MULTI COMPARTMENT MODEL BIOPHARMACEUTICS..pdf

MULTI-COMPARTMENT
MODELS
M. BALASUNDARESAN,
ASSISTANT PROFESSOR,
ARUNAI COLLEGE OF PHARMACY,
TIRUVANNAMALAI.

• Ideally a true pharmacokinetic model should be the one with a rate constant for
each tissue undergoing equilibrium.
• Therefore best approach is to pool together tissues on the basis of similarity in
their distribution characteristics.
• The drug disposition occurs by first order.
• Multi-compartment characteristics are best described by administration as i.v
bolus and observing the manner in which the plasma concentration declines with
time.
The no. Of exponentials required to describe such a plasma level-time profile
determines the no. Of kinetically homogeneous compartmentsinto which a
drug will distribute.
The simplest and commonest is the two compartment model which classifies the
body tissues in two categories :
1. Central compartment or compartment 1
2. Peripheral or tissue compartment or compartment 2.

ADMINISTRATION:
Elimination from centra
T
lW
com
OpC
arO
tm
M
enP
tARTMENT OPEN MODEL-IV BOLUS
Fig:
• After the iv bolus of a drug the decline in the plasma conc. Is bi-exponential.
• Two disposition processes- distribution and elimination.
• These two processes are only evident when a semi log plot of C vs. Tis
made.
• Initially, the conc. Of drug in the central compartment declines rapidly, due
to the distribution of drug from the central compartment to the peripheral
compartment. This is called distributive phase.
1
Central
2
peripheral

Extending the relationship X= vd C
Dcc = K21 xp – K12 xc – KE xc
Dt vp vc vc
X= Amt. Of drug in the body at any time t remaining to be eliminated
C=drug conc in plasma
Vd =proportionality const app. Volumeof distribution
Xc and xp=amt of drug in C1 and C2
Vc and vp=apparent volumes of C1 and C2
= K12 xc – K21 xp
Vc vp On integration equation gives conc of drug in central and
peripheral compartments at any given time t
Cp = xo [(
Vc
K21 – a)e-at + (K12 – b)e-bt]
b – a a – b
Xo = iv bolus dose

• The relation between hybrid and microconstants is given as:
a + b = K12 + K21 + KE
A b = K21 KE
Cc = a e-at +be-bt
Cc=distribution exponent + elimination
exponent
A and B are hybrid constants for two exponents and can be resolved bygraph
by method of residuals.
A = X0 [K21 - A] = CO [K21 –A]
VC
B = X0
B –A
[K21 - B] =
B –A
CO [K21 – B]
VC A –B A – B
CO = Plasma drug concentration immediately after i.v. Injection

• Method of residuals : the biexponential disposition curve obtained after i. V.
Bolus of a drug that fits two compartmentmodel can be resolved into its
individual exponents by the method of residuals.
C = a e-at + b e-bt
From graph the initial decline due to distribution is more rapid than the terminal
decline due to elimination i.E. The rate constant a >> b and hence the term e-at
approaches zero much faster than e –bt
C = B e-bt
Log C = log B – bt/2.303 C = back extrapolated pl. Conc.
• A semilog plot of C vs t yields the terminal linear phase of the curve having
slope –b/2.303 and when back extrapolated to time zero, yields y-intercept log
B. The t1/2 for the elimination phase can be obtained from equation
• t1/2 = 0.693/b.
• Residual conc values can be found as-
Cr = C – C = ae-at
Log cr = log A –at
2.303
A semilog plot cr vs t gives a straight line.

Ke = a b c
A b + B a
K12 = a b (b - a)2
C0 (A b + B a)
K21 = A b + B a
C0
• For two compartment model, KE is the rate constant for elimination of drug
from the central compartment and b is the rate constant for elimination from
the entire body. Overall elimination t1/2 can be calculated from b.
Area under (auc) = a + b
The curve a b
X0 = X0
App. Volume of central =
compartment C0 KE (AUC)

App. Volume of = VP = VC K12
Peripheral compartment K21
Apparent volume of distribution at steady state or equilibrium
Vd,ss = VC +VP
Vd,area= X0
BAUC
Total systemic clearence= clt = b vd
Renal clearence=clr = dxu = KE VC
Dt
The rate of excretion of unchanged drug in urine can be represented by :
dxu = KE A e-at + KE B e-bt
Dt
The above equation can be resolved into individual exponents by the method of residuals.

TWO – COMPARTMENT OPEN MODEL- I.V.
NF
I USION
The plasma or central compartment conc of a drug when administered as constant rate (0 order) i.V.Infusion is
given as:
C = R0 [1+(KE - b)e-at +(KE - a)e-bt]
VC KE b – a a - b
At steady state (i.E.At time infinity) the second and the third term in the bracket becomes zero and the equation
reduces to:
Css = R0
Vc ke
Now VC KE = vd b
Css = r0 = r0
Vdb clt
The loading dose X0,L = css vc = R0
Ke
1
Central
2
Peripheral

TWO-COMPARTMENT OPEN MODEL-
• First - E
ord
X
erT
abs
R
orp
A
tio
V
n :ASCULAR ADMINISTRATION
• For a drug that enters the body by a first-order absorption process and
distributed according to two compartment model, the rate of change in drug
conc in the central compartment is described by three exponents :
• An absorption exponent, and the two usual exponents that describe drug
disposition.
The plasma conc at any time t is
C = n e-kat + l e-at + me-bt
C = absorption + distribution + elimination
Exponent exponent exponent
• Besides the method of residuals, ka can also be found by loo-riegelman method
for drug that follows two-compartment characteristics.
• Despite its complexity, the method can be applied to drugs that distribute in any
number of compartments.

CALCULATING Ka using Wagner-
nelson method(Bioavailability
parameters)

WAGNER-NELSONS METHOD
THEORY: The working equations can be derived from the mass balance
equation: Gives the following eqaution with time and mass balance
• Above equation Integratinggives
• Tothe equation amount
absorbed VERSUS TIME

• Taking this to infinity where cp equals 0
• Finally (Amax - A), the amount remaining to be absorbed can also be
expressed as the amount remaining in the GI, xg
• Wecan use this equation to look at the absorption process. If, and onlyif,
absorption is a single first orderprocess

• Example data for the method of wagner-nelson kel (from IV data) = 0.2 hr-
Time
(hr)
Plasma
Concentratio
n
(mg/L)
Column
3
ΔAUC
Column
4
AUC
Column 5
kel * AUC
A/V
[Col2 +
Col5]
(Amax - A)/V
0.0 0.0 0.0 0.0 0.0 0.0 4.9
1.0 1.2 0.6 0.6 0.12 1.32 3.58
2.0 1.8 1.5 2.1 0.42 2.22 2.68
3.0 2.1 1.95 4.05 0.81 2.91 1.99
4.0 2.2 2.15 6.2 1.24 3.44 1.46
5.0 2.2 2.2 8.4 1.68 3.88 1.02
6.0 2.0 2.1 10.5 2.1 4.1 0.8
8.0 1.7 3.7 14.2 2.84 4.54 0.36
10.0 1.3 3.0 17.2 3.44 4.74 0.16
12.0 1.0 2.3 19.5 3.9 4.9 -
∞ 0.0 5.0 24.5 4.9 4.9 -

• The data (Amax-A)/V versus time can be plotted on semi-log and linear
graph paper

• Plotting (Amax-A)/V versus time produces a straight line on semi-log graph paper and a
curved line on linear graph paper. This would support the assumptionthat absorptioncan be
described as a single first process. The first-order absorption rate constant, ka, can be
calculated to be 0.306 hr-1 from the slope of the line on the semi-log graph paper.
ADVANTAGES:
• The absorption and elimination processes can be quite similar and accurate determinations of
ka can still be made.
• The absorptionprocess doesn't have to be first order. This method can be used to investigate
the absorption process.
DISADVANTAGES:
• The major disadvantage of this method is that you need to know the elimination rate constant,
from data collected following intravenous administration.
• The required calculations are more complex.

RESIDUAL METHOD OR
FEATHERINGTECHNIQUE
• Absowhen a drug is administered by extravascular route, absorption is a
prerequisite for its therapeutic activity.
• The absorption rate constant can be calculated by the method of
residuals.
• The technique is also known as feathering, peeling and stripping.

φ It is commonly used in pharmacokinetics to resolve a
multiexponential curve into its individual components.
φ For a drug that follows one-compartment kinetics and
administered extravascularly, the concentration of drug
in plasma is expressed by a biexponential equation.
C=
𝐾𝑎𝐹𝑋0
𝑉𝑑(𝐾𝑎−𝐾𝐸)
[e-KEt – e-Kat] (1)
If KaFX0/Vd(Ka-KE) = A, a hybrid constant, then:
C = A e-KEt – A e-Kat (2)

φ During
almost
the elimination phase, when absorption is
over, Ka<<KE and the value of second
exponential e-Kat approaches zero whereas the first
exponential e-KEt retains some finite value.
φ At this time, the equation (2) reduces to:
𝐶−
= 𝐴𝑒−𝐾𝐸𝑡(3)
φ In log form, the above equation is:
Log C−
= log A -
𝐾𝐸𝑡
2.303
(4)

Where ,
C− = back extrapolated plasma concentrationvalues
φ A plot of log C versus t yield a biexponential curve with a
terminal linear phase having slope –KE/2.303
φ Back extrapolation of this straight line to time zero yields y-
intercept equal to logA.
70

Plasma conc.-Time profile after oral administration of a single dose of a drug

φ Subtraction of true plasma concentration values i.e.
equation (2) from the extrapolated plasma
values i.e. equation (3) yields a series
concentration
of residual
concentration value Cτ.
(C− - C) = Cτ = A e-Kat (5)
φ In log form , the equation is:
τ 2.303
log C = log A - 𝐾𝑎𝑡
(6)

φ A plot of log Cτ versus t yields a straight line with slope -
Ka /2.303 and y-intercept log A.
φ Thus, the method of residual enables resolution of the
biexponential plasma level-time curve into its two
exponential components.
φ The technique works best when the difference between
Ka and KE is large (Ka/KE ≥ 3).

MULTI COMPARTMENT MODEL BIOPHARMACEUTICS..pdf

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