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MBW:A mathematical model comparing solute kinetics in low- and high-BMI hemodialysis patients

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In this work, we reproduce the results of an article titled " A mathematical model comparing solute kinetics in low- and high-BMI hemodialysis patients " by D.Cronin-Fine, F.Gotch, N. Levin, P. Kotanko, M.Lysaght published in the International Journal of Artificial Organs. Vol. 30/no.11,2007/pp.000



Hemodialysis is necessary to remove waste products from the blood when a patients kidneys are not functioning properly, for information please click here, Acute Kidney Injury. Dialysis works [1]on the theory of diffusion of solutes and ultrafiltration of fluid across a semi-permeable membrane. Blood flows by one side of a semi-permeable membrane, and dialysis solution on the opposite side. Smaller solutes and fluid pass through the membrane. The blood flows in one direction and the solution flows in the opposite. The counter-current flow of the blood and solution maximizes the concentration gradient of solutes between the blood and solution, which helps to remove more urea and toxin from the blood. The concentrations of solutes are undesirably high in the blood, but low in the dialysis solution and constant replacement of the solution ensures that the concentration of undesired solutes is kept low on this side of the membrane. For more information on Hemodialysis please follow this link, Hemodialysis.

Figure 1: Schematic of the Dialysis
Figure 2: Bolld-Red, Semipermeable-Yellow, Dialysis Fluid-Blue.

Statement of the Problem

Physically smaller patients have higher mortality risks than the larger patients on maintenance hemodialysis. But the larger patients are more vulnerable to cardiovascular and diabetes diseases and so should have a higher mortality risks, but this is not the case for hemodialysis.

Two explanations for these irregularities during the dialysis have been given by the experts.

  1. The rate of generation of the uremic toxin is controlled by the higher metabolic rate compartments (HMRC). In smaller patients, the HMRCs are in larger proportions compared with their body mass. As a result, larger patients have lower uremic toxin levels compared with their body size and smaller patients have relatively higher levels of uremic toxins.
  2. In the larger patient's increased muscle mass and the adipose tissue hold the uremic toxin from the extracellular tissue, acting as a buffer, and so reduce the toxin's concentration in the extracellular fluid as compare to the smaller patients.


The biological system being studied is the concentration of the uremic toxins in both high and low body mass index patients, and how they are controlled by the two mechanisms described above. The goal is to more thoroughly understand why smaller patients have a higher mortality rate. In order to examine the two mechanisms proposed above, researchers used a three pool kinetic model. It is a molecular system that uses mass-action dynamics summarized by three ordinary differential equations. The first describes the generation of toxins by the high metabolic rate compartment and its release into extracellular fluid. The second shows the release of toxins being stored in muscle and adipose tissue to the extracellular fluid. The third equation shows the changing concentration of the toxins in the extracellular fluid as it comes in from the two sources and leaves via the dialysis taking place.

The Model

The model consists of three compartments.

  1. The organ mass compartment OM,
  2. the muscle mass and adipose tissue compartment MMAT,
  3. the extracellular fluid compartment E .

Figure 3:Schematic of the Model.

Differential equations for the three compartments are formed on the basis of one dimensional concentration gradients i.e., the toxin flows from a compartment with high concentration to a compartment with low concentration. They are

  1. 350px
  2. 450px
  3. 500px


350px is the steady state solution.

Analytical Considerations

As we are primarily interested in quantifying the effects of the BMI which is closely related to MMAT compartment during the dialytic and inter-dialytic phases, therefore to avoid the complexities of the solution of the three differential equations, we suppose that the OM compartment is in steady state.

Dialytic Interval

Steady state of the OM compartment gives


which after substituing in equation(3) gives the following system of differential equations for the dialytic interval.



Inter-Dialytic Interval

Taking the dialytic clearance term 80px to be zero in the extracellular compartment E, we get


From the mass-balance law of the toxin generated during the inter-dialytic interval, we get


Sunstituting this value of G in the preceding equation, we obtain the differential equation for the concentration of toxin in the extra-cellular compartment E for the inter-dialytic interval.


Detail of Variables & Parameters

The detail of variables and parameters used in the model is given below.


The parameters' values are given in the following table.


Evaluation of Parameters' Values

We evaluated the patients' parameters using the simple linear regression primarily correlating the percentage of the body weight of the high metabolic rate compartment (HMRC{%BW}) and the body weight in kilograms (BW) values taken from the data in the Sarkar et al [2].

Then by the simple analysis we get the following linear equation for the parameters.

First the linear regression between the (HMRC{%BW}) and BW is given as


The volume of the OM compartment is therefore given by just multiplying the right side by the BW as


The protein catabolic rate (PCR) is found by the linear regression between (PCR) in grams and the mass of the (HMRC) in kilograms as


We assume that the middle molecule generation rate (G) is directly proportional to (PCR), therefore









Numerical Implementations

Using the matlab function ode45, we solve the differential equation system for the dialytic interval with initial conditions as 250px taken from the steady state of the OM compartment for the three patients classified as

  • small patients
  • medium patients
  • large patients

with the value of G from the table and dialysis clearance (Kd) rate of 200ml/min. The dialytic interval is 4hrs.

Then we solve the differential equation for the interdialytic interval with the initial conditions taken from the values of the extracellular concentrations for the three patients where the dialytic interval finishes.

Also we evaluated the time average concentration (TAC) of the solute during inter-dialytic interval for different values of the MMAT mass-transfer coefficients.

The inter-dialytic intervals are 200 minutes, 500 minutes, 1000 minutes, 1500 minutes, 3312 minutes(2.3 days)-to compare with given simulations and 4000 minutes.


Original simulations

Dialytic Interval 450px

Inter-Dialytic Interval 650px

TAC 450px

Our Simulations

We simply classify our simulations in three categories namely

  1. dialytic interval,
  2. inter-dialytic interval and the
  3. time average concentration(TAC).

We then see the effects of changing the muscle mass adipose tissue (MMAT) parameter over the concentration of the hypthetical intermediate molecular weight species (IMWS) in extra-cellular compartment.

  • For dialytic interval,

We took the silumations for different values, very large and small, of the MMAT coefficient and got the almost same results. Two of them with the given MMAT coefficients are given in the figures. The figures show that the small patient has lower final concentration than the medium and the large patient. By increasing the MMAT-Mass transfer coefficient, the concentration in the smaller patient reaches its steady state fastly as compared to lower value of this coefficient.

350px 350px

  • For inter-dialytic interval.

Here we see the behaviour of the extra-cellular concentration of the (IMWS) soon after dialytic interval for different intervals upto 4000 minutes by also changing the values of the MMAT-Mass transfer coefficient .

  • Upto 200 minutes. For the initial phase of the inter-dialytic interval. The concentration of the IMWS solute with high value of the MMAT-Mass transfer coefficient reaches rapidly to steady state as compared to other patients. But for the lower value of MMAT-Mass transfer coefficient, all three concentrations do not reach to steady states in this phase i.e., time required to reach the steady state with smaller value of the mass-transfer coefficient is more than that of higher values.

350px 350px

  • upto 500 minutes. Behaviour of the representative concentrations for the smaller, medium and the large patients is almost identical. In the beginning of the inter-dialytic interval, the concentration for the smaller patients remains high for all values of the MMAT-Mass transfer coefficient.

350px 350px

  • upto 1000 minutes.

350px 350px

  • upto 1500 minutes.

350px 350px

  • upto 3312 minutes-to compare with the given simulations in the original model.

350px 350px 350px 350px

  • upto 4000 minutes. So by increasing the value of MMAT-Mass transfer coefficient, we observed that the concentration of the IMWS solute for smaller patient reaches more rapidly to the steady steady states as compared to medium and large patients. The time required for reaching the steady states in smaller patients is much less than that required by the medium and large patients.

350px 350px

  • Time Average Concentration (TAC). The results obtained in this case are in accordance with our discussed analysis.



The given paper has been studied. The simulation of the papers have also been given. Our simulations are different from those of original model. In the dialytic phase, maybe the difference may not be so important. For example the simulation of the original model and our own for dialytic interval are given above with MMAT- mass transfer coefficient 0.1 ml/min, but they describe the same situations for the model. However, for the inter-dialytic interval, the simulations in both the cases are more different. This difference is more evident in the figures above. Same is the case for the graph of the TAC.

Matlab Code

In this section, we provide the matlab code to regenerate the above picture. media:dialysis.m

Related Article

A recent paper which references the above summarized article is “A Mathematical Model Comparing Slute Kinetics in Low- And High-BMI Hemodialysis Patients” by Kappel et al. This paper is investigating the same biological system and hypothesis as above. They are looking at the creation of uremic toxins in the body and its concentrations in muscular tissue, adipose tissue, and the extracellular fluid. The key difference between this article and the one discussed in this wiki page is that this new model treats the adipose tissue and muscular tissue as separate compartments for the uremic toxins. The article discussed above treated these as a combined compartment. Again they are using mass action terms to model the dynamics of the toxin as it moves through the various compartments. The organ mass, muscular tissue and adipose tissue compartments all interact with the extracellular compartment via a bidirectional transfer of the toxin. In this article they also have an additional storage compartment S for use in an extended model. The compartment S has a saturation level associated with it and only connects with the adipose compartment. Further they assume that the rate of transfer from the adipose compartment to S is larger than the reverse transfer. They compare the systems for small, medium and large BMI patients and adjust the compartment capacities accordingly. They begin by examining the simpler model (without S) over three cycles, each cycle going from an interdialytic to a dialysis phase, for each of the three BMI categories. The model shows a clear increase in the concentration of toxins in all compartments for the smaller BMI compared with the medium and the larger BMI. When including the S compartment they find that after a few cycles the S compartment stays at its maximum concentration and no longer affects the model. By then including the degradation of the toxin and taking the model over 10 cycles they show that while the overall concentrations decrease, there is still a relatively lower concentration of toxins in the larger BMI patients compared with the smaller. They go on to find the sensitivities of the concentration of toxins in the extracellular fluid based on several parameters. This sensitivity analysis shows that measurements in the interdialytic phase are not needed to improve the parameter estimation in the model. The conclusion is that the hypothesis proposed by Cronin-Fine et al. (in the above model) can account for the counterintuitive increased survival rate of higher BMI patients. [3]


  1. Wikipedia
  2. Sarkar Sr, Kuhlman MK, Kotanko P, Zhu F, Heymsfield SB, Wang J, Meisels IS, Gotch FA, Kaysen GA, Levin NW. Metabolic consequences of the Body Size and Body Composition in Hemodialysis Patients(Submitted for Publication).
  3. F.Kappel et al. "A Mathematical Model Comparing Solute Kinetics in Low- And High-BMI Hemodialysis Patients, " (March 11, 2009).