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"Is affirmed"
At methodical session of the chair of medical
and biological physics
2007. The report №
chief of chair
Shaplavskij M.V.
Methodical instructions (indication)
for plan students self- preparation
“Simulation of medicobiological process by differential equations ”
Module 1: Mathematical treatment of medical and biological information
Pithy module 2. Mathematical treatment of medical and biological information
Chernivtsi-2007
Objective: To give practice with applying some techniques of integration to solving separable
differential equations, including applications to chemical and pharmacological kinetics.
Introduction:
Models are simply mathematical constructs (pictures) which seem to explain the relationship
of concentration with time (equations) when drugs are given to a person (or an animal). These
models are useful to predict the time course of drugs in the body and to allow us to maintain drug
concentration in the therapeutic range (optimize therapy). The simplest model is the one used to
explain the observations.
Background:
1. Notion of function. Exponential function
2. Notion of derivative.
3. Notion of differential equation.
Basic question:
1. Determination of solution of differential equation.
2. Types of differential equation.
Practical exercise:
1. To analyze the basic typical tasks on simulation of medico-biological process and to learn a
methods of their decision.
2. To find solution of differential equation, which simulate biological processes offered by teacher
as attached to decision on board so and independently to clear up herewith the incomprehensible
questions and decisions receptions and to clear up them by the medium of teacher.
Final knowledge and skills:
Students should know:
1. Governed finding derivative of basic functions.
2. Governed finding of derivative sum, work, particle and complicated function.
Students should be able:
1. To find solution of modeled differential equation.
2. To apply modeling for pharmacokinetic processes.
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Contents of theme
Separable differential equations often appear in chemical kinetics, which is the study of rates
of chemical reactions. Determining rates of chemical reactions is a crucial first step in the design of
any commercial chemical process, so this is an area of particular interest to chemists, medical
chemists and chemical engineers.
1. Separable differential equations in chemical kinetics.
2. Separable differential equations Pharmaceutical kinetics.
-
linear pharmacokinetic model
-
half-life elimination
-
intravenous infusion of a drug, (Model with a dropper)
-
intramuscular infusion of a drug, (Linear suction pharmacokinetic model)
3. Law of reproduction of bacteria in time.
Literature
1. Raymond A.Serway, Jerry S. Faughn Colloge physics, Fifth Edition.-Saunders College
Publishing.- 1999.-1029 p.
2. Marzeniuk V.P., Diduch V.D., Vakulenko D.V. et al. Biophysics and medical informatics.Ternopil: Ukrmedkniha, 2004.-480p.
3. Hugh D. Young, Roger A. Freedman University physics with modern physics, 11 th edition.Pearson education.-2004.-1714p.
4.Frank J. Blatt Modern Physics .-McGRAW-HILL, INC.-1992.- 517 p.
5. Methodical instructions (indication) for student’s preparation, lecture.
Questions for self-preparation of students:
1. Separable differentiable equations
2. Separable differential equations in chemical kinetics.
3. Law of reproduction of bacteria in time
4. Pharmaceutical kinetics
- intravenous injection
- model with a dropper
- linear suction pharmacokinetic model (first order linear differential equation).
Tests.
1. half-life is
a) the time it takes for the concentration of the drug to fall to half
b) the time it takes for the concentration of the drug to increase twice
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c) the concentration if time increase twice
2. At favorable condition the increase of quantity of bacteria in time takes place on
a) an exponential law
b) an sum law
c) an power law
d) an logarithmical law
3. the following pharmacokinetic scheme
Is
a) linear pharmacokinetic model
b) Model with a dropper
c)intramuscular infusion of a drug.
Class independent work
1. Find decreases law of drug in human organism, if in 1 hour after infusion of 10 mg of drug its
mace decreases twice. How much drug remains in human organism in 2 hours?
2. Initial mace of enzyme -1 g, in 1 hour -1,2g. Find mace in 5 hours after the beginning of
fermentation. Consider speed of increase of enzyme proportional to its available quantity.
3. Rate of disintegration of some drug is proportional to the available quantity of drug. In 1 hour in
human organism remained 31,4 g of drug, in 3 hours – 9,7 g. Find
1. initial mace of drug
2. time, when in organism remained 1% from initial mace.
4. Dissolving rate of salt is proportional to difference of concentration of saturated y0 and real x
solutions. Find law of variation of salt concentration, if for t  0 x  x0 .
5. Rate of enzymatic reaction sometimes is given by formula
k (  x) ,
dx
 1
dt 1  k 2 (  x)
where x is concentration
of product at the moment t, a is initial concentration of reagent. Find concentration dependence of
time.
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