Application of Differential Equations in Medical Science

 

K. T. N. Jyothi*1, P. S. R. Subrahmanyam2, A. Ch. Sravanthi3

Department of Mathematics, Ideal College of Arts and Sciences, Kakinada.

*Corresponding Author E-mail: ktnjyothi9@gmail.com

 

Abstract:

It is an incontrovertible fact that differential equations form the most important branch of mathematics and in fact, the position at the centre stage of both pure and applied mathematics. It also obvious that differential equations from the basis of applied mathematics. Keeping it into observation in this paper let us discuss about an epidemiology problem like leukemia. It is impossible to describe all the roles played by differential equations in pure and applied mathematics in single paper. Recently there has been significant activity in the mathematical community, aimed at developing quantitative tools for studying leukemia. Mathematical models have been applied to evaluate existing therapies and suggest novel therapies. This paper reviews the contributions of mathematical modeling to leukemia. These modeling have great potential in this field.

 

KEY WORDS:Mathematical modeling, leukemia, differential equations, bone marrow, growth.

 


INTRODUCTION:

Leukaemia (blood cancer) is a non infectious and non genetic disease in humans. This may be caused by smoking, ionization, some dangerous chemicals but the exact cause of leukaemia is unknown. Leukaemia is a group of cancer that usally begins in the bone marrow and result in high number of abnormal white blood cells. These white blood cells are not fully developed and are called blasts or leukaemia cells. Symptoms may include bleeding andbruisingproblems,feeling tired, fever, and an increased risk of infections. These symptoms occur due to a lack of normalblood cells. Diagnosis is typically made byblood testsorbone marrow biopsy. There are many types in leukaemia the common observation in a leukaemia affected person is the count of white blood cells, which are extraordinarily high, when compared to a healthy person.

 

 

Damage to the bone marrow, by way of displacing the normal bone marrow cells with higher numbers of immature white blood cells, results in a lack of bloodplatelets, which are important in theblood clottingprocess. This means people with leukemia may easily becomebruised,bleedexcessively, or develop pinprick bleeds (petechiae). White blood cells, which are involved in fightingpathogens, may be suppressed or dysfunctional. This could cause the patient's immune system to be unable to fight off a simple infection or to start attacking other body cells. Because leukemia prevents the immune system from working normally, some patients experience frequentinfection, ranging from infected tonsils,sores in the mouth, ordiarrheato life-threateningpneumoniaoropportunistic infections.

Finally, the red blood cell deficiency leads toanemia, which may causedyspneaandpallor.

 

The wordleukemia, which means 'white blood', is derived from the characteristic high white blood cell count that presents in most afflicted patients before treatment. The high number of white blood cells is apparent when a blood sample isviewed under a microscope, with the extra white blood cells frequently being immature or dysfunctional. The excessive number of cells can also interfere with the level of other cells, causing further harmful imbalance in the blood count.

 

Growth Rate of White Blood Cells in Leukemia:

In a non infectious disease like leukaemia the rate of growth of disease depends up on white blood cells. In this case assume that the death rate of white blood cells may be neglected. Because the growth of white blood cells is very high compare to the death rate in leukeamia affected person.

 

In leukaemia the rate of growth white blood cells is depends upon time. In this assume that w(t) is the count of white blood cells in the blood, which is dependent on time t. And the rate of change of white blood cells directly proportional to the number of white blood cells

                                                                          

i.e.,,Wherekisaparameter.

Forparticularpatienttherateofgrowthmayalsochangefromtimetotime.Therateofgrowthmayalsochangefrompatienttopatient.Thenthekchangesfrompatienttopatient.

 

Heretheparameter‘k’hasvitalroleintheequationasitschangesarenoteasilypredictable.

Solutionforthisdifferentialequationis,wherecisconstant.

 

Applyinginitialconditionsw=,t=.

Then

 

CONCLUSIONS:

In this review paper my intension is to estimate time period of tumor growth. This solution may help us in estimating the status of white blood cells at any particular time ‘t’ in future. Depending upon siviority of condition, the doctor can go ahead by varying the mode of treatment. 

 

ACKNOWLEDGEMENT:

Our sincere thanks to our beloved madam Dr. Smt. P. ChiranjeeviniKumari, M.A., Ph.D., Secretary and Correspondent, Director of P.G. Courses, Ideal College of Arts and Sciences, Kakinada

 

REFERENCES:

1.     Differential Equations and their applications by Zafar Ahsan

2.     The definitions, concepts and scientifically established facts relating to medical science is barrowed from different sources including internet. We acknowledge our sincere thanks.

 

 


 

 

 

 

Received on 18.08.2017       Modified on 21.09.2017

Accepted on 29.09.2017      ©A&V Publications All right reserved

Research J. Science and Tech. 2017; 9(3):425-426.

DOI: 10.5958/2349-2988.2017.00074.2