two opened compartment model

1. Week 4 - Biopharmaceutics and
Pharmacokinetics
Pn. Khadijah Hanim bt Abdul Rahman
School of Bioprocess Engineering
University Malaysia Perlis

2. Multicompartment models:
intravenous bolus administration
• Pharmacokinetic models- represent drug
distribution and elimination in the body.
• A model should mimic closely the physiologic
processes in the body
• In compartmental models, drug tissue
concentration is assumed to be uniform within a
given hypothetical compartment.
• All muscle mass and connective tissues may be
lumped into one hypothetical tissue
compartment that equilibrates with drug from
the central (or plasma) compartment.

3. • Since no data is collected on the tissue mass,
the theoretical tissue concentration is
unconstrained and cannot be used to forecast
actual tissue drug levels.
• However, tissue drug uptake and tissue drug
binding from the plasma fluid is kinetically
simulated by considering the presence of a
tissue compartment.

4. • Multicompartment models were developed to
explain and predict plasma and tissue
concentrations for the behavior of these
drugs.
• In contrast, a one-compartment model is used
when the drug appears to distribute into
tissues instantaneously and uniformly.

5. • Central compartment
– These highly perfused tissues and blood make up
the central compartment.
• Multicompartment drugs
– multicompartment drugs are delivered
concurrently to one or more peripheral
compartments composed of groups of tissues
with lower blood perfusion and different affinity
for the drug.

6. Two Compartment Open Model
• Many drugs given in a single intravenous bolus
dose demonstrate a plasma level–time curve
that does not decline as a single exponential
(first-order) process.
• The plasma level–time curve for a drug that
follows a two-compartment model shows that
the plasma drug concentration declines
biexponentially as the sum of two first-order
processes—distribution and elimination.

7. • A drug that follows the pharmacokinetics of a
two-compartment model does not equilibrate
rapidly throughout the body, as is assumed for
a one-compartment model.
• In this model, the drug distributes into two
compartments, the central compartment and
the tissue, or peripheral compartment.

8. • Central compartment:
– represents the blood, extracellular fluid, and highly
perfused tissues. The drug distributes rapidly and
uniformly in the central compartment.
• Second compartment,
– known as the tissue or peripheral compartment,
contains tissues in which the drug equilibrates more
slowly.
• Drug transfer between the two compartments is
assumed to take place by first-order processes.

9. General Grouping of Tissue According
to Blood Supply

10. Distribution phase- represents the
initial, more rapid decline of drug from
the central compartment into tissue
compartment (line a)
• distribution phase- drug elimination
and distribution occur concurrently
•Net transfer of drug from central to
tissue compartment
•Fraction of drug in the tissue
compartment during distribution
phase increases to max.
•At max. tissue conc. – rate of drug
entry into tissue = rate of drug exit
from tissue.
•Drug in tissue compartment-equilibrium
with drug in central
compartment (distribution
equilibrium)
•Drug conc in both compartment
decline in parallel and more slowly
compared to distribution phase
The decline is 1st order process and
called elimination phase or β phase
(line b)

11. Two Compartment Models
• There are several possible two-compartment
models

12. • compartment 1 is the central compartment
and compartment 2 is the tissue compartment.
• The rate constants k12 and k21 represent the
first-order rate transfer constants for the
movement of drug from compartment 1 to
compartment 2 (k12) and from compartment 2
to compartment 1 (k21).

13. Relationship between drug
concentrations in tissue and plasma
• The maximum tissue
drug concentration may
be greater or less than
the plasma drug
concentration.

14. • the rate of drug change in and out of the tissues:

15. • The relationship between the amount of drug in each
compartment and the concentration of drug in that
compartment is shown by:
where
– DP = amount of drug in the central compartment,
– Dt = amount of drug in the tissue compartment,
– VP = volume of drug in the central compartment, and
– Vt = volume of drug in the tissue compartment.

16. • Rate equation

17. • Drug concentration in blood and tissue
• Amount of drug in blood and tissue

18. • The rate constants for the transfer of drug between
compartments are referred to as microconstants or
transfer constants, and relate the amount of drug
being transferred per unit time from one
compartment to the other.
• The constants a and b are hybrid first-order rate
constants for the distribution phase and elimination
phase, respectively.

19. • Equation
• Constants a and b- rate constant for distribution phase
and elimination phase
• Can be write as
• The constants A and B are intercepts on the y axis for
each exponential segment of the curve

20. • Intercepts A and B are hybrid constants

21. Method of residuals
Method of residual- feathering or peeling,
useful for fitting a curve to the
experimental data of drug when drug does
not follow one compartment model.
E.g: 100 mg of drug administered by rapid
IV injection to a 70-kg healthy adult male.
Blood sample were taken periodically and
the following data were obtained:

22. When data is plotted, a curved
line is observed. The curved-line
relationship between logarithm
of the plasma conc and time
indicates that drug is distributed
in more than one compartment.
From these data, biexponential
equation, may be derived
The rate constant and intercepts
were calculated by method of
residuals
As shown in biexponential curve,
the decline in initial distribution
phase is more rapid than
elimination phase.
Rapid distribution phase
confirmed with constant a being
larger than constant b.
at some later time Ae-at will
approach 0, while Be-bt still have
value.

23. • Therefore,
• In common logarithms,
• From equation above, rate constant can be
obtained from the slope (-b/2.3) of a straight
line representing the terminal exponential
phase.

24. • The t1/2 for elimination phase (beta half life)
can be derived from the following
relationship:
• From Eg. b was found to be 0.21 hr-1. from this
info the regression line for terminal
exponential or b phase is extrapolated to the y
axis; y intercept = B or 15um/mL.

25. •Values from the extrapolated line are then substracted from the original
expremental data points and a straight line is obtained. This line represents
the rapidly distributed a phase
•The new line obtained by graphing the logarithm of residual plasma conc
(Cp- C’p) against time represents the a phase. The value for a is 1.8 hr-1 and
y intercept is 45ug/mL. elimination half life, t1/2 computed from b, has the
value of 3.3 hr.

26. • A no of pharmacokinetic parameters may be
derived by proper substitution of rate
constants a and b and y intercepts A and B to
following equations:

27. Apparent Volumes of distribution
• VD- parameter that relates plasma conc (Cp) to
the amount of drug in the body (DB)
• Drugs with large extravascular distribution/
high peripheral tissue binding- the VD is
generally large
• Polar drugs with low lipid solubility- VD is small

28. Volume of the Central compartment
• Useful to determine drug conc. after IV
injection
• Also refered as Vi = initial VD as the drug
distributes within plasma and other body
fluids
• Vi- generally smaller than terminal VD after
drug distribution to tissue is completed
• Vol of central compartment- generally greater
than 3L

29. • For polar drugs, initial Vol of 7-10 L- interpreted
as rapid drug distribution wthin plasma and
extracellular fluids
• E.g: Vp of hydromorphone about 24 L- possibly
becoz of rapid exit from plasma into tissues even
during initial phase.
• As in the case of one-compartment model- Vp
determined from the dose and instantaneous Cp.
• Vp useful in determination of drug clearance if k is
known

30. • In two-compartment model, Vp considered as
mass balance factor governed by mass
balance between dose (D) and Cp
• Ie. Drug conc x vol of fluid = dose at t=0
at t=0, no drug eliminated, Do= VpCp
• At t=0, all of drug in the body is in central
compartment

31. • Cp0 can be shown to be equal to A and B by
following equation:
• At t=0, e0=1, therefore
• Vp is determined from this equation by
measuring A and B after feathering the curve

32. • Alternatively, the vol of central compartment
may be calculated from the similar to
calculation of VD for one compartment model
• for two-compartment model
=

33. Apparent volume of distribution at
steady state
• At steady-state- the rate of drug entry into
tissue compartment from central
compartment = rate of drug exit from tissue
compartment into the central compartment
• Amount of drug in central compartment, Dp =
VpCp,

34. • Total amount of drug at steady state = Dt + Dp
• The apparent vol of drug at steady state (VD)ss
• Substitute of Dt and expresses Dp as VpCp
• (VD)ss- function of transfer constants, k12 and k21=
rate constants of drug going into and out of tissue
compartment

35. Extrapolated volume of distribution
• Where B= y intercept obtained by
extrapolation of the b phase of plasma level
curve to y axis. Can be calculated by this
equation:

36. Volume of distribution by area
• (VD)area = (VD)β – obtained through
calculations similar to those used to find Vp
except that rate constant b instead of
elimination rate constant, k.
• (VD)β calculated from total body clearance
divided by b (influenced by drug elimination in
the beta, or b phase).

37. • Total body clearance = , (VD)B may be
expressed as
• By substitution of kVp :

38. Drug in the tissue compartment
• Vt= apparent volume of the tissue compartment may
be calculated from knowledge of the transfer rate
constants and Vp:
• Calculation of amount of drug in tissue compartment
does not involve the use of Vt
• Vt
-provides an estimate for drug accumulation in the
tissues
-Vital in estimating chronic toxicity and duration of
pharmacologic activity to dose

39. • To calculate the amount of drug in tissue
compartment, Dt

40. Drug Clearance
• Clearance- vol of plasma that is cleared of
drug per unit time
• Clearance may be calculated without
consideration of compartment model
• Cl in the two-compartment model is the
product of (VD)β and b

41. Elimination rate constant
• In two-compartment, the elimination rate constant, k
represents the elimination of drug from the central
compartment
• b represents drug elimination during the beta or
elimination phase when distribution is mostly complete
• Plasma-drug level curve declines more slowly in b phase-redistribution
of drug out of tissue compartment
• b is smaller than k
• k- true elimination rate constant
• b- hybrid elimination rate constant- influenced by rate
transfer of drug in and out of tissue compartment

42. Three compartment open model
• Three compartment- two compartment model +
deep tissue compartment
• Central compartment- distributed most rapidly-highly
perfused tissues
• Compartment 2- less rapidly
• Compartment 3- very slowly- poorly perfused
tissues, i.e. bone/ fat

43. • Rates of flow of drug into and out of the
central compartment:
• A, B and C – y intercept of extrapolated lines
for central, tissue and deep tissue
compartment
• a, b and c – 1st order rate constant

44. • Elimination rate constant, k
• Vol of central compartment
• Area