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The steady-state volume of distribution
186
Pharmacological Research, Vol . 25, Supplement 2, 1992
THE STEADY-STATE VOLUME OF DISTRIBUTION Bianca Maria Bocchialini School of Pharmacy, University of Parma, Parma . Key words : Volume of Distribution, Steady-State, Dilution Factor . In 1919 Widmark(1) published in Sweden a paper about the elimination of ethanol and acetone from blood introducing the concept of Volume of Distribution . We shall call Apparent Volume of Distribution the ratio : Vapp = Q(t) / C(t) where Q(t) is the amount of the drug in the body and C(t) the plasma concentration, usually the only site accessible to sampling . The above ratio changes with the time, because Q(t) depends on the elimination and C(t) depends on both elimination and distribution . The Apparent Volume of Distribution varies between the Initial Volume(2) and the Steady-State Volume. The former is defined by : Yin = Dllim C(t)
t->o
where D is the dose given as a bolus at time t=0 and the limit in the denominator is the value of C(t) extrapolated at t -> 0 Ignoring the values of the concentration taken before the drug is uniformly mixed in the plasma ; Vi n is obviously equal to plasma volume . The Steady-State Volume is defined by : = lim Q (t)1C(t) YSS
t->
Now we introduce a new parameter, called Dilution Factor, 0 (3), equal to the ratio between the total amount of drug in the body and the amount X1(t) in the sampling compartment (plasma compartment),
8 = limo (t)1X t (t) t-> The ratio X1 (t) / Vin is equal to C(t) ; therefore 0 = Vss/Vin For a monocompartmental sistem, the total amount of drug in the body is equal to the amount in the sampling compartment, therefore 0 =1 and Vss=Vi n . We shall now consider a two-compartment system with a single bolus . The equations of this system are *1 =- K1 x1 + k12 x2 X2=k12 x1 K2x2 XI(o) - D ; x2(o) - 0
If this system Is closed, then k21 =K2, k1 2=K1, and at Steady-State, x2/x1 = K1/K2 Therefore 0 = (xl+x2)/x1 = 1 + ( K1/K2)
Vss has a finite value Vss = (1 + t1/t2) Vin or an infite one if K2=0 . If the system is open, then the solution of the differential equations is :
1043-6618/92/25110186-02/$03 .00/0
© 1992
Pharmacological Research, Vol . 25, Supplement 2, 1992
x1
187
a [(K2- a) e-0d
(t) =
X2 (t) -
12 D
+
( 8-1(2) e -tit]
(e -cd - e -8 t)
Feeding and sampling in the first compartment 0
= 1 +
[k12/(K2 - a)]
where a is the smaller (in absolute value) exponent . Also in an open sistem
Vss
has a finite value :
Vss _ [1 + k12/(K2 - a)) Vin or an infinite one if K2= a . Feeding in the first compartment and sampling in the second one : 0 =1 + (K2 - a)/k12 Vss
has a finite value, but different from the last one Vss = [1 + (K2 - a)/k12J Vin
or an infinite value if k12=0 . We have a special case when a = B . In this case k21 k12=0, Kt-K2, and the equations become : Kit xi (t) = De X2 (t) = ki2De -K
tt
it follows that : 8= 1+limk12 t=+oo t-> .
then
Vss
has an infinite value . With a costant infusion and sampling in the first compartement : 0 = 1 + k12/K2
then : Vss = (1 + k12/K2)
Vin
This value becomes infinite if K2=0 . If sampling in the second compartment, then 0=
1
+ K2/k12
and : Vss _ (1 +
or Vss =
+ a*
K2/k12) Vin
if k12=0 . With more compartments the result are analogous .
Conclusion We can conclude that Vss is a very important parameter indicating the extent of distribution of drug in others organs . However, as we have shown here, Vss depends on the sites of sampling . References (1) Widmark EMP . Studies in the concentration of indifferent narcotics in blood and tissue . Acta Med . Scand. 1919, 52 :87-164 . (2) Resciprto A, Bocchialini BM . Pharmacokinetics : unfolding of a concept . In : Rescigno A, Thakur AK, eds. New Trends in Pharmacokinetics . New York : Plenum Press, 1991 :1-25 . (3) Mordenti J, Rescigno A . Estimation of Permanence Time, Exit Time, Dilution Factor, and SteadyState Volume of Distribution . Pharm . Res . 1992, 9 :17-25 .
Pharmacological Research, Vol . 25, Supplement 2, 1992
THE STEADY-STATE VOLUME OF DISTRIBUTION Bianca Maria Bocchialini School of Pharmacy, University of Parma, Parma . Key words : Volume of Distribution, Steady-State, Dilution Factor . In 1919 Widmark(1) published in Sweden a paper about the elimination of ethanol and acetone from blood introducing the concept of Volume of Distribution . We shall call Apparent Volume of Distribution the ratio : Vapp = Q(t) / C(t) where Q(t) is the amount of the drug in the body and C(t) the plasma concentration, usually the only site accessible to sampling . The above ratio changes with the time, because Q(t) depends on the elimination and C(t) depends on both elimination and distribution . The Apparent Volume of Distribution varies between the Initial Volume(2) and the Steady-State Volume. The former is defined by : Yin = Dllim C(t)
t->o
where D is the dose given as a bolus at time t=0 and the limit in the denominator is the value of C(t) extrapolated at t -> 0 Ignoring the values of the concentration taken before the drug is uniformly mixed in the plasma ; Vi n is obviously equal to plasma volume . The Steady-State Volume is defined by : = lim Q (t)1C(t) YSS
t->
Now we introduce a new parameter, called Dilution Factor, 0 (3), equal to the ratio between the total amount of drug in the body and the amount X1(t) in the sampling compartment (plasma compartment),
8 = limo (t)1X t (t) t-> The ratio X1 (t) / Vin is equal to C(t) ; therefore 0 = Vss/Vin For a monocompartmental sistem, the total amount of drug in the body is equal to the amount in the sampling compartment, therefore 0 =1 and Vss=Vi n . We shall now consider a two-compartment system with a single bolus . The equations of this system are *1 =- K1 x1 + k12 x2 X2=k12 x1 K2x2 XI(o) - D ; x2(o) - 0
If this system Is closed, then k21 =K2, k1 2=K1, and at Steady-State, x2/x1 = K1/K2 Therefore 0 = (xl+x2)/x1 = 1 + ( K1/K2)
Vss has a finite value Vss = (1 + t1/t2) Vin or an infite one if K2=0 . If the system is open, then the solution of the differential equations is :
1043-6618/92/25110186-02/$03 .00/0
© 1992
Pharmacological Research, Vol . 25, Supplement 2, 1992
x1
187
a [(K2- a) e-0d
(t) =
X2 (t) -
12 D
+
( 8-1(2) e -tit]
(e -cd - e -8 t)
Feeding and sampling in the first compartment 0
= 1 +
[k12/(K2 - a)]
where a is the smaller (in absolute value) exponent . Also in an open sistem
Vss
has a finite value :
Vss _ [1 + k12/(K2 - a)) Vin or an infinite one if K2= a . Feeding in the first compartment and sampling in the second one : 0 =1 + (K2 - a)/k12 Vss
has a finite value, but different from the last one Vss = [1 + (K2 - a)/k12J Vin
or an infinite value if k12=0 . We have a special case when a = B . In this case k21 k12=0, Kt-K2, and the equations become : Kit xi (t) = De X2 (t) = ki2De -K
tt
it follows that : 8= 1+limk12 t=+oo t-> .
then
Vss
has an infinite value . With a costant infusion and sampling in the first compartement : 0 = 1 + k12/K2
then : Vss = (1 + k12/K2)
Vin
This value becomes infinite if K2=0 . If sampling in the second compartment, then 0=
1
+ K2/k12
and : Vss _ (1 +
or Vss =
+ a*
K2/k12) Vin
if k12=0 . With more compartments the result are analogous .
Conclusion We can conclude that Vss is a very important parameter indicating the extent of distribution of drug in others organs . However, as we have shown here, Vss depends on the sites of sampling . References (1) Widmark EMP . Studies in the concentration of indifferent narcotics in blood and tissue . Acta Med . Scand. 1919, 52 :87-164 . (2) Resciprto A, Bocchialini BM . Pharmacokinetics : unfolding of a concept . In : Rescigno A, Thakur AK, eds. New Trends in Pharmacokinetics . New York : Plenum Press, 1991 :1-25 . (3) Mordenti J, Rescigno A . Estimation of Permanence Time, Exit Time, Dilution Factor, and SteadyState Volume of Distribution . Pharm . Res . 1992, 9 :17-25 .