A COMPARISON OF DIFFERENT SOLUTION APPROACHES FOR A MATHEMATICAL MODEL OF HEMODIALYSIS

R. Hussain 1 , A. Ali 1 , Mohammad Munir 2 and Sadia Nasar 1 . 1. Mirpur University of Science and Technology (MUST), Mirpur-10250(AJK ), Pakistan. 2. Government Postgraduate College No 1, Abbottabad, Pakistan. ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History

Mathematical models were developed to gain a better understanding of many biological processes. The basic aim of present study was to establish a new technique in order to solve problems in the field of mathematical modeling. A mathematical model for hemodialysis with some modification was reviewed in this study. Different mathematical techniques, Laplace Transforms, Eigenvalue, and integrating factor for linear differential equations of the model were used, in all cases unique solution were obtained for dialytic interval.

…………………………………………………………………………………………………….... Introduction:-
Kidney is the major organ to clean blood by removing excess fluid, minerals and wastes. Moreover, it produces a hormone that keeps the bones strong. When the kidney fails harmful wastes build up in the body, due to which blood pressure may rise and the body may retain excess fluid which has to be removed. Hemodialysis is the most common method used to treat advanced and permanent kidney failure. In hemodialysis blood is allowed to flow through a special filter that removes wastes and extra fluids. The clean blood is then returned back to the body.
Mathematical modeling for dialysis has not long history, first bi-compartmental model of a typical profiled heomodialysis session for setting suitable sodium profiles was proposed in 2000 (Stephen et al., 2000) . In the same year another model known as one-and two-compartment model for hemodialysis with aim to analyze the course of treatment and to predict the effect of dialysis procedures was also published (Ziolkoet al., 2000). But the results on modeling of uric acid concentrations have not been published till 2007. Present study is based on the Mathematical model for solute kinetics in hemodialysis patients proposed by Cronin-Fine in 2007 including the modeling of uric acid concentration.

Material And Methods:-
The motivation for this research was to modify the model equations presented by previous researchers in the field of mathematical modeling for hemodialysis and present different solution approaches for Mathematical model of hemodialysis.

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Set of three 1 st order differential equations of the model were Equations 1-3. Equation no 4 and 5 were for dialytic interval whereas equation no 6 and 7 were for inter-dialytic interval. Solution of the model was presented by three different approaches Laplace transform, Eigenvalue and integrating factor of linear differential equations.

Refining the Governing Equations:-
To make the solution more manageable, Organ Mass (OM) was assumed to be at a steady-state, d(C OM )/dt=0.
The dialysis clearance term (K d C E ) in equation (5) was eliminated to modified the equations for interdialytic interval, yeilding equation (6). It was still impossible to find an explicit solution for the pair of equations (4 and 5) because these equations could be combined as one equation with two variables, which was by definination unsolvable. Therefore an additional equation (7a) was developed, which described the solute mass-balance for the interdialytic interval. This additional Equation resulted in a pair of equations (6 and 7a) for the inter-dialytic interval with manageable solutions as given below.
This equation was found to be inadequate and need modification. Equation (7a) was corrected to a new version of equation (7b) and given below: This study's focus was to determine whether modeling could provide an explanation for the level of toxin concentration in extracellular compartment during dialysis and in between dialysis treatment. So in this section the different solution procedures for dialytic and inter-dialtic intervals were presented in detail to comprehend the mathematical model of hemodialysis which is as under:

Explicit Solution (Dialytic interval)
Consider equations (4) and (5) Where Equation (10) is the explicit solution for dialytic interval where r 1 and r 2 were constants of integration. Obtained solution indicates how concentration of toxins in extracellular compartment C E variedover time. The constant of integrations were obtained by assuming C E at t=0 was equal to C MMAT due to steady state kinetics.

Explicit Solution (Inter-dialytic interval)
Eq (11) Eq (7b) became (12) Here integrating factor was , then by multiplying above equation by integrating factor as follows: 394 Integrating further solution obtained were given below (13) Equation (13) represented the explicit solution in case of the inter-dialytic interval, where c was constant. The constant was obtained by assuming C E at t = o and was denoted by C E0.

Solution by Laplace Transform (Dialytic Interval) Consider
Taking Laplace Transform on both sides of the above equation

Conclusion:-
The comparative study of three different solution approaches for modified model of hemodialysis were presented with an aim that in all three cases unique solutions were obtained and were exactly the same results as were published in previous study, that the extracellular solute (toxin concentration) increases more rapidly in case of inter-dialytic interval. This study can be extended further for stability analysis of the model.