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Basic Pharmacokinetics REV. 99.4.25 11-1Copyright 1996-1999 Michael C. Makoid All Rights Reserved http://kiwi.creighton.edu/pkinbook/

CHAPTER 11 Multicompartment Modeling

Author: Michael Makoid

Reviewer: Phillip Vuchetich

OBJECTIVES

1. This chapter is not completed.


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Multicompartment Modeling


11.1 Executive Summary


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11.2 Equations


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11.3 PHARMCOKINETICS: MAMMILLARY MODELS

For many drugs the equilibrium between drug concentrations in different tissues is

not achieved rapidly. Thus, one of the assumptions of the one-compartment open

model sometimes becomes invalid. A more complex mammillary open model is

often necessary to describe mathematically the plasma concentration data (for

example) seen after the administration of some drugs. The simplest mammillary

open model is a two-compartment open model: for example:

Compartment One (central compartment) can be sampled through the blood (or

plasma, or serum). It may consist of organs or tissues which, being highly perfused

with blood, are in rapid equilibrium distribution with the blood.

Compartnent Two (peripheral compartment) cannot normally be sampled. It may

consist or organs or tissues which, being poorly perfused with blood, are in slow

equilibrium distribution with the blood.

The Body is the sum of both compartments.


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1. Biexponential Properties of Two-Compartment Open Model

Following an intravenous bolus injection, the plasma concentration against time

profile has two phases:

a. Initial phase - ( - phase)

b. Terminal phase - ( - phase)

On semilogarithmic paper the terminal phase is linear, indicating that initial distri-

bution has been completed and that equilibrium has been attained. The terminal

half-life ( ) can be measured from the terminal phase.

2. Intravenous Bolus Administration: Plasma Concentration Data

For a one-compartment open model,

(EQ 10-26

i.e., the concentration of drug in the plasma declines exponentially with time

For a two-compartment open model,

t1 2

Cp Cp( )oekt

=


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(EQ 10-27

i.e., the concentration of drug in the plasma declines biexponentially with time

2.1 Symbols

and are intercept constants

and are hybrid rate constants

is the apparent volume of unchanged drug distribution in compartment one

, are micro rate constants

2.2 Relationships (for reference, except Eq. 3)

Cp A1et

B1et

+( )=

A1 B1 M L3( )

T 1( )

V1

L3( )

k10 k12 and, , k21 T1( )

0.5 k10 k12 k21+ +( ) k10 k12 k21+ +( )2

4k10k12+[ ]=


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(EQ 10-28

2.3 Obtaining Pharmacokinetic Parameters by Feathering

By convention, >

a. Plot against t on semilogarithmic paper

b. Find from the linear terminal phase: see Intravenous Administration

section A1.4a

c. Calculate the terminal hybrid rate constant ; in reality it contains both dis-

tributive ( and ) and elimination factors.

(EQ 10-29

0.5 k10 k12 k21+ +( ) k10 k12 k21+ +( )2

4k10k12[ ]=

A1D

V1------

k21( )

( )----------------------=

B1D

V1------

k21 ( )

( )---------------------=

A1 B1 Cp( )o=+

Cp

t1 2

( )k12 k21 k10( )

0.693t1 2

-------------=


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d. Draw a straight line through the linear terminal elimination phase and extralpo-

late this line to t = 0. The intercept is equal to .

e. Read estrapolated plasma concentrations from the plot at times equal to

those given for values of which are prior to the terminal phase.

f. At each of these times calculate:

g. Plot against t (see Eq.8) on semilogarithmic paper. The is a feath-

ered line and should decline linearly.

h. Find the half-life of the plot. It wil refer to the initial phase. Calculate,

i. Measure the intercept of the feathered line; it will equal to (Note that usu-ally , even theoetically).

B1

Cp( )

Cp

Cp( )diff Cp Cp=

Cp( )diff

0.693half life-------------------------=

A1

A1 B1=


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j. Calculate from Eq. 3

k. Calcultate by

(EQ 10-30

.

Theory

When t is large, < . Hence, Eq. 2 becomes

(EQ 10-31

i.e., when t is large, the concentration of the drug in the plasma declines exponen-

tilly with time.

The extrapolated plasma concentrations are

(EQ 10-32

Substituting from Eqs. 2 and 7a into Eq. 5,

Cp( )o

V1

V1Xo

Cp( )o-------------

D

A1 B1+------------------==

et

et

Cp B1et

=

Cp B1et

=


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(EQ 10-33

i.e., the difference between observed and extrapolated drug concentrations in the

plasma declines exponentially with time.

Note (for reference only)

It is usually not informative to determine the microrate constant; but see one use

under the note on dosage regimens.

2.4 Clearance and Volume

If model-independant equations can be used to define these terms, this is preferred

a. Systemic Clearance (Cl) may be calculated by,

Cp( )diff A1et

=

k21B1 A1+

A1 B1+--------------------------=

k10 k21=

k12 k10 k21+=


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(EQ 10-34

b. The volume terms are complex than in a one-compartment open model. There

are two terms of interest:

The apparent volume of distribution in compartment one

This is calculated using Eq. 6.

The apparent volume of distribution at pseudo-distribution equilibrium

This volume may be defined only in relation to the terminal phase ( phase), when

initial distribution has been completed.

As requires calculation of the total area under the plasma concentration against

time curve it is sometimes known as .

c. Comparing Eqs. 9 and 10,

(EQ 10-35

ClD

AUC( )o

-----------------------=

V1( )

V( )

VD

AUC( )o---------------------------=

V

Varea

Cl V=


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It may also be shown that,

(EQ 10-36

This follows as systemic clearance is always given by the elimination rate constan

out of the body multiplied by the apparent volume of distribution in the compart-

ment from which drug leaves the body. Comparing Eqs. 11 and 12,

(EQ 10-37

Note that (the elimination rate constant) is not the same as (the terminal

hybrid rate constant).

2.5 Bioavailability

Find using trapezoidal rule and, if necessary, the calculation for the ter-

minal area.

(EQ 10-38

This is a model-independent equation.

Cl k10V1=

V

k10

-------V

1

=

k10

AUC( )0

AUC( )o

AUC( )ot Cp

------+=


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2.6 Dosage Regimens

The maintenance dose (D) is given by the same model-independent equation asbefore,

Where has its same previous definition.

The loading dose achieves a steady-state condition quite rapidly, but only

after initial distribution has been completed. It is given by the previous equation.

(EQ 10-39

As may be expected, equations relating and to are as

before,

Note (reference only)

All dosage regimen equations strictly apply only when,

D Cp( )ssCl =

Cp( )ss

DL( )

DLD

1 e0.693N

---------------------------=

Cma x( )ss Cmin( )ss Cp( )ss


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For digoxin Eq. 17 has a value of 0.947

For warfarin Eq. 17 has a value of 0.990

For cephalexin Eq. 17 has a value of 0.846

This is why, despite the fact that an open two-compartment model is better descrip-

tion of the pharmacokinetic of these drugs, a simple open-compartment model mayoften be assumed for dosage regimen purposes.

3. Intravenous Bolus Administration: Compartment Two

It is not normally possible to measure drug concentrations in compartment two.

However, the mass of drug can be predicted based on the ddrug concentrations

observed in compartment one.

2.6 Dosage Regimens

(EQ 10-40

Where

k10------- 1

k12

k21-------+

1=

X2 B2 et

et

( )=

B2k12D

( )-----------------=


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Note that the equations forms bears a similarity to that seen for plasma concentra-

tions after oral administration inot a one compartment open model.

When t is large, < . Hence, Eq. 18 becomes,

(EQ 10-41

This is comparted to the mass modification of Eq. 7,

(EQ 10-42

Thus, when t is large the masses of drug in each compartment decline exponen-

tially, and in parallel, with time. This indicates that initial distribution has been

completed and equilibrium attained.

If the value of reflects drug concentrations at the active site, the time of maxi-

mum concentration (and maximum pharmacological effect) is:

(EQ 10-43

4. Others Dosage Forms

The equations become complex and it is therefore difficult to obtain useful param-

eter values without th eaid of a computer. Fortunately, because the complexity of

the equations is greater than the experimental accuracy of the assays warrants

drugs that strictly require a mammillary model can be described adequately by an

open one compartment for the purposes of calculating dosage regimens.

4.1 Intravenous Infusion

The plasma concentrations at first rise faster than an open one compartment model

profile would suggest. Later, the rise is slower. The decline, following the cessa-

tion of infusion, is biexponential.

4.2 Oral Administration

et

et

X2 B2et

=

X1 V1B1et

=

X2

tmax ( )ln

---------------------=


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At a time just after the plasma concentration may exhibit a nose, when

compared to the profile of an open one-compartment model.

SELECTED REFERENCES

Riegelman, S., Loo, J.C.K., and Rowland, M., Shortcomings in pharmacokinetic

analysis by conceiving the body to exhibit properties of a single campartment, J

Pharm . Sci., 57, 117-123 (1968).

Riegelmen, S., Loo, J.C.k., and Rowland, M., Concept of a volume of distribution

and possible errors in evaluation of this parameter, J. Pharm. Sci., 57, 128-133

(l968).

Benet, L.Z. and Ronfeld, R.A., Volume terms in pharnacokinetics, J. Pharm. Sci.,

58, 639-641 (l969).

Gibaldi, M Nagashima, R., ant Levy, G., Relationship between drug concentra-

tions in plasma or serum and amount of drug in the body, J. Pharm. Sci., 58, 193-

197 (1969).

Metzler, C.M Usefulness of the two-compartent open model in pharmacokinetics

J. Amer. Stat. Assn., 66, 49-54 (1971).

tmax


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Gibaldi, M. and Perrier, D., Drug eliminatin and apparent volume of distribution in

multicompartment systems, J. Pharm. Sci., 61, 952-954 (1972).

Gillette, J.R., The importance of tissue distribution in pharmacokinetics, J. Phar-

macokinetics. Biopharm., 1, 497-520 (1973).

DRUG DISPOSITION: VOLUME TERMS

As apparent volumes of distribution are proportionality constants, and not physio-

logical volumes, more than one term is of value.

1. Apparent Volume of Sampled Compartment

This relates the concentration of drug in the sampled compartment with the mass

of drug present in that compartment.

It may be measured after an intravenous bolus dose:

(EQ 10-44

or (EQ 10-45

It may be measured after an intravenous infusion by:

(EQ 10-46

V1( )

V1D

Cp( )o-------------=

V1D

K AUC ( )o

---------------------------=

V1

X1( )ssCp( )ss

---------------Q

K Cp( )ss-------------------==


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2. Apparent Volume at Pseudo-Distribution Equilibrium

This volume term (sometimes known as the apparent volume of distribution of the

drug in the body) requires the assumption that the drug is evenly distributed

throughout the body. The assumption is not true in practice. Thus can only be

defined in relation to the terminal phase ( -phase) when equilibrium has been

attained; the equation is analogous to Eq. 2.

(EQ 10-47

3. Relationships Between Apparent Volumes

By secondary alebraic definition, a clearance (Cl) is always given by the first-order

rate constant for removal of drug from the body multiplied by the apparent volume

of distribution on the drug in the compartment from which the drug leaves the

body:

(EQ 10-48

(EQ 10-49

(EQ 10-50

However, systemic clearance is measured by

(EQ 10-51

Comparing Eqs. 4 and 8, and rearranging,

V( )

V

V D AUC( )o

---------------------------=

Clr kuV1=

Clm kmV1=

Cl s KV1=

Cl sD

AUC( )o

-----------------------=


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(EQ 10-52

Comparing Eqs. 7 and 9,

(EQ 10-53

Selected References

Riegelman, S., Loo, J.C.K., and Rowland, M., Concept of a volume of distribution

and possible errors in evaluation of this parameter, J. Pharm. Sci., 57, 128-133

(1968).

Benet, L.Z. and Ronfeld, R.A., Volume terms in pharmacokinetics, J. Pharm. Sci.

58, 639-641 (1969).

Gibaldi, M., Nagashima, R., and Levy, G., Relationship between drug concentra-

tions in plasma or serum and amount of drug in the body, J. Pharm. Sci., 58, 193-

197 (1969).

Perrier, D. and Gibaldi, M., Relationship between plasma or serum drug concentra-

tion and amount of drug in the body at steady state upon multiple dosing, J. Phar-

macokin. Biopharm., 1, 17-22 (1973).

Oie, S. and Tozer, T.N., Effect of altered plasma protein binding on apparent vol-

ume of distribution, J. Pharm. Sci., 68, 1203-1205 (19793).

Li;vxrX LLrLLxLS

Cl s V=

V K----V1=


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Drug is usuallo sampled from the centr->l compartment, designated compartmentone.

1. Laplace Transforn for Compartment One

As,l

(in)(dS 1)

where As,l is Laplace Transform for mass of drug in comPartment one

s is the Laplace Operator in is the input function

dS,l is ehe disposition function for compartment one

2 InDut Functions

. .

N ehat input need not necessarily be to compartment one.


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2.1 IV Bolus

(in) - D

where D is the dose

2.2 IV Infusion

Q (l-e sb)

where Q is the zero-order infusion rate, b-t when e


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(s+kr) (s+ka)

where kr is tirst-order dissolution rate constant.

2 5 Dissolution and Absorption (tvpe 2)

(la)

ka(l-e sb)

s(s+k>)

where ko is zero-order dissolution rate, ceasing at time T.

2-6 Others

These may be formed by adtition of functions 2-1 through 2-5

e.g., (in) - D + Q(l-e~Sb)

This denotes the simultaneous commencement of an I.V. bolus and infusion.


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3. Disposition Function for Comcartment One

A driving force compartment has one or more exit rate constanes; for

.

instance, in compartment i, ehe sum of ehe first-order exit rate constants

is Ei.

As,~

n~

kca a +

where q is the compartment into which input occurs, n is the number of driving

force compartments,

i,j, and m are counters (maximum value of n),

kql is the first-order rate constant for transfer of drug from input compartment eo

compartment one,


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kl; and kjl are the first-order rate constants for drug transfer from compartment 1 to

compartment j, ant vice-versa.

3-1 Using the disposition function

(a) If q-l, ehen kqlsl

(b) Tr (Pi) and fT (Pm) are coneinued produces. ~e value of Pi (or Pm) equals one

when thc counter i (or m) takes on a forbidden number. For example, i-l is forbid-

den in the numerator, ant m-l and m-j are forbidden in the denominaeor. 3-Z

E.camples

(a) one-compartmene open model (n-l,q=l)

ds,l = 1 (Eq.l)

(b) Two-compart:.ent open models (n-2.q=1)

(s+E~)

ds , 1

(s+El) (s+E2) - kl2kx

(c) Three-compartmene open models(n-3,q-2)


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dS,l

k21 (s+E3)

(Eq.2)

(s+El) (s+E2) (s+E3) - kl2k21 (s+E3) - kl3t31 (5tk2)

(Eq.3)

3-3 Simplifying the Denominator The number of exponeneial terms in ehe final

ineegrated equation will be equal eo the number of driving force compartments

(n). This is also equal to the maximum power to which the Laplace operator (s)

would nppear if

the denominator were multiplied out. Hence, the denominator is simplified n to

become iXl (S+ki), where ki is a composite first-order rate constane.

(a) ds,l

(b) d5 l

(c) ts.l


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(s+kl)

(s+E2)

(s+kl) ts I k2)

k21 (s+E3)

( s+kl ) ( s+k2 ) ( s+k3 )

(Eq.la)

(Eq.2a)

(Eq.3a)

The exact meaning of [i for any model depends on the equalities evident in the

denominaeors. Example for (b):

(s+kl) (s+k2) - (s+El) (s+E2) - kl2k


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4. Method of Partial Fractions Ihis method is used to solve (integrate) a Laplace

Iransform providing there are no repeating factors in the denominator. Example

no 52 or (s+ki)2

l 1 Prepare che Laplace Transform

Example: I.V. bolus into compartment one of two compartment model

( s+kl ) ( S+k2 )

4-2 Obtaining the Roots of Denominator Factors

If the factor is s, the rooc is zero

If ehe faceor is (s+ki), the root is

4-3 Ridden-Hand Method

(a) Deal with each factor of ehe tenominator in turn.

(b) Cover the factor with a finger, and remember its root.


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(c) Wherever the Laplace operator(s) occurs in the uncovered transform, subseitute

the root for s.

(d) Multiply the resule by eses again substituting ehe rooe for s

(e) After doing (b) through (d) for each factor, simplify.

Example:

X1 D(-kl+E2)e-klt - + D(-k2+E2)e~k2t

(-kl+k2)(-k7+kl )

or C1 - D (kl-Ez)e kl + D (E2-k2)e 2

V1 (kl k2 )

or C1 - Ale 1 + A2e-k2t

In this example the meaning of A1, A2, kl, k2, and E2 depend on the form

of the two-compartmental model.


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5. Laplace Transform for Peripheral Compartments

This is obtained by the following procedure, which is analogous eo that

employed when using the Laplace Transform table.

(a) Write the differential rate equation.

(b) Take the Laplace Transform of each side of the differential rate equation, using

the table where necessary.

V1 (kl-k2)

(c) Algebraically manipulate the transformed equation until an equation having

onlv one transformed dependent variable on the left-hand side is obtained.

(t) Substitute for anv known transformed dependent variables on tlle right-lland

sidc of the equation.

(e) Solve (integrate) bv the method of partial fractions (tlle hidden 1land), and

simplifv.


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6. btethod if Denominator Contains the Factor 52 This may apply to termina

compartments, such as urine, following an I.V. infusion. The hidden hand

method cannot be used for the factor 52 in the denominator as it has no simple

root.

6-1 Example (n=2, q^l, exit from compartmenr one):

aS,U - kloQ.(l-e~sb)(s+Es)

52(5+kl) (s+k2)

where klo is the first-order excretion rate constant from compartment one.

xu ~ kloQ.E2b + ......

klk2

where Xu is the cumulative mass of drug excreted into the urine.

The other factors can be used as before in the hidden-hand method.

6-2 Example (n-3, q-l, exit from compartment one)

aS u kloQ.(l-e 5b)(S+E2)ts+E3)

s2(s+kl)(s+kv)(s+k3)

Xu t kloQ- E2Elb I


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klk2k3

REFERENCES

L.Z. Benet, General treatment of linear mammillary models with elimination from

any compartment as used in pharmacokinetics, J. Pharm. Sci., 61, 536-541 (1972)

D.P. Vaughan, D.J.H. Mallard, A. Trainor, and M. Mitchard, General pharmacoki-

netic equations for linear mammillary models with trug absorption into periphera

compartments, Europ. J. Clin. Pharmacol., 8, 141-148 (1975).

D.P. Vaughan and A. Trainor, Derivation of general equations for linear mammill-

ary models when ehe drug is administered by different routes, J. Pharmacokin

Biopharm., 3, 203-218 (1975).

Two-Compartment Model-l

Prior inputs focuset on one-compartmcnt models, but many drugs arc charactetizet

bettcr by multicompartmcnt motek. In the following three inputs, we shall bricfly

tiscuss multico~_nt motek ant prcstnt a few apB plicadons. Multicompartnent mot-

cis are not uset as fo quentlg u the one-compartment model in therapeutic trug

monitonng, panly because they arc more tifficult to construct ant apply.

Gencially, muldeaw models arc appliet when th,e natural log of plasma drug con-

sentration vcrsus time is not lincar afier an intravenous tose or when thc plasma

concentration versus time psfilc cannot bc chu~ by a single cxpooential function

(i.c., C, - CO e~~). Wben the In of plasma concentration vcrsus timc is not a


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stnight line, a multicompartmcnt model must bc constructet to tescribe the change

in concentrations over time.

Of the mul

models, tbe two-compartment motel is tnost fxqucntly uset. lunis model usually

of thc weU-perfia

tissues ant penpbexal compartment of less weU Erfuset dssues (such as muscle

ant fat). hgure 23^ shows a diagram of thc two-compartmcnt model afir an intrave-

nous bolus tose, where:

consists of a central

Xfamount of diug in centnl comva XP s amount of drug in psipheral cowt

K,2rate const nt for transfcr of drug from cd-compartment zo petipheral com-

partment rne subsaipt 12 irldicses tr nsfcr from thc first (cd) to the second

(peripheral) compattments.

K2, - rate constant for tgansfcr of drugfrov peripheral computment to central comprtment lbe subscript 21 indicales tr nsfcr from Ulc second (periphaal) to the fint

Xo~


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Kl2

K2l

+ Klo

zgre23~ Gr phic reprtsentation of a zwbcompattment model.

(centri) compartments. (Nott h Kz2 and K2,calkd micxnts.)

fintvder climinX aue consunt (similar to tbe Jr uxd paviously), iting elimiXn of

dmg out of tbe caul ~ into urine, feces,

esc~

A log plasma conscatration versus time curve for a two-compattment model shows

a curvilinear profilea atrved potoon followed by a straight li=. This biexponen-

tial curve c n bc described by two expoKntial tcrms (Flgure 23B). lEc phases of

the curve may reprcstnt rapid d1stributioo to organs with high blood flow (central

compuenent) and slower distnbution to organs with Ess blood flow (penphcnl

compartmcnt).

Mer thc intovenous injection of a drug that follows a t_ model, thc drug consentra-

tions in all fluids and dmms associated with tbc central compartmcnt declinc morc

rapidly in tbc distributioo phasc thao during the post-diwibubon phasc. ARcr sornc


34/131



tirnc, a pseudocquihEutn is attained betwcen thc ccotral compartmcot and thc

dssucs atid fluids of toc pc ipheral compO thc pl sma coocentration vcrsus timc

ptnfile is thco chaserized by a linear pnmcess.

For many drugs, suco as aminoglycosides, thc distributdoo phast is vcry shott (e.g.

1920 mtn). If serum consentradons are measured after this phase is compited, toe

ceotral compartmcot can be ignorcd and a one-compartrKnt model adequatcly

repttsents the serum coocentratioos observed. However, for drugs such as vanco-

mycin, thc-distribution phase lasts 1-2 hr after an intravenous dosc. If plasma con-

centrations of vancomycin are determined within the first hour after a dose is

given, thc nonlincar (multiexponential) decline of vancomycin concentrations

must bc considered.

REVIEW PROBLEMS

23.1. In the twocompartment model. Xt wpresents the

23.2. The log plasms concentration vcrsus time curve for a two-compartment

model is reprcsented by a (bicxponential or monoexponential) cuNe. (Sclect one.)

23.3. The first portion of the log plasma concentration ver. sus timc cune. where

the log concentration r;tpidly declincs. is lomwn as the

phasc.

23.4. The final. Iinear portion of the curve is the phase.


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11- 1

~1

I>Pur 2t 61

70 [ So

~i

~m

~ .

F*wf 238 Four st ges of drug distribution nd eliminatioo following rapid intrave-

nous injectiott. Points I, U, m, nd tv (ript) corrcspond to the points oo the plasmx

concentntion curve (leR). Point 1: The injection has just becn compicted, and drug

density io the cd compartment is hipcst. Drug distribution and elimination hve just

begun. Poin~ 11: At midway through tbe distributioo process, the drug density in

thc central compartment is falling r pidly, dulioly owing to rapid drug distributioo

out of the centd ccrnpartment into the peripheral compartment. The density of drug

io the peripheral compartment has not yet mched tht in the central compartmcnt

Poixt 111: Distribution equilibrium h s been attained, and drug densitics in the

centd and periphed compartments arc appgoximately equal. Drug distribution inboth directioos contioucs to talze place, but the ratio of drug quantitics in toe centel

and peripheral compartments remains constant. At this point, the major determi-

nant of drug disappearance from thc central compartmcnt becomes the elimination

process; previously, drug disappearance W&S determined mainly by distribution

Poi/s : During this elimination phase, the drug is being -drained from both com-


36/131



partmcnts out of the body (via the central companment) at approximatcly the same

rate. (Reproduced, with permission, from Grecnblatt DJ &nd Shader Rl, Phrma-

colsinetics in clinical practice, W.B. Saunders, Philadelphia. PA, 198S.)

TwoW M~

lo this input, wc soall apply mathematicaS principles to toe two-compartmcnt

model to calculatc useful poarmacokinetic parameters.

horo tiscussioo of the ooe-comparaneot model, we koow that the climination ztecoostant (J[) is estsmated fxm the slope of the lo pbsma coocentration vcrsus ame

curve. However, in a two-canzrg tnodel, wose the lo plasma coocentration versus

time curve is curvilinear, the slope varics, tepcndiog on waich porioo of toe curve

is cxamined (Flgurc 24A).

In a two compartment model, the tenninal slope from the pos,t-digributive phase of

the curve may bc backextrapolata~ to axnc zero (T). The oegative slope of this line

is teferret-to as beta (O, aot ,B is the tennioal eliminatton tate coostant of the trug.

The iotesept of this lioe on the In plasn coocentration axis is koown as B and isuset in vanous two~cotnpeneot equations.

Bc~ is similtr to K in to t it vsents the tsminal elim meion r te constant. From it, a

half-life can be cal~ culS

T%, 0.693

is


37/131



which is refenet to as the beta haSf-Ute.

lEroughout the ame th_t trug is present in the boty, tistribuiion takes place betweenthe central ant peripheral compartmenu. We can calculate a ratc of tistribuaion

using the mct rcsidxals. This methot estim tes the cffect of distribution on the

ovcrall plasma concentration curvc and uses thc diZfcrcncc between thc cffect of

climination and thc actual plasma consentrations to determinc thc distribution rate.

In the In concentration versus vime curvc in Flgure 24A, the slope of the initial

portion is determined prim--arily by the distribution rate while thc tenninal portion

is determined primarily by thc climination rate.

E


38/131



The methot of residuals may be used for calculating phamiacoMinctic parameters

of thc two-compartmcnt model. FuSst, b cl:-extrapolate the terminal straight-line

portion of tbe curve (Flgure 24B). If w, s. y, and 2 are actual, detennined concen-

tration time points, let w, ~, y, and 2 xpresent points on the new (extopolated)

line at the same times that tbe aaual points were obscrved. These newly generatedpoints xpresent the cffect of clim~ ination alone, as if distribution had been instan-

taneous. Subtnction of the extrapolated points from the corresponding actual

points (ww, X~, etc.) yields a new set of pbsma concenti-ation points for

each time point. If we plot tbese new points with the appmpriate times, we gener-

ate a new line, the residual line (hgure 24C).

The slope of tbe xsidual line isst, and alpha (a) is the distribution rate constant

for the two-compartment system. The intrcept of the residual line is A. Therefore

witb the coocept of residuals, we attempt to separate the two pwocsscs of diseribu-tion and climin~ jon.

Ist us now pn~cood through an exampic, applying thc metbot of xsidtis. Draw tbc

plot for thc following cxampb on somilog gnph paper. A dosc of dnag is ad

10.0

50

:!

Z c 10

o

fL Q


39/131



Tlmo F~241R Mcthod of xsiduals.

E

100

5e

Y

Bz

~

\ Sbpo = a

s -R

Timo

F; 24C Dctermination of the rcsidual linc.


40/131



62

InPtJT 24 63

)

ministered by rapid intravenous injection, and the tollowing concentrations result:

Tkne afttr

Dose (hr)

0.2S

O.S

1.0

I.S

2.0

4.0

8.0

12.0

16 0


41/131



Plasms

Concentration

(u~/ml)

43

32

20

14

11

6.S

2.8

1.2

0.S2

A linc is trawn connecting the last four points and intcnecting the y-axis. Then, for

the first five points, cxtrapolated values can be cstimated at cach time (0.2S, O.S,

1.0,l.S, and 2.0 hr). If the extrapolated values from the actual plasma concentra-

tions are subtracted, a new set of points is generated (resitual concentration points)

as fts w~

Tlnse dir

Doz (hr)

0.2S

O.S


42/131



1.0

1.5

20

Pbsma Concentratlon (>g/ml)

^>e

Exupol ted Residud

14.S 28.S

13.S 18.S

12.3 7.7

1 1.0 3.0

!O.0 I.D

The zsidual concentrations are then plotted (on semilog paper) versus time, and the

slope of that plot equals 1.8 hr~t. When the negative is dropped, this slope

equals sx; we observe from the plot that the intercept (A) of the line is 4S Fg/ml.

We also can estimate a from the slope of the terminal straight-line portion (equal to

0.21 hr~ ~) and 8 (equal to IS 1lg/ml).

Alpha (ex) must be greater than beta (a), indicating that drug removal from plasma

by distribution into tissues proceeds at a greater rate than does drug removal from


43/131



plasma by eliminating organs (e.g., kidncys and liver). rhc initial portion of the

plot is steeper than the terminal portion.

REVIEW PR08LEMS

24.1. Dnw a log pbsma concentntion versus timc profile for a drug Oinimed by the

intravenous bolus nmute and best durizZ by a two-companrnent modeJ (Figure

24D).

242. Tbe slope of the tenninal phase of the above. plot equals

243. Tbe inucept of thc tenniial portion on tbe In pbsma concentstion axis is tenned

>.~ sca(g)~ tbe tenninal const nt of the dmg s it leaves the body.

24.S. One w y to calculatc a distnbution zate is to use tbe metbod of

24.C. Tbe fint step in the metaod of residuals is to.

- the telminal straight-line portioa of the curve. 24.7. The extryolatd points awsubtmed from tbe actuaJ observed at the correspoading times.

24.8. Tbe slope of the residual line equals


44/131



24.9. A is tbeof tbe In

plasnu concenamtion axis by tbe

line.

2410. Tbe coocept of residuals attempts to separate tbe two

processes ofand

100

50

ca e

Z c 10

F Q

Fkwe 24D

Tim

l j g


45/131



Two-Compartment Model-3

The estimations of A, 8, ss, ant t perforrned in tbe last input are useful for predic-

ing plasrrs concentotions of dmg ch~nzi by a two-compartrnent model. For

awrst rnodel (Flgureo2SA), we know th t thc plaSsma concentration (C) t any

time (t) can be dessnbet by

Cf ^ Ce e t

where CO is the initial concentration and g is ttne climination rate. Thae two-com-

partment rnodel (Flgure 2SB) is thc surn of two linear components, reprcsenting

distnbution ant elimination (Flgurc 2SC).

In thc sarnc w y, we can dctrminc dnag consentration (C) at ny tinx (t) by iding thc

two linear components. In cach casc, A or B i-s uset for CO, ant ex or z is used for

XY. Therefe

C, s +- 8 e~*

ThiSs equation is called a biexponential cquation bccausc two cxponents rc

iwaled. With thc onos


46/131



wrunt rnodcl (intravenous), wherc:

C. s Ce e ~

2 B

100

501

CO

10

S

Timo Fzwe 25A Pls dmg concentrations with a one com putment model aher an

insvenous bolus dose (first order elimination).

100

50

, o

z -, 10

5


47/131



~ c

o

Timc ft 25B Plasma drug concentrations with 3 two compartment model after an

intnvenous boluXs dose sfirst-order elimination) .

100

50

E; tO

Sz

Tlmo

Fw 25C lOr azmpo~s of a twozxpo~ (>

t) model.

~nentid becausc thc linc is

t tnodel, diffcrcnt volume of

thc equation is


48/131



describet by onc exponents

E or thc twowrat dWibution psramctrs cxist thc centol volurnc {V), thc cxtrapo-lated volunc (V,~ ,), tbc volunc by arca (V,,, also lcnown as V~|), ant thc stcadys-

tatc vohunc of diwibuX (V,,). Each of thcsc voluncs rclste to diffcrcat undertying

assumptioos.

As in thc onc-cornpenent rnotcl, a volunc can bc calculated by

V dose dox

,~ + B Co

For thc two-compartrnent model, this volurne would bc cquivalent to thc volumc

of the central compartment (V). Thc Ve rclates the amount of drug in thc-central

compartmcnt to the concentration in the central compartment. In thc two-compar-

trnent model, CO is determined by cxtrapolating back to thc y-axis from the upper

or initial straight-line portioo of the plot.

When we calculate the extrapolated volume of distribution (V,x,,,p), we assume

that instantaneous distribution has occurred. The effect of the iniial distribubon

phase is ignored:

da

B

V_


49/131



wherc B is the w-intetcept of thc line extrapolated from the terminal portion of the

curve. This volume of distribution determination may not provide a useful volume

term since it ovcrsimplifics the two-compartmcnt model and disregards thc distri-bution phasc.

Another volumc (V,,O or V~) is detcrmined from the area under the plasma con-

centration vcrsus time curvc and thc tcrminal climination rate constant. This vol-

umc is related as follows:

114

lNvts S 65

vffi

dox CL

=

ffi x AUC3

va


50/131



This calculation is not subject to thc ovcrsimplification of V,S,,p, but it is affected

by changes in clearance. The V,,,, relates the amount of drug in the body to the

concentration of drug in plasrna in the post-absorpion and postwdisttibuiion phase

A ffnal volume tenn is the volume of disttibution at steady state (V,,). Although it

is not affected by changes in drug eliminadoo or ckarance, it is more difficult to

calculate. One way to estimate Vs, is to use the two compartment microconstants:

V,Ve + ~2vf

21

or it may be estimated by more complicated methods using AUC.

Since-different methods can be used to calculate the various volumes of distribu-

tion of a two-compartment model, you should always specify the metbod used

When reading a pharrnacolcinetic study, pay particular attenion to the method forcalculating thc volume of distribution.

REVIEW PROBLEMS

2S.1. The terminal eliniination rate constant tn a twoaconF putment model is

2S.2. For the two-compurtment model. complete the equo tion describing the

relazionship of plasma concentration with time: C, -


51/131



25.3. (True or False) The equation describing elimination afer an intravenous bolus

dose of a drug charauerized by a two-compartment model lequires two exponential

terms.

2!;.4. A patient is given a 500-mg dose of drug by intravenous injection and the

following plasma concentrations result:

Plssms

Time snerConcentrstion

Dese (hr)ItsSml)

Oo ss

0.7S

l S

3

,6

72.0

46.0

33.0

26.3

20.0


52/131



16.6

1r.2

9.0

5.0

_.7

n Rr

Plot the points on semilog paper Ithree cycle) and deterTnine the following: a. ~. b.

B. c. Residual concentrations for the first five points. d. A.

. t.

Prodictod Dlssma concentration at 1.2 hr after the

dose.

S- V,.

h. V,, (if AUC = 131.S mg/L x hr and dose SOO mg).

~ A#=W{~ SGS 2

PRi4CTlCE SET 2


53/131



The following pxbiems are for your xview. Dcfinitions of symbols ant Icey equa-

tions = pxvitet hc~:

k ~w climinton nte cowt

C, s pbSllk tn~ _jUSt Aher a single inuvenous injectioQ

e - bese for the nxl log f~|eion - 2@71S

4 - nte of tose ~ion (msy be cxptesset as

milligsams per bour in the sense of a coQtinuous

infusion or u trug tose divitot by itifusion tune

for intemtittent infusions)

V voluxne of tistribution

C_t - pcalc p1ssfu a ocentntis

C_ tnmugn plsCOOCCD

a steaty state


54/131



a stcady st te t duFn of intsvenous inAlsion

For muldpk dose, intennittent, intnvenous bolus injeciiOQ g stF st~

k V (I - e~t)

C_Cw e~t

For muletplffbse~ intctmiu, intnvenow itafi~ioo

C - t (I ~ e t~)

~ VK (Ie~tD

C_ - Cp e~tt~4

Fot contitiuous infusion before stee st te is reacbed:

C - V^r (I - C-t)

For continuous infusion t stcaty stac

C.4. 4

Vt Ws


55/131



PS2*1. A 60-kg patient is begun on a continuous intravenous infusion of theophyl-

line at 40 mgtbr. a Forty-eight hours after beginning ttte infusion, the plasma con-

centniion is 12 ~/ml (12 mg/L). If we assume that tbis COncentRtiOD is tbe stcady

state, wb t is the tbeoobylline clearansc?

b. If the volume of distribution is cstimated to be 30 L, what are the X and half-

life?

c. Since we kww V and t. what would the concentntion bc 10 hr aftcr beginning the

infusion?

d. ff the idision i continuet for 3 days ant then discondauet, what woult tD plamS

consentm tion be 12 hr ~r stopping the infwiont

e. If tne infusion is continuct for 3 d ys at 40 mg/ hr ant the stcaty-statc pluma con-

centtation is 12 Fg/tnl, WDat rate of trug infusion would liltely xsult sn a concen-

tradon of 18 ~/ml?

f. ARcr the iocosed infusion nte above is begun, how bog would it tic to tuch a plu

coo~ cenudon of 18 Fgiml?

PS2< A 60-log p ekat is st rted on 80 mS of gentunicin ewety 6 br in I*br infu-

siagL

L If this pSt is us led to have an avenge V of 15 L ult a normal half-life of 3 hr

wtuat will be the pealc plm cocenttadon at stcady stuc?


56/131



b. After thc fifth tese. a peak plasma concentratlon (dsawn at thc ent of thc infu-

sion) is S Ag/ml ant thc ttnugh consentration (drawn right befott thc sixth tosc) is

0.9 Ag/ml. What is thc patients xtual gentamicin half-life? What is thc xtual vol-

ume of disvribution?

c. For this patient, what dosc should bc administen:d to reach a new steady-state

peak gentamicin concentration of 8 Fgiml? At this dosc. what will bc thc steady-

state trouQh concentration!

II_q

tFoD~ sU

~ Tvo-ConPart-ent Open Hodel

tatlente ufferln~ chronle tenal fellure often require h _ odiolyele. Drug~ cy be

~dxinletered by Injeetlon Into the wenoue lde of the he odtcl~ser ~chine

Such e rituetlon vax ~ecerlbed by L tourneeu-Scheb t 1 (Int. J. Clln. tharv col IS

116-120 {1972)) for lx pctlente vho received n introvenoue done ot gent _ letn

(90 ng). The neen etru- concentrctleno et sent~nicin (C~) hovvd blexponenttel

deellne vith tl t (t).

Ce t


57/131



(-~lllter)(oln)

o 60 10

5 75 20

5 25 lO

4 80 40

50 50

3 95 90

3 40 150

3 10 180

2 90 240

2 55 28S

(

o. Caleelete the ter lnal half-llfe (t~) the hybrid rete eonetento

nd O nd the coefflelente (~1 3 Sl) for gent- leln to theeo he-odlulyal~ putlente

b. tor ubJecte vith noreel renal functlon. the yete-le elearence (Cl~) of gentanletn

le pproalaetel1 0.041 llter/ In. Sec uee 98X ef Cl~ le due to excretlon ofunchanged drug. renel tciluro utll ~rkedly effect ~ent~lcin clecr~nce.


58/131



Show vhether the petlent~ undergoinfS he odlelyele xhlbit o nor-ol gentcnicln

yett-le elexr~nce.

Calculete the two pperent volune ot dletrtbut10n terwe (V nd V )

nd the ell fnetlon rete eonetont. Uhet frectton of tho gt ted eln ln the body fter 3

hr nlght be la the peripherel ee part~ent7 At vh t tbae doee the gent _ Icin in the

peripherel ee pert~ent rench

~t

4. Under the eondittons of h odSxlyxte. coleulete the doxe tD) of

sentcoleln tlSch vould be dolaletered very S hr {1) In order to ~alnteln n

everogew teady-ctete erus concentretlon of 4 g/llter.

. Wlthout the hc odielyrer two ~ale petlente ach xhlbited creetinine eleorence

(Cle ) of 5 llwin; the norx~l velue le 117 t 20 I/nin. A~using that the deereFeed

Clcr {e due to decresce In glo-eruler filtretloo rcte vhst vould h-ve beeD the renel

clecrxnce (Clr) of gent~elcia In there two patlentn hed they rz cined vithout the he-

odlolysert Uhct done vould then need to be dsinletered every hr to n Intoin (C~ t

4 tert Co pcro your eouer wStb tho thy~telea e Deek Refer neo.

Vancomycin is an antibiotic used in the treatment of endocarditis in patients aller-

gic to penicillin. It is poorly absorbed orally and acute pain is associated with inter-

muscular injection. I.V. is the route of choice. After a 1 gm I.V. dose the following

data is observed:


59/131



Time (hours)Concentration (mcg/ml)

.5 51

1.0 36

1.5 28

2.0 23

3.0 18.5

4 o 16

6.0 12.5

8.0 9 9

12.0 6.25

FIND: a) A1 ;

b) alpha

c) Clearance ;

d) * of drug in peripheral compartment at equilibrium .

Clearance:


60/131



In normals, Erythromycin is cleared 90% by metabolism and 10% by the kidney

Also, it is 90% protein bound at therapeutic levels. Welling and Creig (JPS 67

1057-9,1978) reported an increase in the half life from 2.0 to 2.3 hours while also

reporting an increase in clearance from 275 to 485 ml/min and an increase in Vdss

from 57 to 100 litres when comparing normals to uremic patients. (Uremic patientssuffer from inveased concentration of urea as a result of severe renal failure.) A

physician has just called you and asked you to explain how he could have seen an

increase of 15% in the half life and a 75% increase in clearance at the same time

and what is the impact of this on antibiotic therapy for his uremic patient. In clear

concise English (not techno-bable), prepare a short written answer to this request

Keep in mind that the man who requested the information is a medical professionat

intelligent, and very busy.

The following data was collected from a normal patient in the revious study (Well-ing, op. cit.) following an IV bolus injection of 500 mg of erythromycin (as lacto-

bionate salt ).

time (hr)concentration (mcg/ml)timeconcentration

0 12 4 3 2.6

1 6.4 4 1.9

2 4.0 6 1.2

8 0.4

~ .

Fmd the peak time in the peripheral compartment.the fraction of the drug in the

peripheral compartment at four hours.


61/131



Two compartment model

Spectinomycin (TROBICIN^TR^T) is an aminocyclitol antibiotic shown to beactive against most strains of NEISSERIA GONORRHOEAE at a minimum

inhibitory concentration of 20 mcg/ml. The usual adult dose is 2 g (4 g in areas of

known resistance) given I.M. through a 20 gauge needle. Initial studies were done

by the company to determine the pharmacokinetic parameters of the drug. The data

from a single IV Bolus dose of 0.5 g is as follows.

Time (minutes)Concentration

10 63

20 51

30 43

45 35

60 30

120 183

240 7.6

360 3.2

FIND:

(5 points) a) A^V1^V =

(5 points) b) B^V1^V =

(5 points) c) Clearance =


62/131



(5 points) d) t max in the peripheral compartment =

(10 points) e) % of drug in peripheral compartment at equilibrium =

Spectinomycin (TROBICINR) is an aminocyclitol antibiotic shown to be active

against most strains of NEISSERIA GONORRHOEAE at a minimum inhibitory

concentration of 20 mcg/ml. The usual adult dose is 2 g (4 g in areas of known

resistance) given I.M. through a 20 gauge needle. Initial studies were done by the

company to determine the pharmacokinetic parameters of the drug. The data from

a single IV Bolus

dose of 0.5 g is as follows.

Time (minutes)Concentration

10 63

20 51

30 43

45 35

60 30

120 18.3

240 7.6

360 3.2

b} B c) Ciearance d) t max in the peripheral compartment e) % of drug in periph-

eral compartment at equilibrium


63/131



The following information is offered from a 142 mg IV bolus dose of grisiofulvin

given to a 73 Kg man.

Time mcg/ml

1 1.67

2 1.22

3 .97

4 .83

6 .66

8 .56

12 .42

18 .27

24 .17

30 .11

Find A1, B1, alpha, beta, K10, K12, K21, Peak time in the peripheral

compartment, % in the peripheral compartment at equilibrium.

Can you assume that this drug can be estimated by a one compartment

model upon multiple dosing ?


64/131



Quinidine is current-ly used to treat ventricular and supraventricular arrythmias.- It

is available as a sulfate, gluconate, and polygalactouronate which contain 83%

62* and 60* by weight free base (pRa 8.6). Qunidino sulfate is available in 200,

260, 300 and 325 mg tablets, while quinidine gluconate is available in 325 mg tab-

lets. The following pharnacokinetic parameters are reported by Ueda:

PARAMETER

C1 renal (ml/min/kg)

C1 hepatic (ml/min/kg) V1 (L/kg)

V beta (L/kg) F (oral)

t 1/2 alpha (min) t 1/2 beta (hr)

MTC (mic/ml) NEC (mic/ml)

Population Ave (+ SD)

0.93 (0.52)

3.26 (1.74)

0.66 (0.38)

2.61 (1.10)

0.71 (0.16)


65/131



6.69 (4.03)

6.44 (1.63)

8.53 (0.81)

* , 1.85 (0.19)

% bound to protein80.00(2.50)

During the last pharmacokinetic exam you noticed ome cardiac problems.

Wh n you ch-cked with your physician, He prescribes quinidine. What

dose should you be on?

Later, he diagnoses mononucliosis from lack of sleep and poor ating h bits s weli

as cardiac arrythmias. Your liver funcion has dropped to 60* of normal. He pre-

scribes quinidine for you. What is the dose that you should be on ?

ie. -

ll-3

Quinidine is currently used to treat ventricular and supraventricular arrythmias. It

is available as a sulfate, gluconate, and polygalactouronate which contain 83%

62% and 60% by weight free base (pKa 8.6). Qunidine sulfate is available in 200,


66/131



260, 300 and 325 mg tablets, while quinidine gluconate is available in 325 mg tab-

lets. The following pharmacokinetic parameters are reported by Ueda:

PARAMETERPopulation Ave (+ SD)

C1 renal (ml/min/kg)0.93 (0.52)

C1 hepatic (ml/min/kg)3.26 (1.74)

V1 (L/kg)0.66 (0.38)

V beta (L/kg)2.61 (1.10)

F (oral)0.71 (0.16)

t 1/2 alpha (min)6.69 (4.03)

t 1/2 beta (hr)6.44 (1.63)

MTC (mic/ml)8.53 (0.81)

MEC (mic/ml)1.85 (0.19)

% bound to protein 80.00 (2.50)

During the last pharmacokinetic exam you noticed some cardiac problems. When

you checked with your physician, He prescribes quinidine. What dose should you

be on?

Later, he diagnoses mononucliosis from lack of sleep and poor eating habits aswell as cardiac arrythmias. Your liver funcion has dropped to 60% of normal. He

prescribes quinidine for you. What is the dose that you should be on ?


67/131



Methotrexate is a cytolytic used for the treatment of acute leukemia and other

forms of cancer. After a a 400 mg/kg dose the following data was recorded for a 12

y/o boy.

Time (hours)

6

12

18

36

48

60

72

90

Concentration (mcg/ml) 360 70 15

2

1.2

0.46

0.36

0.15


68/131



bA) B c) Clearance d) % of drug in peripheral compartment at equilibrium

PROBLEM SET

zumper w~wFS

t L) v , ~ s D fv

Patients suffering chronic renal failure often require hemodialysis. Drugs may be

administered by injection into the venous side of the hemodialyzer machine.

Such a situation was described by L*tourneau-Saheb et al (Int. J. Clin. PKinma-

col., 15, 116-120 (1977)) for six patients who receivet an intravenous dose 7 gen-

tamicin (90 mg). The mean serum concentrations of gentamicin (Cs) showed a

biexponential decline with time (t).

(~9cyml) (mRn)

6.60 10

5-75 20

5.25 30

4.80 40

4.50 50

3.95 90

3.40 150


69/131



3.10 180

2.90 240

2.55 - 285

. .

(a) Calculate the values of t1/29 B, Bj, ~, A~, and V1. LFtourneau

Saheb et al reported,

tl/2 = 5.50 t 0.77 hr (mean i SD)

B s 0.0022 t 0.0004 min 1

B1 s 4.76 f 0.62 vg/ml

v 0.053 t 0.009 min

A1 a 3.47 s 1.01 pg/ml

V1 s 11.3 ffi 2.0 litre

(b) Calculate C1 and VB

LFtourneau-Sahleb et al reported,

C1 s 40.8 t 7.8 ml/min

VB s 19.2 h 2.7 litre


70/131



_1

11-33

98X of the systemic clearance of gentamicin is composed of renal clearance-of

unchanged drug, so that renal failure woult severely alter gentamRcin systemic

clearance. In this case, the use of the hemodialyzer gave a systemic clearance close

to the 41.0 ml/min seen in patients with normal renal function.

(c) Under the conditions of hemodialysis, calculate the dose (D) of gentamicin

which would be administered every 8 hr (t) in order to maintain an Waverage

steady-state serum concentration of 4 ug/ml. What would be (Cmax)ss and

(Cmin)ss?

(d) Without the hemodialyzer two male patients each exhibited a creatinine clear-

ance (Clcr) of S ml/min; the noW l value is 117 + 20 ml/min. Assuming that the

decreased ClCr is due to a decrease in glomerular filtration rate, what would havebeen the renal clearance (Clr) of gentamicin in these two patients had they

remalned without the hemodialyzer? What dose would then need to be adminis-

tered every 8 hr to maintajn (~s) at 4 ug/ml? Compare your answer with the Physi-

cians Desk Reference (5677).

Your third patient comes from Edelman et al (Clin Pbarmacol Therap 35:382-6

(1984)). He studied metotrexate is artbritic patients. The pharmacokinetic parame-ters gleened from the 10 mg IV data arc:

A1 (mg/L) 0.663 Alpha (hr-1) 059 B1 (mg/L) 0.073 Beta (hr-1) 0.097


71/131



15) What is the AUC (0 to inf) (mg / L * hr) ?

a) 0.75 b) 1.08 c) 1.12 d) 1.88e) 9.1

16) What is V1 (L) ?

a) 13.6 b) 15.1 c) 253 d) 36.1e)103.1

17) What is the clearance (L/hr) ?

a) 53 b) 8.9 c) 133 d) 15.4e) 17.2

18) What is V(beta) (L) ?

a) 13.6 b) 253 c) 36.1 d) 45e 55

19) What is the t(max) in the perepheral compartment (hr) ? a) 0 b) 1.8 c) 3.7 d) 5.1e)6.1

20) What percent of the drug is in the perepheral compartment at cquilibrium ? a)

25 b) 50 c) 75 d) 100 e) 125

Two compartment: It has been proposed (Ionescu et al. Clin Pkin 14:178-

186,1988) that morphine injectcd dirtctly into thc spioal chord would give signifi-cant analgesia. The following is a CSF concentration - time profile resulting from

05 mg/lcg IV bolus dose of Morphine:


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time conc. time conc.

(min) (mg/L) (min) (mg/L)

2 251 120 323

5 181 240 173

10 142.5 360 82

20 1043 480 2.4

40 75.1 720 1.2

80 48.8

1 a) fraction of remaining drug contained in peripheral compartment at equilib-

rium. Lb) Can this drug be approximated by a one compartment model ? Support

your contention with calculations. Lc) Calculate a rcasonable dosage regimen for

the above patient to maintain the concentration within the therapeuticwindow of

50 to 5 mg/L.

Two compartment:

(1) It has been proposed that diazepam has anticonvulsant properties above 350

nanograms per milliliter. The following is a concentration - time profile resulting

from 10 mg IV bolus dose of diazepam :

time conc. time

(hrs) (ng/ml)(hrs)

0.25 480 6.0


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0-50 400 8.0

l.o 300 10.0

2.0 170 16.0

3.0 120 24.0

o n llo

l.a) fraction of remaining compartment at equilibrium.

n

~ _

conc.

(ng/ml)

100

90

85

70

53

Dlezzewm


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UA tD 20A AD

Time

drug contained in peripheral

k21 = (1.05 * 116.8) + (0.0325 * 487.6) / (487.6 + 116.8)

= 138.5 / 604.4

= 0.239

klO = (1.05 * 0.0325) / 0.239

= n ls

kl2 = 1.05 + 0.0325 - 0.15 - 0.239

= n 7

B2 = (0.7 * lOOOOmic) / (1.05 - 0.0325)

= 6880

V1 = lOOOOmic / 604.4 mic/l

= 16.55

X2 / total = 6880 / t6880 + (16.55 * 116.8mic/1)]


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= 0.78

l.b) Can this drug be approximated by a one compartment model? Support yourcontention with calculations.

AUC = (487.6 / 1.05) + (116.8 / 0.0325)

= 464.4 + 3594.8

= 4058.2

Since the beta phase contributes more than 80% of the total AUC, the model can

be collapsed to a one compartment model.

3594.8 / 4058.2 = 0.886 = 89%

l.c) Calculate a reasonable dosage regimen for the above patient to maintain the

concentration within the therapeutic window of 350 to 1100 nanograms per ml.

K = 1/MRT = 1 / (111007 / 4058.2) = 1/27.35 = 0.0253 per hr

t 1/2 = 0.693 / 0.0253 = 27.39 hr

2^N^ = 1100 / 350 = 3.14

N = ln 3.14 / ln 2 = 1.65


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Tau max = 27.39 * 1.65 = 45.19 hr drop the dosing interval to 24 hours now need

to find a new N

N = 24 / 27.39 = 0.876

Vss = lOOOOmic / (4058.2mic*hr/1 * 0.0253/hr) = 97.25L

To find dose:

l.lOOmg/L = (D / 97.25 L) * tl / (1 - 0.5^0.876^)]

D = 49 mg daily (aggressive therapy)

0.350mg/L = (D / 97.25L) * t 1 / (1 - 0.5^0.876^)](0.5^0.876^)

D = 29 mg daily (conseervative therapy)

Therefore, any dose between 30 and 50 mg a day can be given.

Two Compartment:

We have used theophylline as a test drug in many of our calculations in class. We

assumed that it was a one compartment model. Look at the data from Mitenko

(Clin Pharmacol and Ther,14p509 1974) for intravenous theophylline (Dose 5.6mg/kg):

sos


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theophylilne

.

i ,_

1.

O D 2D 4wD 6D8S tD

Tlm~

Time (hr.)

0.167

0.333

0.500

0.833

1.0

1.5

2.0

3.0

4.0

6.0


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8.0

Conc. (mg./L.)

24.7 Estimates of Pharmacokinetic parameters

20.3 are as follows:

18.1 A1 (mg/L)16.1

16.1 B1 (mg/L)17.9

15.6 Alpha (hr-l)4.8

14.3 Beta (hr-l)0.15

13.3 AUC (O to inf) 122.7

11. AUMC (O to inf) 797.6

9.8

7.3

s.4

1) What is the volume of the central compartment (L/Kg) ?


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*A) 0.165 B) 0.30 C) 0.35 D) 2.9 E) 6.1

2) What is the clearance of theophylline (L/Kg/hr) ?

A) 0.007 *B) 0.046C) 2.7 D) 22 E) 142.4

3) What is the Volume of

distribution in the beta phase (Vbeta)

(L/Kg) ?

A) 0.165 *B) 0.30C) 0.35D) 2.9E) 6.1

4) What percent of the equilibrium ?

A) 12 B) 23

dose is in the peripheral compartment at

*C) 46 D) 69 E) 92 treated as a one compartment model for

5) Can theophylline be dosing purposes ?


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A) No, a larger percentage of the drug is in the peripheral compartment.

B) NO, A1 and C) Yes, alpha D) Yes, the approximately

*E) Yes, the approximately

B1 are about the same value. is bigger than beta contribution of the alpha phase

AUC is the total AUC. contribution of the beta the total AUC.

phase AUC is

6) What is the mean residence time (MRT) for the IV dose (hr) ?

A) 4.25 B) 5 C) 5.75*D) 6.5E) 7.25

For the same dose of an oral product information was obtained:

AUC (O to inf)

AUMC (O to inf)

7) What is the MRT for the


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A) 8.0 B) 8.75

122

1400

(TheoDur^TQ^T) the following

oral dose (hr) ?

C) 9.5 D) 10.25*E) 11.5

8) What is the mean absorption time (MAT) for the oral dose (hr)?

A) 1 B) 2 C) 3 D) 4 *E) 5

9) What is the absorption rate constant for theophylline in TheoDur

(hr^T-l^T) ?

A) 0.14 B) 0.17 *C) 0.2 D) 0.25 E) 0.3


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Two compartment: The following pharmacokinetic information was obtained from

EA, a

45 y/o, 70 kg, healthy male following an800 mg IV dose of

vancomycin:

A1 60 mg/L

alpha 1.33 hr^-l

B1 20 mg/L

beta 0.129 hr^-l

AUC (mg/L * hr)200

AUMC (mg/L)1237

5) What is your patients vancomycin clearance(L/hr)?

a) 0.25 *b) 4 c) 24.74 d) 45.1 e) 155

6) What is your patients V(beta)(L)?

a) 10 b) 24.74 *c) 31 d) 45.1 e) 60


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7) What is your patients V1(L)?

*a) 10 b) 24.74 c) 31 d) 45.1 e) 60

8) What is your patients MRT(iv) (hr)?

a) 0.112 b) 0.162 c) 1.39 d) 4.29 *e) 6.19

9) What is your patients effective rate of elimination (hr^-l)? a) 0.112 *b) 0.162 c)

0.5 d) 4.29 e) 6.19

10) What is your patients Vss (L)?

a) 10 *b) 24.74 c) 31 d) 45.1 e) 60

11) Following an IM dose of vancomycin, the MRT(im) was calculated to be 8.185

hr. What was the absorption half life (hr)? a) 0.112 b) 0.162 *c) 1.39 d) 2 e) 6.19

RP s angina was controlled on 40 mg TID of propranolol. You calculated his phar-

macokinetic parameters to be: Vd (L) = 125; T1/2 (hr) = 3.1; Qh (L/hr) = 33; Bio-

availability (f) = .7; Bound(%) = 95. Propranolol is essen^_ tially 100%

metabolized.

12) What is his Total body clearance (L/hr)? a) 1.4 b) 14 *c) 28 d) 33 e)40.3


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13) What is his hepatic extraction ratio?

a) 0.042b) .42 *c) .85 d) 1 e) 1.22

14) Assuming propranolol to be rapidly absorbed, what is his Cpss

max free concentration (ng/mL)?

*a) 13.4 b) 19.2 c) 188 d) 268 e) 355

15) What is his Cpss max total(bound and free) (ng/mL)?

a) 13.4 b) 19.2 c) 188 *d) 268 e) 355

16) What is his Cpss min free (ng/mL)?

*a) 2.2 b) 3.2 c) 31.4 d) 44.8 e) 59.3

He is now suffering from renal failure. His half life went up to 3.8 hr while his

binding went up to 98.3% because of an increase in AAG, a plasma protein to

which propranolol binds.


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17) What is his new clearance (L/hr)?

a) 14.4 *b) 21.7 c) 24 d) 28 e) 36.2

18) What is his new volume of distribution (L)?

a) 79 *b) 119 c) 132 d) 153 e) 199

19) What will his new Cpss max free be if we keep him on the same

regimen (ng/mL)?

*a) 5.26 b) 7.5 c) 310 d) 442 e) 554

20) What is his new Cpss max total(ng/ml) ? a) 5.26 b) 7.5 *c) 310 d) 442 e) 554

21) What is his new Cpss min free (ng/mL)?

*a) 1.23 b) 1.75 c) 72.3 d) 103 e) 129


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22) The physician sees that the clearance has dropped and consequently the total

plasma concentrations have gone up. He wants to decrease to dose to 40 mg / BID.

What would you recommend? a) Sounds good to me. That will get the Cpss max

total concentration back to it was before he was sick. b) His new clearance was

marginally changed because the drug is cleared by the liver. Id leave it alone. c) Ithink the change from TID to BID is a bit much. How about lowering it to 30 mg

instead of 40 mg TID. *d) We need to increase the dose, not lower it. Id recom-

mend 40 mg QID. e) We need to increase the dose, not lower it. Id recommend 80

mg QID.

Two compartment

Methotrexate is a cytolytic used for the treatment of acute leukemia and other

forms of cancer. After a a 400 mg/kg dose the following data was recorded for a 12

y/o boy.

Time (hours)Concentration (mcg/ml)

6 360

12 70

18 15

36 2

48 1.2

60 0.46

72 036

Q(} 0.15


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11.4 Begin


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11.5 Problems


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Aspirin (Problem 11 - 1

Fu, C., Melethil, S., and Mason, W., "The pharmacokinetics of aspirin in rats and the effect of buffer", Journal of Pharmacokinetics

and Biopharmaceutics, Vol. 19, (1991), p. 157 - 173.

Aspirin is an analgesic/ antipyretic commonly used to relieve minor pain and is used in such conditions as rheumatic

fever, rheumatoid arthritis, and osteoarthritis. The major metabolite of aspirin is salicylic acid. The following set o

data was collected using rats which weighed 250 - 300 g.

1. What is ?

2. What is ?

3. What is your patient's clearance?

4. What is your patients MRT?

5. What is your patient's ?6. What is your patient's V1?

7. What is your patient's ?

8. What is ?

9. What is ?

10. What is ?

11. What is ?

12. What is ?

13. What is ?14. What is the in the peripheral compartment?

15. What percent of the dose is in the peripheral compartment at equilibrium?

16. Can this drug be treated as a one-compartment model for dosing purposes?

17. If this drug can be treated as a one-compartment model, what is K ?

Problem Submitted By: Maya Leicht AHFS 00:00.00

Problem Reviewed By: Vicki Long GPI: 0000000000

PROBLEM TABLE 11 - 1. Aspirin

weight of rat 275 g

Dose 5 mg/kg IV

A1 8.58

a 1.07

B1 7.24

b 0.2

AUC 38.8

AUMC 116.0


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Buprenorphine (Problem 11 - 2

Ohtani, M., et al., "Pharmacokinetic analysis of enterohepatic circulation of buprenorphine and its active metabolite, norbu-

prenorphine, in rats", Drug Metabolism and Disposition, Vol. 22, (1994), p. 2 - 7.

Buprenorphine is a morphine derivative which has twice the duration of action and 30 times the potency of morphine.

Buprenorphine is partially metabolized to norbuprenorphine which is also active in the body. In this study, buprenor

phine was given to rats weighing 280 - 300 g.

1. What is the ?

2. What is the ?

3. What is the AUC?

4. What is your patient's clearance?5. What is your patients MRT?


7. What is your patient's V1?

8. What is ?

9. What is ?

10. What is ?

11. What is ?

12. What is ?13. What is ?

14. What is the in the peripheral compartment?





PROBLEM TABLE 11 - 2. Buprenorphine

Weight of rat 290 g

Dose 0.06 mg/kg IV

A1 41

a 3.89

B1 10

b 0.271

AUC 48.3

AUMC 135.24


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17. If this drug can be treated as a one-compartment model, what is K ?


95/131



Caffeine (Problem 11 - 3

Shi, J., et al., "Pharmacokinetic-pharmacodynamic modeling of caffeine: Tolerance to pressor effects", Clinical Pharmacology

and Therapeutics, Vol. 53, (1993), p. 6 - 14.

This study looks at the cardiovascular effects of caffeine. Caffeine is known to increase blood pressure upon its with-

drawl. This study looks at how tolerance to caffeine and its pressor effects develops and disappears with time.

1. What is ?

2. What is ?

3. What is ?




7. What is ?

8. What is ?

9. What is ?

10. What is ?

11. What is ?


13. What is ?






PROBLEM TABLE 11 - 3. Caffeine

Patient weight 80 kg

Dose 4 mg/kg oral

A1 10.55

a 4.9

B1

9.1

b 0.23

f 98.4 %


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Cefazolin (Problem 11 - 4

Nightingale, C., et al., "Changes in pharmacokinetics of cefazolin due to stress", Journal of Pharmaceutical Sciences, Vol. 64,

(1975), p. 712 - 714.

Cefazolin is a cephalosporin antibiotic used in the treatment of many types of infections. This study looks at the effec

of stress on the pharmacokinetics of cefazolin. The following data was approximated from the graph given in this arti-

cle.

1. What is ?

2. What is ?

3. What is ?




7. What is ?

8. What is ?

9. What is ?

10. What is ?

11. What is ?

12. What is ?






PROBLEM TABLE 11 - 4. Cefazolin

Patient weight 56.3 kg

Dose 1 g IV

A1 206.48

a 4.832

B1 122.96

b 0.573


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