# Non-Linear Pharmacokinetic Models

## Dosing Approaches

### First dose

One approach is to use the population values for Vm and Km. For phenytoin we could use population values of Vm = 7 mg/kg/day and Km = 5 mg/L. Aiming at 15 mg/L for Cpaverage with a patient weight of 80 kg Equation 21.5.1. can be used to estimate the 'first' dose. Equation 21.5.1 Dosing Rate versus Cpaverage Probably better to start out low since toxicity is more probable above 20 mg/L.

### Second dosing regimen

That is after giving a continuous dose regimen to steady state, measure plasma concentration and adjust dose. For example if after 420 mg/day, Cpaverage is 20 mg/L then a downward adjustment would be necessary. If we assume that the Km is close to the average value of 5 mg/L we can estimate Vm from the equation above or   thus a new dose rate can be calculated approximately 400 mg/day. Note: A reduction in dose of 20 mg/day (5 %) is calculated to give a 5 mg/L change (25 %) in Cpaverage.

Another approach at this point could be the use of the "Orbit Graph" method described by Tozer and Winter, 1992, redrawn from Vozeh, S. 1981. This method uses data previously derived from a patient population to construct shapes, orbit representing 50, 75, 90,95 and 97.5% of the population values of Vm and Km. Plotting this information with one data point allows the estimation of a second doing regimen.

### Third dosing regimen

If we already have two steady state plasma concentrations after two different dose rates we can solve Equation 21.5.1 for the two parameters, Vm and Km. This assumes that the patient is fully compliant. using simultaneous equations.

With Cpaverage, 1 = 8.0 mg/L and Cpaverage, 2 = 27.0 mg/L for R1 = 225 mg/day and R2 = 300 mg/day 225 • Km + 225 • 8 = 8 • Vm (1)

and

300 • Km + 300 • 27 = 27 • Vm (2)

or multiplying (1) x 300

300 • 225 • Km + 300 • 225 • 8 = 300 • 8 • Vm (3)

and multiplying (2) x 225

300 • 225 • Km + 300 • 225 • 27 = 225 • 27 • Vm (4)

subtracting (4) - (3)

300 • 225 • (27 - 8) = (225 • 27 - 300 • 8) • Vm and With these Vm and Km values we can now calculate a new, better dosing regimen.

Calculator 21.5.1 Calculate Vm and Km from Two Steady State Cpaverage Values

 First Dose Rate: First Average Cp: Second Dose Rate: Second Average Cp: Km is: Vm is:

Error Message Value is not a numeric literal probably means that one of the parameter fields is empty or a value is inappropriate.

### Graphical methods

There are a number of graphical methods which have been described for when you have data from two or more dosing intervals. Basically these rely on converting the equations mentioned above into a straight line form which can be plotted to give the Vm and Km as a function of the intercept and/or slope. Again, steady state, average concentrations are required with the patient fully compliant. Some of these methods have been reviewed by Mullen and Foster, 1979. They found that iterative computer analysis was the best method, followed by a plot of Cpaverage versus Cpaverage/Dose (Equation 21.5.2), and this was followed by a direct linear plot method. Since the direct linear plot method was the least complicated and required no calculations these authors thought it should be quite useful. Equation 21.5.2 Equation for Cpaverage versus Cpaverage/Dose

#### Direct Linear Plot Method

This method, described by Mullen, 1978, involves plotting Dose and Cpaverage data points on linear graph paper. The y-axis represent Dose and Vm while the x-axis represents Km in the positive direction and Cpaverage in the negative direction. This is shown in Figure 21.5.1. Figure 21.5.1 Linear plot of Dose, Vm versus Km, Cpaverage

After a dose of 256 mg/day and 333 mg/day the steady state Cpaverage values were measured to be 7 and 20 mg/L, respectively. These data are represented as the red and blue lines, respectively. These lines intercept at 4,400 which provide a value of Km and Vm of 4mg/L and 400 mg/day, respectively. If we were aiming for a Cpaverage value of 15 mg/L this point on the x-axis could be connected with the intersection of the red and blue lines by drawing the green line. The intersection of the green line with the y-axis provides the required dose of 316 mg/day. Various dose and expected Cpaverage could be estimated from the intersection of the red and blue lines.

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References
• Tozer, T.N. and Winter, M.E. 1992 Chapter 25, "Phenytoin" in Applied Pharmacokinetics, 3rd. ed., Evans, W.E., Schentag, J.J., and Jusko, W.J. ed., Applied Therapeutics, San Francisco, CA Figure 25-11, p 25-31
• Vozeh, S. et al. 1981 Predicting individual phenytoin dosage, J. Pharmacokin. Biopharm., 9, 131-146
• Mullen, P.W. and Foster, R.W. 1979 Comparative evaluation of six techniques for determining the Michaelis-Menten parameters relating phenytoin dose and steady-state serum concentrations, J. Pharm. Pharmacol., 31, 100-104
• Mullen, P. W. 1978 Optimal phenytoin therapy: a new technique for individualizing dosage. Clin. Pharmacol. Ther., 23(2), 228–232

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