CL = rate of elimination/C, where C is
the concentration of drug in blood or plasma

rate of elimination = CL· C

CL = Vd x k_{el}
where V_{d} = volume of
distribution and k_{el }is the
elimination rate constant

CL = Vd · (0.693/t_{1/2})
where
0.693 = ln 2 and t_{1/2} is the
drug elimination half-life

note that plasma clearance
CL_{p} include renal (CL_{r})
and metabolic (CL_{m}) components

Renal
Clearance

CL_{r} = (U · C_{ur}) / C_{p}
; where U is urine flow (ml/min); C_{ur}
is urinary drug concentration and C_{p}
is plasma drug concentration.

Steady-State Drug Plasma Concentration (C_{ss})

The calculation required to determine
being steady-state drug plasma concentration illustrates
the sensitivity of the plasma concentration to number of
factors, in this case for a drug taken orally.

First look at the overall form of
the equation:

equation 1: C_{ss}=
1/(k_{e}*V_{d}) * (F*D)/T

The drug elimination rate
constant,k_{e}
is related to the drug half-life ( t_{1/2} = 0.693/k_{e}) and thus can be
calculated from knowledge of the drug half-life.

The plasma steady-state drug
levels also dependent on the dose, D, as well as a fraction of
the drug that's actually absorbed following ingestion
(F).

"T" is the dosing
interval, so the once-a-day dosing would be 1 day or to keep
the units consistent, 24 hours.

The steady-state level will
also be dependent on the apparent volume of distribution (V_{d})

Now let's take an example using
the drug phenytoin (Dilantin) which is used to manage
epilepsy.

The once-a-day dose is 200
mg.

The drug half-life is 15
hours

For the once-a-day dose,
the dosing interval (T) is 24 hours [to keep the units the
same as the drug half-life will use "hours"]

Let's say that about 60% of
the ingested does is in fact absorbed, giving us a value
of 0.6 for "F" in equation 1 above.

The volume of distribution
for phenytoin (Dilantin) is 40,000 mls (40 liters)

k_{e}
= 0.693/15 hours = 0.0462/hr

Let's now
compute the results:

equation 1:
C_{ss}= 1/(k_{e}*V_{d}) *
(F*D)/T or

C_{ss}=
1/(0.0462/hour*40000 ml) * 0.6 (200 mg)/24 hours
or

Css =
0.0027 mg/ml or 2.7 ug/ml

Time to Steady-State

Let's consider the above
problem from a little different point of view, that is, How
long would it take to reach 50% of the C_{ss}
(no bolus).

Consider the dose is 300
mg/24h (dosing interval is 24 h or T; dose is 300 mg)
but for convenience we'll represent it as 12.5 mg/hr, such
that T is now 1 hr. The equation is:

f = 1 - e ^{-k}e^{TN}
or 0.5 = 1 - e ^{-k}e^{TN
}where k_{e} is the
elimination half-time of 0.0462/hr, T = 1 and N is the
number of doses needed to reach 50% of C_{ss}.

Rearranging, 0.5 = e ^{-0.0462/hr
* 1 hr * N} --(note time (hour) units cancel)
so taking antilogs,

-0.693 = -0.0462 * N or N =
-0.693/-0.0462 = 15

15 doses at an interval of
1 hour/dose gives the time to 50% of C_{ss}
equal to 15 hours--a predictable time since drugs reach 50%
of their steady-state value in 1 half-life

Constant Infusion Dosing

Next, let's consider the
case by which drugs are administered by constant infusion.

The infusion rate is Q or
in this example, 150 ug/min and for simplicity, the drug is
again phenytoin with a k_{e} of
0.0462/hr; t_{1/2} of 15 hrs and a
V_{d} of 40000 mls

[note that we have been
careful to use the same units for k_{e}
and Q, i.e. 0.0462/hr = 0.0462/60 min]

Holford, N. H.G. and Benet, L.Z.
Pharmacokinetics and Pharmacodynamics: Dose Selection and
the Time Course of Drug Action, in Basic and Clinical
Pharmacology, (Katzung, B. G., ed) Appleton-Lange, 1998,
pp 34-49.

Benet, Leslie Z, Kroetz, Deanna
L. and Sheiner, Lewis B The Dynamics of Drug Absorption,
Distribution and Elimination. In, Goodman and Gillman's
The Pharmacologial Basis of Therapeutics,(Hardman, J.G, Limbird, L.E,
Molinoff, P.B., Ruddon, R.W, and Gilman, A.G.,eds) TheMcGraw-Hill Companies, Inc.,1996, pp. 3-27

Pazdernik, T.L. General
Principles of Pharmacology, in ACE the Boards, (Katzung,
B. G., Gordon, M.A, and Pazdernik, T.L) Mosby, 1996, pp
22-28

Edward J. Flynn, Ph.D. Professor of Pharmacology, New Jersey
School of Medicine and Dentistry, personal communication, 1980,
1999.