# Calculus and Its use in Medicine

Calculus is the mathematical study of changes (Definition). Calculus is also used as a method of calculation of highly systematic methods that treat problems through specialized notations such as those used in differential and integral calculus. Calculus is used on a variety of levels such as the field of banking, data analysis, and as I will explain, in the field of medicine. Medicine is defined as the science and/or practice of the prevention, diagnosis, and treatment of physical or mental illness (Definition).

The term medicine can also mean a compound or a preparation applied in treatment or control of diseases, mostly in form of a drug that is usually taken orally (Definition). Calculus has been widely used in the medical field in order to better the outcomes of both the science of medicine as well as the use of medicine as treatment. (Luchko, Mainardi & Rogosin, 2011). There has been a strong movement towards the inclusion of additional mathematical training throughout the world for future researchers in biology and medicine.

It can be hard to develop new courses as well as alter major requirements, but institutions should consider the importance of a clear understanding of the function of mathematics in science. However, scientists who have not had the level of mathematical training needed to work in their field often employ creative methods in order to incorporate both math and biology as seen in calculus (Butkovskii, Postnov & Postnova, 2013).

Calculus is used in medicine to measure the blood flow, cardiac output, tumor growth and determination of population genetics among many other applications in both biology and medicine. Although sometimes less obvious than others, Calculus is always being used. One of the clearest examples of the application of calculus in medicine is in the Noyers Whiter Equation.

This equation is used in the calculation of dosage rates. Medical professionals apply calculus in pharmacology in order to determine the proper dosage. It is their job to insure a steady rate of absorption of the drug being administered (Valerio, Machado & Kiryakova, CALCULUS IN MEDICINE 3 2014). For example, they must consider that when a tablet is ingested, it must pass into aqueous (water-based) solution in the stomach and dissolve at the appropriate rate for the medicine to do what it is supposed to do. In this case, dosage forms must be regulated and controlled since the rate of dissolution in each drug is different from another.

In a case where the drug distributed is in a dissolving form or dissolves slowly, it is imperative that the calculations are accurate or the medicine will be ineffective. (Kocher & Roberts, 2014). Therefore, medicine is required to follow the strict rules that are effective and provide the appropriate monitoring for both long releasing medicines as well as immediate releasing medicines. Sharp releases have critical peaks and tend to drop into blood concentration quickly.

As stated in Pathways to Careers in Medicine and Health, the formula used to determine dosage rates in medicine is as follows: dW/dt=DA (Cs-C)/L, whereby dW/dt represents dosage rate, A is surface area of solid drug, Cs represents concentration of solid in the entire dissolution medium, C represents the concentration of solid in diffusion surface that surrounds that solid, D is diffusion coefficient while L is the thickness of the diffusion layer (Fuchs & Miller, 2012).

It is vital to note that the Noyes Whitney equation is a representation of the surface problem. Therefore, the rate of a compound’s dissolution tends to depend on the surface area of the medicine being administered. Most importantly, any change made to the compound’s surface area (ex.Breaking the pill. ) that is exposed to external basic or acidic surrounding will alter the medicine’s effectiveness.

This is just one example of calculus usage that has made determination of dosage rates easier in the field of pharmacology. In the fields of medicine and biology, calculus has been widely applied in allometry. Allometry is the study of size of the body and its influences on the organism’s functions and behavioral patterns. The term “Allometry” was created by Julian Huxley and Georges Tessier when they were studying the extremely large claw of the fiddler crab and how it may have CALCULUS IN MEDICINE 4 developed this trait (Shingleton, 2010).

Allometry has emerged as a vital biological phenomenon to examine relative growth, which contains variables that also need fractional equation in evolution in order to formulate the joint probability density function (PDF). Most importantly, the solutions provided by the fractional equations consist of allometry relations (ARs). Fractional calculus used in allometry is new; therefore a careful review of familiar materials is important before one can apply allometry to the study of biological scaling or other growth processes (Niknejad & Petrovic, 2013).

In this case, the allometry relationship denoted by AR existing between two elements of a living network denoted by X and Y is usually represented by X= aYb whereby one or even two of the variables measure the size as well as the allometry coefficient a along with the exponent b that are fit to that data (Butkovskii, Postnov & Postnova, 2013). For example, a comparison can be made between the growth of the heart and the brain of a child and the growth of the body overall. A simple linear equation can be used to describe the relation of the organs compared to the body: log y = ? log x + log b (Shingleton, 2010).

Calculus plays a huge role in proving that allometry parameters tend to co-vary and that there exists a clear and explicit functional relationship between the two, (organ size and body size). The empirical probability density function is usually determined as the Pareto distribution or the power law plays a role in the establishment of inconsistency of interspecies in allometry relationship (Kocher & Roberts, 2014). The use of probability calculus to determine and establish the scaling of the probability density and its function will eliminate the inconsistencies.

Medical professionals also use calculus, differential calculus in particular, in population genetics. In genetics, population growth models often use calculus. A good example is that of tumor growth as well as the spread of illnesses. Tumor refers to an abnormal cell population created when a natural balance in cell division as well as death is distorted. The simplest model CALCULUS IN MEDICINE 5 used to determine tumor growth falls under calculus as an exponential growth and decay function.

This model is presented as V (t) = Veat, whereby V (t) represents the volume of that tumor at a given period t and the a represents the per capita rate of growth of the tumor (Chernyak et. al, 2014). Despite the fact that exponential growth can be uncontrollable, a lot of data exist in order to support the simple model in the initial phase of tumor growth. A good example is that of Looney and his colleagues who utilized exponential growth as well as exponential decay in order to model the entire growth of rat tumors that went untreated and they ended up radiating the tumors.

Most importantly, linear regression can be used to measure the per-capital rates of growth for the relevant non-radiated as well as heavily radiated tumors (Fuchs & Miller, 2012). It is through those estimates that one can be able to compute doubling time for untreated tumors as well as half-life of heavily radiated tumors. Whilst exponential growth can give reasonable descriptions of population growth whenever there is a large population, it can not be maintained indefinitely. It is clear that calculus is a vital field of study since it helps to analyze changes in scientific settings through different mathematical tools and models.

Calculus has been applied widely in both biological and medical fields especially in determining changes. Institutions have introduced courses that provide students with knowledge of application of calculus in daily life events. Calculus has been applied in many fields including chemistry, physics and other life sciences. In this case, the analysis has focused on medicine that has incorporated biological studies.

Medicine is a field of life sciences that is highly dependent on calculus as a technique to analyze different aspects. Different models have been developed in calculus, including regression analysis, linear models, Noyers Whiter Equation, joint probability density function in determination of allometry variations and many others that are widely used in medical analysis. CALCULUS IN MEDICINE 6 These models have played a huge role in research and development in medicine since they have enabled simplified analysis.

#### References

• Butkovskii, A. G. , Postnov, S. S. , & Postnova, E. A. (2013). Fractional integro-differential calculus and its control-theoretical applications. II. Fractional dynamic systems: Modeling and hardware implementation. Automation and Remote Control, 74(5), 725- 749.
• Chernyak, V. Y. , Chertkov, M. , Bierkens, J. , & Kappen, H. J. (2014). Stochastic optimal control as non-equilibrium statistical mechanics: calculus of variations over density and current.
• Journal of Physics A: Mathematical and Theoretical, 47(2), 022001. Definitions. (n. d. ). Retrieved June 6, 2014, from Merriam Webster website: http://www. merriam-webster. com/dictionary
• Fuchs, B. A. , & Miller, J. D. (2012). Pathways to careers in medicine and health. Peabody Journal of Education, 87(1), 62-76.
• Kocher, R. , & Roberts, B. (2014). The calculus of cures. New England Journal of Medicine, 370(16), 1473-1475.
• Luchko, Y., Mainardi, F. , & Rogosin, S. (2011). Professor Rudolf Gorenflo and his contribution to fractional calculus. Fractional Calculus and Applied Analysis, 14(1), 3-18.
• Niknejad, A. , & Petrovic, D. (2013). Introduction to computational intelligence techniques and areas of their applications in medicine. Medical Applications of Artificial Intelligence, 51.
• Shingleton, A. (2012) Allometry: The study of biological scaling. Nature Education Knowledge, 3(10):2.
• Valerio, D. , Machado, J. T. , & Kiryakova, V. (2014). Some pioneers of the applications of fractional calculus. Fractional Calculus and Applied Analysis, 17(2), 552-578.

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