory walls, due to complex fluid-structure interactions. These forces and stresses play an important role in the onset and progression of many acquired and congenital cardiovascular diseases, such as arterial atherosclerosis and aneurysms, and airway ailments such as obstruction in obstructive sleep apnea and narrowing in asthma.

The blood flow in the arterial system and the airflow in the respiratory system diseases have a common physical problem which is change in the fluid flow passage as "restriction" or "enlargement" of fluid flow passage. Atherosclerosis, for example, involves the accumulation of plaque in the intima of the arterial wall, which reduces arterial lumen and increases local arterial stiffness. There is substantial evidence on the localization of these plaque deposits at sites with hemodynamic conditions commonly characterized by low wall shear stress (Caro et al. 1971; Taylor 1959). Aneurysms, on the other hand, involve the degradation of local arterial wall tissues, resulting in lowering of local arterial stiffness and enlargement of local vessel cross section. If, in extreme cases, the wall stress due to the transient fluid-structure interactions exceeds the strength limit of the dilated artery wall, it causes vessel rupture leading to death from internal hemorrhage which has been reported to be between 80 and 90% of the cases (Scotti et al. 2005). Aneurysms are common in locations with secondary...