Kharkov National Medical University MEDICAL INFORMATICS Mathematical Modelling

Kharkov National Medical University МЕДИЧНА ІНФОРМАТИКА MEDICAL INFORMATICS

Mathematical Modelling in Medicine

Lecture plan n n n n 1. Modelling classification. 2. «Predator-Prey» mathematical model. 3. Mathematical modelling in immunology. 4. Mathematical model «The Growth of Bacterial Populations» . 5. Mathematical model «The spread of infectious diseases at settlement» 6. The exponential law of reproduction. 7. Logistics growth model 8. Pharmacokinetic models

Modelling is the one of basic research methods in medicine. Modelling process involves three elements: n subject (researcher), n object of investigation (prototype), n model that defines or represents relations between the cognizing subject and the object to be known.

MODEL n n n artificial object of any kind, which replaces or restores the real object of investigation in order to obtain new information a bout it. Model investigation allows estimate the behavior of the modelling object in new conditions or under different influences that are not possible to be checked with real object (investigation of people, for instance) or can be verified with difficulties (for example, costly experiments or experiments with negative consequences). Model always more restricted than the real object. It always represents only some main characteristics (depending on the task which we are developing the model for).

Models classification Models physical material biological energy informational language mathematical

Physical models classification material Reproduce the structure of the object and relations between its parts energy Are used for modelling of functional relations in investigated objects

Inforamtional models classification language Description of the object of investigation or process by words and symbols mathematical Description of the relation between initial data and desired quantities by mathematical formulas

Mathematical modelling Mathematical models: description of real object processes by mathematical equations, formulas, functions. For the developing of the mathematical model it is possible to use physical laws obtained during the experiment with the modelling object. determined It is a system of algebraic or differential equations These are the models where we have one-to-one correspondence between variables which describe the object or phenomena stochastic Represent the data of mathematical statistics They are ground on the results of experimental detection of dynamic object parameters on basis of mathematical statistics methods

«Predator-Prey» model Modelling task is next. There are two kinds of animals living in some ecologically isolated district (for example, lynxs and rabbits). Rabbits (preys) eat vegetable food, which is always enough. Lynxs (predators) can eat only rabbits. Define, how the amount of predators and preys are changeable during the time.

Let us define variable N as preys, variable М as predators N = АN t ВN t СМN t М = QNM t PM t, А, Q – the birth rate, В, P – natural death rate, С – coefficient of the violent death rate (predators and preys encounters).

«Predator-Prey» model in immunology Immune system is a complex of organism reactions to intrusion of antigens (foreign objects): molecules, cells, tissues, etc. where Х is amount of antigens; Y is amount of antybodies; Subclinical form - 1 Z is amount of plasma cells, which produce antybodies. Acute form - 2 Chronic form - 3 Lethal form - 4

The growth of bacterial population model А = 2, 5; В= 0, 001; Yнач. = 50. where Y is amount of cells in colony; t is time; d. Y/dt – rate of the cells amount changing; А is a coefficient that depends on the mean of reproduction period; В is a coefficient, that takes into account the death rate.

The spread of infectious disease in settlement Q – amount of people in the settlement; А – averaged amount of people, which are being infected every day by one sick person; R – averaged duration of the disease, in days; Х – amount of healthy people; Y – amount of sick people; Z – amount of people who have gained immunity after disease

Exponential law of reproduction g - birth rate, h – death rate The law of exponential growth : x(0)=x 0 – initial conditions if g > h , then population increases during the time, if g = h, then population amount is a constant value, if g < h, then population decreases during the time.

Logistics law of reproduction Model of population amount changing in view of competition between individuals (Verhulst model ) Logistics law of reproduction During the time variable х becomes stationary хst. x(0)=x 0 – initial conditions

Pharmacokinetic model Pharmacokinetic camera is a part of organism, where the investigated drugs are evenly distributed. Set of processes, which cause the decrease of the drugs in organism with the time is called elimination Integral equation of linear unicameral pharmacokinetic model (М - the mass of the drugs in camera, М 0 - initial mass of the drugs, kel – elimination constant) c - the concentration of the drugs in camera, c 0 - initial concentration of the drugs

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