Figure 4.4.1 Scheme for a One Compartment Model, Intravenous (IV) Bolus Administration
Developing the Differential Equation
From the previous page on Figure 4.3.2 we estimated the slope of the Cp versus time line at various times. Plotting ΔCp/Δ (the slope) versus Cp produced a straight line plot (Figure 4.3.3). Thus the rate of change of Cp versus time is proportional to the concentration remaining to be eliminated, Cp. The slope of this line. The proportionality constant can be defined as kel, the elimination rate constant. If we measure the slope over very small time intervals we are calculating the tangent to the line. We can now say that the rate of change of Cp versus time is the differential of the concentration with respect to time as Δt approaches 0; ΔCp/Δt approaches dCp/dt which gives:
Equation 4.4.1 Rate of Change of Concentration versus Concentration
Equation 4.4.1 is a differential equation for the one-compartment model after an IV bolus administration. By taking very small time steps we are going from the gross or large time interval term ΔCp/Δt to the continuously varying dCp/dt term. Note, the negative sign in front of the kel term. The slope or tangent is decreasing or negative for positive concentration values.
Equation 4.4.1 relates the rate of change of Cp versus time. We can also relate the rate of elimination to the concentration remaining. Previously, we developed the required differential equations by looking at the arrows leaving or entering a component of the model. In Figure 4.4.1 there is one arrow leaving one component. Thus we could write the differential equation for the model shown above as Equation 4.4.1.
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