Various specific mathematical methods are applied to areas of biology and medicine such as taxonomy, ecology, epidemic theory, genetics, medical diagnostics, and health service organization.

Including classification methods as applied to problems of biological taxonomy and medical diagnostics, models of genetic linkage, epidemic spread and population growth, the use of operations research methods in organizational matters related to health care,

Mathematical models are also used for biological and physiological phenomena in which probabilistic aspects play a subordinate role and which are associated with the apparatus of control theory or heuristic programming.

Essentially, the important question is in which areas the mathematical methods are applicable. The need for a mathematical description arises in any attempt to conduct a discussion in precise terms and that this applies even to such complex areas as art and ethics. We will take a closer look at the areas of application of mathematics in biology and medicine.

So far, we have been referring mainly to those medical research that require a higher level of abstraction than physics and chemistry, but are closely related to these latter. Next, we turn to problems related to animal behavior and human psychology, that is, to the use of applied sciences to achieve some more general goals. This area is rather vaguely referred to as operations research. For now, we will only note that we will talk about the application of scientific methods in solving administrative and organizational problems, especially those that are directly or indirectly related to medicine.

In medicine, there are often complex problems associated with the use of drugs that are still at the testing stage. The physician is morally obliged to offer his patient the best available remedy, but in fact he cannot make a choice. Until the test is over. In these cases, the use of well-designed statistical test sequences can reduce the time required to obtain final results.

Ethical problems are not removed in this case, however, such a mathematical approach somewhat facilitates their solution.

The simplest study of recurring epidemics by probabilistic methods shows that this kind of mathematical description makes it possible to explain in general terms an important property of such epidemics - the periodic occurrence of outbreaks of approximately the same intensity, while the deterministic model gives a number of damped oscillations, which is not consistent with the observed phenomena. If more detailed, realistic models of bacterial mutations or recurring epidemics are to be developed, this information obtained from preliminary simplified models will be of great value. Ultimately, the success of the entire line of scientific research is determined by the capabilities of the models built to explain and predict real observations.

Any doctor or healthcare professional will confirm that they have used the same multiplication table, or the rules for calculating rational numbers, more than once.

Mathematics solves the problems of chemistry, physics, sociology and many other sciences. For a long time medicine developed "in parallel" with mathematics. Let's turn to history. An outstanding Italian physicist and astronomer, one of the founders of exact natural science, Galileo Galilei (1564-1642) said that "The book of nature is written in the language of mathematics." Almost two hundred years later, the founder of German classical philosophy, Immanuel Kant (1742-1804), asserted that "There is as much truth in any science as there is mathematics in it."

Mathematics is needed in medicine so as not to be mistaken in the doses of drugs, when you donate blood for analysis, laboratory assistants calculate the results to write, for example, how much hemoglobin in the blood you need to calculate, calculate, for this they use mathematics to calculate. Mathematics is needed everywhere: in the laboratory, in medicine, in computing. cardiology and so on.

Leonardo Da Vinci (1452-1519) Trying to find a mathematical basis for the laws of nature, considering mathematics a powerful means of knowledge, he applies it even in such a science as anatomy. He studied every part of the human body with the greatest care. Leonardo can be considered the best and greatest anatomist of his era. And, moreover, he is undoubtedly the first who laid the foundation for the correct anatomical drawing. The works of Leonardo in the form in which we have them at present are the result of the enormous work of scientists who deciphered them, selected them by topic and combined them into treatises in relation to the plans of Leonardo himself. Working on the depiction of human and animal bodies in painting and sculpture awakened in him the desire to learn the structure and functions of the human and animal body, led to a thorough study of their anatomy.

At present, mathematical methods are widely used in biophysics, biochemistry, genetics, physiology, medical instrument making, and the creation of biotechnical systems. The development of mathematical models and methods contributes to: expanding the field of knowledge in medicine; the emergence of new highly effective methods of diagnosis and treatment, which underlie the development of life support systems; creation of medical equipment.

IN last years the active introduction of mathematical modeling methods into medicine and the creation of automated, including computer systems, have significantly expanded the possibilities of diagnosing and treating diseases.

Mathematical statistics occupies an important place in modern medicine. Statistics (from the Latin status - state of affairs) - the study of the quantitative side of mass social phenomena in numerical form.

At the beginning, statistics were applied mainly in the field of socio-economic sciences and demography, and this inevitably forced researchers to study more deeply the issues of medicine.

The Belgian statistician Adolphe Quetelet (1796-1874) is considered the founder of the theory of statistics. He gives examples of the use of statistical observations in medicine: two professors made an interesting observation about the pulse rate - they noticed that there is a relationship between growth and the number of pulses. Age can affect the pulse only with a change in height, which in this case plays the role of a regulating element.

The number of beats of the pulse is thus inversely related to square root growth. Taking 1.684 m for the height of an average person, they assume the number of heartbeats to be equal to 70. With this data, it is possible to calculate the number of heartbeats in a person of any height.

The most active supporter of the use of statistics was the founder of military field surgery N.I. Pirogov. Back in 1849, speaking about the successes of domestic surgery, he pointed out: "The application of statistics to determine the diagnostic importance of symptoms and the dignity of operations can be regarded as an important acquisition of the latest surgery."

Gone are the days when the use of statistical methods in medicine was questioned. Statistical approaches underlie modern scientific research, without which knowledge in many areas of science and technology is impossible. It is also impossible in the field of medicine. Medical statistics should be aimed at solving the most pronounced contemporary problems in the health of the population. The main problems here, as you know, are the need to reduce morbidity, mortality and increase the life expectancy of the population. Accordingly, at this stage, the main information should be subordinated to the solution of this problem.

Mathematics is widely used in cardiology. Modern devices allow doctors to “see” a person from the inside, correctly diagnose and prescribe effective treatment. The creation of such devices is carried out by engineers using the apparatus of physical and mathematical research. Heart rhythms and movement mathematical pendulum, the growth of bacteria and the geometric progression, the DNA formula are all examples of the application of mathematical calculations in medicine.

Simulation is one of the main methods to speed up technical process, reduce the time for mastering new processes. Nowadays, mathematics is increasingly called the science of mathematical models. Models are created for different purposes - to predict the behavior of an object over time; actions on the model that cannot be performed on the object itself; presentation of an object in an easy-to-view form and others. A model is a material or ideal object that is built to study the original object and which reflects the most important qualities and the parameters of the original. The process of creating models is called modeling. Models are divided into material and ideal. Material models, for example, can be photographs, layouts of building areas, etc. ideal models often have an iconic shape.

Mathematical modeling belongs to the class of sign modeling. Real concepts can be replaced by any mathematical objects: numbers, equations, graphs, etc., which are recorded on paper, in the computer's memory. Models are dynamic and static. The time factor is involved in dynamic models. In static models, the behavior of the modeled object depending on time is not taken into account. So, modeling is a method of studying objects, in which, instead of the original (the object of interest to us), the experiment is carried out on the model (another object), and the results are quantitatively extended to the original. Thus, based on the results of experiments with the model, we must quantitatively predict the behavior of the original in working conditions. Moreover, the extension to the original of the conclusions obtained in experiments with the model should not necessarily mean a simple equality of certain parameters of the original and the model. It is enough to get a rule for calculating the parameters of the original that are of interest to us. There are two main requirements for the modeling process.

First, the experiment on the model should be simpler, faster than the experiment on the original.

Secondly, we must know the rule according to which the parameters of the original are calculated based on the model test. Without this, even the best model research will be useless. Statistics is the science of methods of collecting, processing, analyzing and interpreting data characterizing mass phenomena and processes, i.e. phenomena and processes that affect not individual objects, but whole sets. A distinctive feature of the statistical approach is that the data characterizing the statistical population as a whole is obtained as a result of generalizing information about its constituent objects. The following main directions can be distinguished: methods of data collection; measurement methods; methods of data processing and analysis. Data processing and analysis methods include probability theory, mathematical statistics and their applications in various fields of technical sciences, as well as the sciences of nature and society.

Mathematical statistics develops methods of statistical processing and analysis of data, is engaged in the justification and verification of their reliability, efficiency, conditions of use, resistance to violation of the conditions of use, etc. In some areas of knowledge, applications of statistics are so specific that they are singled out into independent scientific disciplines: reliability theory - in technical sciences; econometrics - in economics; psychometry - in psychology, biometrics - in biology, etc. These disciplines consider industry-specific methods for collecting and analyzing data.

Examples of the use of statistical observations in medicine. Two renowned professors from the Strasbourg Faculty of Medicine, Rameau and Sarru, made an interesting observation about the rate of heart rate. Comparing observations, they noticed that there is a relationship between growth and heart rate. Age can affect the pulse only with a change in height, which in this case plays the role of a regulating element. The number of heartbeats is thus inversely related to the square root of growth. Assuming 1.684 m for the height of an average person, Rameau and Sarru assume the number of heartbeats to be equal to 70. With this data, it is possible to calculate the number of heartbeats in a person of any height. In fact, Quetelet anticipated dimensional analysis and allometric equations in relation to the human body... Allometric equations: from the Greek. alloios is different.

In biology, a large number of morphological and physiological parameters depend on the size of the body; this dependence is expressed by the equation: y = a * xb.

Biometrics is a branch of biology, the content of which is the planning and processing of the results of quantitative experiments and observations by methods of mathematical statistics. When conducting biological experiments and observations, the researcher always deals with quantitative variations in the frequency of occurrence or the degree of manifestation of various signs and properties. Therefore, without special statistical analysis it is usually impossible to decide what are the possible ranges of random fluctuations of the studied quantity and whether the observed differences between the variants of the experiment are accidental or reliable. The mathematical and statistical methods used in biology are sometimes developed independently of biological research, but more often in connection with problems arising in biology and medicine. The use of mathematical and statistical methods in biology is the choice of a certain statistical model, verification of its compliance with experimental data and the analysis of statistical and biological results arising from its consideration. When processing the results of experiments and observations, 3 main statistical problems arise: estimation of distribution parameters; comparison of parameters of different samples; identification of statistical relationships.

GOU SPO "Moscow Medical School No. 21"

Mathematics in medicine

Completed: student 111gr.

Sorokina Natalia

Checked by: Kadochnikova

Lydia Konstantinovna

Moscow 2011

Plan:

Introduction

The value of mathematics to the healthcare professional

Mathematical Methods and Statistics in Medicine

Examples of

Conclusion

Bibliography

Introduction

The role of mathematics education in vocational training medical professionals is very large.

The processes taking place at the present time in all spheres of the life of society place new demands on the professional qualities of specialists. The modern stage of development of society is characterized by a qualitative change in activities medical staff, which is associated with the widespread use of mathematical modeling, statistics and other important phenomena that take place in medical practice. mathematics health worker statistic

At first glance, medicine and mathematics may seem like incompatible areas of human activity. Mathematics, admittedly, is the "queen" of all sciences, solving problems of chemistry, physics, astronomy, economics, sociology and many other sciences. Medicine, for a long time developing "in parallel" with mathematics, remained practically an unformalized science, thereby confirming that "medicine is an art."

The main problem is that there are no general health criteria, and the set of indicators for one particular patient (conditions when he feels comfortable) may differ significantly from the same indicators for another. Often doctors are faced with common problems formulated in medical terms in order to help the patient, they do not bring ready-made problems and equations to be solved.

When applied correctly, the mathematical approach does not differ significantly from the approach based simply on common sense. Mathematical methods are simply more precise and employ clearer formulations and a broader set of concepts, but ultimately they must be compatible with, though they probably go beyond, conventional verbal reasoning.

1. The value of mathematics to the healthcare professional

Currently, according to the requirements of state standards and current training programs in medical institutions, the main task of studying the discipline "Mathematics" is to equip students with mathematical knowledge and skills necessary to study special tasks using mathematical methods. This situation cannot but affect the results of the mathematical training of physicians. The level of professional competence of the medical staff depends to a certain extent on these results. These results show that, studying mathematics, in the future, health workers acquire certain professionally significant qualities and skills, and also apply mathematical concepts and methods in medical science and practice.

Professional orientation of mathematical training in medical educational institutions should ensure an increase in the level of mathematical competence of medical students, an awareness of the value of mathematics for the future professional activity, the development of professionally significant qualities and techniques of mental activity, the development by students of the mathematical apparatus that allows them to model, analyze and solve elementary mathematical professionally significant problems that take place in medical science and practice, ensuring the continuity of the formation of the mathematical culture of students from the first to senior courses and raising the need for improving knowledge in the field of mathematics and its applications.

2. Mathematical Methods and Statistics in Medicine

At the beginning, statistics were applied mainly in the field of socio-economic sciences and demography, and this inevitably forced researchers to study more deeply the issues of medicine.

The Belgian statistician Adolphe Quetelet (1796-1874) is considered the founder of the theory of statistics. He gives examples of the use of statistical observations in medicine: Two professors made an interesting observation about the rate of the pulse. Comparing my observations with their data, they noticed that there is a relationship between growth and heart rate. Age can affect the pulse only with a change in height, which in this case plays the role of a regulating element. The number of heartbeats is thus inversely related to the square root of growth. Taking 1.684 m for the height of an average person, they assume the number of heartbeats to be equal to 70. With this data, it is possible to calculate the number of heartbeats in a person of any height .

The most active supporter of the use of statistics was the founder of military field surgery, N.I.Pirogov. Back in 1849, speaking about the successes of Russian surgery, he pointed out: The application of statistics to determine the diagnostic importance of symptoms and the merit of operations can be considered an important acquisition of the latest surgery. .

In the 60s of the XX century, after the obvious successes of applied statistics in technology and the exact sciences, interest in the use of statistics in medicine began to grow again. V.V. Alpatov in the article On the role of mathematics in medicine wrote: The mathematical assessment of the therapeutic effects on a person is extremely important. New therapeutic measures have the right to replace measures that have already entered into practice only after justified statistical tests of a comparative nature. ... The statistical theory can be used enormously in the formulation of clinical and non-clinical trials of new therapeutic and surgical measures.

Gone are the days when the use of statistical methods in medicine was questioned. Statistical approaches underlie modern scientific research, without which knowledge in many areas of science and technology is impossible. It is also impossible in the field of medicine.

Medical statistics should be aimed at solving the most pronounced modern problems in the health of the population. The main problems here, as you know, are the need to reduce morbidity, mortality and increase the life expectancy of the population. Accordingly, at this stage, the main information should be subordinated to the solution of this problem. Data should be carried out in detail, characterizing from different sides the leading causes of death, morbidity, frequency and nature of contacts of patients with medical institutions, providing those in need with the necessary types of treatment, including high-tech ones.

3. Examples of

Objective 1.According to the doctor's prescription, the patient is prescribed a drug 10 mg, 3 tablets per day. He has a 20 mg drug available. How many tablets should the patient take without violating the doctor's instructions?

Solution:

10 mg. - 1 tablet 10 * 3 = 30 mg per day.

The dosage is 2 times exceeded. (20: 10 = 2)

20 = 10 mg is not enough

Thus, the patient should drink 1.5 x 20 mg instead of 3 x 10 mg, without violating the prescribed dose.

Objective 2.The course of air baths starts from 15 minutes on the first day and increases the time of this procedure by 10 minutes each next day. How many days should you take air baths in the specified mode in order to reach their maximum duration of 1 hour and 45 minutes?

NS 1= 15, d = 10, x n = 105 minutes

NS n = x 1+ d (n - 1).

NS n = 15 + d (n - 1) x n = 15 + 10n - 10.

n = 100. n = 10 Answer. 10 days

Problem number 3

The child was born 53cm tall. How tall should he be at 5 months, 3 years?

Solution:

The growth for each month of life is: in the 1st quarter (1-3 months) by 3 cm. for every month,

In the 2nd quarter (4-6 months) - 2.5 cm, in the 3rd quarter (7-9 months) - 1.5 cm, in the 4th quarter (10-12 months) - 1 , 0cm.

The height of a child after a year can be calculated using the formula: 75 + 6n

Where 75 is the average height of a child at 1 year, 6 is the average annual increase, n is the age of the child

Child's height at 5 months: X = 53 + 3 * 3 + 2 * 2.5 = 67cm

Child's height at 3 years old: X = 75+ (6 * 3) = 93cm

Conclusion

Recently, a friend and I observed the following picture at the City Clinical Hospital: two nurses were solving the following arithmetic problem: "One hundred ampoules of five in a box - how many boxes will there be? Okay, let's write 100 ampoules, and then let them count themselves." We laughed for a long time: how is that? Elementary things!

Medical science, of course, does not lend itself to total formalization, as happens, say, with physics, but the colossal episodic role of mathematics in medicine is beyond doubt. All medical discoveries must be based on numerical ratios. And the methods of the theory of probability (taking into account the statistics of morbidity depending on various factors) are absolutely necessary in medicine. In medicine, you can't take a step without mathematics. Numerical ratios, for example, taking into account the dose and frequency of medication. Numerical consideration of related factors, such as: age, physical parameters of the body, immunity, etc.

My opinion is firmly on the fact that doctors should not close their eyes to at least elementary mathematics, which is simply necessary to organize fast, accurate and high-quality work. Each student should note for himself the importance of mathematics from the first year of study. And understand that not only in work, but also in Everyday life this knowledge is important and makes life much easier.

Bibliography:

www..aspx“Mathematics in Medicine. Statistics"

10.02.2018

One can treat mathematics as the “queen of all sciences” in different ways: it is easy for some, while others should sweat to achieve a result.

If you initially look at such two sciences as mathematics and medicine, then you are not sure to find something in common. However, medical professionals need to be well versed in mathematical issues, since the roles of these disciplines are complementary.

History of the development of mathematics and medicine

Historically, astronomy and physics were closely related to mathematical calculations. Medicine, on the other hand, was developed by the side, and for a long time was not recognized formally. After its formation as a science, the connection between mathematics and medicine became inseparable.

Galileo argued that the whole essence of nature depends on mathematics. Kant and Leonardo da Vinci were of the same opinion. The Italian artist applied the methods of mathematics in order to study all aspects of anatomy. The first connected chains between the two sciences were found in the drawing "Vitruvian Man", which depicts a man, a circle and a square. It clearly illustrates the canonical proportions, the ratio of body parts.

The legendary creation of Leonardo da Vinci

The importance of mathematics in medicine

The role of mathematics in medicine is to help in carrying out diagnostic procedures, using a computer, and medical equipment. To date, the methods of treatment and diagnostics have expanded: the majority medical centers use methods of mathematical modeling, which helps to establish a more accurate diagnosis.

Knowledge of the basics of mathematics is used by doctors to describe the processes occurring in the human body. This is necessary, since it allows you to distinguish between a diseased organism and a healthy one by the taken pictures and monitor screens. Most educational institutions along with basic medical disciplines, students study mathematics. It is believed that healthcare professionals should be able to solve professional problems using mathematical methods.

What mathematics has gained from medicine

Do not think that doctors need math more than she needs them. These two sciences played an important role in joint development, complemented each other. Under the influence of biomedical problems, new computational algorithms and mathematical concepts have emerged. For instance:

• theory of automata;
• mathematical statistics;
• theory of probability;
• optimal control methods;
• game theory.

According to history, medicine plays an important role in the development of mathematics. Specialists were able to learn a lot thanks to the influence of medicine. New knowledge was successfully applied in other disciplines, technology, science.

The use of mathematics in medicine: examples

One of the striking examples of the combination of these two sciences is statistics. Adolphe Quetelet is the founder of the theory of statistics. The scientist gave the following example of using statistical data to solve a medical problem.

Some professors have made conclusions about the rate of heart rate. Quetelet compared their observations with his own, and found that there is a relationship between heart rate and height. Age has an effect when the amount of growth changes. Heart rate is inversely related to the square root of growth.

If a person's height is 1.68 m, then the heart rate will be equal to 70. Thus, this allows you to determine the pulse of a person of any height.

The role of statistical observations is quite important: they can be used anywhere and in any way. For example, in the news you can often hear such phrases "according to statistics, the incidence increased by 30%" - these conclusions are made on the basis of mathematics.

Other examples of using math:

1. Reading X-ray tomography and other diagnostic methods.
2. Calculation of the dosage of drugs.
3. Collection and compilation of statistics.
4. Prognosis of improvement or worsening of the condition.
5. Working with computer equipment, filing reports.

Math saved lives

You can better understand why mathematics is in medicine by reading not only Interesting Facts, but also a life story about how she saved the girl's life. Vicki Alex was a 14-year-old schoolgirl. Suddenly she began to have problems with respiratory system... Her family could not understand what was the matter until the doctors announced the diagnosis - blood cancer.

The girl was prescribed a long course of treatment, which really helped her until Vicki began to feel the symptoms of a cold. Further, a lump jumped on the back, which doctors diagnosed as a boil. The specialists prescribed antibiotics.

Such drugs strongly affect even the strongest person, not to mention a child with a weakened immune system. The body could not get rid of the infection, and it was decided to put Vicki into a coma in order to be able to use the drugs. But the doctors said right away that even if the medication worked, the girl had no chance of returning from a coma. After the drug course, the doctors tried to return the girl to consciousness, but nothing worked. Another way to get out of a coma is the voices of family members. The parents were allowed into the room, and they spent days talking with their daughter about everything in the world. But nothing helped.

Suddenly, the father remembered an interesting fact from the life of her biography: his daughter was very fond of counting. He began to ask simple things, for example, how much would 1 plus 1. And then, his daughter's lips barely perceptibly moved, and the father asked: “Two?”. The patient nodded slightly. Gradually, the father began to give his daughter more difficult tasks and Vicki woke up the same day.

This is certainly not an example of the absolute participation of mathematics in saving a person, but it shows its role in improving health. The case clearly illustrates how the brain loves to solve peculiar math problems.

The left hemisphere of the brain helps solve math problems

There are no less interesting facts describing the connection between mathematics and medicine. So, the mathematician was able to calculate when he would die. As an old man, he found himself sleeping more. Each day, the duration of sleep increased by 15 minutes. Thanks to an arithmetic progression, he calculated the date when sleep reaches 24 hours.

Other interesting facts about medicine that could not have been determined without the use of mathematics:

1. When talking, 72 muscles are tense.
2. The brain needs only 10 watts of energy to function.
3. The human skeleton consists of 206 bones, 25% of which are located in the lower extremities.
4. The capillary chain of the lungs would exceed 2400 km in length.
5. Filtration in the kidneys is as follows: 1.3 liters of blood for 60 seconds and urine output of 1.4 liters daily.
6. The heat generated by the human body will boil 2 liters of water.
7. Growth increases by 8 mm during sleep, but after waking up, it returns to its previous figures. The law of gravity is to blame.

Department of Education g. About. Saransk

Municipal educational institution

"Lyceum No. 43"

Research work

Methods of mathematical analysis in medicine

Completed by: Ulanov Kirill

student of grade 11 B

MOU "Lyceum No. 43"

Saransk.

Supervisor:

mathematic teacher

biology teacher

Academic Supervisor: Professor

Saransk 2012

Introduction ................................................. .................................................. ........................... 3

Chapter 1. Theoretical basis studying the state of the cardiovascular system

1.1. Morphology of the human heart and its internal structures …………………….… 7

1.2. The concept of the human circulatory system, blood pressure ... ................. 10

1.3. Theoretical foundations of the study of the work of the heart. Research methodology

daily monitoring of the electrocardiogram ………………………… ....… .... 14

1.4. Theoretical foundations of the study of blood pressure. Methodology

studies of daily monitoring of blood pressure ……………………………………………………………………………………… ..... 15

Conclusions for chapter # 1 ............................................. .................................................. ..........sixteen

Chapter 2. Practical part.

2.1 Description of the experimental procedure ............................................. ............................... 17

2.2. Results and mathematical analysis of indicators of daily monitoring

electrocardiograms ………. …………………………… .. …………………………… 18

2.3. Results and mathematical analysis of indicators of daily monitoring

blood pressure ………………………………………………. ………… .. …… 20

Conclusions on Chapter 2 ............................................. .................................................. .......... 22

Conclusion................................................. .................................................. ...................... 23

Bibliography................................................ .................................................. .......... 24

Applications ................................................. .................................................. ..................... 25

Introduction

Medical science does not lend itself to total formalization and the colossal role of mathematics in medicine is beyond doubt. All medical discoveries are based on numerical ratios. Mathematical methods in medicine are a set of techniques for studying the processes occurring in living organisms, their populations, in the field of health protection, using quantitative methods for describing phenomena and objects of biomedical nature, as well as the connections between them.
In medicine, mathematical methods are used to establish the degree of reliability and generalize information obtained in the course of clinical, biomedical, and laboratory research. The need to attract mathematics to medicine is associated with the lack of other opportunities to overcome the difficulties inherent in the study of biological objects: the high variability of individual indicators of the state of organs, physiological systems, biochemical processes of the whole organism in health and disease. In addition, mathematical statistical methods are important in medicine as a means of accumulating and systematizing information; they make it possible to put forward and test, confirm or refute, the meaningfulness of hypotheses about the connection of the processes and phenomena under study by quantifying the relationships.
The purpose of mathematical methods in medicine is to increase the reliability and objectivity of decisions made by specialists. An important direction in this area is associated with the choice of the most convenient presentation of information for a specialist.

[Schmidt, V. M. 1985]

Mathematical methods include a wide variety of approaches and directions. Well-known methods of systematization and presentation of biomedical data (tables, graphs, nomograms, histograms) are complemented by extremely visual forms of visual presentation of information using computer programs. Mathematical methods cover a range of biomedical problems that lend themselves to mathematical description, in the form of equations based on experimental and clinical observations or theoretical considerations. A set of equations, often very complex, describing various aspects of the functioning of an object (organism, biological system) or interacting objects are mathematical models. Mathematical models are most effectively used to study the impact of therapeutic or damaging factors on the body and its individual systems, to predict the development of individual areas of medicine. [, 2002]

Many researchers are interested in the role of the relationship between such a branch of medicine as cardiology and mathematics. Cardiology is a direction in medicine that studies the structure and functions, diseases of the heart and blood vessels, studies the causes and mechanisms of development of diseases, clinical manifestations and diagnostic methods. In this area, systems for the development of methods of treatment and prevention of heart diseases, issues of rehabilitation of patients with cardiovascular diseases are involved. Cardiovascular diseases account for 57% of the total mortality in Russia. There is no such high indicator in any developed country in the world! A year from cardiovascular diseases in Russia, 1 million 300 thousand people die - the population of a large regional center.

As you know, the circulatory system is important for humans. It delivers oxygen, water, proteins, carbohydrates, fats, minerals, vitamins to organs and tissues and removes carbon dioxide, etc. harmful products exchange formed in the process of life; provides heat regulation and humoral regulation in the body, is an important factor in immunity.

Blood movement through the vessels arises due to the pumping function of the heart. The aorta and arteries of the body are a pressure reservoir in which blood is under high pressure. The heart throws blood into the arteries in separate portions. At the same time, the elastic walls of the arteries are stretched, therefore, during diastole, the energy accumulated by them maintains the blood pressure in the arteries at a certain level, which ensures the continuity of blood flow in the capillaries. The blood pressure level depends on the resistance of the peripheral vessels. The work of the mechanisms that regulate blood circulation is primarily aimed at satisfying the oxygen demand of organs and tissues.

Significant progress in the study of the complex circulatory system has been achieved through the use of mathematical methods in the study of the cardiovascular system. The correct interpretation of mathematical indicators in the study of the heart and blood vessels plays an important role in the early diagnosis of complex diseases and helps the doctor to achieve good results in the treatment of the patient. Mathematical indicators of the heart and blood pressure can act as integral markers of the functional state of the cardiovascular system and the whole organism.

The relevance of this work is that in a holistic assessment of health and the state of the body's adaptive processes, the state of the cardiovascular system plays the main role. Assessment of the functional state of the body is quite complicated and requires a comprehensive examination of all organs and systems, which cannot always be carried out in full. From these positions, the mathematical indicators of the heart and blood pressure can act as integral markers of the functional state of the cardiovascular system and the whole organism.

Purpose of the study is to determine the relationship of mathematical indicators of heart rate and blood pressure

Object of study: human cardiovascular system (heart, blood vessels).

Subject of study: indicators of the heart and blood pressure.

Hypothesis research: methods of mathematical analysis help to identify violations of the heart and blood pressure .

Tasks:

1. To study the literature and find out the theoretical foundations of the methods of mathematical analysis.

2. To characterize the indicators of the heart and blood pressure. heart and blood vessels

3 To get acquainted with the methods of research of the heart and blood vessels.

4. Determine the methods of mathematical analysis in heart studies (daily monitoring of the electrocardiogram - Holter - ECG)

5. Determine the methods of mathematical analysis in the study of blood pressure (daily monitoring of blood pressure - ABPM).

6 Conduct a comparative analysis of the use of mathematical methods of indicators of the work of the human heart and blood vessels.

6. Establish and study the relationship between the mathematical indicators of the heart and blood pressure and the functioning of the cardiovascular system .

7. Based on the results obtained, build comparative tables and diagrams.

Scientific novelty The study consists in a systematic analysis of the use of mathematical methods in the study of heart and blood pressure. It is proposed to apply together a number of mathematical indicators. The prospect of using mathematical indicators of the heart and blood pressure has been proven.

Theoretical significance. Scientifically based data are of interest from the point of view of methods of mathematical analysis in medicine due to the openness and relevance of this issue on the present stage development of mathematics and medicine.

Practical significance the research consists in identifying patterns between the mathematical indicators of the heart and changes in blood pressure; in the possibility of using the research results in optional classes and lessons in mathematics, biology in mainstream schools, mathematical and medical faculties UNIVERSITIES.

Research methods:

1. Theoretical - the study of literature, setting goals and objectives.

2. Experimental - detailing measurements of the work of the heart and blood vessels, approbation, testing of the studied phenomena in controlled and controlled conditions, obtaining the required information.

3. Empirical - observation, description, interpretation and explanation of the results of the experiment.

4. Analytical method - analysis of individual sides, signs, properties and tracking the dynamics of the phenomenon under consideration for a certain period.

5. Statistical method

6. Comparative analysis.

Chapter 1. Theoretical foundations of studying the state of the cardiovascular system.

1.1. Morphology of the human heart and its internal structures.

Among the indicators of the state of the body, the most important are data on the activity of the cardiovascular system.

The human heart is a cone-shaped hollow muscular organ, which receives blood from the venous trunks flowing into it, and pumps it into the arteries that adjoin the heart. The heart cavity is divided into 2 atria and 2 ventricles. The left atrium and the left ventricle together form the "arterial heart", so named after the type of blood passing through it, the right ventricle and the right atrium - the "venous heart". Contraction of the heart is called systole, relaxation is called diastole. The shape of the heart is not the same in different people. It is determined by age, gender, physique, human health. In simplified models, it is described by a sphere. The measure of the elongation of the heart is the ratio of the largest longitudinal and transverse linear dimensions of the heart. With a hypersthenic body type of a person, the ratio is close to 1.0 and asthenic -1.5. The length of the heart of an adult varies from 10 to 15 cm, the width at the base is 8-11 cm and the anteroposterior size is 6-8.5 cm. The average heart mass in men is 332 g, in women - 253 g.

In relation to the midline of the body, the heart is located asymmetrically - about 2/3 to the left of it and about 1/3 to the right. There are transverse, oblique and vertical position of the heart. Performing pumping functions in the circulatory system, the heart constantly pumps blood into the arteries.

The heart is located on the left side of the chest in the pericardial sac - the pericardium, which separates the heart from other organs. The wall of the heart consists of three layers - epicardium, myocardium, endocardium. The epicardium consists of a thin (no more than 0.3-0.4 mm) lamina of connective tissue, the endocardium - of epithelial tissue, and the myocardium - of the cardiac striated muscle tissue.

The heart is made up of four separate chambers called chambers: left atrium, right atrium, left ventricle, right ventricle. They are separated by partitions. The hollow veins enter the right atrium, and the pulmonary veins enter the left atrium. The pulmonary artery (pulmonary trunk) and the ascending aorta leave the right ventricle and left ventricle, respectively. The right ventricle and the left atrium close the pulmonary circulation, the left ventricle and the right atrium complete the large circle. The wall of the left ventricle is 3 times thicker than the wall of the right ventricle, since the left ventricle pushes blood into the systemic circulation. In addition, the blood resistance in the systemic circulation is several times higher, and the blood pressure is higher than in the small circle. [, 2010]

There is a need to maintain blood flow in one direction, otherwise the heart could be filled with that very blood. The flow of blood in one direction is regulated by valves that open and close at the appropriate moment, allowing blood to flow through or blocking it. The valve between the left atrium and the left ventricle is mitral or bicuspid, since it consists of two lobes. The valve between the right atrium and the right ventricle is tricuspid, consisting of three lobes. The heart also contains the aortic and pulmonary valves. They control the flow of blood from both ventricles. (Annex 1)

Every cell of the heart muscle must have a constant supply of oxygen and nutrients. This process ensures the heart's own blood circulation, that is, the coronary circulation of the two arteries, which, like a crown, entwine the heart. The coronary arteries branch off from the aorta. [, 1976]

In one cycle of the heart, three phases are distinguished:

1) The blood-filled atria contract. In this case, blood is pumped into the ventricles of the heart through the open bicuspid and tricuspid valves. The contraction of the atria begins from the place where the veins fall into it, so their mouths are compressed and the blood cannot get back into the veins.

2) There is a contraction of the ventricles with a simultaneous relaxation of the atria. The tricuspid and bicuspid valves that separate the atria from the ventricles rise, close, and prevent blood from returning to the atria, and the aortic and pulmonary valves open. The contraction of the ventricles pumps blood into the aorta and pulmonary artery.

3) A pause (diastole) is a relaxation of the whole heart, or a short period of rest. During a pause, blood from the veins enters the atria and partially flows into the ventricles. When a new cycle begins, the blood remaining in the atria will be pushed into the ventricles - the cycle will repeat.

One cycle of work of the heart lasts about 0.85 seconds, of which the time of atrial contraction is 0.11 seconds, of the ventricles - 0.32 seconds, the rest period is 0.4 seconds. The heart of an adult at rest works in the system for about 70 cycles per minute, that is, the heart rate is 70 beats per minute, and the stroke volume of blood is 70 ml per beat. The heart pumps about 5 liters of blood per minute. This figure is determined by the need for oxygen in the myocardium and the body. During the maximum load, the stroke volume of a trained person's heart can exceed 200 ml, the pulse can exceed 200 beats per minute, and the blood circulation can reach 40 liters per minute.

A certain part of the heart muscle specializes in issuing control signals to the rest of the heart in the form of corresponding electrical impulses. These parts of muscle tissue are called the excitatory-conducting system. Its main part is the sinus-atrial node, called the pacemaker, placed on the right atrial fornix. It controls the rate at which the heart works by sending out regular electrical impulses. The electrical impulse through the pathways in the atrial muscle enters the atrioventricular atrio-ventricular node. The excited node sends an impulse to the bundles of His and Purkinje fibers further, to individual muscle cells, causing them to contract. The excitatory-conducting system ensures the rhythmic work of the heart with the help of synchronized contraction of the atria and ventricles.

Thus, the following main functions of the heart are distinguished:

Automatism is the ability of the heart to generate impulses that cause excitement. Normally, the sinus node has the greatest automatism.

Conductivity - the ability of the myocardium to conduct impulses from the place of their origin to the contractile myocardium.

Excitability - the ability of the heart to get excited under the influence of impulses. During arousal, there is electricity, which is recorded by a galvanometer in the form of an electrocardiogram.

Contractility - the ability of the heart to contract under the influence of impulses and provide a pump function.

Refractoriness - the inability of excited myocardial cells to reactivate when additional impulses occur. Refractoriness is divided into: absolute (the heart does not respond to any excitement) and relative (the heart responds to very strong excitement).

The regulation of the frequency and strength of heart contractions is carried out by the nervous and endocrine systems. Sympathetic nervous system causes an increase in the contractions of the heart muscle, parasympathetic - weakens. The main gland for secreting hormones is the adrenal glands, which secrete adrenaline and acetylcholine, whose functions relative to the heart correspond to those of the sympathetic and parasympathetic systems. [, 2007] The same work is performed by the ions Ca and K. [, 1995]

1.2. The concept of the human circulatory system, blood pressure.

All the main functions of blood are realized due to its constant circulation in the body through the circulatory system, which consists of a pumping organ - the heart, which acts as a pump, and vessels that deliver blood to various organs and tissues. Blood circulation occurs along two main paths, called circles: small and large circle of blood circulation.

In a small circle, blood circulates through the lungs. The movement of blood in this circle begins with a contraction of the right atrium, after which blood enters the right ventricle of the heart, the contraction of which pushes blood into the pulmonary trunk. Blood circulation in this direction is regulated by the atrioventricular septum and two valves: a tricuspid valve (between the right atrium and the right ventricle), which prevents blood from returning to the atrium, and a pulmonary artery valve, which prevents blood from returning from the pulmonary trunk to the right ventricle. The pulmonary trunk branches out to a network of pulmonary capillaries, where the blood is saturated with oxygen through ventilation of the lungs. Then blood through the pulmonary veins returns from the lungs to the left atrium.

The systemic circulation supplies organs and tissues with oxygenated blood. The left atrium contracts simultaneously with the right and pushes blood into the left ventricle. From the left ventricle, blood enters the aorta. The aorta branches into arteries and arterioles, going to various parts of the body and ending in a capillary network in organs and tissues. Blood circulation in this direction is regulated by the atrioventricular septum, the bicuspid (mitral) valve, and the aortic valve.

Blood moves through the systemic circulation from the left ventricle to the right atrium, and then along the pulmonary circulation from the right ventricle to the left atrium. The movement of blood through the vessels is carried out due to the pressure difference between the arterial system and the venous system. This statement is completely true for arteries and arterioles; auxiliary mechanisms appear in capillaries and veins. The pressure difference is created by the rhythmic work of the heart, pumping blood from the veins to the arteries. Since the pressure in the veins is very close to zero, this difference is taken for practical purposes, equal to the arterial pressure. The right half of the heart and the left one work synchronously.

The cardiac cycle includes general diastole (relaxation), atrial systole (contraction), and ventricular systole. During general diastole, the pressure in the heart cavities is close to zero, in the aorta it slowly decreases from systolic to diastolic pressure. Normally, a person's blood pressure is 120 and 80 mm Hg, respectively. Art. Because the pressure in the aorta is higher than in the ventricle, the aortic valve is closed. The pressure in the large veins (central venous pressure) is 2-3 mm Hg. Art., slightly higher than in the cavities of the heart, so that blood enters the atria and, in transit, into the ventricles. The atrioventricular valves are open at this time.

During atrial systole, the circular muscles of the atria squeeze the entrance from the veins into the atria, which prevents the backflow of blood, the pressure in the atria rises to 8-10 mm Hg. Art., and the blood moves to the ventricles.

During the subsequent systole of the ventricles, the pressure in them becomes higher than the pressure in the atria (which begin to relax), which leads to the closure of the atrioventricular valves .. Then the pressure in the ventricle exceeds the aortic, as a result of which the aortic valve opens and the expulsion of blood from the ventricle into the arterial begins. system. The relaxed atrium fills with blood at this time. The physiological significance of the atria mainly consists in the role of an intermediate reservoir for blood coming from the venous system during ventricular systole.

At the beginning of the total diastole, the pressure in the ventricle drops below the aortic (closing of the aortic valve), then below the pressure in the atria and veins (opening of the atrioventricular valves), the ventricles begin to fill with blood again.

Arteries, which contain almost no smooth muscle, but have a powerful elastic membrane, perform mainly a "buffer" role, smoothing out the pressure differences between systole and diastole. The walls of the arteries are elastically extensible, which allows them to receive an additional volume of blood, "thrown in" by the heart during systole, and only moderately, by 50-60 mm Hg. Art. raise the pressure. During diastole, when the heart is not pumping anything, it is the elastic stretching of the arterial walls that maintains the pressure, preventing it from dropping to zero, and thereby ensures the continuity of blood flow. It is the stretching of the vessel wall that is perceived as a pulse beat. Arterioles have developed smooth muscles, thanks to which they are able to actively change their lumen and, thus, regulate the resistance to blood flow. It is the arterioles that account for the greatest pressure drop, and it is they that determine the ratio of blood flow and blood pressure. Accordingly, arterioles are called resistive vessels.

Capillaries are characterized by the fact that their vascular wall is represented by one layer of cells, therefore they are highly permeable to all low molecular weight substances dissolved in blood plasma. Here, the exchange of substances between tissue fluid and blood plasma takes place. When blood passes through the capillaries, blood plasma is completely renewed 40 times with interstitial (tissue) fluid. The volume of diffusion through the common exchange surface of the body's capillaries is about 60 l / min or l / day. The pressure at the beginning of the arterial part of the capillary is 37.5 mm Hg. Art. The effective pressure is about (37.5 - 28) = 9.5 mm Hg. Art .. The pressure at the end of the venous part of the capillary, directed outward of the capillary, 20 mm Hg. st .. Effective reabsorption pressure is about (20 -28) = - 8 mm Hg. Art.

From the organs, blood returns through the postcapillaries to the venules and veins into the right atrium along the superior and inferior vena cava, along the coronary veins.

Venous return is mediated by several mechanisms.

1) basic mechanisms due to the pressure difference at the end of the venous part of the capillary directed outward of the capillary about 20 mm Hg. Art, effective reabsorption pressure directed into the capillary, about (20 -28) = minus 8 mm Hg. Art .;

2) for the veins of skeletal muscles, it is important that when the muscle contracts, the pressure "from the outside" exceeds the pressure in the vein, so that the blood is "squeezed" from the veins of the contracted muscle. The presence of venous valves determines the direction of blood flow in this case - from the arterial end to the venous end. This mechanism is especially important for the veins of the lower extremities, since here the blood rises through the veins, overcoming gravity. Thirdly, the suction role of the chest. During inhalation, the pressure in the chest drops below atmospheric pressure (which we take for zero), which provides an additional mechanism for blood return. The size of the lumen of the veins, and accordingly their volume, significantly exceed those of the arteries. In addition, the smooth muscles of the veins provide a change in their volume over a wide range, adapting their capacity to the changing volume of circulating blood. Therefore, the physiological role of veins is defined as "capacitive vessels".

Stroke volume of the heart(Vcontr) - the volume that the left ventricle ejects into the aorta (and the right one into the pulmonary trunk) in one contraction. In humans, it is 50-70 ml.

Minute blood flow(Vminute) - the volume of blood passing through cross section aorta (and pulmonary trunk) in a minute. In an adult, the minute volume is approximately 5-7 liters.

Heart rate(Freq) - the number of heartbeats per minute.

Arterial pressure- blood pressure in the arteries.

Systolic pressure- the highest pressure during the cardiac cycle, reached by the end of the systole.

Diastolic pressure- the lowest pressure during the cardiac cycle is reached at the end of the ventricular diastole.

Pulse pressure- the difference between systolic and diastolic.

Mean arterial pressure(Pmean) is easiest to define as a formula. So, if blood pressure during the cardiac cycle is a function of time, then

where tbegin and tend are the start and end times of the cardiac cycle, respectively.

The physiological meaning of this value: it is such an equivalent pressure that, if it were constant, the minute volume of blood flow would not differ from that observed in reality.

Total peripheral resistance- the resistance that the vascular system has to blood flow. It cannot be measured directly, but it can be calculated based on the minute volume and mean arterial pressure.

(3) Minute blood flow is equal to the ratio of mean arterial pressure to peripheral resistance.

This statement is one of the central laws of hemodynamics.

The resistance of a single vessel with rigid walls is determined by Poiseuille's law:

(4)

where is the viscosity of the liquid, R is the radius, and L is the length of the vessel.

For vessels connected in series, the resistances add up:

(5)

For parallel, add conductivities:

(6)

Thus, the total peripheral resistance depends on the length of the vessels, the number of vessels connected in parallel, and the radius of the vessels. It is clear that there is no practical way to find out all these quantities, in addition, the walls of the vessels are not rigid, and the blood does not behave like a classical Newtonian fluid with constant viscosity. Because of this, as noted by V. A. Lishchuk (1991) in the "Mathematical theory of blood circulation", "Poiseuille's law has an illustrative rather than a constructive role for blood circulation." However, it is clear that of all the factors that determine peripheral resistance, greatest value has the radius of the vessels (the length in the formula is in the 1st degree, the radius in the 4th), and that this same factor is the only one capable of physiological regulation. The number and length of the vessels are constant, while the radius can vary depending on the tone of the vessels.

Taking into account formulas (1), (3) and the nature of peripheral resistance, it becomes clear that mean arterial pressure depends on volumetric blood flow, which is determined mainly by the heart and vascular tone, mainly arterioles. [AND. P. Pavlov, 2002]

1.3. Theoretical foundations of the study of the work of the heart.

Research methodology for daily monitoring of the electrocardiogram

Acoustic phenomena called heart sounds can be heard by placing your ear or stethoscope on your chest. Each heart cycle is normally divided into 4 tones.

In the 19th century, it became clear that the heart, during its work, produces a certain amount of electricity, which causes the appearance of an electromagnetic field around the working organ. The electrical activity of the heart can be recorded using special electrodes applied to specific areas of the body. With the help of an electrocardiograph, an electrocardiogram (ECG) is obtained - a picture of changes over time in the potential difference on the body surface. (Appendix 2)

The first electrocardiograms were recorded by Gabriel Lippmann using a mercury electrometer. Lippmann's curves were monophasic in nature, only vaguely resembling modern ECGs. The experiments were continued by Willem Einthoven, who designed a device (string galvanometer) that made it possible to record a true ECG. He came up with modern designation EKG teeth and described some irregularities in the work of the heart, for which in 1924 he was awarded the Nobel Prize in Medicine.

Electrocardiography is a technique for recording and studying the electric fields generated during the work of the heart. Electrocardiography is a valuable method of electrophysiological instrumental diagnostics in cardiology. The direct result of electrocardiography is an electrocardiogram (ECG) - a graphical representation of the potential difference arising from the work of the heart and conducted to the surface of the body. The ECG reflects the averaging of all vectors of action potentials that arise at a certain moment of the heart.

Daily ECG monitoring, Holter monitoring, or long-term ECG recording is a method of electrophysiological instrumental diagnostics proposed by the American biophysicist Norman Holter in 1961, which was called the Holter ECG study. The study is a continuous recording of an electrocardiogram for 24 hours or more (48, 72 hours, sometimes up to 7 days). ECG recording is carried out using a special portable device - a recorder (recorder), which the patient carries with him (on a belt over his shoulder or on a belt). Recording is carried out on 2, 3, or more channels (up to 12 channels). Until now, the most common are 2- and 3-channel recorders. In some cases, it is possible to obtain a mathematically reconstructed ECG of 12 channels with three-channel recording, which can be useful in the topical diagnosis of extrasystoles. [, 2003] Disposable adhesive electrodes are used to make contact with the patient's body. During the study, the patient leads his usual way of life (works, walks, etc.), noting in a special diary the time and circumstances of the occurrence of unpleasant symptoms from the heart, taking medications and changing types of physical activity. (Appendix 3).

1.2. Theoretical foundations of the study of blood pressure.

Research methodology for 24-hour blood pressure monitoring

Daily blood pressure monitoring is an automatic measurement of blood pressure throughout the day. A blood pressure cuff connected to a portable monitor is put on the patient's shoulder. The device is mounted on a belt or on a shoulder strap. Measurements are performed on an outpatient basis, under conditions of normal patient activity. The device provides automatic measurement of the pulse, systolic and diastolic blood pressure at set time intervals using the oscillometric method, that is, by analyzing the pulse phenomena in the pneumo cuff. The monitor is programmed before installation on the patient using a computer. The measurement results are stored. After the end of the study, the monitor is connected to the computer to process and display the measurement results. , 1998] During the examination, the patient is advised to keep a diary, which notes the state of health, complaints, type of activity, physical activity, taking medications, wakefulness and sleep time. (Appendix 4)

Conclusions for chapter # 1

1) Among the indicators of the state of the body, the most important are data on the activity of the cardiovascular system

2) The work of the human heart is a coordinated contraction of two atria and two ventricles. The cardiac cycle includes general diastole (relaxation), atrial systole (contraction), and ventricular systole.

3) The heart has the functions of contractility, excitability and automatism.

4) The heart provides the movement of blood in the small and large circle of blood circulation, as a result of which oxygen-rich blood enters all organs, tissues and cells.

5) The heart rate (HR) in a healthy person at rest has a constant value of 70 beats per minute, and the stroke volume of blood is 70 ml per beat.

6) Blood moves in the systemic circulation from the left ventricle to the right atrium, and then along the pulmonary circulation from the right ventricle to the left atrium.

7) Mean arterial pressure depends on volumetric blood flow, which is determined by the heart and vascular tone.

8) Electrocardiography - a technique for registering and studying electromagnetic fields generated during the work of the heart

9) Daily monitoring of the electrocardiogram is a method of electrophysiological instrumental diagnostics proposed by the American biophysicist Norman Holter in 1961.The study is a continuous recording of an electrocardiogram for 24 hours or more (48, 72 hours, sometimes up to 7 days).

10) Daily monitoring of blood pressure is an automatic measurement of pulse, systolic and diastolic blood pressure at set time intervals throughout the day