1. MEDICAL INFORMATICS Lectures KHARKIV – 2018/2019
MINISTRY OF HEALTH OF UKRAINE KHARKIV NATIONAL MEDICAL UNIVERSITY DEPARTMENT OF MEDICAL AND BIOLOGICAL PHYSICS AND MEDICAL INFORMATION SCIENCE
MEDICAL INFORMATICS
Lectures
KHARKIV – 2018/2019

2. УДК 61: ](07.07)
ББК я7
М42
Authors:
L.V. Batyuk – Ph.D., Associate professor of Department of Medical and Biological Physics and Medical Information Science
V.G. Knigavko – Dr.Sci., Prof., Chief of Department of Medical and Biological Physics and Medical Information Science
G.O. Chovpan – Ph.D., Associate professor of Department of Medical and Biological Physics and Medical Information Science
Kharkiv National Medical University, 2018/2019

3. MATHEMATICAL MODELING IN BIOLOGY AND MEDICINE Lecture 2
MINISTRY OF HEALTH OF UKRAINE KHARKIV NATIONAL MEDICAL UNIVERSITY DEPARTMENT OF MEDICAL AND BIOLOGICAL PHYSICS AND MEDICAL INFORMATION SCIENCE
MATHEMATICAL MODELING IN BIOLOGY AND MEDICINE Lecture 2

4. Classification of modeling methods
Model is like a mentally imagined or materially realized system which reflects or represents a research object.
There are many model classifications, the most general of them divides all the models in material, energy, mathematic and information ones.

5. Under material models it is common to understand those models which reproduce the object structure and relations between its parts.
The example of such models in medicine can be different artificial limbs, which look very similar to real limbs they replace.

6. Artificial Kidney - Dialysis
Energy models are used to model functional interrelations in investigated objects.
These models, according to their appearance, don't resemble modeled objects, but their aim is the execution functions of these objects.
For example, in medicine, such systems as the artificial kidney or medical ventilation apparatus are widely used.
Artificial Kidney - Dialysis
Blood is made to flow into the dialysis machine made of long cellulose tubes coiled in a tank having a having a dialysing solution. Waste substances diffuse out of blood into tank. The cleansed blood is pumped back into patient.
A medical ventilator
The illustration shows a standard setup for a ventilator in a hospital room. The ventilator pushes warm, moist air (or air with increased oxygen) to the patient. Exhaled air flows away from the patient.

7. Biooperated artificial limbs, artificial lens, recent developments in the field of artificial heart belong to such models.
In the case of liver model, a polyvinyl alcohol was used to simulate the wetness and texture of a human liver.
Using CT and MRI images of patients’ organs generated life-size models of kidneys, livers and other organ that are medically accurate.
When printing on machines that can handle multiple materials, the outer structure of models can be rendered transparent, helping diagnose cancer, find scar tissues and build a plan for surgical operations.

8. In distinction from the first two models, the information models - are the object description.
In medical-biological researches till the recent time mainly verbal models had been used to describe the work of biological systems.

9. Mathematical models are divided into deterministic and probabilistic.
In the deterministic models variables and parameters are supposed to be constant and they are described by deterministic functions.
Deterministic mathematical models more often represent the system of algebraic or differential equations.
In probabilistic models, variables and parameters, characterizing it, are random functions or random values.
Probabilistic models are built according to results of experimental determination of dynamic characteristics of objects on the basis of mathematical statistics methods.

10. Alfred J. Lotka (1880-1949) and Vito Volterra (1860-1940).
"Lotka-Volterra" model
For the first time in biology the mathematical model of periodical change of antagonistic animal species quantity was offered by the Italian mathematician V. Volterra with his collaborators.
So this classic model is known as the "Lotka-Volterra" model (resources and consumers, predators and preys, pathogens and their hosts - all these models belong to "predator-prey" model ("Elements of physical biology“, 1924).
Alfred J. Lotka ( ) and Vito Volterra ( ).

11. "Lotka-Volterra" model
In some ecologically secluded area animals of two species live (for example, lynxes and hares). It is to be determined how the quality of preys and predators will change with time.
Let's mark the quantity of preys as N, and the quantity of predators as M.
N and M values are functions of time t.

12. the rate of preys‘ reproduction in the specified conditions;
Let during some time Δt the quantity of preys and predators will change:
(ΔN)1 = A∙N∙Δt
where A is the proportionality coefficient, which characterizes
the rate of preys‘ reproduction in the specified conditions;
(ΔN)1 is the result of natural reproduction of preys.
(ΔN)2 = - B∙N∙Δt
where B is the proportionality coefficient, which characterizes the rate of prey’s dying;
(ΔN)2 is the result of natural dying of preys.
The frequency of meetings of predators with preys is proportional to the quantity of preys and quantity of predators, i.e. to their product MN:
(ΔN)3 = -C∙N∙M∙Δt
where C is the proportionality coefficient, which characterizes the frequency of encounters of preys and predators.

13. (A - B - C ) (Q - P ) Δ M= Q∙N∙M∙Δt - P∙M∙Δt, i.e.
The preys' quantity change:
ΔN = A∙N∙Δt - B∙N∙Δt - C∙M∙N∙Δt or
dN/dt = A∙N - B∙N - C∙M∙N
The growth of quantity of predators as a result of natural reproduction
at the sufficient quantity of food (preys), ΔM1 =Q∙N∙M∙Δt
and it decreases due to the natural dying out, ΔM2= - P∙M∙Δt,
Δ M= Q∙N∙M∙Δt - P∙M∙Δt, i.e.
dM/dt =Q∙N∙M - P∙M
(A - B - C )
(Q P )

14. Mathematical modeling in immunology
Immunity is a complex aggregate of organism reactions at the intrusion of antibodies - foreign objects or cells, tissues, proteins which regenerated.
The specific immune reaction at the molecular level starts from the fact that the specialized plasmatic cells produce in big quantity protein molecules - antibodies, which neutralize antigens.

15. where X - is the quantity of antigens;
Mathematical model of immune reaction during infectious diseases is three interdependent differential equations:
where X - is the quantity of antigens;
Y - is the quantity of antibodies;
Z - is the quantity of plasma cells which produce antibodies.

16. C and L – the coefficients of natural decay of antigens and antibodies;
N – the coefficient of the natural dying out of plasma cells;
XY – the interaction antigen-antibody in the agglutination reaction;
D – the coefficient of the inflow of antigens to blood;
The function F(X) – the rate of plasma cell generation;
M – the coefficient, is proportionally dependent on the temperature (M = M(T)).

17. Mathematical model of immune reaction is used in clinical practice for treatment of acute pneumonia and other diseases.
At the figure there are possible cases of immune reaction dynamics (X - is the quantity of antigens, Y - is the quantity of antibodies, t - is the time)
Subclinical form 1 takes place without physiological disorders in an organism and without visible manifestations.
Acute form 2 - in this case an organism is attacked by an unknown antigen in big quantities.
Chronical form 3 - the dynamic equilibrium between antigens and antibodies is set. The permanent sick condition appears.
Lethal form 4 - immune response delays too much and big quantity of antigens provokes irreversible changes in an organism.

18. Mathematical model of bacteria population growth
where Y - the quantity of cell in a colony; t is the time;
dY/dt - the rate of cell quantity change;
A - the coefficient which depends on the average value of the generation period;
B - the coefficient which takes into account the death-rate.
At fig. - the example of the investigation of the mathematical model for such coefficient values:
A = 2,5; B = 0,001;
Ystart = 50 is shown.

19. Mathematical modeling of infectious disease distribution in a population aggregate
where X(t) – the quantity of healthy people;
Y(t) – the quantity of sick people;
Z(t) – the quantity of people who recovered already and got the immunity;
Q – the quantity of citizens;
XY – the number of contacts of healthy people with the sick;
A – the quantity of those infected by every sick person during one day;
R – the average length of disease counted in days.

20. Exponential reproduction law
G=g∙X∙Δt
where G – the new-born;
X – the population quantity;
g – the averaged relative number of the new-born during the time
unit, is called the birth rate.
For example, g=0,054 (in 1 year) means 54 new-born at 1000
citizens during one year.
The number of the dead:
H=h∙X∙Δt
where h – the averaged relative quantity of the dead during the
time unit is called death-rate.
Then: ΔX = G-H = g∙X∙Δt - h∙X∙Δt = (g-h)∙X∙Δt
where (g - h) = const - the relative increase.

21. the population grows with time,
The solution of this equation is called the exponential growth formula:
x(t) = x0e (g-h)t ( )
Analyzing the obtained connection, we see that
the population grows with time,
if g>h;
and stays at the same level if g= h;
or decreases if g<h.

22. Pharmacokinetic model
Pharmacokinetics studies out the distribution of the investigated biologically active substance in an organism and its concentration change in time.
In pharmacokinetics an organism is divided into chambers. A pharmacokinetic chamber is the part of an organism where the investigated drug is distributed uniformly.
The aggregate of processes which set conditions for the decrease of the drug content in an organism with time is called elimination.

23. One-chamber linear pharmacokinetic model
If the change rate of the drug dM/dt is proportional to the first degree of the drug mass M in a chamber, this process belongs to the linear pharmacokinetic models;
The differential equation of one-chamber linear pharmacokinetic model is defined as follows:
where kel is the elimination constant, i.e. the coefficient of the drug extraction from an organism.
The integral equation of one-chamber linear pharmacokinetic model is defined as follows:
M=M0e-kelt
where M is the remedy mass in a chamber at time t=0.

24. where c0 is the drug concentration in the initial time period.
If V is the chamber volume, then the remedy mass is
defined according to the concentration: M = V*c.
The equation for the remedy concentration is defined as:
c(t) = c0e-kelt
where c0 is the drug concentration in the initial time period.
The elimination (excretion) of biologically active substances from an organism in practice is studied out according to the decrease of their concentration in blood.
The blood is the basic test-tissue.

25. Thank You for Attention!