Understanding the Human Machine - A Primer for Bioengineering - Max E. Valentinuzzi

Chapter 7

Feedback: The Need of Mathematical Models

Theoretical studies in biology and physiology may sound weird, especially to the young student, as they frequently involve appalling oversimplifications. However, mathematical physics also makes use of amazing oversimplifications. The works of Maxwell and Einstein once were considered purely speculative. Yet it was Maxwell’s work that made our present day electrical industry possible. It was Einstein’s speculations about four-dimensional space-time that made the atomic age possible. The thousands of ungenerous patents that followed them, mostly buried in oblivion and many giving healthy profits to their inventors and more comfortable life to the users, represent the practical arm. Biology as yet has not had its Maxwell or its Einstein … but they may come … perhaps sooner than expected … perhaps they are in front of these modest lines. Paraphrased from Nicholas Rashevsky (1964).

7.1. Introduction

The line of thought of this undergraduate introductory textbook started with definitions and general concepts of bioengineering/biomedical engineering (Chapter 1), it proceeded to describe and explain the sources of physiological signals (Chapter 2), entering thereafter into the signals themselves (Chapter 3). Chapter 4 dealt with the way a signal is picked up while Chapter 5 went into considerable details of the first step in signal processing, that is, amplification, acting as some kind of magnifying lens so that the interpreter (Chapter 6) is better equipped to read the signal making use of a number of algorithms mostly processed by computer aid. If one takes a look at the recording channel, we easily realize that it has given us the blueprint for the book. Now feedback is needed, as the stage to test our reading and interpretation, where mathematical models play a central role trying to improve the quantification process. Many

349

times the model must be disposed of or modified because its results did not fit acceptably the real situation, but this is a calculated risk and should not bring discouragement. Thus, this Chapter 7 signals the time to look back, review, revise, study, search, and experiment again, initiating an endless feedback loop to improve knowledge and understanding. The chapter speaks of the need of mathematical models but it does not contain mathematics, as the preceding one spoke of signal processing but did not enter into the subject matter proper because that is the task of future courses. The student now is getting an overall view of bioengineering, as anticipated in the introduction of the book.

Ever since the early 17th century, there has been a growing explosion of the physical sciences. Biology could not stay as a mere observer or as an outsider to such fascinating phenomenon and it started to try enlightening its own problems by adopting physical and mathematical perspectives, so definitely entering into the quantification stage. Descartes’ mechanical views were a beginning followed by several other scientists, even though earlier and more isolated attempts can be found in the history of science. Electricity and magnetism also influenced experimental and theoretical biology for long periods. During the 20th century, the work of Lotka represents another set of physical arguments and analogies to produce biologically significant understanding of several biological events. Differential equations and linear algebra, field and bifurcation theories, computation and game theories, differential geometry and group theory, are background knowledge applicable to modern theoretical biology (Lumsen, Brandts & Trainor, 1997), which should be considered a relatively new branch slowly but firmly setting its feet in the scientific scenario.

If there is a good theory, predictions can be made, and that means better knowledge of the subject in question. Thus, in experimental research the mere collection of facts is not sufficient. The Human Genome Project dramatically exemplifies the concept; in it, the amount of data has reached huge quantities while quantitative models and clear interpretation still stay quite behind and definitely not keeping pace, at least for the time being. Perhaps we should credit Nicholas Rashevsky, a Russian born physicist, as the founder and propeller of modern Mathematical Biology (also called Biomathematics), when in the 1930–1940 decades cre-

ated a strong working group, later a department (which was called “committee”), at the University of Chicago. Besides, he founded the Bulletin of Mathematical Biology, still published, where outstanding contributions saw the light. The amount of theoretical material collected in it over more than 65 years is, no doubt, a consultation and inspirational source.

One of the purposes of a theory is to determine certain quantities that cannot be measured directly, thus, it may provide an indirect measurement. Another aim of a theory is to explain a given set of facts answering question such as “how does it happen” and “why does it happen”, one step ahead of the simpler “what happens”, usually explained by the experimental act. A third and more provocative outcome of a theory is the suggestion of further experimental studies, perhaps even predicting the existence of new events or effects. However, no theory in biology or physiology or medicine has been developed to such an extent as the theory of celestial mechanics, but there is no reason to doubt, however, that predictions of this type will eventually be made. Advances in the last 20 years have shown possible avenues and anticipated significant discoveries. Nontheless, even though optimism well dresses up a good attitude, always remember that no theory can describe a set of phenomena in its entire complexity and no theory should be condemned because it explains certain phenomena only approximately. The requirement that a theory should be realistic in all its aspects is itself unrealistic. Such a requirement is against the very nature of theoretical thinking. When a theory has stood the acid test of numerous successful comparisons with experiments or observations, we can rely on it within a wide range, even though this range is always limited. Models born in the biomathematics cradle, old or new, whatever simple or complex, linear, non-linear or statistical, offer the background theoretical material for the signal processing employed by the interpreter (Rashevsky, 1938).

7.2. Linear versus Non-Linear Models

A system is termed linear when its output signal exactly reproduces the input signal, except for a magnifying or demagnifying constant factor. Such factor takes many times the form of a relationship called the trans-

fer function which characterizes the system. If the output signal is distorted, the system becomes non-linear. The system is, in any case, deterministic, because given an input there is always an output.

A system is classified as statistical when, after an input signal is applied, the output signal shows up with a given probability; thus, it may not appear. If it shows up, it may be either linear or non-linear, but it is not deterministic as above.

A system is called oscillatory when, given an input, the output is periodic with time; it may be either a sustained or a fading oscillation. Obviously, the system is non-linear, but it may be treated as linear; it is also deterministic. However, some systems may not even need an input to produce an oscillation; thus, they are true biological oscillators (as, for example, the heart). Structurally and functionally, periodicity is a frequent characteristic in biology and physiology; striated muscle, the Ranvier nodes sequence in mielinated nerves, the repetition of bone laminae, the periodic DNA organization, are clear examples of the former spatial type, while the latter shows itself with respect to time, as in the heart and respiration, in the cerebral cortex, sexual cycles, circadian rhythms, migrational seasons in birds and fish, and others.

The definitions given above are oversimplified in their wordings but briefly summarize concepts frequently handled in modeling and signal processing. The concepts of linearity and non-linearity are at times a bit elusive and may lead to long discussions, perhaps more philosophical in themselves than real. Contributions amount to thousands of papers if somewhat arbitrarily we take 1930 as the starting modern date. Textbooks are also many (perhaps 40 or 50), and some of them must be referred to because they contain fundamental and basic material.

Something about the available literature of models in biology: An early treatise was published in 1938 (Rashevsky, with updated editions in 1948 and 1960). It truly is an almost (or partially) successful attempt of building-up a systematic mathematical biology, similar in its structure and aims to mathematical physics. The book looks for physical interpretation of the biological phenomenon, sometimes with a grain of naivety when looked at from a 50years after perspective, but straight and highly stimulating, even in controversial subjects. A simplified, more accessible and practical version of biomedical mathematics appeared a few years later (Rashevsky, 1964); it is recommended for the beginner, for it has good examples in the cardiovascular, endocrine and nervous systems, including pharmacological applications. A posthumous, more advanced and rather speculative piece saw the light the

year of Rashevsky’s death (Rashevsky, 1972). As one of his disciples said, “he made a step towards unity of science by the construction of the conceptual structure which he called 'world set', of which physics, biology and sociology are subsets”. Defares and Sneddon (1961) produced a nice didactic book very much in the line of thought established by Rashevsky but of a more elementary level. Its applications section is also recommended for the beginner. Thereafter, we find a series of more engineering oriented books that can be considered as the biological control branch of engineering; they contain several excellent illustrating examples of different levels of complexity. Mostly, they are based on linear models with the Laplace Transform being the principal mathematical tool (Riggs, 1963, 1970; Grodins, 1963; Milhorn, 1966; Milsum, 1966). Curiously, Rashevsky’s contributions are not recognized in this literature, perhaps because he and his school were considered too theoretical. Bailey (1967) is less engineering-like, somewhat more mathematically biased and with a hint of clinical engineering; it is one of the very few referring to Rashevsky’s. Blesser (1969) produced a good integrative attempt where very basic definitions are introduced. Schwan (1969), Clynes and Milsum (1970), and Brown, Jacobs and Stark, (1971) inaugurated the edited multiauthored work covering selected lists of subjects and, thus, as any text of this kind, lacking a comprehensive nature although with outstanding chapters while Jones (1973), Gold (1977) and Finkelstein and Carson (1979), already well in the 70’s, went back to a more integrative bioengineering look. Edelstein–Keshet (1988) aimed at presenting instances of interaction between biology and mathematics following a line of thought that encompasses more biology at large than the way we have focused it herein; its contents must be considered as of advanced level. There are books dealing with models in specific areas, as compartmental analysis (Rescigno & Segre, 1966; Welch, Potchen & Welch, 1972; Jacquez, 1972), which represents an important concept and rather successful analytical technique. Transport phenomena in the cardiovascular system, —say, within blood vessels or across capillaries or among body compartments—, were well handled by Middleman (1972) and, thus, it becomes a good complement and expansion to deepen the pure descriptive physiological event. Mechanical systems of different types (Ghista, 1979) and the organs of equilibrium, as superb control devices (Valentinuzzi, 1980; Highstein, Cohen & Büttner–Ennever, 1996), are appealing and attractive for the advanced student. Other books have unique or more focalized approaches, as Lieberstein (1973) or Schneck (1990), and are in some respects of a higher level. This brief review does not pretend to be exhaustive since other texts perhaps could have been included.

Characterization of a linear behavior is usually carried out in any of four equivalent ways: impulse response, step response, frequency response, and transfer function. All require an input or forcing function to elicit an output or response (Glantz, 1979). As time advanced, the nonlinear approach gained more and more predominance, especially because computer power and versatility increased significantly and at tremendous pace making tractable problems that before could only be set as a challenge to solve in the future; well, we seem to already be witnessing the

future. Khoo (2000) is a relatively recent example that didactically proceeds from the simple traditional linear approach to the more complex and sophisticated non-linear case.

Is linearity bad or something that we should be ashamed of? Not at all, and do not shy away from it; quoting and paraphrasing from Grodins (1963): “Linear differential equations are usually backing up the linear systems. These equations constitute a minute fraction of the totality of differential equations and they are the only ones for which a complete analytical theory exists and for which general analytical solutions can be obtained. It is fortunate that many physical systems are sufficiently linear to permit their satisfactory description by linear differential equations.” Besides, most of the times the parameters involved are physically concentrated in space; the so-called lumped-constant systems. In other words, time is the only variable. Space coordinates are not considered and, if they are, partial differential equations must be used. This is the case of the distributed-parameter system. However, non-linearity is the rule and not the exception in biology. As first approximation, linear models work surprisingly well in many instances, but one can find other instances in which the non-linear features are critical for the proper understanding and description of the system. The principle of superposition is not applicable in non-linear systems and that appears as clear disadvantage (Khoo, 2000). The impulse response is no longer usable as local solutions cannot be extrapolated to global scale. Sometimes, nonlinearity can be treated by means of the piece-wise analysis, that is, any curve is always decomposable in sufficiently small straight lines; in that way, addition of several linear equations can account for the phenomenon.

Let us now suggest a series of illustrating tasks or exercises that the student should try to work out checking, thereafter, with the indicated bibliography. Some other simple examples have previously been developed in Chapter 2, as in the case of the kidney equations and the calculation of total and partial blood flow.

Student task 1: Read, study and discuss linear and non-linear models of lung mechanics as described by Khoo (2000) in its Chapter 9, pp 229–230. It briefly shows how in some instances the linear model fails or is not accurate enough.

Student task 2: The glucose concentration-time curve in blood during continuous intrave-

nous injection of glucose represents a clear case of clinical interest. Assuming that the concentration rate increases with the rate of infusion and decreases by regular elimination, a simple linear differential equation is obtained which leads to the typical exponential growth. See Defares & Sneddon, 1961, Chapter X, pp 527–9.

Student task 3: The form of the arterial pulse can also be modeled with a linear differential equation considering first the systolic phase and, thereafter, the diastolic run-off. See Defares & Sneddon, 1961, Chapter X, pp 529–533.

Student task 4: The three exercises above serve as introduction to the uptake of radioactive potassium by human erythrocytes, growth of an isolated population and growth of a nonisolated one. See also Defares & Sneddon, 1961, Chapter X, pp 533–540.

Student task 5: Blair’s Theory of nervous excitation. By now, the student is sharp enough to go into this a little more complicated example. The source is the same, that is, Defares & Sneddon, 1961, Chapter X, pp 593–597. The theory has been largely superseded by new knowledge, however, it represents an excellent step in the teaching-learning process. The motivated student, already full of enthusiasm, would not waste his/her time if the whole Chapter X in Defares & Sneddon’s book is taken up; if not, he/she may proceed to the next task.

Student task 6: The reflex regulation of pupil area by the iris is a typical example of a biological control system. For this subject see now Chapter 2, pp 24–49, in Milsum, 1966. Student task 7: About the semicircular canals. Higher animals possess sensitive transducers of translational and rotational skull motion, including Coriolis effects. They have profound actions on the postural control of the body. See Chapter 8, pp 186–191, in Milsum, 1966. It is a beautiful example of second-order linear lumped model. The student should also find out about Coriolis acceleration and the meaning of second order systems. Experiment on yourself the Coriolis effect: Stand up, start rotating around your longitudinal axis (feet to head), as when dancing. After a few turns and having reached a little speed, tilt your head to one side. What do you feel? Be careful, because the dizziness may even cause your fall, especially if you are not fit and well trained. After recovery, explain. Who was Coriolis? For a more advanced description, see Valentinuzzi (1980).

7.3. Characteristics of a System Model

Linearity (or non-linearity) could have been listed as another characteristic of a system, perhaps the most important of them all. However, we preferred to introduce it as a classification criterion. Once the system is placed in the proper group, we have to determine the other different characteristics in order to better predict its performance. Let us define and briefly discuss them:

− Stability

This concept is more or less intrinsic to us. Anyone knows that a normal person stands and walks with stability while a one year old toddler is unstable in its standing and walking trials and, very likely, will eventually fall down; thus, it is unstable. However, as engineers, we need to do better than this. A system may show oscillations that fade out either as underdamped, overdamped or critically damped responses after stimulation with an impulse. All three represent stable cases. A fourth situation may give off an undamped or sustained oscillation termed conditionally or marginally stable. Still another situation may occur when the impulse stimulus elicits a growing oscillation, that is, the system output never returns to its original operating point prior to stimulation. The latter is clearly unstable. Hence, after the preceding examples, we can say that a stable dynamic system is one that will respond to a bounded input with a bounded response (Khoo, 2000). All these cases are based on a wellknown mathematical model and are unequivocally described in those terms.

− Identifiability

Once again, the common language understanding of the word offers a useful hint: there is need to know the identity of what we are studying, as the police searches for data to clearly establish who the suspect is; and again, something more quantitative is required by the bioengineer.

First type of identification: Say that the equations modeling a given system are known based on physical or physiological principles and that, after a specific input function, the output response is predicted by means of those equations. This type of analysis is called the forward or prediction problem (Khoo, 2000). Predictions tell whether the model provides an accurate description of the process under study, especially if the results are compared against actual experimental records.

Second type of identification: Greater challenge is posed by the inverse problem, which arises when, having a model and the measured output, the input is not observable and must be deduced. Often it requires the application of deconvolution (opposite of convolution).

Third type of identification: It is the most difficult and elusive and usually referred to directly as the identifiability of the system. This is the case of systems about which knowledge is very limited or when only

some knowledge is available. There is no established strategy for these situations, not easy to handle, and experience plays a significant role; specialists speak often of black box or gray box approaches, according to the actual level of knowledge of the system one has. Identification is subject matter of advanced courses.

− Optimality

Before fully accepting a model after preliminary tests, its response to a given input must be compared to the response of the actual physiological system to the same input. Thus, a criterion is to be chosen in order to evaluate the goodness of fit between the two time series (measured and calculated or real and theoretical). In other words, by minimizing the deviations between both series we are actually optimizing parameters.

Study subject: Find the precise definitions of damped, undamped, overdamped, underdamped, and critically damped oscillations. Recall the mechanical system formed by a spring, a mass and a friction.

Study subject: Find at least one example in the literature in which the model parameters have a one-to-one correspondence with the underlying physiological entities. Black box and gray box approaches may not show a similar correspondence. Find out what convolution and deconvolution are, as mathematical operations.

Student task 8: Neuronal dynamics and the Hodgkin–Huxley Model. See Chapter 9, pp 257–260, in Khoo, 2000.

7.4. Partial Final Remarks

The chapter is just an outline of what the modeling stage is or may be. Lots of questions must and should be flooding the young mind, feeling perhaps some frustration and even getting angry with the author while asking “why not more, why not getting deep into the subject”. The answer is simple, it takes time, you are just at the beginning and this is the primer to start. Besides, there are excellent texts fully devoted to mathematical models in biology and their computer implementation.