The word ‘mathematics’ has been derived from the Greek word, ‘mathemata’ which means the ‘subject of study’, is one of the most ancient types of knowledge known to mankind. It was primarily invented, since the Old Stone Age, to solve the practical problems with the help of concepts, modernly known as arithmetic and geometry. The use of arithmetic operations was to counting, computing, marking the calendar and the function of geometry in defining the parameters in the constructions. Soon in early 1900’s, a British mathematician, G. Hardy though his exceptional contribution in the field of mathematics, found great implications of this practical knowledge in Biology, particularly Genetics.
He gave the famous law with a German physician, Weinberg, known as Hardy-Weinberg’s Equilibrium. This law emphasize on transmission of Mendelian characters, finds its significance in studying the Rh-factors as well as the treatment of hemolytic disorders among the newborns.
Through applied and pure mathematics, scientists have become able in estimating the effectiveness of the treatment and prescription of dosage, finding significant implications in epidemiology, diagnosis, ECG and other medical imaging etc. The most widely used concepts of mathematics which are used in medicine are pre-algebra, geometry, statistics and probability.
Mathematics and Diagnosis
Diagnosis refers to the identification of a disease or a medical condition, for which mathematical tools can be used for the correct diagnostic or medical testing such as; to test the diseases such as heart attack or diabetes etc. This involves the measurement of the quantity or quality of certain substances in blood samples or different types of samples as well as cultures. This brings a range of results such as:
- True positives which mean that the test is positive and disease is present.
- False positives which mean that the test is positive even when no disease is present.
- True negatives which mean that the test is negative and disease is absent.
- False negatives which mean that the test is negative even when disease is present.
Through diagnostic accuracy, a test can discriminate the difference between a healthy or a infected or diseases patient with the help of following equation:
Whereas the sensitivity of a test helps in correctly identifying the true positives, and the specificity of a test helps in correctly identifying the true negatives. For example, the negative result in a test which gives 100% of sensitivity increase the diagnostic accuracy of the true negatives.
Furthermore, precision is also very important to check the reliability of the results which is calculated as:
Mathematics and Epidemiology
Epidemiology is the study of the spread, distribution, causal and preventive factors of diseases as well as disorders or any other health related condition in a population such as coronavirus, cancer, heart disease etc. For this in-depth information, epidemiologists have to identify the current level of infection. Therefore, the incidence rate gives the number of new disease victims in a certain time within a population. Whereas, the prevalence rate gives the proportion of a population which has caught disease in one time.
R0 (r naught) is a term, which gives the basic reproduction rate of an infectious organism (virus, bacteria), thereby, estimating the average number of cases infected by one case. If R0 < 1 indicates that an infected person, on average, will infect < 1 another person; If R0 > 1, the disease will spread by manifolds, e.g., R0 = 3 will result in one person infecting 3 others, and 3 infected persons infecting 3 others resulting in 27 infected persons more.
This is termed as exponential growth and may get uncontrollable unless R0 is reduced. This indicates that, the mathematical concept of basic reproduction rate is important in epidemiological study of a disease.
Mathematics and ECG
Electrocardiograph (ECG) is a method which is used to measure the electrical activity of a heart in order to understand its correct functioning, if suspected or reported unwell. ECG is instrumental in measuring the size and rhythm of the electrical signals in the pumping heart, when electrodes are places on a patient’s arms, chest and legs. The electric information is then mapped onto a graph paper and represented by different waves. It has a distinctive shape and letters are assigned to the specific highs (peaks) and lows (troughs). The P wave refers to the contraction of atria (top chambers of heart).
The QRS wave shows the functioning of the ventricles (lower chambers of heart). The R wave is large since it maps the big ventricles of heart; therefore, the gap between the two R waves shows the heart rate whereas the final end of the cycle is presented by T wave. Abnormalities in the length or width of these waves indicate the underlying pathology or infection.
Mathematics and Medical Imaging
Medical imaging is a technique through which inside of the body can be seen without any cut or incisions. A digital image is formed which is the collection of points or pixels with the help of two coordinates (x, y). Each pixel has its intensity known as gray level which ranges from white to black. In terms of mathematics, the coordinates (x, y) function to assign each pixel a number according to its gray level and create black and white images. An image which has many variations in the gray levels are tend to be sharper than those with smaller variations (dull images).
CT-Scans and X-Rays have been used in which a patient is placed between an X-ray source and a coated film sensitive to the energy of X-rays whereas in digital radiography, the X-rays are digitalized through digital devise after passing through the patient. The intensity of X-rays wary as they pass through the body and fall onto the devise. Similarly, in Angiography, X-ray contrast medium is injected into patient through catheter which helps in magnifying the image of blood vessels to identify the blockages.