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Euler's Method

Euler's method is a point-slope numerical integration method. Using a single point (initial condition, X1(0) or Dose for example) and the slope (the differential equation) it is possible to estimate a new point value. This new value and slope can be used to calculate the bext value. Seems simple enough and really is quite simple. The method can be presented graphically for a one compartment model.

Plot of point slope method

Figure 10.2.1 Linear plot of amount versus time

We known from an earlier Chapter (Chapter 4) that the 'real' curve is not a straight line. Thus, Euler's method provide only an approximation to the real answer. However, if you use short enough steps the approximation can be useful.

Euler's method is a point slope method

The point (starting point) is the initial value for the first calculation
The slope (differential equation is the result of calculating (evaluating) the differential equation value at the starting point.

Thus, for this example of a one compartment model after an IV Bolus dose;

Point = X1(0)
and Slope = - k1 • X1(0)

Euler's method with stepsize of 0.1

Figure 10.2.2 Calculation of the next data point

Thus, with X1(0) = 100 mg, k1 = 0.25 hr-1 and stepsize = 0.1 the new value is 97.5 mg. For comparison, the concentration using the exact equation of Dose • e-k1 • 0.1 = 100 • e-0.25 • 0.1 = 97.53. Not too bad, about 0.03% error.

We can extend the calculation out for as long as we like. The table below provides the calculation out for 0.5 hours.

Table of calculations using Euler's method

Table 10.2.1 Table of Calculations using Euler's Method

Note the exact answer is Dose • e-k1 • 0.5 = 100 • e-0.25 • 0.5 = 88.25. An error of 0.17%.

Euler's method calculations are based on the equation for the differential equation, the slope.

Euler's method Equations

Figure 10.2.3 Equations used in Euler's Method Calculation

The equations in Figure 10.2.3 can be repeated out to what ever time is required. Remember this is a straight line approximation to a line that is usually curved. The method is only accurate if the step size is not too big. We can see this with another example. Using parameters Cp0 = 100 and kel = 0.3 hr-1 we can explore the effect of step size. In the table below either 1, 2, 4, or 10 steps are taken to get from time 0 to time 1.

Effect of stepsize on Euler's method calculation

Table 10.2.2 Effect of stepsize on accuracy with Euler's method

Note the error drops from over 5 % to 0.05 % by decreasing the stepsize by 10. After 10 steps the error with the smaller stepsize would be approximately 0.5 %. A 10 fold improvement for a 10 fold increase in computational cost.

The effect of the larger stepsize, in this case 1, on the calculation is illustrated below.

Plot of Cp versus time

Figure 10.2.4 Plot of Amount versus Time calculated exactly and using Euler's method

Notice the divergence between the slope at each time point and the exact curve, as time progresses.

Euler's method is quite straight forward and easy to program but it requires small sizes for an accurate calculations. The smaller the stepsize the large the number of steps and time and computational cost required to achieve a satisfactory answer. Other point-slope methods, such as Runge-Kutta, tend to be more efficient, giving more accurate answers with relatively fewer steps.

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