Onecompartmentmodeling

ONE COMPARTMENT MODEL
(Instantaneous Distribution Model)
Department of Pharmacy Practice

One compartment
2Department of Pharmacy Practice

One compartment

More than one compartment
4

More than one compartment

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Assumptions:- (one-compartment open model)
 The one-compartment open model is the simplest model. Owing
to itssimplicity, it is based on following assumptions-
1. The body isconsidered asasingle, kinetically homogeneous unit
that has no barriers tothe movementof drug
2. Final distribution equilibrium between the drug in plasma and
other body fluids (i.e. mixing) is attained instantaneouslyand
maintainedatall times. This model thus applies only to those drug
that distribute rapidly throughout the body
3. Drugs move dynamically, in (absorption) andout
(elimination) of thiscompartment
4. Elimination isa firstorder(mono exponential) process
with first order rateconstant

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5. Rate of input (absorption)> rate ofoutput(elimination)
6. The anatomical reference compartment is plasma and
concentration of drug in plasma is representative of drug
concentration in all body tissues ie. Any change in plasma drug
concentration reflects a proportional change in drug
concentration throughout thebody
However the model does not assume that the drug concentration
in plasma is equal to that in other bodytissues

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One compartment:

ONE COMPARTMENT “OPEN MODEL”
 The term
(availability)
open indicates
and output (elimination)
that the input
are
unidirectional and that the drug can be
eliminated from the body.
8

 One – compartment open model is generally used to describe
plasma levels following administration of a single dose of a
drug.
Blood andother
BodytissuesDrug
Ka
Input
(absorption)
Ke
output
(Elimination )
Metabolism
Excretion

 Depending upon the rateof input, Following one
compartmentopen modelscan be defined:
1. One –compartment open model, I. V.bolus
administration
2. One –compartment open model, continuousI.V.
infusion
3. One-compartment open model, E.V.Administration,
zero orderabsorption
4. One compartment open model E.V.Administration,
first orderabsorption

One-compartment open model
12
Intravenous Bolus Administration:-
 When drug thatdistributes rapidly in the body is given
in the form of a rapid intravenous injection, it takes
about one to three minutes for complete circulation
and therefore the rate of absorption is neglected in
calculations.
 The model can be depictedas
Blood and other
Body tissues
Ke

dX/dt= Rate in (availability)- Rateout (elimination)----(1.1)
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Where KE= Firstorderelimination rateconstantand
X= amountof drug in the bodyatany time t remaining to be
eliminated
 Negativesign indicates that thedrug is being lost from the body
 Thegeneral expression forrateof drug presentationtothe body is
dX/dt= - Rateout -----(1.2)
dX/dt= -KE X -------------(1.3)
Sincerate inorabsorption isabsent, theequation becomes
If rateoutorelimination follows firstorder kinetics then

Estimation of pharmacokinetic parameters –IV Bolus
Administration
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 Fora drug that followsonecompartment kinetics and
administered as rapid IV injection, the decline in
plasma drug concentration is only due to
elimination of drug from the bodyand notdue to
distribution, the phase being called as elimination
phase. Elimination phase can be characterized by 4
parameters-
1. Elimination rateconstant
2. Apparent volume ofdistribution
3. Elimination half life
4. Clearance

Elimination rate constant (KE
)
 Elimination rate constant represents thefraction
of drug removed per unit oftime
 K has a unit of reciprocal of time (e.g.minute-1,
hour-1, and day-1)
 With first-order elimination, the rate of
elimination is directly proportional to theserum
drug concentration

Elimination rate constant
16
 The equation forelimination rate is
now integrating this equation
Where X0= amount of drugat time t = zero
Aboveequationcanalso bewritten in the following monoexponential
formatas:-
X= X0 e-K
E
t -----(1.5)
8
dX/dt= -KE X -----(1.3)
lnX= lnX0- KE t -----(1.4)

 Transforming equation (1.5) intocommonlogarithimwe
get:-
 Since it is difficult to determine directly the amount of drug in the body X,
advantage is taken of the fact that a constant relationship exists between
drug concentration in plasmaC and X thus
WhereVd = proportionality constant popularly known asthe apparent
volume ofdistribution
It is a pharmacokinetic parameter that permits the use of drug concentration
in place of amount of drug in the body.
17
Log x = log x0 –KEt/2.303 -----(1.6)
8
X= Vd C -----(1.7)

18
8
Then the equation 1.6 therefore becomes:
Log C = log C0 –KEt/2.303 -----(1.6)
Where Co = plasma drug concentration immediately after i.v. injection
Equation 1.6 is that of a straight line and indicates that a semilogarithmic
plot of log C versus t will be linear with Y-intercept log Co .
The elimination rate constant is directly obtained from the slope of the line
(Figure 1.a).
 It has units of min-1.
 Thus, a linear plot is easier to handle mathematically than a curve which in
this case will be obtained from a plot of C versus t on regular (Cartesian) graph
paper (Fig. 1.b).

Figure 1-(a) Cartesian plot of a drug that follows one-compartment
kinetics and given by rapid i.v. injection, and (b) Semi logarithmic plot
for the rate of elimination in a one-compartment model.
14
8

Thus, Co, KE (and t 1/2 ) can be readily obtained from log C versus t graph.
The elimination or removal of the drug from the body is the sum or
urinary excretion, metabolism, biliary excretion, pulmonary excretion, and
other mechanisms involved therein.
Thus KE is an additive property of rate constants for each of these
processes and better called as overall elimination rate constant.
8
KE = Ke + Km + Kb + Kl+ -----(1.7)

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The fraction of drug eliminated by a particular route can be evaluated if the
number of rate constants involved by and their values are known.
For example, if a drug is eliminated by urinary excretion and metabolism only,
them, the fraction of drug excreted unchanged in urine Fe and fraction of drug
metabolized Fm can be given as :
Fe = Ke
KE -----(1.7)
Fm = Km
KE -----(1.8)

Elimination Half-Life:
 It is defined as the time taken for the amount of
drug in the body as well as plasma concentration
to decline by one-half or 50% its initial value.
8
Elimination half-life can be readily obtained from the graph of log C versus t
as shown in Fig. 1.b
-----(1.9)

Today , increased physiologic understanding of
pharmacokinetics shows that half-life is a secondary parameter
that depends upon the primary parameters clearance and
apparent volume of distribution according to following
equation
8
Clearance and apparent volume of distribution are two
separate and independent pharmacokinetic characteristics of a
drug.

 The parameters that are closely related with
the physiologic mechanisms in the body, they
are called as primary parameters.
 1. Apparent volume of distribution, and
 2. Clearance.
16

8
Apparent Volume of Distribution (Vd)
Apparent volume of distribution may be
defined as the hypothetical volume of
body fluids into which a drug is
distributed.

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 The volume of distribution represents a volume that must be
considered in estimating the amount of drug in the body from the
concentration of drug found in thesampling compartment
 In general, drug equilibrates rapidly in the body. When plasma or
any other biologic compartment is sampled and analyzed for drug
content, the results are usually reported in units of concentration
instead ofamount
 Each individual tissue in the body may contain a different
concentration of drug due to differences in drug affinity for that
tissue. Therefore, the amount of drug in a given location can be
related to its concentration by a proportionality constant that
reflects the volume of fluid the drug is dissolvedin.
8

The real Volume of Distribution has physiological
meaning and is related to body water
Plasma
Interstitial
fluid
Intracellular
fluid
Total body water 42L
Plasma volume 4 L
Interstitial fluid volume 10 L
Intracellular fluid volume 28L
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Apparent Volume of Distribution
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 Drugs which binds selectively to plasma proteins, e.g. Warfarin
have apparent volume of distribution smaller than their real
volume ofdistribution
 Drugs which binds selectively to extravascular tissues, e.g.
Chloroquines have apparent volume of distribution larger than
their real volume of distribution. The Vdof such drugs is always
greater than 42 L (Total bodywater)
8

The Extent of Distribution and Vdin a 70 kg Normal Man
Vd, L
%
Body
Weight
Extent of Distribution Examples with volumeof
distribution in litre
5, low 7 Only in plasma Warfarin-7,
5-20,
medium 7-28 In extracellular fluids
ibuprofen-10
20-40,
High 28-56 In total body fluids. Theophylline -50
>40,
very
high
>56 In deep tissues; bound to
peripheral tissues
Ranitidine-500,
chloroquine-15000
8

 Lipid solubility of drug
 Degree of plasma proteinbinding
 Affinity for different tissueproteins
 Fat : lean bodymass
 Disease like Congestive Heart Failure (CHF),uremia,
cirrhosis
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 In order todetermine theapparentvolumeof distribution of
a drug, it is necessary to have plasma/serum concentration
versus timedata
dose

X0
initial conc. C0
Vd 
8

Significance of Vd
 It simply indicates how widely the drug is distributed in the
tissues compared toplasma
 For example Vdof paracetamol is 0.950 l/kgbody weight
 It means that 0.950 l of tissue is expected to contain the same
concentration of paracetamol as that contained in the blood
on the basisof average kg body weight.
 It does not mean that the remaining tissue contains zero
drug concentration. It is conceptually assumed and
expressed in thismanner.
8

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 Higher the Vd of a drug, more extensive is its distribution in
the tissue
 If the plasma drug concentration is low, it can beinferred
that the Vd is higher for a given dose
 If Vdis small then the drug concentration is morein
plasma and less distributed intissue.
 If Vd is 100% of body weight, then it may beassumed that
the drug concentration in certain tissue compartments
 If a drug is restricted to the vascular spaces and can freely
penetrate erythrocytes, the drug has a volume of
distribution of 6 litre.
 If the drug cannot permeate the RBC’s the available space is
reduced to about 3litre
8

 Clearance is a measure of the removal of drugfrom
the body
 Plasma drug concentrations are affected by the
rateat which drug is administered, thevolume in
which it distributes, and itsclearance
 A drug’s clearanceand thevolumeof distribution
determine its half life
 It is the most important parameter in clinical drug
applications and is useful in evaluating the
mechanism by which a drug is eliminated by the
wholeorganism or bya particularorgan
Clearance (Cl)

Clearance (Cl)
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 Clearance (expressed as volume/time) describes the removal of
drug from a volume of plasma in a given unit of time (drug loss
from the body)
 Clearance does not indicate the amount of drug being removed.
It indicates thevolumeof plasma (or blood) from which the drug
is completelyremoved, orcleared, in agiven time period.
 Figures in the following two slides representtwoways of
thinking about drugclearance:
 In the first Figure, theamountof drug (the numberof dots)
decreases but fills the same volume, resulting in a lower
concentration
 Anotherwayof viewing thesame decreasewould be tocalculate the
volume that would be drug-free if the concentration were held
constant as resented in thesecond Figure

Clearance (Cl)
The amount of drug (the number of dots) decreases but fills the
same volume, resulting in a lowerconcentration
37

Clearance (Cl)
 The most general definition of clearance is that it is ‘‘a proportionality
constant describing the relationship betweenasubstance’s rateof elimination
(amountperunit time) at a given time and its corresponding concentration in
anappropriate fluid at thattime.’’

Clearance (Cl)
Clearance is defined as the theoretical volume of body fluid containing
drug from which the drug is completely removed in a given period of
time.
It is expressed in ml/min or liters/hour.
Rate of eliminatio n
Clearance 
Plasma drug concentration
Vd = X0
KE AUC
Clearance is usually further defined as blood clearance (Clb), plasma
clearance (Clp) or clearance based on unbound or free drug
concentration (Clu) depending upon the concentration measured for the
equation
dX/dt
or Cl = ----------
C

Total Body Clearance : Elimination of a drug from the body involves
processes occuring in kidney, liver, lungs, and other eliminating
organs.
Clearance at an individual organ level is called as organ clearance.
It can be estimated by dividing the rate of elimination by each organ
with the concentration of drug presented to it. Thus,

The total body clearance. ClT also called as total systemic clearance, is an
additive property of individual organ clearances.
Hence, Total Systemic Clearance :
Because of the additivity of clearance, the relative contribution by any
organ in eliminating a drug can be easily calcualted.

Clearance by all organs other than kidney is sometimes known as nonrenal
clearance ClNR
It is the difference between total clearance and renal clearance.
Cl T = ClR + ClNR
ClNR = Cl T - ClR
According to cleareance definition
Substituting dX/dt = KEX above equation, we get:

Since X/C = V d above equation can be written as:
Parallel equations can be written for renal and hepatic clearances as:
Since KE = 0.693/ t1/2, clearance can be related to half-life by the following equation:
Equation shows that as ClT decreases, as in renal insufficiency, t1/2 of the drug
increases. As the ClT takes into account Vd, changes in Vd as in obesity or
edematous condition will reflect changes in ClT.

Identical equations can be written for ClR and ClH in which cases the t ½
will be urinary excretion half-life for unchanged drug and metabolism
half-life respectively.
The noncompartmental method of computing total clearance for a drug that
follows one- compartment kinetics is:
For drugs given as i.v. bolus, For drugs administred e.v.
For a drug given by i.v. bolus, the renal clearance ClR may be estimated by determine
the total amount of unchanged drug excreted in urine, Xu and AUC

49
One –compartment open model,
continuous I.V. Infusion

50
 IV infusion isadministered when thedrug has potential to
precipitate toxicity or when maintenance of a stable
concentration oramountof drug in the body is desired.
 In such a situation, the drug for eg. Theophylline,
procainamide, antibioticsetc is administered at aconstant
rate(zero order) by IVinfusion.
 Advantages of zero order infusion of drugsinclude:
a. Easeof control of rateof infusion to fit individual patient
needs
b. Prevents fluctuating maxima and minima plasmalevel
c. Otherdrugs, electrolytes and nutrientscan beconveniently
administered simultaneously by the same infusion line in
critically ill patients

51
Intravenous infusion administration of fluids into a vein by means of a steel
Needle or plastic catheter.
This method of fluid replacement is used most often to maintain fluid and
electrolytebalance, or to correct fluid volume deficits after excessive loss of body
fluids, in patients unable to take sufficient volumes orally.
An additional use is for prolonged nutritional support of patients with
gastrointestinal Dysfunction.
Intravenous infusion administration of drug through the vein into the systemic
circulation at a constant rate over a determined period of time

52
 Model can be represent as : ( i.v infusion)
At any time during infusion, the rate of change in the amount of drug
in the body, dX/dt is the difference between the zero-order rate of drug
infusion Ro and first-order rate of elimination, -KEX:
Integration and rearrangement of above equation 1 yields:
-----(1)
-----(2)

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Since X = V d C, the equation (2) can be transformed into concentration terms as
follows:
At the starting of constant rate infusion, the amount of drug in the body is
zero and hence, there is no elimination.
As time passes, the amount of drug in the body rises gradually ( elimination rate less
than the rate of infusion) until a point after which the rate of elimination equals the rate
of infusion i.e. the concentration of drug in plasma approaches a constant
value called as steady-state, plateau or infusion equilibrium (Fig.- 10.3 ).
-----(3)

54

55
42
If adrug is given at a more rapid infusion rate, a higher SS drug
concentration isobtained but the time to reach SS is the same.

56
42
At steady-state, the rate of change of amount of drug in the body is zero,
hence the equation 1 becomes:
(or)
Transforming to concentration term and rearranging the equation:.
(---4)
(---5)
Where Xss and Css are amount of drug in the body and concentration of drug
in plasma at steady –state repectively.

57
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The value of KE (and hence t1/2) can be obtained from the slope of straight line
obtained after a semilogarithmic plot (log C versus t) of the plasma concentration-
time data generated from the time when infusion is stopped (Fig. 1)
 Alternatively, KE can be calculated from the data collected during infusion to
steady-state as follows:
Substituting R0/Clt = Css from equation 5 in equation 3 we get:
(---6)
(---7)
Rearrangement Yields:-

42
Transforming into log form, the equation 7 becomes:
(---8)
A semilog plot of (Css - C)/C55 versus t results in a straight line with
slope -KE /2.303 (Fig. 10.4).
Fig:-10.4 Semilog plot to compute KE from infusion data upto steady-state

59
42
The time to reach steady-state concentration is dependent upon the
elimination half-life and not infusion rate. An increase in infusion rate will
merely increase the plasma concentration attained at steady-state (Fig.
10.3).
If n is the number of half-lives passed since the start of infusion rate,
equation equation 6 can be written as:
(---9)

42
The percent of Css achieved at the end of each t1/2 is the sum of Css at previous
t1/2 and the concentration of drug remaining after a given t1/2 ,tabulated as
follows:-

61
42
For therapeutic purpose, more than 90% of the steady-state drug
concentration in the blood is desired which is reached in 3.3 half-
lives. It take 6.6 half-lives for the concentration to reach 99% of the
steady-state. Thus, the shorter the half-life (e.g. penicillin G, 30 min),
sooner is the steady-state reached.

62
42
 It takes a very long time for the drugs having longer half-lives
before the plateau concentration is reached (e.g. Phenobarbital, 5
days).
Thus, initially, such drugs have sub therapeutic concentrations.
This can be overcome by administering an i.v. loading dose large
enough to yield the desired steady-state immediately upon
injection prior to starting the infusion. It should then be followed
immediately by i.v. infusion at a rate enough maintain this
concentration (Figure 10.5 ).
Infusion Plus Loading Dose

63
42
A loading dose is an initial higher dose of a drug that may be administered
at beginning of a course of treatment before dropping down to a lower
maintenance dose.
Figure 10.5 Intravenous infusion with loading dose
As the amount of bolus dose remaining in the body falls, there is a
complementary rise Resulting from the infusion

64
42
Importance of Loading dose

65
42

66
42
Recalling once again the relationship X = VdC, the equation for computing the loading
dose Xo,L can be given:
Xo,L = CssVd
Substitution of Css =Ro/KEVd from equation 5 in above equation yields
another expression for loading dose in terms of infusion rate:
(--------10)
(--------11)

67
42
The equation describing the plasma concentration-time profile following
simultaneous i.v. loading dose (i.v. bolus) and constant rate i.v. infusion is the sum
of two equations describing each process
(--------12)
If we substitute CssVd for Xo,L (from equation 10 ) and CssKEVd for Ro (from
equation 5) in above equation and simplify it, it reduces to C = Css indicating
that the concentration of drug in plasma remains constant (steady) throughout
the infusion time.

68
42
Assessment of Pharmacokinetic Parameters
The first-order elimination rate constant and elimination half-life can be
computed from a semi log plot of post-infusion concentration-time data.
Equation 8 can also be used for the same purpose.
Apparent volume of distribution and total systemic clearance can be
estimated from steady state concentration and infusion rate (equation 5).
These two parameters can also be computed from the total area under the
curve (Fig 10.3 ) till the end of infusion:
Where, T = infusion time.
The above equation is a general expression which can be applied to several
pharmacokinetic models.
(--------13)

Conclusion
 In contrast to short duration of infusion of an i.v bolus
(few second) ,the duration of constant rate infusion is
usually much longer than half life of drug.
 The time course of drug conc determined after its
administration by assuming the body as single well
mixed compartment.
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Onecompartmentmodeling

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