Metabolism and Excretion - Parallel Pathways

Scheme or diagram

Figure 12.7.1 Diagram Illustrating Multiple Elimination Pathways with a One Compartment Model

In Figure 12.7.1 ke is the excretion rate constant and km is the metabolism rate constant. Here we have two parallel pathways for elimination (with others as a shadow). We can write the differential equations for the four components shown in this diagram (X, U, M, Mu). There could be more pathways. It may be necessary to specify excretion by exhalation, in sweat, or as is commonly the case, more than one metabolic pathway.

The Differential equation

Equation 12.7.1 Rate of Change of the Amount of Drug in the Body

Equation 12.7.1 includes terms for excretion and metabolism. The number (and type) of these elimination processes can be changed to accommodate a variety of possible routes of excretion or metabolism. Some of these processes may not be first order, however many can be represented by first order parameters.

The elimination rate constant, kel, represents the sum of all the ('first-order') rate constants so we can substitute kel for (ke + km) in Equation 12.7.1

Equation 12.7.2 Rate of Change of the Amount of Drug using Clearance Parameters

In Equation 12.7.2 the rate of elimination of drug from the central or plasma compartment is expressed in clearance terms, here, renal (CL_R) and metabolic (CL_M) clearance. The total body clearance (CL) is equal to the sum of the clearance terms in the model. In Equation 12.7.2 total body clearance, Cl = CL_R + CL_R.

With more elimination pathways we sum all these process parameters to arrive at the elimination rtae constant, kel, or total body clearance, Cl. Thus the equation for rate of elimination, either with rate constant parameters or clearance parameters, is the same as before with one elimination pathway. The integrated equation is the same as before:-

Equation 12.7.3 Cp versus time with kel as the elimination rate parameter

Equation 12.7.4 Cp versus time with CL as the elimination rate parameter

When we move to the equation for the cumulative amount of drug excreted into urine there is a significant difference. The elimination rate constant or total body clearance is replaced with the excretion rate constant, ke, or the renal clearance, CL_R.

Equation 12.7.5 Rate of Change of Cumulative Amount Excreted into Urine

Notice we have ke here in place of kel and CL_R in place of CL.

Then substituting for Cp (= (Dose/V) • e^{-kel • t} or = (Dose/V) • e^{-CL • t / V}) we get

Equation 12.7.6 Rate of Excretion of Unchanged Drug into Urine

after integrating using Laplace transforms we get:

Equation 12.7.7 Cumulative Amount Excreted as Unchanged Drug versus Time

Again, note: ke or CL_R are in the numerator of Equation 12.7.7 and not kel or CL

For M, the amount of drug which has been metabolized the equations are:-

Equation 12.7.8 Rate of Change of Amount of Metabolite in the Central Compartment

AND

Equation 12.7.9 Rate of Excretion of Metabolite into Urine

After integrating Equation 12.7.9 using Laplace transforms we get:

Equation 12.7.10 Cumulative Amount Excreted as Metabolite versus Time

At t = ∞; e^{-kel • t} approaches 0

From 12.7.7 setting the e^{-kel • t} term to zero gives:

Equation 12.7.11 Total Amount Excreted as Unchanged Drug into Urine

From 12.7.10 setting each e^{-kel • t} term to zero gives:

Equation 12.7.12 Total Amount Excreted as Metabolite into Urine

NOTE:-

Equation 12.7.13 Mass Balance - Total Amount Eliminated equals Dose

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Copyright 2001-3 David W. A. Bourne (david@boomer.org)