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Mathematical Journeys I
Lab 9
Hemodiaylsis
Introduction
The kidneys provide the essential function of removing toxins from the body. The incidence of renal (kidney) failure is increasing, in part due to the increasing number of older people. Medical costs impact federal, state and private funding sources as well as personal finances.
At this point in time, renal transplant and dialysis are the only medical alternatives. The number of kidneys needed is greater than the number of available healthy organs. Thus, dialysis is frequently used to counteract the effects of renal failure. There are two types of dialysis – hemodialysis and peritoneal dialysis. Hemodialysis will be the focus of this laboratory.
In hemodialysis, the patient is connected to a hemodialysis machine. The patient’s blood is directed to flow through the machine, which filters away toxins. The blood is then transferred back to the body with the toxins removed. Patients usually receive dialysis treatments three times a week. Each treatment can last 3 to 4 hours.
Figure 1
A dangerous side effect of dialysis is that blood clots can form. This not only poses a direct risk to the patient, but also impedes the flow of blood through the dialysis machine. Anticoagulants such as heparin are used to prevent blood clots. Medical staff must monitor the dialysis treatment, balancing the danger from clot formation with the danger from premature termination of the treatment.
Technology Problem
As a medical expert, you have been called in as a consultant regarding dialysis treatment for a kidney patient. You will need to determine the appropriate heparin loading dose and constant heparin infusion rate for a 4.5- hour dialysis treatment. Unfortunately, the process to identify the appropriate doses is very experimental. Your job will be to develop a model to calculate the correct initial and continuing doses of heparin. This is important for the health of the patient because heparin modeling has made it possible to reduce the amount of heparin required for the dialysis treatment and yet achieve the desired effect.
To solve this problem, you will apply the scientific method. The scientific method has five steps
- problem, question, hypothesis, test, and decision. The first two steps of the scientific method (problem, question) depend on exploring the situation and developing a better understanding of the problem. You may want to meet with your professor and develop a timeline with specific objectives and deadlines. You will probably begin exploring the problem with some bibliotechnology research.
Bibliotechnology Research
To begin to understand this problem, you will first need to establish a baseline of general knowledge about hemodialysis. Some questions you may want to explore include the following.
What is end-stage renal disease (ESRD)? What is the demographic profile of an ESRD patient? What is the treatment for ESRD? What is the cost of ESRD treatment? What is the mortality rate for an ESRD patient?
Mathematics Tools
In the investigation of this problem, you will need to select mathematics tools that best fit the problem, that help explain the problem, and that provide a solution to the problem. You may wish to review functions, exponential functions, integration by substitution, differential equations, and statistics.
Model Portfolio
As part of the solution to the technology problem, you will need to create a model portfolio. The model portfolio should contain a thorough description of the technology problem and its identifying characteristics. You need to explain the mathematics, technology, and science you use to solve the problem. In addition, you need to include the mathematical and/or physical models you create. It is important that you keep a careful record of your work. It should be complete enough so that someone else could follow your work step by step. The model portfo lio will be very helpful when you present your thesis defense. Remember that you are a medical consultant and may need to explain your recommendations to other medical personnel, to the patient, or to the patient’s family and support system.
Model Development
The third step of the scientific method is to develop a hypothesis about the technology problem. You can think of the model(s) you create as a hypothesis. Your model constitutes a proposal that describes and explains the technology problem.
In order to develop your model, you will need to explore two aspects of dialysis − the amount of time it takes for blood clots to form and the elimination rate constant for a given rate of heparin infusion.
Section I Coagulation time
You will need to determine a measure of the clotting rate of blood for your patient. Two different measures in use are the activated partial thromboplastin time (aPTT) and the whole blood activated partial thromboplastin time (WBPTT ). In this laboratory, you will use the WBPTT approach.
To determine a patient’s WBPTT, test tubes are inserted into a heating block maintained at
37 o^ C. Each tube contains 0.2 mL of a particular clotting reagent such as heparin. A tube is selected, and 0.4 mL of whole blood drawn from the patient is added to the tube containing the reagent. The reagent and whole blood are mixed and then returned to the heating block to be timed with a stopwatch. The tube is then checked every 5 seconds until a clot forms, at which time the stopwatch is stopped. The time taken for the clot to form is defined to be the WBPTT.
Heparin affects the clotting mechanism very soon after it is administered. When given intravenously, it is well distributed in the circulating blood within three minutes. The coagulation time depends on several variables. To determine an equation for coagulation time, begin with the following variables.
BL = Baseline WBPTT, i.e. the coagulation time without heparin.
D = Dose of heparin administered, measured in international units (IU).
R = Response, measured in seconds. Response is defined to be the difference in WBPTT with and without heparin. Thus, response R is described by R = WBPTT (with heparin ) −BL (1)
S = Sensitivity, measured in seconds per IU. This value indicates how quickly heparin is absorbed by an individual patient.
The response R can be mathematically related to the dose D and sensitivity S with a simple linear model. The results of experimental research indicate that a scatter plot of data consisting of various doses of heparin and corresponding response times exhibits a strong linear relationship between response time and dose. In statistical terms, it can be said that the correlation coefficient is very close to the number 1. Thus, the theoretical model for the relationship between response and dose is described by
R = S⋅ D (2)
Note that the sensitivity of the individual patient serves as the slope of the straight- line relationship between dose and response.
In practice, the sens itivity for an individual patient can be determined as follows.
- Measure a baseline WBPTT without heparin
- Administer a dose D of heparin, e.g. D = 2000 IU
- After 3 minutes (when the heparin is uniformly distributed in the circulating blood), measure the WBPTT again.
- Substitute the values for the dose and response into Equation (2) and calculate S.
Exercise 1
Determine the sensitivity of a patient with a baseline of 70 seconds who has a WBPTT of 160 seconds after a dose of 2000 IU of heparin.
Exercise 3
Construct a table of values for the concentration of a drug every hour for 6 hours following administration of a dose that achieves an initial drug-plasma concentration of 15 mg/liter. Assume that the elimination rate cons tant is 0.6. Plot the graph of drug-plasma concentration over time.
Part B A pharmacokinetic model for heparin clearance
The simple model of drug concentration described above, for which the drug-plasma concentration decreases at a rate proportional to the concentration present, results in a model of exponential decay. By contrast, the first-order model for heparin clearance asserts that the rate of change in the amount of heparin in the blood during continuous infusion at a constant rate is the difference between the rate at which it is infused and the rate at which it is absorbed. The latter is proportional to the drug-plasma concentration, as was shown in the simple pharmacokinetic model. To model continuing infusion, you will need the following va riables. Let
M = amount of circulating heparin at any time t
V = plasma volume (volume of plasma in the patient’s bloodstream)
C = drug-plasma concentration (concentration of heparin in the patient’s plasma)
I (^) R = constant infusion rate
k = clearance constant (elimination rate constant)
Thus, heparin elimination during continuous infusion at a constant rate is modeled by the first- order differential equation
d(M) dt
= IR − kC
Observing that M = C ⋅ V, it follows that
d(C ⋅ V) dt
= IR − kC (3)
Since M is equivalent to D in Equation (2), we see that
M = D = R/S
So that Equation (3) may be written
R
d R R I k dt S V S
or defining a new elimination rate constant to be
K =
k V
we obtain
dR dt
= IR ⋅ S − K⋅ R (4)
During the course of a dialysis, the plasma volume V can be assumed to be constant, so that K is constant. Equation (4) can be solved to obtain the result
R = R 0 e−Kt^ +
IRS
K
(^1 −^ e−^ Kt) (5)
Exercise 4
Verify that Equation (5) is the general solution to Equation (4).
To emphasize that R is a function of t in the case of continuous infusion, replace R in Equation (5) with R (^) t to obtain
Rt = R 0 e−Kt^ +
I RS
K
(^1 −^ e−^ Kt) (6)
Part C Constant infusion rate
The exponential relationship expressed by Equation (6) implies that R (^) t will gradually decline to a steady state. It is easily verified that as time increases without bound the steady state value of R (^) t will be
IRS
K
The patient for whom you are consulting will be administered an initial dose of heparin (loading dose) followed by continuous infusion of heparin at a constant rate. The primary physician will specify a desired coagulation time, i.e. a value for the WBPTT. The desired loading dose to ensure this degree of anticoagulation can be determined from Equations (1) and (2) using the patient’s measured sensitivity and the desired response RD ( RD = Specified WBPTT – BL,
RD = SD). During continuous infusion, the response time will be maintained at a constant value, Rt = RD = R 0.
Exercise 6
Verify that Equation (6) can be solved algebraically to obtain Equation (7).
2) Infusion discontinued
Note that IR = 0 when infusion is discontinued, so that Equation (6) becomes
Rt = Rt − 1 e−^ Kt
Exercise 7
Solve this equation for K. This will give you the formula to determine the elimination rate constant K when infusion has been discontinued.
Using the above information, you can develop your model (hypothesis), the third step of the scientific method. This will provide the basis for the dosage information you provide as a consultant for your patient.
Test the model
You are now ready for the fourth step of the scientific method, test the model. You will need the following information for your patient: sensitivity, S = 0.045 s/IU, baseline WBPTT = 70 seconds. To determine an initial estimate for the elimination rate constant K, the following patient information has been collected from a 4.5-hour dialysis treatment with an initial loading dose of 2000 IU (D 0 = 2000) and a continuous infusio n rate of 1500 IU/h.
Table 1 Measured Values of WBPTT
Time from Start of Dialysis (hours)
WBPTT
(seconds)
Response (seconds)
Heparin Infusion (IU/hour)
Elimination Rate Constant (per hour) 0 155 1500 1 140 1500 2 130 1500 3 125 1500 4 (Infusion discontinued) 125 1500 4.5 100
Exercise 8
Complete Table 1 by calculating values of the response and elimination rate constant. The response values at each time t can be determined from Equation (1) written as R = WBPTT – BL, where BL = 70. Note that for t = 1, 2, 3, and 4 you will need to use Equation (7) to calculate values for K. This equation can be solved intrinsically using technology such as a TI-86 or TI- 92 graphing calculator or a symbolic algebra system such as Derive or Maple. For t = 4.5, you will need to use the formula for K when infusion has been discontinued.
Exercise 9
Calculate the mean value and the standard deviation for K from the values in Table 1. The mean will be used as an initial estimate of K.
You are now ready to determine an individualized heparin dose for your patient’s next dialysis treatment.
Exercise 10
Your patient’s primary physician has prescribed a WBPTT of 125 seconds for the next dialysis treatment. Use the mathematical model you developed to determine the loading dose and the constant infusion rate. The result of Exercise 9 can be used for the value of the elimination rate constant. Recall that the baseline is still BL = 70.
You have now determined the values of D and IR to use for the next dialysis treatment for this patient. This process may be repeated once or twice more, depending on how stable the patient is. Thereafter, for relatively stable patients, the modeling procedure need only be repeated once every two or three months. For less stable patients, the procedure may be applied more frequently, depending on the physician’s clinical assessment of the patient.
Decide to Retain, Reject or Modify the Model
The fifth step of the scientific method is to make a decision regarding your hypothesis (model). Your action depends on how well the mathematical model actually predicts usable values for patient care. If it does not predict well, you will want to revise the model. You may want to compare your loading dose and infusion rate with those used in common practice. For example, the following facts have been ascertained from statistical analysis of hundreds of dialysis patients.
Mean loading dose is 1700 ± 500 IU (Range: 500 to 4000 IU) Mean infusion rate is 1500 ± 300 IU/h (Range: 600 to 3000 IU/h) (The customary statistical usage “mean ± standard deviation” is employed.)
