Biochemical Reactions and Enzyme Kinetics - Introduction to Biomedical Engineering - page 467

8.3 ADDITIONAL MODELS USING THE
QUASI-STEADY-STATE APPROXIMATION

The quasi-steady-state approximation for Michaelis-Menten kinetics can be used for
more than just enzyme reactions. 2 The quasi-steady-state approximation is useful when
describing the elimination of substances from the body with capacity-limited rates, such
as renal excretion and metabolism, and even for linearized models of muscle using the Hill
equation. While the quasi-steady-state approximation was developed for enzyme reactions
with variables q S , q E , q ES , and q P , we will use it as a nonlinear transfer rate in a compartment

model, where the substrate,

q S ,inEq. (8.47) is the quantity in a compartment. Here, we first
consider a one-compartment model with a variety of inputs and then a two-compartment
model.

8.3.1 One-Compartment Model
Consider a one-compartment model in which the elimination is characterized by the
quasi-steady-state approximation

q S ¼ V

max
q S þ K M

Þ q S þ f ð t Þ

ð 8 : 51 Þ

ð

Impulse Input
Consider first an impulse input. Previous examples have used , where z is the

strength of the impulse function—a problem that is handled most simply by a change in the
initial condition. To solve Eq. (8.51) with an impulse input, we have

f ð t Þ¼ zd ð t Þ

q S ¼ V

max
q S þ K M

Þ q S

ð 8 : 52 Þ

ð

with q S ð 0 Þ¼ z : As before, Eq. (8.52) is rearranged to give

þ K M
q S

1 dq S ¼ V dt

max

and after integrating, we have

1 z

q S

t ¼ z q S þ K M ln ð 8 : 53 Þ

V

max

The same comments about the solution of Eq. (8.43) apply to Eq. (8.53).

2 Material in this section based on Godfrey.