From Big Medical Encyclopedia

MATHEMATICAL METHODS in medicine — set of the mathematical approaches used for the obtaining quantitative dependences, creation of models of patterns of any processes or the phenomena occurring in live organisms, and also relating to the organization of health service and health protection.

In spite of the fact that M. are applied by m practically in all areas of human knowledge, the role and value M. of m in certain specific areas of science are various. So, if rather simple forms of the studied phenomena are described with sufficient completeness (e.g., in the equipment), then at their research there are problems relating to area of actually mathematical methods — such as development of specific systems of symbolical record, algorithms of problem solving, ways of quantification of characteristics of the studied phenomena etc. In the same cases when it is necessary to deal with big and difficult organized objects as it takes place in medicine and biology, the main difficulty of a research consists already not so much in development of mathematical theories and the device of a research how many in the choice of specific premises and initial positions for the subsequent mathematical processing, and also in interpretation of the results received by means of M. by m.

In all cases M.'s use is subordinated by m to problem solving of specific spheres of activity of the person that considerably enriches the theory and practice in these areas. It is obvious that reasonable use of M. of m in medicine, and also in adjacent areas (biochemistry, physiology etc.) gives the real chance to lift researches in these areas to the level corresponding to their value in life of modern society.

At implementation of quantitative methods of a research in the field of medicine and biology receiving enough complete and reasonable descriptions of the processes and the phenomena given in language and in the terms answering to specifics of specific solvable objectives is necessary. Complexity consists first of all here in identification and assessment of multiple interdependence since the analysis of multidimensional representations at the level of their subconscious comprehension is extremely complicated, and in some cases is almost impossible. The modern medical science faces such complex challenges in the analysis fiziol, processes in an organism, at problem solving of diagnosis and treatment of diseases. At the solution of some private tasks with success various graphic descriptions are applied (schedules, charts, nomograms, etc.). So, description of blood as physical. - it is more convenient to carry out chemical systems by means of nomograms — multidimensional schedules with 8 — 10 coordinates. If, e.g., on such schedule to draw a straight line through two points corresponding to at the same time measured sizes pO2 and pCO2, then on it there will be all sizes which are functionally connected with these values (pH of blood serum, percent of hemoglobin, pH of cellular fraction etc.)

When it is possible to obtain sufficient quantitative data, use more exact ways of the mathematical description funkts, dependences, i.e. build the equations connecting among themselves separate measured (and in some cases and not measured) variables in an organism. Calculations of stroke and minute output of heart according to the measured data of heart rate and a form the curve ABP can be an example. Creation of such dependences in experimental conditions is made on the basis of statistical methods, napr, a method of least squares (see. Leastsquares method ).

The wide spread occurance for the description of the variables and processes changing in time was received by the differential equations so one or several such equations express ratios between changes of the main variables. The so-called model of an elastic tank — the linear differential equation of type can be an example of the description of a course of processes in cardiovascular system:

(1/k) * (dP/dT) = P/R + W(t),

where a variable P — an instantaneous value of the ABP; parameters R and k — respectively the general resistance of a circulatory bed to a blood flow and coefficient of elasticity of an aorta; W(t) — instant rate of volume flow of emission of blood from heart. When the studied situation is described by system of three-four and more differential equations, their decision requires use of the COMPUTER (see. Electronic computer ).

The highest step of use of M. of m in biology and medicine is the analysis of systems (see. Systems analysis ) and them mathematical modeling (see). In this case at the decision practical medical and biol, tasks there is a possibility of assessment of a current status of an organism or other analyzed systems, forecasting of a tendency of change and a prediction of different results of corrective influences. Information necessary for this purpose on a large number a component of system and their relationship is presented usually in the form of the equations. Besides, development of some general conceptual ideas or structural representations playing a role of a framework to Krom it is required numerous specific characteristics and quantitative descriptions of the analyzed processes and the phenomena can be attached.

At this most difficult stage of implementation of M. of m in medicine and biology the principal value is got by methods of the theory of management (see. Cybernetics , Cybernetics medical ), theories of mass service (see. Mass service theory ), theories of games (see. Modelling ), theories of decisions, and also methods of the theory of information (see. Information theory ). Direct implementation of these methods in a wedge, medicine and practice medical - biol, researches happens within medical cybernetics, the main directions of development a cut are: development of the automated systems of collecting, processing and storage of medical information (including development of methods of creation of the automated medical services and archives, data banks, methods of the analysis of results of inspection of the patient, etc.); creation of diagnostic systems for different types of diseases using the COMPUTER (see. Diagnosis machine ); development and use of methods of mathematical modeling and systems analysis of various systems of an organism are normal also in the conditions of pathology — including tasks of management of treatment. The last direction is adjoined by works on modeling various epidemiol, processes and a research in the field of mathematical modeling and the analysis of systems of the organization of health care.

A starting material for M. of m in medicine and biology are, as a rule, judgments of experts in the field, the quantitative data obtained at measurement morfol., fiziol, and biochemical, variable in an organism. Set of methods and methods of data processing in biology and medicine sometimes consider as specific area of quantitative methods of data collection and processing — biometrics (see).

For the strict and adequate description biol, and the medical objects which are characterized by considerable accidental fluctuations probabilistic approaches, and for disclosure of sense of these phenomena — methods of probability theory are used (see. Probabilities theory , Correlation analysis ). For the description of the real phenomena by means of probability theory use the term probabilistic (statistical) model. The important section of probability theory is the mathematical statistics, the purpose a cut consists in studying of compliance between theoretical model and reality and check of adequacy of probabilistic model.

At a stage of obtaining initial information about biol, and the medical phenomena the correct statement of experiments is important they led to essential conclusions, to saving of time, labor and materials, could be easy and are unambiguously interpreted, would yield clear results. The section of statistics studying ways of the organization and carrying out observations in an experiment is called planning of experiments (see. Experiment ).

At problem solving of planning of experiments widely use methods factor analysis (see), the purpose to-rogo is definition of that contribution which brings each of the factors influencing its outcome in the general variability of results of an experiment.

Methods of probability theory and mathematical statistics were widely adopted in practice of medico-experimental and a wedge, researches, napr, during the processing a lab. and a wedge, data (including in the analysis of an ECG and EEG, receiving distributions of microobjects in optikogeometrichesky parameters in gistol, drugs etc.), in the course epidemiol. researches, in to sanitary statistics (see), pharmacy chain etc.

Use of quantitative methods at mathematical modeling demands accurate information of a task, initial assumptions and hypotheses, and also means systematization of the consecutive steps conducting to required conclusions and results. Besides, the research problem by preparation for modeling shall follow logically from the current state of the explored area and consider the restrictions imposed by opportunities and availability of methods of measurement, processing of the obtained data and the subsequent analysis.

Process of mathematical modeling includes the following main stages: the choice of structure of model and the formulation of the laws connecting its elements; the analysis of the received description (verification), i.e. check of proximity of the processes received on model, and real processes and definition of area of adequacy to the received model; obtaining new data and modernization of model. During the modeling of processes in a human body concepts and methods of cybernetics and the theory of management, such as were of particular importance feed-back (see), stability, reliability (see), sensitivity (see) etc. These concepts are extremely important for the formal description fiziol, and medical concepts (a homeostasis of an organism, adaptation and compensation, a stress) and quantification of processes of a disease and treatment.

Works on M. of m of the solution of physiological medicobiological and medico-experimental tasks endure the period of rapid development. So, in Ying-those cybernetics of AN of USSR one of the largest models of a complex fiziol, systems of an organism, allowing to study at the same time processes is developed breath (see), blood circulations (see), vodnosolevy exchange (see) and thermal control (see). In Ying-those cardiovascular surgery of A. N. Bakulev models of cardiovascular system are successfully applied in a wedge, practice. In Ying-those problems of management together with Ying volume of transplantology and artificial organs of M3 of the USSR methods of mathematical modeling of artificial internals in their interaction with various fiziol, systems of an organism are developed. Successfully work on mathematical modeling of system of public health care in scales of the country develops. In Moscow, Minsk, Voronezh and other cities of the country work on the analysis of processes of management of treatment is developed. Perspective range of application of M. of m is the .issledovaniye of processes pharmacokinetics (see) and pharmacodynamics (see), and also modeling and the analysis of various types patol, and protective processes in a human body (modeling of a diabetes mellitus, early stages of an idiopathic hypertensia, immune responses, process of cellular growth of malignant cells, etc.).

In medicine M.'s penetration the m occurs hl. obr. through statistics, biol, and medical cybernetics (see. Cybernetics medical ). At the same time the methods used in biol and medical cybernetics, in many respects match, and these disciplines are inseparably linked among themselves.

In general adequate use of M. of m is a perspective method of the analysis medical and biol, the phenomena; their use in medicine promotes progress in medico-experimental and a wedge, areas and helps the doctor, increasing his creative opportunities.

Bibliography: Adler Yu. P., Markova E. V. and Granovsky Yu. V. Planning of an experiment by search of optimal conditions, M., 1971; Bailey N. Mathematics in biology and medicine, the lane with English, M., 1970, bibliogr.; B y x about Sunday to and y M. L. and In and sh N of e in with to and y A. A. Cybernetic systems in medicine, M., 1971, bibliogr.; Lasted L. B. Introduction to a problem of decision-making in medicine, the lane with English, M., 1971, bibliogr.; Lisenkov A. N. Mathematical methods of planning of multifactorial medicobiological experiments, M., 1979, bibliogr.; Modeling of physiological systems of an organism, under the editorship of B. V. Petrovsky, M., 1971, bibliogr.; Novoseltsev V. N. Theory of management and biosystem, M., 1978, bibliogr.; Petrovsky A. M. Systems analysis of some medicobiological problems connected with management of treatment, Automatic equipment and telemechanics, No. 2, page 54, 1974; Sidorenko G. I. Cybernetics and therapy, M., 1970; Statistical methods of a research in medicine and health care, under the editorship of L. E. Polyakov, L., 1971; Theoretical researches of physiological systems, under the editorship of H. M. Amosova, Kiev, 1977, bibliogr.

V. N. Novoseltsev.