Medical and Biological Physics 3rd ed. / Медична і біологічна фізика, вид. 3 / Чалий О. В. (за ред.)

MEDICAL AND BIOLOGICAL PHYSICS

MEDICAL AND BIOLOGICAL PHYSICS Edited by Prof. A. Chalyi

Ministry of Health Care of Ukraine O. O. Bogomolets National Medical University

MEDICAL AND BIOLOGICAL PHYSICS Edited by Prof. Alexander V. Chalyi

Recommended by the Ministry of Health Care of Ukraine as a textbook for the students of higher medical establishments of the IV accreditation level

Third edition

Vinnytsia Nova Knyha 2017

UDC 577.3(075.8) BBC 28.071я73 M42 Recommended by the Ministry of Health Care of Ukraine as a textbook for the students of higher medical establishments of the IV accreditation level (letter № 08.01-47/2735 by 25.12.2009) Authors: A. V. Chalyi – Head of the Department of Medical and Biological Physics, Corresponding member of the Academy of Pedagogical Sciences of Ukraine, Dr. Sci. (Phys. & Math.), Professor; Ya. V. Tsekhmister – Deputy Rector of the National Medical University, Head of the Department of Biomedical Engineering, Dr. Sci. (Pedagogy), Cand. Sci. (Phys. & Math.), Professor; B. T. Agapov – Associate Professor, Dr. Sci. (Biology); N. V. Stuchynska – Associate Professor, Dr. Sci. (Pedagogy.); A. V. Melenevska – Associate Professor, Cand. Sci. (Biology); M. I. Murashko – Associate Professor, Cand. Sci. (Tech.); N. F. Radchenko – Associate Professor, Cand. Sci. (Chem.); I. F. Margolych – Associate Professor, Cand. Sci. (Phys. & Math.); V. V. Pashchenko – Associate Professor, Cand. Sci. (Pedagogy); D. V. Lukomskyi – Assistant Professor; E. V. Zaitseva – Assistant Professor, Cand. Sci. (Phys. & Math.); E. N. Chaika – Assistant Professor, Cand. Sci. (Phys. & Math.); N. L. Grytsenko – Assistant Professor.

Reviewers: L. A. Bulavin – Head of the Department of Molecular Physics (Taras Shevchenko National University of Kyiv), Academician of the National Academy of Sciences of Ukraine, Dr. Sci. (Phys. & Math.), Professor.

N. S. Myroshnichenko – Head of the Department of Biophysics (Taras Shevchenko National University of Kyiv), Dr. Sci. (Biology), Professor.

V. I. Dotsenko – Professor of the Department of Medical Biology (National Stomatological Medical Academy), Dr. Sci. (Phys. & Math.), Professor.

L. Ya. Avrakhova – Head of the Department of Foreign Languages (National Medical University), Associate Professor. М42

Medical and Biological Physics : textbook for students of higher medical institutions of the IV accreditation level / Chalyi A. V., Tsekhmister Ya. V., Agapov B. T. [et al.] ; Edited by A. Chalyi. – 3rd ed. – Vinnytsia : Nova Knyha, 2017. – 480 pp.

ISBN 978-966-382-639-4 The book is aimed at elucidating the most important aspects of the medical and biological physics in accordance with the program asserted by the Ministry of Health Care of Ukraine, and written for the students of higher medical institutions as well as for teachers, scientific researchers and all those readers interested in modern problems of the medical and biological physics.

UDC 577.3(075.8) BBC 28.071я73

ISBN 978-966-382-639-4

© Authors, 2013 © Authors, 2017 © Nova Knyha, 2017

CONTENTS PREFACE ....................................................................................................9 MODULUS 1. Mathematical processing of medical and biological data .12 LECTURE SECTION 1..............................................................................12 CHAPTER 1.1. Mathematical methods of computing medical and biological information (principles of calculus) ...........12 1.1.1. Elements of differential calculus ...........................................12 1.1.2. Elements of integral calculus ................................................22 1.1.3. Elements of theory of differential equations. ........................28 CHAPTER 1.2. Fundamentals of the theory of probability and mathematical statistics .............................................32 1.2.1. Fundamentals of the theory of probability. ...........................32 1.2.2. Sampling method.

Finding of characteristics of distribution...............................48 1.2.3. Elements of theory of correlation.

Correlation and statistical dependence ................................50 PRACTICAL SECTION 1 ..........................................................................52 PRACTICAL WORK 1.1. Elements of differential calculation..........52 PRACTICAL WORK 1.2. Elements of integral calculation ..............66 PRACTICAL WORK 1.3. Elements of the theory of differential equations ........................................................75 PRACTICAL WORK 1.4. Elements of the probability theory...........80 PRACTICAL WORK 1.5. Elements of mathematical statistics ........86

MODULUS 2. Principles of biological physics..........................................90 LECTURE SECTION 2..............................................................................90 CHAPTER 2.1. Essential principles of biomechanics...............................90 2.1.1. Mechanical properties of biological tissues ..........................92 2.1.2. Deformation of biological tissue ...........................................98 CHAPTER 2.2. Fluidity of viscous fluids in biological systems ..............100 2.2.1. Fluid viscosity .....................................................................101 2.2.2. Blood viscosity ....................................................................103 2.2.3. Viscoelastic properties of biological tissues .......................104 2.2.4. Basic equations of fluid flow ...............................................107

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CONTENTS

2.2.5. Criteria of mechanical affinity of flowing fluids ...................115 2.2.6. Pulse waves ........................................................................117 CHAPTER 2.3. Mechanical oscillations ..................................................120 2.3.1. Harmonic oscillations and their principle parameters.........120 2.3.2. Damped oscillations and aperiodic motion ........................124 2.3.3. Forced oscillations ..............................................................127 2.3.4. Resonance phenomenon and auto-oscillations .................128 2.3.5. Addition of harmonic oscillations ........................................130 CHAPTER 2.4. Mechanical waves .........................................................133 2.4.1. Wave equation. Longitudinal and transverse waves ..........134 2.4.2. Wave energy flow. Umov vector .........................................135 CHAPTER 2.5. Acoustics. Elements of hearing physics.

Fundamentals of audiometry .........................................136 2.5.1. Sound nature, its main characteristics (objective and subjective) ...................................................137 2.5.2. Weber-Fechner law .............................................................141 2.5.3. Ultrasound ..........................................................................144 2.5.4. Infrasound ...........................................................................146 CHAPTER 2.6. Structure and properties of biologic membranes ..........146 2.6.1. Passive transport of uncharged molecules.........................152 2.6.2. Passive transport of ions ....................................................155 2.6.3. Active transport...................................................................159 CHAPTER 2.7. Biological potentials.......................................................161 2.7.1. Nernst equilibrium membrane potential..............................163 2.7.2. Diffusion potential ...............................................................164 2.7.3. Donnan’s potential. Donnan’s equilibrium ..........................166 2.7.4. Stationary potential of Goldman-Hodgkin-Katz..................169 2.7.5. Action potential. Mechanism of generation and propagation of nerve impulse ......................................172 CHAPTER 2.8. Elements of dental material science ..............................178 PRACTICAL SECTION 2 ........................................................................191 LABORATORY WORK 2.1. Measurement of hearing threshold by audiometric method ......................................................................191 LABORATORY WORK 2.2. Study of elastic properties of biological tissues ............................................................198 LABORATORY WORK 2.3. Determination of dependence of liquid's surface tension coefficient on temperature and surface-active substances ......................204

CONTENTS

5

LABORATORY WORK 2.4. Measurement of coefficient of viscosity ..........................................................................210 LABORATORY WORK 2.5. Measurement of concentration potential using compensation method ...............................216 COMPUTER SECTION 2........................................................................222 COMPUTER PROGRAM 1. Haemodynamics ...............................222 COMPUTER PROGRAM 2. Structure and transport properties of membranes ...................................................229 COMPUTER PROGRAM 3. Rest and action electrical potentials of membranes ....................................................244

MODULUS 3. Principles of medical physics ..........................................258 LECTURE SECTION 3............................................................................259 CHAPTER 3.1. Electrostatics .................................................................259 3.1.1. Major characteristics of electric field ..................................259 3.1.2. Electric dipole .....................................................................262 3.1.3. Dielectrics, dielectric polarization .......................................264 3.1.4. Dielectric properties of biological tissues ...........................268 3.1.5. Piezoelectric effect..............................................................270 CHAPTER 3.2. Continuous (direct) current. Conductivity of biological tissue. Alternating current and impedance of biological tissues .............................271 3.2.1. Characteristics of electric current .......................................271 3.2.2. Conductivity of biological tissues and fluids.......................272 3.2.3. Action of electric current on living organism ......................275 3.2.4. Equation of electric oscillations ..........................................277 3.2.5. Forced electric oscillations, alternating current ..................279 3.2.6. Total resistance of alternating current circuit (impedance). Ohm’s law for alternating current circuit .......282 3.2.7. Impedance of biological tissues .........................................284 3.2.8. Electromagnetic waves. Bias current .................................287 3.2.9. Maxwell’s equations ........................................................289 3.2.10. Plane electromagnetic waves. Umov-Poynting vector ....291 3.2.11. Electromagnetic spectrum ...............................................293 CHAPTER 3.3. Magnetic field ................................................................296 3.3.1. Magnetic field in vacuum and its characteristics ................296

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CONTENTS

3.3.2. Biot-Savart-Laplace’s law ...................................................298 3.3.3. Action of magnetic field on movable electric charge.

Ampere force, Lorentz force ...............................................299 3.3.4. Magnetic properties of substances ....................................302 3.3.5. Magnetic properties of biological tissues, physical bases of magnetobiology. ...................................306 CHAPTER 3.4. Medical electronic equipment........................................308 3.4.1. General information of medical electronic equipment (MEE) ................................................308 3.4.2. Classification of medical electronic equipment ..................309 3.4.3. MEE performance specification ..........................................310 CHAPTER 3.5. Physycal principles of optical microscopy, refractometry and polarimetry .......................................313 3.5.1. Geometrical optics ...........................................................313 3.5.2. Ideal centered optical system ..........................................313 3.5.3. Optical microscopy ..........................................................316 3.5.4. Light polarization..............................................................319 3.5.5. Light polarization at reflection and refraction ..................320 3.5.6. Polarization at double refraction in crystal.......................321 3.5.7. Light polarization at passing through an absorbing anisotropic substance ....................................323 3.5.8. Plane-of-polarization rotation by an optically active substance ..............................................................324 3.5.9. Interaction of light with substance. Light absorption.......326 3.5.10. Light scattering ................................................................329 CHAPTER 3.6. Physical foundations of thermography, laws of heat radiation .....................................................331 3.6.1. Heat (temperature) radiation ...............................................331 3.6.2. Kirchhoff’s law of spectral radiation ...................................333 3.6.3. Planck’s radiation law .........................................................334 3.6.4. Stefan-Boltzmann’s law ......................................................335 3.6.5. Wien displacement law .......................................................336 3.6.6. Infrared radiation .................................................................338 3.6.7. Ultraviolet radiation .............................................................339 CHAPTER 3.7. Notations of quantum mechanics ..................................339 3.7.1. The place of quantum mechanics in scientific system of body motion .......................................................339 3.7.2. De Broglie’s hypothesis ......................................................341 3.7.3. Heisenberg’s uncertainty relation .......................................344

CONTENTS

7

3.7.4. Fundamental equation of quantum mechanics – Schrödinger equation .........................................................345 3.7.5. Schrödinger equation for hydrogen atom ...........................347 CHAPTER 3.8. Energy radiation and absorption by atoms and molecules ................................................349 3.8.1. Atomic spectrum..............................................................349 3.8.2. Molecular spectra ............................................................351 3.8.3. Luminescence ..................................................................355 3.8.4. Luminescence types ........................................................355 3.8.5. Photoluminescence. Stokes law ......................................356 3.8.6. Luminescence mechanisms ............................................358 3.8.7. Induced radiation .............................................................360 3.8.8. Equilibrium and inverse dependence...............................360 3.8.9. Structure and principle of laser’s operation.....................362 3.8.10. Electronic paramagnetic resonance, nuclear magnetic resonance and their medico-biological application .........363 3.8.11. Method of electron paramagnetic resonance ..................364 3.8.12. Method of nuclear magnetic resonance ..........................368 CHAPTER 3.9. The nature and generation of X-ray beams ...................372 3.9.1. Deceleration X-ray radiation ............................................374 3.9.2. Characteristic X-ray radiation, its nature .........................375 3.9.3. Radioactivity, its properties .............................................377 3.9.4. Principle law of radioactive decay, half-life period, activity .....................................................380 3.9.5. The rules of shift, spectra specific feature at the time of radioactive decay ......................................384 3.9.6. Exposure dose, its rate, units ..........................................387 3.9.7. Absorbed dose, its rate, units ..........................................389 3.9.8. Equivalent dose, its rate, units .........................................391 3.9.9. Dosimeters .......................................................................393 3.9.10. Primary physical mechanisms of interaction of X-ray radiation with substance ....................................395 3.9.11. Primary mechanisms of radioactive radiation and particle flux effect on substance ...............................399 PRACTICAL SECTION 3 ........................................................................403 LABORATORY WORK 3.1. Operation of electrocardiograph .......403 LABORATORY WORK 3.2. Operation of rheograph .....................411 LABORATORY WORK 3.3. Study of electrical impedance of biological tissues ............................................................419

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CONTENTS

PRACTICAL LESSON. Interaction of electromagnetic field with biological tissues .................................................425 LABORATORY WORK 3.4. Physiotherapeutic equipment operation ...........................................................435 LABORATORY WORK 3.5. Study of microscope and measurement of microobjects .....................................446 LABORATORY WORK 3.6. Study of solution concentration by refractometric method ............................450 LABORATORY WORK 3.7. Study of solution concentration by polarimetric method ................................454 LABORATORY WORK 3.8. Study of laser operation ....................460 LABORATORY WORK 3.9. Measurement of linear damping coefficient of gamma radiation ............................464 INDEX OF SUBJECTS ...........................................................................471 INDEX OF NAMES .................................................................................474

PREFACE

9

PREFACE The problems of evolution of living and non-living nature and the whole Universe still attract great interest and agitate inquisitive human mind. It becomes possible to look into micro- and macrocosms due to a materialistic scientific approach (we will not touch on theology, which may exist since the number of faithfuls is much bigger than the number of atheists on the Earth).

Really, on one hand, modern telescopes give us an opportunity to study very distant regions of the Universe, which keeps expanding with the speed rising at the periphery due to Hubble’s law. These distant regions of the Universe are now at the distance of three billion light years, i.e.

2.8·1025 m, and move with the speed 90,000 km/s which is about one third of the light velocity. It is interesting to mention that the speed of expansion of far galaxies may become almost equal to the light velocity but the intensity of their radiation will approach zero at that time, due to modern theory of the Universe evolution. These very far regions of the Universe will be at the distance of about 10 billion light years and will have the speed of expansion about 300,000 km/s. Just these regions determine the boundary (the radius) of the Universe as per our perception. Thus, this Universe is finite, due to a materialistic scientific approach.

On the other hand, the most powerful elementary particle accelerators give us an opportunity to study the processes taking place inside the atomic nucleus, i.e. at the distances less than 10–15 m. Nowadays we know even about the internal structure of so-called elementary particles (electrons, protons, neutrons, etc), namely about quarks and quanta of quark field – gluons.

The understanding of processes of cell origin (their typical sizes are about 10–6 m) came from the study of evolution processes in the living nature at the distances intermediate to the farthest (more than 1025 m) and the smallest (less than 10–15 m). The cells emerged due to creation of plasmatic membranes creation. It is the process concerned with so-called selfassembly of lipid molecules in the aquatic environment, which is absolutely known, i.e. its probability equals 1.

The biophysics of membrane processes (and also molecular biophysics, biophysics of compound systems) gained much success in close association with other exact natural sciences. One must use both precise

10

PREFACE

experimental methods and modern theoretical concepts to understand deeply complex processes in living systems occurring on a molecular level.

It is combination of qualitative and quantitative levels of understanding of medical and biological processes in cells and in the whole organism, undoubtedly, the direction of the evolution of medicine in XXI century.

The new requirements to the content, methodology and organization of teaching of various disciplines in the higher medical school emerged at the modern stage of higher medical education evolution. This fully relates to the teaching of the discipline “Medical and Biological Physics”. Many achievements of medicine are combined with progress in physics, biology, computer science, informatics, and medical instrument engineering. That is why the students of higher medical educational institutions need to obtain special knowledge in medical and biological physics.

Unfortunately, until now there has not been any textbook for Englishspeaking students studying in Ukrainian higher medical educational institutions which would contain the main information on the discipline of “Medical and Biological Physics” and given in a simple and compact form.

This suggested textbook is likely to correct this fault. It is based on the lectures and practical courses that have been taught to students of O. O. Bogomolets National Medical University for many years. It is this textbook which gives an opportunity to master the discipline “Medical and Biological Physics”, which is both difficult and very important for the medical education of physicians working in XXI century.

Each of three moduli contains not only theoretical lectures but also practical or laboratory works as well as tasks for self-study. The authors tried to combine the “triad” (lectures, laboratory works and problems) in one textbook with quite a small, in our opinion, volume.

The materials of well structured manuals prepared by teachers of the Department of Medical and Biological Physics of O. O. Bogomolets National Medical University were used in this textbook. These materials were approved while teaching the discipline “Medical and Biological Physics” for the students of the first year of study.

The authors are greatly indebted to all the pedagogical staff that have worked and are still working at the Department of Medical and Biological Physics of O. O. Bogomolets National Medical University for many years. Our special thanks to Associate Professors, Candidates of Sciences A. V. Govorukha and A. I. Yegorenkov.

We are sincerely indebted to reviewers of our textbook: Head of the Department of Molecular Physics (Taras Shevchenko National University of

PREFACE

11

Kyiv), Academician of the National Academy of Sciences of Ukraine, Dr. Sci. (Phys. & Math.), Professor L. A. Bulavin; Head of the Department of Biophysics (Taras Shevchenko National University of Kyiv), Dr. Sci. (Biology), Professor N. S. Myroshnichenko; Professor of the Deparment of Physiology (National Stomatological Medical Academy), Dr. Sci. (Phys. & Math.), Professor V. I. Dotsenko; Head of the Department of Foreign Languages (National Medical University), Associate Professor L. Ya. Avrakhova.

The authors will be grateful for any critical remarks and hope that the textbook “Medical and Biological Physics” will be useful not only for students of higher medical establishments, but also for students, post-graduate students, teachers of other higher educational institutions and all those readers interested in ideas and successes of modern medical and biological physics.

Prof. A. V. Chalyi

MODULUS 1 MATHEMATICAL PROCESSING OF MEDICAL AND BIOLOGICAL DATA “In next years I could not forgive myself that lack of tenacity which kept me from getting over mathematics to gain an understanding of its great basic principles. People who mastered these principles have the extra sense body comparing to mere mortals”.

Charles Darwin

Lecture Section 1 CHAPTER 1.1. Mathematical methods of computing medical and biological information (principles of calculus)

1.1.1. Elements of differential calculus Derivative and differential of function Derivative of function y = f (x) in the point x0 (symbolically f '(x0)) is the limit of ratio of an increment of function y to an increment of argument x provided that the increment of argument is infinitesimal (x  0), i.e.

Chapter 1.1. MATHEMATICAL METHODS OF COMPUTING MEDICAL AND BIOLOGICAL INFORMATION

13

(1.1) Derivative is denominated as: f '(x), y'х, y x ,

df ( x) dy , . dx dx

If the function y = f(x) has a derivative in some permanent point, then the function in this point is infinitesimal.

An inverse assertion is not true, i.e. not every function infinitesimal in the point х has a derivative in this point.

If the function f(x) is differentiated in each point of some interval, then we may speak about differentiation of function on this interval. If, in addition, the derivative f '(x) is infinitesimal, then we may say that the function is infinitesimally differentiated on this interval.

The procedure of finding a derivative is called differentiation of function.

If the function y = f(x) is differentiated, then its increment y may be represented in the form: y = y'x +  (х, x) x.

Product of  (х, x) x is an infinitesimal of the higher order infinitesimal relative to x, since α ( x, x)x (1.2) lim  lim α ( x, x)  0. x  0 x  0 x

differential of this function to the differential of an argument.

LECTURE SECTION 1

Thus, an increment of differentiated function consists of two parts (addends), first of which tends to zero “more slowly” as compared with the other and is linear relative to x. This part of increment is called principal.

The principal part of function increment y, i.e. is equal to the product of the derivative y' by the increment of argument x is called differential of function and is denoted as dу: dy = y'x.

If у = х, then dy = dx = x' x, but x' = 1. From this dх = х, i.e. differential of an independent variable equals to its increment. Finally we have: differential of function equals to the product of function differential by differential of an independent variable dy = y' dx.

(1.3) dy From (1.3) we have y   , i.e. the function derivative equals to the ratio of dx

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MATHEMATICAL PROCESSING OF MEDICAL AND BIOLOGICAL DATA

MODULUS 1

It follows that at small x the approximate equality is true y  dy, (1.4) or f(x + x)  f(x) + f '(x) dx.

(1.5) These equalities are used for approximate calculus and in the theory of errors; they allow reducing calculus of function increment to calculus of derivative (differential), that, in general, is a simpler problem.

Geometrical explanation of derivative and differential. Geometrical explanation of derivative is closely connected with the notion of tangent. The tangent to function graph y = f(x) in the point М(x0,y0) is a boundary position of secant MN (Fig. 1.1) at unbounded approximation of the point N to the point M. It is simply to y notice from the Fig. 1.1 that angular coefficient of the secant is equal to: tg α  . x If the point N tends to M, then the secant MN occupies position for which α  φ at х  0. Thus, the value of derivative y' = f '(x0) in the point х0 determines angular coefficient of tangent drawn to the function graph y = f (x) in the point М(x0, y0), i.e. lim tg α  tg ϕ or y'(х0) = tg . x  0

So, this is the point for geometric explanation of a derivative. The equality of tangent to the function graph у = f(х) in the given point М(х0, у0) has the form:

LECTURE SECTION 1

y  y0  f   x0  x  x0  .

Fig. 1.1

Tangent of a tangent inclination angle (y' = tg φ) is a very important and convenient characteristic of function behaviour in the point: if it is positive (angle  is sharp), then the function is probably increasing in the neighbouring space of the point under study; if tg φ < 0 (angle φ is obtuse), then it is decreasing; by the module tg ϕ  y  we may predetermine the rate of increase (decrease) of function. Later we shall come back to this problem and study it in details. Here we can only assert

Chapter 1.1. MATHEMATICAL METHODS OF COMPUTING MEDICAL AND BIOLOGICAL INFORMATION

15

that the nature of a derivative may be completely determined by the character of the input function y = f(x).

As it is seen on Fig. 1.1, АВ = tg φ · x since tg φ = y'(x0), and x = dx > 0, AB = dy, i.e. differential dy equals to the increment of tangent ordinate drawn in the point M to the graph of a given function у = f (х).

Physical explanation of derivative and differential. In each point where the function y=f (x) is differentiated, the derivative y' = f '(x0) is the rate of alteration of function in this point relative to the argument x. Substitution of function increment by its differential allows considering the process of function alteration as linear relative to sufficiently small alterations of argument.

General rules of differentiation Assume that u = f (x) and v = f (x) are differentiated functions of the independent variable x; c – some constant, then the following assertions are true.

1. Derivative of the constant y = c equals to zero: c' = 0.

2. Derivative of sum (difference) of two functions equals to sum (difference) of derivatives of these functions: (u  v)' = u'  v'.

3. Constant multiplier may be brought out of the sign of the derivative: (cu)' = c u'.

4. Derivative of the functions’ product may be determined by the formula: (uυ)' = u'υ + uυ' 5. Derivative of quotient  v  v u  u v , provided that u ≠ 0.    u2 u

The table of derivatives of principal elementary functions Calculus of a derivative and hence a differential in accordance with the definition is rather complicated. Let us show the number of results, which make it possible to simplify greatly the procedure of differentiation.

LECTURE SECTION 1

Rules for differential calculus are the same as the rules for derivative calculus (from this we have the term “differentiation”). To obtain the differential of function it is necessary to multiply the derivative by the differential of an independent variable.

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MATHEMATICAL PROCESSING OF MEDICAL AND BIOLOGICAL DATA

1. (xn)' = n xn–1.

2. (sin x)' = cos x.

1 .

4. (tg x)' = cos 2 x

3. (cos x)' = – sin x.

5. (ctg x)' = 

1 . sin 2 x

7. (arccos x)' = –

1

6. (arcsin x)' =

1

1 x 1 .

9. (arcctg x)' = – 1  x2

2

.

8. (arctg x)' =

1  x2

.

1 .

1  x2

10. (ax)' = ax ln a.

11. (ex)' = ex.

13. (loga x)' =

MODULUS 1

12. (ln x)' =

1 . x

log a e 1  (x > 0, a > 0). x ln a x

The rule of differentiation of composite function If y = f (u), and u = φ (x), i.e. y = f [φ (x)] is a composite function, and functions y(u) and u(x) are differentiated, then the composite function y = f [φ (x)] is also differentiated, and y ( x)  y (u ) u ( x) , (1.6) or in other symbols dy dy du  . dx du dx For example, if y = sin2x3, then

LECTURE SECTION 1

y' = 2 sin x3 (sin x3)' = 2 sin x3 cos x3 (x3)' = = 3x2 2 sin x3 cos x3 = 3x2 sin 2x3.

Derivatives and differentials of the higher order dy The derivative f ( x)  may be a continuous function in itself, in this case dx we may introduce the notion of a second order derivative. A second order derivative or second derivative of function is the derivative of its derivative:

Chapter 1.1. MATHEMATICAL METHODS OF COMPUTING MEDICAL AND BIOLOGICAL INFORMATION

(f ' (x))' = f " (x) = y"= i.e. y   lim

x  0

17

d2y , dx 2

y (x 0  x)  y (x 0) . x

For example, acceleration according to the definition is the second derivative of the path by time a(t) = s" (t). If the second derivative (function f " (x)) is differentiated, then we may find the third derivative f ''' (x).

The derivative of the n-order of the function y = f(x) is denoted as y(n)(x) or n d y and may be received as a result of differentiation by n time of the function dx n

LECTURE SECTION 1

y = f(x). For calculus of a derivative of the higher order we use the same rules as for calculus of a derivative of the first order.

Acceptable feature of monotony. Let us consider the function y = f (x), continuous and differentiated on some segment a ≤ x ≤ b. If in this case the inequity is true in each point of some segment of the said interval: y' > 0, (1.7) then the function increases on this segment.

If y' < 0 (1.8) on some segment, then the function decreases on this segment.

Maximum and minimum of function. One supposes that the function f (x) has maximum (minimum) in the point x0 if there is the environment of this point (x0 – , x0 + ) that for each x of this environment the inequality f (x) < f (x0) (f (x) > f (x0)) is true.

In other words, the function f (x) has maximum (minimum) in the point x0, if for a rather small increment x (of any sign) the inequality f (x0 + x) < f (x0) (f (x0 + x) > f (x0)) is true.

Maximum or minimum of a function is its extremum.

Requirements for existence of extremum. If the function differentiated on some interval (a; b) has an extremum in the point x0  (a; b), then its derivative in this point is equal to zero f '(x0) = 0.

Inner points of the domain of function where derivative exists and is equal to zero, or does not exist, is called critical. Equality of the derivative to zero is requirement but not sufficient condition of extremum. For example, for the function y = x3 the derivative y' = 3x2 transforms into zero at x = 0. But at x = 0 the function y = x3 has no extremum.

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MATHEMATICAL PROCESSING OF MEDICAL AND BIOLOGICAL DATA

MODULUS 1

LECTURE SECTION 1

Sufficient condition for extremum existence. If the derivative of the function f ' (x) transforms into zero in the point x0 and when crossing through this point in the direction of x increase it changes the sign “+” (“–”) into “–” (“+”), then in the point x0 this function has maximum (minimum). If the first derivative at crossing through the point x0 does not change its sign, then the function f (x) has no extremum in this point.

Maximum or minimum value of function on a segment. First let us note that one should not confuse maximum (minimum) with the largest (smallest) function on a segment. In accordance with the definition maximum (minimum) is the point where the function acquires the largest (smallest) value as compared with the value of function in points rather proximate to the extremum. The function may have several maximums and minimums, however, the largest value, if it exists, is common.

It also may be applied to the least value. If the function f (x) is determined on the segment [a; b], then it reaches the least and the largest value on this segment, so to determine them it is required to find all critical points of the function, which belong to this segment as well as to add to them values at the end of the segment (x = a; x = b), to find values of function at all these points and to chose the largest and the least values.

If f (x) has a finite number of break points on the segment [a; b], then it is required to study the behaviour of function in the environment of every break point. In case of open interval we may also study the behaviour of function in unidirectional environments of the interval ends.

Partial derivatives and partial differentials of function of many variables The notion of function of one variable does not cover all functional dependencies, which exist in nature. Most applied problems deal with functions of many variables. For example, the level of radioactive contamination is a function of coordinates and time, gas pressures is a function of volume and temperature.

Let G be an ensemble of points of coordinate space. Let us consider the function f (M), M  G, which associates some real number u = f (M) with each point М of set G. Since the point М is simultaneously determined by its coordinates (x; y), then one may say that u is the function of two variables х and у, so we may write down u  f  x, y  ,  x, y   G .

Thus, function of two variables f (x, y) is the function, which associates some number u = f (x, y) with every pair of numbers (x, y)  G. Let us write the increment of function z = f (x, y), which corresponds to increment x of the argument х at the fixed y = y0:

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19

 z  f  x0   x, y0   f  x0 , y0  .

Let us form the ratio of increment of function z to the increment of argument x: z f ( x0  x, y0 )  f ( x0 , y0 )  . x x If there is a limit of this ratio provided that x  0, then it is called a partial derivative of the function z = f (x, y) by the variable x in the point (x0, у0). We may write it down in symbols:  z z f lim    z x ( x, y ) . x 0  x x x In much the same manner we may introduce a partial derivative by у and dez note it as , z y , f y . y Partial derivative of function of many variables is called derivative of function, which is calculated on the grounds of assumption that only one of the arguments may be changed whereas the others remain constant.

Total differential Let us consider u = f (x, y, z) as the function of three independent variables, which is certain and differentiated in some domain. The part of function increment, principal and linear relatively to х, y, z, is called total differential du of function of three variables х, y, z: u u u du  dx  dy  dz  f x' dx  f y' dy  f z' dz .

(1.9) x y z

LECTURE SECTION 1

The product of a partial derivative of function by the differential of a correspondent independent variable is called partial differential of function n of independent variables.

Thus, total differential of function is equal to the sum of its partial differentials.

During study of function behavior in the given point of space a special interest in physics is given to the problem of direction of maximum increase of function in the given point. Vector, the module of which equals to the largest rate of function increase u = f (x, y, z) in the given point Р, and direction of which coincides with the direction of maximum increase is called gradient of function. The gradient has the point Р as its origin, and its projections will be values of partial derivatives of function u(х, у, z) in the point Р

20

MATHEMATICAL PROCESSING OF MEDICAL AND BIOLOGICAL DATA

grad u 

u  u  u  i j k , x y z

MODULUS 1

(1.10)

   where i , j , k are unit vectors of axes Ox, Oy and Oz correspondingly.

Application of function differential for calculus of errors Let us remind shortly that measurements may be direct and indirect (mediate).

In direct measurements we determine the proper sought quantity: temperature by thermometer, mass by balance, time by stop-watch, etc. Indirect measurements are purposed for measurement of sought quantity using calculus by other measuring m is calculated, mass m and volume V is quantities. For example, density ρ  V F measured. Acceleration a  is calculated by the measured force F and mass m. m Accuracy of measurement is characterized by error. Absolute error is the difference between measured xi and exact value x0 of the quantity 

xi  xi  x0 ,

i  1, 2, ..., n,

LECTURE SECTION 1

where n is the number of measurement.

Relative error is the ratio of an absolute error to exact (true) value of the measured quantity: x . ε x0 We differentiate three types of errors: random error, systematic error, crude error. Random errors are errors, which change their quantity and sign in every experiment accidentally. Systematic errors are errors, which are constants or regularly changing in repeated experiments keeping their sign and sometimes their value invariable. Reasons: inaccuracy of measurements and devices, incorrect exposure and calibration of instruments, imperfect measurement methods. Crude errors are blunders, which considerably exceed those expected under the given conditions.

Reasons: negligent calculus, incorrect connection of devices, etc.

We shall restrict our consideration to random errors. Since the accurate value x0 is unknown (if it is known there is nothing to measure), then we most frequently accept arithmetic average value x as x0 x0 

x1  x2  ...  xn 1 n   xi n n i 1 .

MODULUS 2 PRINCIPLES OF BIOLOGICAL PHYSICS “Mechanical motion in animals’ organism is governed by the same regularities as the motion of abiotic organism. Therefore, it is obvious that the problems how and to what extent blood flow in vessels depends on muscular and elastic forces of the heart and vessels, come to the problems of highly specialised sections of hydraulics”.

Thomas Young

Lecture Section 2 CHAPTER 2.1. Essential principles of biomechanics Biological tissue is complicated by its texture and heterogeneous as to its composition. Its structure and properties are specified by the functions it performs in living organisms. There are several types of tissues in morphology: epithelial, tissue of internal environment (blood and lymph), connective, muscular and neural. As a rule, all of them have cellular texture, a complex structure. To a greater or lesser extent all types

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of tissue are subject to mechanical motion beginning from intracellular micro motions of contraction protein fibres to macromotions of specific organs and systems. Some tissues are intended to performance musculoskeletal functions, and they experience heavy mechanical burden in the process of vital activity. Biomechanics studies different forms of mechanical motion in living systems. Foundations of biomechanics as a science of laws of mechanical motions in biological systems were laid down in time of Aristotle, Leonardo da Vinci, Borelli, Galilei, Descartes, Hooke, Euler, Bernoulli, Young, Helmholtz, Poiseuille, etc. (Note, that the latter four were professors of Medicine).

In study of specific mechanical properties of biological tissues, it is convenient to imagine them as continuum, without examining their microstructure and abstracting from their cellular structure. Medium may be considered as continuous if the distance where its averaged properties change (for example, density, viscosity, etc) considerably exceeds the size of particles (in our case – cells, regular elements) composing the medium. In this case the real tissue can be divided into a number of elementary volumes, size of which greatly exceeds the size of cells, and to each of them we may apply laws of mechanics to describe different mechanical phenomena such as fluidity or deformation of medium.

The part of mechanics that studies fluidity and deformation of continuum is called rheology. The task of biorheology is to study the motions mentioned above in biological systems. Now we shall examine some important notions of rheology.

Let us denote elementary volume V with mass m in continuum. Forces F acting in continuum may refer to mass (volume) unit or surface area unit.

Let’s denote the force acting on the mass unit of substance as f  F / m; the quantity of the force acting on volume unit f = F/V, the so-called volume force, is defined in a similar way. E.g. volume forces of inertia and gravity are correspondingly equal to f = m · a/V =  · a;

f = m · g/V =  · g.

LECTURE SECTION 2

It follows from these expressions that size of volume forces does not depend on size and mass of a body and may be defined only by average properties of bodies (density ) and characteristics of their mechanical motion (acceleration a). They force simultaneously on all elementary volumes of substance. It is convenient to use them for description of fluidity and deformation of real continuum. Thus, for example, the use of these forces allows writing the equation of motion of different fluids, blood included. (it should be stressed that for the description of motion of different media not only volume forces are used but also volume density of energy w = W/V which characterises the quantity of energy per volume unit).

Different parts of medium may interact with each other on boundary; in this case it is convenient to use surface forces, i.e. forces acting on surface area

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MODULUS 2

unit. Assume that two parts of body I and II bound with each other by surface AB (Fig. 1.1).

Fig. 1.1. Media boundary acting forces

LECTURE SECTION 2

Let’s mark small area dS on surface АВ influenced by force dF at some angle to the normal (Fig. 1.1a). In this case stress magnitude  equal to the force that operates in the area unit is characteristic of surface forces:  = dF/dS [N/m2]. It is convenient to introduce two components  with respect to vector n of the normal to the surface element dS: normal component n, that acts perpendicularly to plane, and tangential component , directed to a tangent to the surface dS (Fig. 1.1b). The first component has scalar in itself, i.e. pressure Р that is equal to the ratio of force quantity and surface area quantity: P = F/S. Another example of action of surface forces can be the phenomenon of surface tension, which is characterised by coefficient of surface tension . This coefficient is numerically equal to the force dF acting on length unit of arbitrary contour dL on the surface and directed to the surface tangent (Fig. 1.1c):  = dF /dL [N/m].

Surface forces are used for the description of deformation phenomena, flowing of viscous media, plasticity, creepage, surface tension, etc. which may be observed in functioning of biological tissue.

2.1.1. Mechanical properties of biological tissues Let us discuss the most important mechanical properties of biological tissues, owing to which different mechanical phenomena take place. They are functioning of musculoskeletal system, deformation processes of tissue and cells, elastic wave

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propagation, contraction and relaxation of muscles, flow of fluid and gas-like media. Among these properties we can distinguish: – resiliency – ability of bodies to renew dimensions (form or volume) after relieving of stress; – rigidity – ability of material to resist outside stress; – elasticity – ability of material to change dimensions under the action of outside stress; – durability – ability of bodies to resist destruction under the influence of outside forces; – plasticity – ability of bodies to hold (in full or partially) change of dimensions after relieving of stress; – fragility – ability of material to be destroyed without further noticeable residual deformations; – viscosity – dynamic property that characterises ability of a body to resist changing of its form under the action of tangential stress; – fluidity – dynamic properties of medium that characterises ability of some of its layers to move with some velocity in space relatively to other layers of medium.

Outside force acting on the part of body becomes due to elastic force acting on this part from the direction of neighbouring one. Physical magnitude equal to elastic force per sectional area unit of body is called stress, as it was mentioned earlier: dF σ  el . dS

LECTURE SECTION 2

Elastic properties of bodies. Deformation All real bodies may be deformed. Deformation is a change of form or volume of the body under the influence of outside forces. Deformations may be elastic and plastic. Elastic deformation is the deformation, which disappears completely after stop of action of outside forces. Renewal takes place owing to internal forces – elastic forces, originating in the body as a result of deformation. In case of plastic deformation the body keeps the state of strain after stop of action of outside forces.

Quantitative measure of body deformation is called absolute and relative deformation. If under body deformation some quantity that characterises dimensions or form of the body (e.g. length or volume), takes the value X, then change of this quantity Х = Х – Х0 under the action of applied forces is called absolute deformation. The ratio of absolute deformation to initial value Х0 is called strain: X ε .

(1.1) X0

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MODULUS 2

English physicist R. Hooke proved experimentally that the stress in elastostrained body is directly proportional to its strain (Hooke’s law): σ  E ε , (1.2) where E – elastic modulus, its quantity is defined by properties of material the body is made of. Depending on the type of deformation the elastic modulus has different names, notations and numerical values. It should be stressed that Hooke’s law is true only for rather small relative deformations.

Any complex body deformation may be represented as a result of overlap of simpler deformations: longitudinal stretching (tensile) or compression, uniform stretching or compression and shift.

Longitudinal stretching (tensile) or compression deformation Let us examine body deformation when one of its ends is fixed and the external force F stretching the body is applied to the other end.

Fig. 1.2. Bar tensile deformation Strain in this case is equal to:

LECTURE SECTION 2

ε

l , l0

where l is the sample length under the external force, l0 is the initial length of the sample.

In static state external force F becomes balanced by elastic forces Fel, appeared in the body under deformation (Fig. 1.2).

Hooke’s law is defined as: l σ E  Eε , (1.3) l0

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where σ  F S – is direct (normal) stress since effecting force is perpendicular to the sectional area of sample S. Elastic modulus E is called Young’s modulus. From the Hook’s law it follows that Е = , if  = 1, i.e. if l = l0. In other words, Young’s modulus E is equal to direct stress, which would appear in the sample with its length increased twice as much if the Hook’s law was true for such big deformations.

Note, that under compression of a sample the Young’s modulus corresponds to the stress at which the sample’s length tends to zero. Stretching (or compression) of a sample is always accompanied by their cross contraction (expansion), i.e. by change of their transverse dimensions: d = d – d0.

The ratio of relative change of cross dimension to relative change of longitudinal dimension is called Poisson’s ratio: d d 0 μ .

(1.4) l l0 Considering that d < 0 at l > 0, then  > 0. For poorly compressed materials  ≈ 1/2. Almost all biological materials including walls of vessels are hardly compressed, that is why for them  ≈ 1/2. Further we shall discuss only isotropic media, rheological properties of which are equal in all directions.

Uniform stretching (tensile) deformation or compression (volume deformation) Volume deformation appears at uniform distribution of compression or stretching (tensile) forces over the body surface (Fig. 3.3a).

In this case Hooke’s law will look as:

LECTURE SECTION 2

Fig. 1.3. Types of deformation: a) volume deformation; b) shear deformation; c) bending

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PRINCIPLES OF BIOLOGICAL PHYSICS

σ χ

MODULUS 2

V , V0

where  is the volume elastic modulus, V and V0 – change of volume of the body and initial volume correspondingly. Transmural pressure may be an example of stress that causes volume deformation. Transmural pressure is equal to difference of pressures inside and outside of a vessel Ptp  Pi  Po . Then Hooke’s law will be: Ptp  χ

V .

V0

(1.5)

Shear deformation Shear is the deformation of the body when its flat layers displace each other in parallel (Fig. 3b). Share appears under shearing stress: στ 

Fτ .

S

(1.6)

LECTURE SECTION 2

In accordance with Hooke’s law  = G, where G – shear modulus,  ≈ tg = CC'/CD – angle of share that is also called relative share (Х = СС' – absolute share is equal to share of one layer relative to the other, and CD – distance between these layers).

Torsional deformation Torsional deformation appears in a sample when one of its sections is immobile whereas the other is influenced by a pair of forces, the moment of which is directed along the sample’s axe. This deformation is used in torsion balance.

For each type of deformation stated above proportional dependence may be observed between stress and relative deformation within the bounds of elastic reaction. Coefficients of proportionality – elastic modulus – may be noted through the Young’s modulus (E) and Poisson’s ratio () of material, i.e. for elastic deformation of isotropic bodies E and  completely define the sample’s reaction to the applied stress. For example, volume elastic modulus of a thin wall of a vessel may be presented as follows:  = 2hE(1 – 2)R, (1.7) where h and R – wall thickness and radius of the vessel correspondingly, h << R. Experimentally received stress dependence on relative deformation appeared in a sample under deformation is called deformation diagram. A standard view of the tensile stress-strain diagram for a metal sample is presented on Fig. 1.4.

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Fig. 1.4. Tensile stress-strain diagram for steel

LECTURE SECTION 2

The depicted curve may be conditionally sectioned into five zones. OA zone is called a zone of proportionality. The Hooke’s law is true within the limits of this zone. OB zone is an elasticity zone where the body restores its dimensions and form after removal of stress. BC zone is a zone of general fluidity. Lengthening of a sample in this zone takes place with no noticeable increase of stress. CD zone is a zone of strengthening. In this zone lengthening of a sample is accompanied by an increase of stress. Now in a sample we may notice the place of future break called a neck, formation of which (point D) is accompanied by a process of local fluidity in DE zone and break of a sample. If load in BC zone decreases, then a correspondent graphic  = f () will be in parallel to OA and cross all abscissa in some point O1.

Segment OO1 defined residual deformation ext that characterises plastic deformation of a sample. Deformation diagram allows to define a number of the most important points and magnitudes correspondent to them: – proportionality limit prop – the highest stress for which the Hook’s law is true; – stress limit el – the highest stress at which there are no residual deformations; – fluidity limit fl – the highest stress when there is an increase of deformation with no noticeable increase of stress; – strength limit st – the highest stress tolerated by a sample.

In bodies under deformation the viscoelastic properties often appear, which prove that stress depends not only on deformation () but also on the rate of its change in the course of time, i.e. on derivative ε  .

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PRINCIPLES OF BIOLOGICAL PHYSICS

MODULUS 2

2.1.2. Deformation of biological tissue Let us consider deformation diagrams of those biological tissue and organs, which are subject to considerable stress in the process of their functioning. They are, for example, bone tissue, muscular tissue, tendon tissue, tissue of vessels’ walls, etc. It was experimentally proved that for most of the tissues tensile stress-strain or compression diagrams differ sharply from the diagram depicted on Fig. 1.4. As a rule, for biological materials a zone of general fluidity is not expressed, although this property becomes apparent clearly in the process of functioning of biological materials. Deterioration of material takes place also with no noticeable drop of stress typical for a CD zone.

Bone tissue Bone tissue in its mechanical properties is close to wood, concrete, some metals, i.e. materials used in construction. We shall not explain the structure of bone tissue here, but it is worthy to note that its structure is very complicated. Bone tissue is a composite material that is composed of organic and inorganic substances and has anisotropic properties.

Fig. 1.5 shows tensile stress-strain and compressive diagrams along longitudinal axes of samples cut from a hipbone.

LECTURE SECTION 2

Fig. 1.5 Deformation diagrams for bone and collagen It is obvious from the diagrams, as compared with steel, deformation takes place within the considerable limits – up to 10 % under compression and up to 5 % under stretching. Under minor deformation (less than 2 %) bone behaves as a “Hooke’s body”, for which the dependence  = f() is close to linear. Note, that bone “works” better for compression that for stretching – strength limit and size of deformations under compression twice as many of those observed under stretching.

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Collagen fibres Collagen fibre is an important constructive part of connective tissue. Collagen fibres are included into bones, vesselsâ&#x20AC;&#x2122; walls, muscular tunic, etc. They are strong flexible protein fibres formed by aggregation of triple spirals, which become stabilised owing to hydrogen bond that insures high strength of fibres under the work in break. The stretching (tensile stress-strain) diagram for collagen fibres is shown on Fig. 1.5b. This diagram is identical to the diagram of bones. Both of them have similar values of limit deformations, however strength limit in collagen is more than one order lesser as compared with the strength limit of bone.

Elastic fibers Elastin is a rubbery material remarkable for its great tensility and flexibility.

These properties make it indispensable component in structures of those tissues that considerably change their form and dimensions in the process of functioning (walls of vessels, muscles, covering tunics, etc). Flexibility and tensility of elastin are related to properties of its subunits â&#x20AC;&#x201C; globules combined into net structure by rigid chemical bonds (compounds called desmosines). The net is easily deformed without break of these bonds under the influence of outside load. Strength of fibres increases in the process of stretching accompanied by the extension of globules â&#x20AC;&#x201C; subunits of elastin, as it is shown on the diagram (Fig. 1.6a). b)

Tensile stress-strain diagram for vessels Walls of vessels have a complicated structure. There is an essential difference in the structure of walls of aorta, arteries, veins, venules and capillaries. Their elas-

LECTURE SECTION 2

Fig. 1.6. Tensile stress-strain diagram for elastin and wall of artery (aorta)

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PRINCIPLES OF BIOLOGICAL PHYSICS

MODULUS 2

LECTURE SECTION 2

tic properties may be defined by the ratio of composition of three types of fibres: elastin, collagen and muscular. Collagen has a greater Young’s modulus as compared with elastin and unstriped muscle fibres of the almost the same elasticity. In big vessels (aorta, vein) elastin and collagen constitute approximately 50 % of dry weight, in elastomuscular vessels their content decreases to 40 % and even lesser.

Walls of vessels are heterogeneous as to their structure, they differ by their aeolotropic properties. Classic approaches for study of elastic properties in the process of determination of Young’s modulus, elastic limit, strength limit, etc. are hardly suitable for such bodies.

Fig. 1.6b shows the diagram of aorta stretching under transmural pressure P (difference of pressures inside and outside the vessel).

Thus, at increase of pressure (under physical activities, different pathologies) the strength of vessels or their tone increases sharply (see dotted line on Fig. 1.6b).

Physiologic essence of this phenomenon is clear: increasing strength of vessels prevents excessive increase of its volume under increase of pressure, that, in its turn, prevents excessive compression of inner tissue (for example, neural tissue of brain), and allows decreasing the volume of circulating blood under loads.

Biophysical mechanism of this phenomenon is rather complicated and still not clearly understood. One may assume that this mechanism is defined by elastic properties of elastin (increase of strength in the process of stretching) as well as by activation of contractility of unstriped muscular system of vessels under stretching (histomechanical theory). Note, that the role of unstriped muscular systems is extremely important in the process of deformation of vessels; in its absence it is impossible to explain viscoelastic properties of vessels, and, therefore, such phenomena as dilatation and constriction of vessels, change of their tone, depositing and outflow of blood.

CHAPTER 2.2 Fluidity of viscous fluids in biological systems Flow of fluid media (blood, lymph, interstitial and cellular fluids) in biological systems plays an important role and provides conditions for normal vital functions of different physiological systems. The task of biophysics is to study physical properties of fluid media and physical basics of their flow. Fluidity of fluids is caused

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