17 multicompartment pharmacokinetic models.docx

8/20/2019 17 Multicompartment Pharmacokinetic Models.docx

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Multicompartment Pharmacokinetic Models

OBJECTIVES

After completing this chapter you should be able to

• Describe the differences between the one-compartment and the two-compartment

pharmacokinetic models

• Define all the parameters of the two-compartment pharmacokinetic model

• Describe the plasma concentration–time profile after a single IV and oral administration, during

constant rate IV infusion, and multiple drug administration of drugs that follow two-compartmentpharmacokinetic model

• Estimate all the pharmacokinetic parameters of the two-compartment pharmacokinetic model

from plasma concentrations obtained after a single IV administration

• Analye the effect of changing one or more of the pharmacokinetic parameters on the plasma

concentration–time profile and the drug distribution between the central and the peripheral

compartments after administration of drugs that follow the two-compartment pharmacokinetic

model

• Describe the general steps for compartmental modeling and discuss the general approaches

used to e!aluate the goodness of model fit

17.1 IT!O"#CTIO

In the pre!ious discussions, it was assumed that the drug is rapidly distributed to all parts of the

body once it enters the systemic circulation" An immediate e#uilibrium is established between

the drug in the systemic circulation and the drug in all parts of the body" $he drug concentration

in different parts of the body is different because of the differences in drug affinity to the different

tissues" %owe!er, any change in the plasma drug concentration due to drug absorption or drug

elimination is accompanied by a proportional change in the drug concentration in the different

tissues" &o the drug concentration–time profiles in the different parts of the body are parallel to

each other" $he rapid distribution e#uilibrium achie!ed between the different tissues makes the

body act as one homogenous compartment"

$he process of drug distribution to the different parts of the body can be demonstrated by a

beaker that has a coating material co!ering its inner wall and is filled with a li#uid as in 'igure

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()"(" In this e*ample, the li#uid in the center of the beaker represents the systemic circulation

and the beaker wall coating materials represents the tissues" If a drop of dye is added to the

li#uid, the dye is distributed in the li#uid in the center of the beaker first and then to the coating

material of the beaker wall" If the dye is rapidly distributed from the li#uid to the coating material,

the dye distribution e#uilibrium is achie!ed rapidly" $he dye concentration in the li#uid in the

center of the beaker becomes constant immediately after addition of the dye" In this case, the

beaker beha!es as a single compartment despite the difference in dye concentration in the

li#uid and in the beaker wall co!ering" 'or drugs that are rapidly distributed from the systemic

circulation to the tissues, rapid distribution e#uilibrium between the drug in the systemic

circulation and tissues is achie!ed" +hen the drug is eliminated, the drug concentrations in the

systemic circulation and in all tissues decline at the same rate because of the rapid distribution

e#uilibrium" Drugs that follow this beha!ior follow the one-compartment pharmacokinetic model"

+hen these drugs are administered by a rapid IV bolus dose, the plasma drug concentrationdeclines monoe*ponentially and the plasma drug concentration–time profile is linear on the

semilog scale"

$I%#!E 17.1 "ia&rammatic presentation o' the distri(ution o' the d)e 'rom the solution

in the center o' the (eaker to the (eaker *all coatin& material. The distri(ution process

causes decrease in the d)e concentration in solution and increase in the d)e

concentration in the coatin& material. +hen the d)e distri(ution to the (eaker *all

coatin& material ,- B/ is 'ast0 the (eaker (ehaes as a sin&le compartment0 *hile *hen

the d)e distri(ution to the (eaker *all coatin& material ,- B/ is slo*0 the (eaker

(ehaes as i' it consists o' t*o di''erent compartments.

age

'or some drugs, the distribution from the systemic circulation to the different tissues is slow" $hebeaker e*ample mentioned earlier can be used to e*plain this condition" After addition of thedye to the beaker, it distributes in the li#uid in the center of the beaker initially" $hen the dyestarts to distribute slowly to the beaker wall coating material" $he distribution of the dye from theli#uid in the center of the beaker to the beaker wall coating material causes gradual decrease inthe dye concentration in the li#uid and gradual increase in the dye concentration in the beakercoating material as illustrated in 'igure ()"(" $he dye concentration in the li#uid reaches a

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constant !alue when e#uilibrium is achie!ed between the dye in the li#uid and the beaker wallcoating material as illustrated in 'igure ()""

After IV administration of a drug that is slowly distributed to the tissues, the drug is immediatelydistributed in the systemic circulation and other highly perfused tissues" $he drug concentrationin the systemic circulation decreases initially at a fast rate due to drug distribution to the tissuesand also due to drug elimination from the body" +hen e#uilibrium is established between the

drug in the systemic circulation and all tissues, the drug concentration in the systemic circulationand tissues declines at a rate dependent on the rate of elimination, which is slower than (theinitial rate of decline" After IV bolus administration of the drugs that follow this beha!ior, theplasma drug concentration–time profile is cur!ilinear on the semilog scale" $hese drugs followthe two-compartment, three-compartment, or any other multi-compartment pharmacokineticmodel" $his model consists of a central compartment that includes the !ascular space andhighly perfused tissues and one or more peripheral compartments that include the other organsof the body"

$I%#!E 17.2 ")e concentration3time pro'ile in the li4uid *ith the decrease in the d)econcentration representin& the distri(ution o' the d)e 'rom the li4uid in the central o' the(eaker to the (eaker *all coatin& material.

17.2 COMP-!TMET-5 P6-!M-COIETIC MO"E5S

harmacokinetic modeling in general in!ol!es the de!elopment of a model that can#uantitati!ely describe the pharmacokinetic beha!ior of the drug in the body" In compartmentalmodeling, the body is described by one or more interconnected compartments depending on therate of drug distribution to the different parts of the body" $he number of compartments in themodel depends on the rate of drug distribution to the different parts of the body" Each of thecompartments has its own !olume of distribution and the intercompartmental clearances thatgo!ern the drug distribution and transfer between these compartments" $he model includes aninput function that describes the drug entry to the systemic circulation" $he order of theelimination process and the compartment where the elimination process takes place areincluded in the model .(/"0ompartmental pharmacokinetic models differ in the number of compartments, thecompartment1s2 where drug elimination occurs, and the arrangement of these compartments"$he number of compartments in the model depends on the rate of drug distribution to thedifferent parts of the body" If the drug in the systemic circulation is distributed rapidly to all parts

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of the body, the body beha!es as a single compartment and the drug pharmacokinetic beha!iorcan be described by one-compartment pharmacokinetic model, while if the drug is distributedrapidly to some tissues and organs and slowly to other tissues and organs, the two-compartment pharmacokinetic model can be used to describe the pharmacokinetic beha!ior ofthe drug in this case" 3n the contrary, when the drug is distributed to the different parts of thebody at three distinguished rates, for e*ample, rapid, slow, and !ery slow, the three-

compartment pharmacokinetic model can be used to describe this pharmacokinetic beha!ior"$he pharmacokinetic beha!ior of most drugs can be described by one-, two-, or three-compartment pharmacokinetic models4 howe!er, models with more compartments can be usedif the obtained data can support these complicated models"

$I%#!E 17.8 "ia&ram represents di''erent compartmental pharmacokinetic models. ,-/E9amples o' t*o:compartment pharmacokinetic models *ith elimination 'romcompartment 10 compartment 20 or (oth compartments. ,B/ E9amples o' the man)possi(le three:compartment pharmacokinetic models that di''er in the arran&ement o'the compartments and in the compartment,s/ *here dru& elimination takes place.

harmacokinetic models that ha!e the same number of compartments can be different whendrug elimination occurs from different compartments ./" $he different two-compartmentpharmacokinetic models presented in 'igure ()"5A differ in the compartment where drugelimination takes place" 3ne of the models has drug elimination from compartment (, thesecond model has drug elimination from compartment , and the third model has drugelimination from both compartments" 'or the three-compartment pharmacokinetic model, thereare se!en different possibilities for the compartment1s2 where drug elimination takes place" Also,pharmacokinetic compartment models can differ in the way the compartments are arranged" 'or e*ample, the models presented in 'igure ()"56 are e*amples of the three-compartment

pharmacokinetic models"

17.8 T+O:COMP-!TMET P6-!M-COIETIC MO"E5

$he two-compartment pharmacokinetic model with elimination from the central compartment isthe most common model used to describe the pharmacokinetic beha!ior of drugs that followtwo-compartment model" &o this model will be used in the following discussion" After IV bolusdose, the drug is distributed rapidly to the body spaces and tissues that represent the centralcompartment" $hen the drug is distributed by a first-order process from the central compartmentto the other body spaces and tissues that represent the peripheral compartment" 6ecause the

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blood is usually part of the central compartment, the drug can be deli!ered to the eliminatingorgan1s2 once the drug is in the central compartment" &o distribution and elimination occursimultaneously after drug administration, which causes rapid decline in the drug concentration inthe central compartment" After the distribution process is completed and e#uilibrium isestablished between the drug in the central compartment and the drug in the peripheralcompartment, the drug concentration in the central compartment declines at a rate dependent

on drug elimination" $he rate of decline in the drug concentration in the central compartmentdue to drug elimination is slower than the initial rate of decline due to distribution andelimination" &o the plasma drug concentration–time profile that represents the drug profile in thecentral compartment consists of two phases on the semilog scale" An initial distribution phasecharacteried by rapid decline in drug concentration, followed by a terminal elimination phasewith slower rate of decline in drug concentration" $he drug concentration–time profile during theterminal elimination phase is linear on the semilog scale since it declines depending on the rateof drug elimination .5/" A typical plasma concentration–time profile after IV bolus administrationof drugs that follow two-compartment pharmacokinetic model is presented in 'igure ()"7"$he two-compartment pharmacokinetic model assumes that at time ero there is no drug in thetissues representing the peripheral compartment" After an IV dose, the drug is rapidly distributedin the central compartment" $he amount of drug in the central compartment declines rapidly due

to the transfer of drug out of the central compartment to the peripheral compartment and alsodue to drug elimination that occurs simultaneously" $he drug in the central compartment istransferred to the peripheral compartment by a first-order process, and can return back to thecentral compartment also by a first-order process" Initially the amount of the drug in the centralcompartment is larger than the amount of the drug in the peripheral compartment, so the netdrug transfer is from the central compartment to the peripheral compartment" $his means thatinitially the amount of drug in the peripheral compartment increases with time"

$I%#!E 17.; Plasma concentration3time pro'ile o' a dru& that 'ollo*s the t*o:compartment pharmacokinetic model a'ter sin&le IV (olus dose.

$I%#!E 17.< "ru& concentration3time pro'ile in the central and peripheral compartmentsa'ter administration o' a sin&le IV (olus dose o' a dru& that 'ollo*s t*o:compartmentpharmacokinetic model.

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s the amount of drug in the peripheral compartment increases, the rate of drug transfer from the

peripheral to the central compartment approaches that from the central to the peripheralcompartment" +hen these two rates become e#ual, the amount of drug in the peripheralcompartment reaches a ma*imum !alue" 6ecause the drug is continually eliminated from thecentral compartment, the amount of drug in the central compartment decreases and the rate ofdrug transfer from the peripheral to the central compartment becomes larger than that from thecentral to peripheral compartment" $he net drug transfer is from the peripheral to the centralcompartment, and the amount of drug in the peripheral compartment starts to decline parallel tothe decline in the amount of the drug in the central compartment" 'igure ()"8 shows the drugconcentration–time profile in the central and peripheral compartments after administration of asingle IV bolus dose" $he concentration of drug in the central compartment is determined bydi!iding the amount of drug in the central compartment by the !olume of the centralcompartment" 9ikewise, the concentration of drug in the peripheral compartment is determinedby di!iding the amount of drug in the peripheral compartment by the !olume of the peripheralcompartment" &o, the drug concentration in the peripheral compartment can be higher or lowerthan the drug concentration in the central compartment depending on the drug affinity to thetissues"

17.; P-!-METE!S O$ T6E T+O:COMP-!TMET P6-!M-COIETIC MO"E5

$he two-compartment pharmacokinetic model presented by the block diagram in 'igure()": assumes that the drug transport between the central and peripheral compartments followsfirst-order kinetics and that the drug is eliminated from the central compartment by a first-orderprocess"5(75(8

$I%#!E 17.= Block dia&ram that represents the t*o:compartment pharmacokineticmodel *ith 'irst:order transport (et*een the central and peripheral compartments and

'irst:order dru& elimination 'rom the central compartment.

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17.;.1 "E$IITIO O$ T6E P6-!M-COIETIC P-!-METE!S

$he following are the definitions of the pharmacokinetic parameters used in deri!ing thee#uations for the two-compartment pharmacokinetic model;

< is the amount of drug in the central compartment and has units of mass"

= is the amount of drug in the peripheral compartment and has units of mass"

k( is the first-order transfer rate constant from the central compartment to the peripheralcompartment and has units of time>("

k( is the first-order transfer rate constant from the peripheral compartment to the central

compartment and has units of time

>(

"k(? is the first-order elimination rate constant from the central compartment and has unitsof time>("

A and 6 are the hybrid coefficients and ha!e units of concentrations"

@ is the hybrid first-order rate constant for the distribution process and has units oftime>("

is the hybrid first-order rate constant for the elimination process and has units oftime>("

t(B@ is the half-life for the distribution phase and has units of time"

t(B is the half-life for the elimination phase and has units of time"

Vc is the !olume of the central compartment and has units of !olume" $his term relatesthe administered dose to the initial plasma drug concentration 1central compartmentconcentration2 after administration of a single IV dose;

Vdss is the !olume of distribution of the drug at steady state and has units of !olume" $histerm relates the amount of the drug in the body and the plasma drug concentration atsteady state;


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VdorVdarea

is the !olume of distribution during the elimination phase and has units of !olume" $histerm relates the amount of the drug in the body and the plasma drug concentrationduring the elimination phase 1-phase2;

Amount of the drug in the body during the elimination phase C Vd0p-phase 1()"5217.;.2 M-T6EM-TIC-5 E>#-TIO T6-T "ESC!IBES T6E P5-SM- COCET!-TIO3TIME P!O$I5E

$he rate of change of the amount of the drug in any compartment is e#ual to the sum of therates of drug transfer into the compartment minus the sum of the rates of drug transfer out of thecompartment" After a single IV bolus dose and based on the pharmacokinetic model in 'igure()":, the rate of change of the amount of the drug in the central compartment 1


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&ince the distribution process is usually faster than the elimination process, the larger hybridrate constant @ is the rate constant for the distribution process and the smaller hybrid rateconstant is the rate constant for the elimination process as in E#uations ()" and ()"(?" During the distribution phase, the drug distribution rate does not depend only on k(, the transfer rate constant from the central to peripheral compartment" $his is because while the drug isdistributing from the central to the peripheral compartment, there is drug returning back to the

central compartment at a rate dependent on the rate constant k(, and also there is eliminationfrom the peripheral compartment affected by the rate constant k(?" &o the obser!ed rate of thedistribution process is described by the hybrid rate constant @, which is dependent on the threerate constants k(, k(, and k(? as inE#uation ()"" &imilarly, the drug elimination rate does notdepend only on k(?, the elimination rate constant from the central compartment" $his is becauseduring the elimination of the drug from the central compartment, there is drug transfer from thecentral to the peripheral compartment at a rate dependent on the rate constant k(, and drugreturning back to the central compartment at a rate dependent on the rate constant k(" &o theobser!ed rate for the elimination process is described by the hybrid rate constant , which isdependent on the three rate constants k(, k(, and k(?as in E#uation ()"(?" $he first-order rateconstants k(, k(, and k(? are usually termed the micro rate constants, while @ and are termedthe macro rate constants"

Di!iding E#uation ()": by the !olume of the central compartment, Vc, gi!es the e#uation for thedrug concentration in the central compartment, and hence the plasma drug concentration, atany time after a single IV bolus dose;

which can be simplified to

where

and

5()5(E#uation ()"(( and its simplified form E#uation ()"( are the e#uations that describe theplasma drug concentration at any time after a single IV bolus dose of a drug that follows thetwo-compartment pharmacokinetic model .7/" $hese e#uations include two e*ponents; onedescribes the distribution process and includes the larger hybrid rate constant, @, and the otherdescribes the elimination process and includes the smaller hybrid rate constant, " As timeelapses after IV drug administration, the e*ponential term that has the distribution 1larger2 hybrid

rate constant approaches ero and the plasma concentration declines at a rate dependent onthe hybrid elimination rate constant, " &o the plasma drug concentration–time profile after asingle IV bolus dose on the semilog scale has a rapidly declining distribution phase and a linearterminal elimination phase"

17.< "ETE!MI-TIO O$ T6E T+O:COMP-!TMET P6-!M-COIETIC MO"E5P-!-METE!S

$he pharmacokinetic parameters k(, Vc, @, and , in E#uation ()"((, or the parameters A, 6, @,and , in E#uation ()"(, can be estimated from the plasma drug concentrations obtained aftera single IV dose 1D2 of the drug by nonlinear regression analysis utiliing specialied statistical

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programs" $he estimated parameters in both e#uations can be used to calculate all the otherparameters of the model" $he pharmacokinetic parameters allow prediction of the drug steady-state plasma concentration during repeated drug administration and determination of the dosere#uired to achie!e certain drug concentration at steady state" $he pharmacokinetic parametersin E#uation ()"(, A, 6, @, and , can be estimated graphically utiliing the method of residuals"

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e#ual to the coefficient A in the e#uation" $he hybrid rate constant @ can be determined fromthe slope of the line 1slope C >@B"5?52" Also, the t(B@ can be determined directly from the lineby calculating the time re#uired for any point on the line representing the @-phase to decreaseby 8?F" $hen the hybrid rate constant @ is calculated as

• • A is always the y-intercept of the faster process 1the process with shorter t(B, the

distribution process2, and 6 is always the intercept of the slower process 1the process withlonger t(B, the elimination process2"

()"8" DE$EGHIA$I3 3' $%E H3DE9 AGAHE$EG&

3nce the parameters A, 6, @, and are determined from the method of residuals, the othermodel parameters can be calculated .(/"

()"8""( Volume of the 0entral 0ompartment, Vc

After IV bolus administration, the drug is distributed initially in the central compartment" &o the!olume of the central compartment can be determined from the dose and the initial drugconcentration in the central compartment, which is same as the initial plasma drugconcentration" $he plasma concentration at time ero is determined from E#uation ()"( bysubstituting the time by ero, and it is e#ual to 1A J 62;

()"8"" Area under the lasma 0oncentration–$ime 0ur!e, AK0

$he area under the plasma concentration–time cur!e is determined by integrating E#uation()"(, which describes the plasma concentration–time profile from time ? to L;

()"8""5 $otal 6ody 0learance, 09$

$he 09$ is determined from the dose and the AK0 similar to the one-compartment

pharmacokinetic model;

()"8""7 'irst-3rder Elimination Gate 0onstant from the 0entral 0ompartment, k(?

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$he 09$ is the product of the first-order elimination rate constant k(? and Vc" +hen Vc and09$ are known, k(? can be calculated;

"

()"8""8 'irst-3rder $ransfer Gate 0onstant from the eripheral 0ompartment to the 0entral0ompartment, k(

As a result of integrating the differential e#uation to obtain the integrated e#uation for theamount of the drug in the central compartment, the relationship in E#uation ()" has beenobtained 1@ C k(k(?2" 6ased on this relationship, k( can be calculated;

()"8"": 'irst-3rder $ransfer Gate 0onstant from the 0entral 0ompartment to the eripheral0ompartment, k(

Also, while integrating the differential e#uation to obtain the integrated e#uation for the amount

of the drug in the central compartment, the relationship in 5?5(E#uation ()") has been

obtained 1@ J C k( J k( J k(?2" 6ased on this relationship, k( can be calculated;

()"8"5 DE$EGHIA$I3 3' $%E V39KHE& 3' DI&$GI6K$I3 '3G $%E $+3-03HAG$HE$ %AGHA03MIE$I0 H3DE9

In the one-compartment pharmacokinetic model the drug is distributed rapidly to all parts of thebody and the distribution e#uilibrium is established immediately after drug administration" &o thedrugs that follow one-compartment pharmacokinetic model are distributed in the same tissuesonce they enter the systemic circulation and they ha!e only one !olume of distribution"%owe!er, in the two-compartment pharmacokinetic model there is more than one !olume ofdistribution" Initially, after IV drug administration the drug is distributed in the centralcompartment only" $hen the drug distributes from the central compartment to the peripheralcompartments" During the elimination phase the drug !olume of distribution is e#ual to Vd andduring steady state the drug !olume of distribution is e#ual to Vdss" $he !olume of the centralcompartment is the smallest, while Vd is the largest of the three !olumes in the two-compartment pharmacokinetic model" Vdss is larger than Vc and smaller than Vd" Vc can becalculated from the dose of the initial drug concentration as in E#uation ()"()"

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Vdss is the factor that relates the amount of the drug in the body and the plasma drugconcentration during steady state when the drug is administered by constant rate IV infusionand the drug concentration in the body is constant" +hen the drug concentration is constant, therate of drug transfer from the central to the peripheral compartment is e#ual to the rate of drugtransfer from the peripheral to the central compartment" Also, a transient steady state isachie!ed for brief moment after a single IV administration when the drug concentration in the

peripheral compartment reaches its ma*imum !alue, and the net rate of drug transfer betweenthe central and peripheral compartments is e#ual to ero" $he rate of drug transfer can bee*pressed by the drug transfer rate constant and the amount of the drug in each compartment"

At steady state,

&ince the amount of the drug in the central compartment is the product of the plasma drugconcentration and Vc" At steady state, Vdss relates the amount of the drug in the body to the drugconcentration in plasma;

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• a" Ksing the method of residuals, calculate the following parameters;t(B@, t(B, k(, k(, k(?,

Vc, Vd, Vdss, AK0, and 09$• b" +hat will be the amount of drug remaining in the body after (8 hN

-ns*er

• a" $he method of residuals used to sol!e the problem and presented in 'igure ()" can

be performed by the following steps;o • lot the concentration !ersus time on a semilog scale"

o • Identify the best line that represents the drug elimination process"555

$I%#!E 17.A -pplication o' the method o' residuals in solin& the e9ample.

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o • $he y-intercept is e#ual to 6" $he hybrid elimination rate constant 12 and -

half-life can be determined from the line"o • 0alculate the residuals from the difference between the plasma drug

concentration and the !alues on the e*trapolated line during the distribution phase"o • lot the residuals !ersus time, and draw the best line that goes through the

points"o • $he y-intercept of this line is e#ual to A" $he hybrid distribution rate constant @

and @-half-life can be determined from this line" A (? mgB96 8 mgB9


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• b" 0p C Ae>@t J 6e>t Amount of the drug in the body during the elimination phase

C 0p O Vd Amount(8 h C 0p(8 h O Vd C :") mgB9 O 5?") 9 C ? mg

17.= O!-5 -"MIIST!-TIO O$ "!#%S T6-T $O55O+ T6E T+O:

COMP-!TMET P6-!M-COIETIC MO"E5• After oral administration of a drug that follows two-compartment pharmacokinetic model,

the drug is absorbed to the systemic circulation, which is part of the centralcompartment" 3nce in the central compartment, the drug can be eliminated from thebody or distributed to the peripheral compartment" If the drug is rapidly absorbed, thedecline in the plasma concentration–time profile after the end of the absorption phasewill be bie*ponential, reflecting the distribution and the elimination phases" $hebie*ponential decline in the drug plasma concentration after drug absorption will be clear only if the absorption, distribution, and elimination processes proceed at three distincti!e


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rates" %owe!er, if the drug absorption is slow, the bie*ponential decline in the plasmaconcentration after the end of drug absorption may not be e!ident" 'igure()" represents the plasma concentration–time profile after a single oral dose of a drugthat follows the two-compartment pharmacokinetic model" $he decline of the drugconcentration in the post-absorption phase is bie*ponential" $he plasma drugconcentration–time profile after oral administration of drugs that follow two-compartment

pharmacokinetic model can be described by a trie*ponential e#uation that representsthe absorption, distribution, and elimination processes as in E#uation ()"5? ./" $heplasma drug concentrations obtained after single oral administration of drugs that followtwo-compartment pharmacokinetic model can be fitted to E#uation ()"5? to estimate themodel parameters" &pecialied computer programs are usually utilied in this fittingprocess"

•

• 5758

$I%#!E 17. Plasma concentration3time pro'ile 'or a dru& that 'ollo*s t*o:

compartment pharmacokinetic model a'ter administration o' a sin&le oral dose.

•

17.=.1 5OO3!IE%E5M- MET6O" $O! "ETE!MI-TIO O$ a -$TE! O!-5

-"MIIST!-TIO O$ "!#%S T6-T $O55O+ T6E T+O:COMP-!TMETP6-!M-COIETIC MO"E5

• $his method is similar in principle to the +agner–elson method that is used for the

determination of the absorption rate constant as discussed in0hapter " $he 9oo–Giegelman method can be applied to determine the absorption rate constant for drugs

that follow linear kinetics, with ero-order or first-order absorption, and follow two-compartment pharmacokinetic model .8/" $he method depends on calculation of thefraction of the dose remaining to be absorbed at different time points .(-1fraction of doseabsorbed2/ to determine the order of drug absorption process and to calculate theabsorption rate constant" $he amount of the drug absorbed up to any time is the sum ofthe amount of the drug in the body and the amount of the drug e*creted, while the totalamount of the drug absorbed is e#ual to the total amount of the drug e*creted" $hefraction of the dose absorbed at any time is the ratio of the amount of drug absorbed upto this time to the total amount of the drug absorbed" If the absorption process is first

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order, a plot of the fraction remaining to be absorbed !ersus time should gi!e a straightline on the semilog scale" $he slope of this line is e#ual to >kaB"5?5" 3n the contrary, ifthe absorption process is ero order, a plot of the fraction remaining to be absorbed!ersus time should gi!e a straight line on the 0artesian scale" $he slope of this line ise#ual to >ka"

• he only difference is that for the drugs that follow the two-compartment pharmacokineticmodel, the amount of the drug in the body is the sum of the amount of the drug in thecentral compartment and the amount of the drug in the peripheral compartment" $hedrug amounts in the two compartments ha!e to be calculated separately from theconcentration and !olume of each compartment because the drug in the twocompartments is not at e#uilibrium all the time after a single drug administration"0alculation of the amount of the drug in each compartment re#uires the pharmacokinetic

parameters for the two-compartment model that can 585:only be obtained after IV

administration of the drug" &o IV administration of the drug is necessary to obtain theseparameters before the absorption rate constant can be determined after oraladministration" $his limits the application of this method to drugs that can beadministered intra!enously"

• ()") 03&$A$ GA$E IV ADHII&$GA$I3 3' DGKP& $%A$ '3993+ $%E $+3-

03HAG$HE$ %AGHA03MIE$I0 H3DE9

• $he plasma concentration–time profile of drugs that follow two-compartment

pharmacokinetic model during constant rate IV infusion increases gradually until itreaches the steady state" At steady state, the rate of drug administration is e#ual to therate of drug elimination" $he steady-state plasma concentration is dependent on the rateof the IV infusion and the 09$ of the drug as in E#uation ()"5(" $his is similar to thedrugs that follow one-compartment pharmacokinetic model"

• $he time to reach steady state during constant rate IV infusion of drugs that follow two-

compartment pharmacokinetic model is dependent on the drug elimination half-life 1t(B2"It takes fi!e to si* times the elimination half-life of continuous IV infusion to reach thesteady state" $ermination of the IV infusion results in bie*ponential decline in the drugplasma concentration, a rapid distribution phase and a slow elimination phase"

• Administration of a loading dose may be necessary to achie!e faster approach to steadystate especially in emergency situations" In this case, simultaneous administration of anIV loading dose and the constant rate IV infusion is necessary" 0alculation of the loadingdose based on Vc and the desired steady-state concentration 10pss O Vc2 should achie!ethe desired concentration initially" %owe!er, due to drug distribution to the peripheralcompartment the drug concentration declines transiently, possibly to subtherapeuticconcentration, and then increases gradually to the desired steady-state concentration"0alculation of the loading dose based on Vd and the desired steady-state concentration10pss O Vd2 can a!oid this transient decline in plasma drug concentration" %owe!er, this

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loading dose should produce !ery high drug concentration initially, which may be to*icfor drugs with narrow therapeutic range" $he loading dose should be calculated basedon an a!erage !alue for Vc and Vd" Another approach that can be used is to gi!e aloading dose calculated based on Vc initially followed by smaller IV doses to compensatefor the transient decline in drug concentration after the loading dose"

17.A M#5TIP5E "!#% -"MIIST!-TIO

Drugs that follow two-compartment pharmacokinetic model accumulate during repeatedadministration until steady state is achie!ed" At steady state, the plasma concentration will bechanging during each dosing inter!al4 howe!er, 5:5)the ma*imum and minimum plasmaconcentrations will be similar if the drug is administered as a fi*ed dose at e#ually spacedinter!als" $he a!erage steady-state concentration is directly proportional to the dosing rate andin!ersely proportional to the 09$;

where

• Q is the dosing inter!al

• ' is the bioa!ailability

$his relationship is similar for one- and two-compartment pharmacokinetic models" It takes fi!eto si* elimination half-li!es 1t(B2 to reach the steady state during multiple administration of drugsthat follow two-compartment pharmacokinetic model" Administration of a loading dose may benecessary to achie!e faster approach to steady state"

17. !E-5 E@C!ETIO O$ "!#%S T6-T $O55O+ T6E T+O:COMP-!TMETP6-!M-COIETIC MO"E5

'or a drug that follows two-compartment pharmacokinetic model, the amount of the drug in thecentral compartment declines bie*ponentially" &o the renal e*cretion rate 1dAeBdt2 !ersus timeprofile will also decline bie*ponentially;

where AR and 6R are the hybrid coefficients and ha!e units of amountBtime" $he renal clearancecan be determined from the renal e*cretion rate and the a!erage plasma concentration duringthe urine collection inter!al" $his is similar to the drugs that follow one-compartmentpharmacokinetic model"

$he renal clearance can also be determined from the total amount of the drug e*creted in urineand the drug AK0" $his is similar to the drugs that follow one-compartment pharmacokineticmodel"

$he fraction of the drug dose e*creted in urine is determined from the ratio of the renalclearance to the total body clearance"

5)5


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17.1 E$$ECT O$ C6-%I% T6E P6-!M-COIETIC P-!-METE!S O T6E "!#%COCET!-TIO3TIME P!O$I5E $O! "!#%S T6-T $O55O+ T6E T+O:COMP-!TMET P6-!M-COIETIC MO"E5

After a single IV dose, the rate of drug distribution depends on the hybrid distribution rateconstant, while the rate of elimination depends on the hybrid elimination rate constant" Duringmultiple drug administration, the steady state is directly proportional to the administration rateand in!ersely proportional to the 09$, and the time to reach the steady state is dependent on thehybrid elimination rate constant .:/"

17.1.1 P6-!M-COIETIC SIM#5-TIO E@E!CISE

harmacokinetic simulations can be used to e*amine how the pharmacokinetic parametersaffect the plasma concentration–time profile and the steady state drug concentration achie!edafter single and multiple drug administration" $he drug plasma concentration–time profile issimulated using an a!erage !alue of each of the pharmacokinetic parameters" $hepharmacokinetic parameters are changed one parameter at a time, while keeping all the otherparameters constant, and simulation of the drug concentration–time profile is repeated" $heresulting profiles are e*amined to see how the changes affect the drug concentration–timeprofile" $he plotting e*ercise of the two multicompartment pharmacokinetics modules in the

basic pharmacokinetic concept section and the pharmacokinetic simulations section of thecompanion 0D to this te*tbook can be used to run these simulations"

()"(?"("( Dose

Administration of increasing doses results in proportional increase in the plasma concentrations"

$he plasma concentration–time profiles after administration of increasing doses will be parallel,

while during multiple drug administration the a!erage steady-state concentration 55is

directly proportional to the administered dose, if the 09$ is constant"

()"(?"(" $otal 6ody 0learance

$he change in 09$ will result in different elimination rate of the drug if the !olume is kept

constant" If the same dose is administered and the !olume of distribution is kept constant, the

decrease in clearance produces larger AK0 and longer elimination half-life" During multiple drug

administration of the same dose, the decrease in drug clearance will result in higher steady-

state concentration and longer time to achie!e the steady state, if the !olume fd distribution is

constant"

()"(?"("5 Volume of the 0entral 0ompartment

$he change in the !olume of the central compartment is accompanied by a proportional change

in Vdss and Vd" 9arger !olume of distribution results in lower initial drug plasma concentration

after administration of the same dose and similar AK0 if the 09$ does not change" +hen the

!olume of distribution changes while 09$ remains constant, the elimination rate constant will be

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different" Hultiple administration of the same dose should achie!e the same a!erage steady-

state concentration if the clearance is similar, but the time to achie!e steady state will be

different if the !olume is different because the elimination half-life will be different"

()"(?"("7 %ybrid Distribution Gate 0onstant

9arger hybrid distribution rate constant results in faster completion of the distribution process

without affecting the rate of drug elimination" $his is assuming that the three micro rate

constants k(, k(, and k(? change in a way that will change the rate of the distribution process

without affecting the elimination process"

()"(?"("8 %ybrid Elimination Gate 0onstant

9arger hybrid elimination rate constant results in faster drug elimination without affecting the

rate of drug distribution" $his is assuming that the three micro rate constants k (, k(, and

k(? change in a way that will change the rate of the elimination process without affecting the

distribution process"

()"(( E''E0$ 3' 0%APIP $%E %AGHA03MIE$I0 AGAHE$EG& 3 $%E DGKP

DI&$GI6K$I3 6E$+EE $%E 0E$GA9 AD EGI%EGA9 03HAG$HE$&

Drugs that follow the two-compartment pharmacokinetic model ha!e different concentration–

time profiles in the central and peripheral compartments after a single IV administration" During

the elimination phase, distribution e#uilibrium is established and the ratio of the drug

concentration in the central and peripheral compartments is dependent on the transfer rate

constants"

()"(("( D3&E

$he change in dose results in proportional change in the amount and concentration of the drug

in the central and peripheral compartments" %owe!er, the distribution ratio between the central

and peripheral compartments does not change" 0hanging the dose does not affect the ratio of

the amount of the drug in the peripheral compartment to the amount of the drug in the central

compartment when the distribution e#uilibrium is established"

7.11.2 $I!ST:O!"E! T!-S$E! !-TE COST-T $!OM T6E CET!-5 TO T6E

PE!IP6E!-5 COMP-!TMET


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$he change in k( affects the drug distribution rate, elimination rate, and the tissue distribution"

9arger k( results in higher amount of the drug distributing to the 555?peripheral

compartment, faster distribution rate, and slower elimination rate" $he ratio of the amount of the

drug in the peripheral compartment to the amount of the drug in the central compartment at

steady state increases due to the increase in k(Bk( ratio"

17.11.8 $I!ST:O!"E! T!-S$E! !-TE COST-T $!OM T6E PE!IP6E!-5 TO T6E

CET!-5 COMP-!TMET

$he change in k( affects the drug distribution rate, elimination rate, and the tissue distribution"

9arger k( results in lower amount of the drug distributing to the peripheral compartment, faster

distribution rate, and faster elimination rate" $he ratio of the amount of the drug in the peripheral

compartment to the amount of the drug in the central compartment at steady state decreases

due to the decrease in k(Bk( ratio"

17.11.; $I!ST:O!"E! E5IMI-TIO !-TE COST-T $!OM T6E CET!-5COMP-!TMET

$he change in k(? affects the drug distribution rate and elimination rate" 9arger k(? results in

faster distribution rate and faster elimination rate" %owe!er, the drug distribution ratio between

the central and the peripheral compartments does not change" $he ratio of the amount of the

drug in the peripheral compartment to the amount of the drug in the central compartment at

steady state does not change due to the change in k(? because the ratio k(Bk( does not

change"

17.12 T6!EE:COMP-!TMET P6-!M-COIETIC MO"E5

+ith the de!elopment of accurate sampling methods and sensiti!e analytical techni#ues it has

been shown that some drugs follow three-compartment pharmacokinetic model" After drug

administration into the central compartment, the drug is distributed slowly to the tissues"

%owe!er, the distribution of the drug to some tissues is much slower than its distribution to other

tissues" $his results in two distinct rates of distribution and a bie*ponential distribution phase

followed by a terminal elimination phase" 'igure ()"(? is an e*ample of the plasma

concentration–time profile for a drug that follows three-compartment pharmacokinetic model"

$he diagram presented in 'igure ()"(( is an e*ample of a three-compartment pharmacokinetic

model in which the elimination of the drug is from the central compartment, and the two

peripheral compartments are connected to the central compartment"

$I%#!E 17.1 Plasma concentration3time pro'ile 'or a dru& that 'ollo*s three:

compartment pharmacokinetic model a'ter administration o' a sin&le IV (olus dose.

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$I%#!E 17.11 Block dia&ram representin& the three:compartment pharmacokineticmodel *ith the t*o peripheral compartments connected to the central compartment anddru& elimination 'rom the central compartment.


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$he e#uation that describes the plasma concentration–time profile after a single IV bolus doseis trie*ponential with the three e*ponential terms describing the rapid and slow distributionprocesses and the elimination process" E#uation ()"5: is the mathematical e*pression thatdescribes the plasma concentration–time profile for a drug that follows three-compartmentpharmacokinetic model after administration of a single IV bolus dose ./;

$his e#uation can be simplified to

$he pharmacokinetic parameters for three-compartment pharmacokinetic models are usuallyobtained by nonlinear regression analysis utiliing specialied data analysis software"

17.18 COMP-!TMET-5 P6-!M-COIETIC "-T- --5DSIS

$he first step in compartmental pharmacokinetic data analysis is to construct the model that candescribe the pharmacokinetic beha!ior of the drug" $he choice of the compartmentalpharmacokinetic model is usually based on the obser!ed 55(55drug concentrations after drug

administration" &o compartmental modeling is considered data-based modeling" 'or e*ample,when the plasma drug concentrations obser!ed after a single IV bolus dose of the drug declineas a straight line on the semilog scale, this suggests that the one-compartment model is theappropriate model to describe the drug pharmacokinetic beha!ior, while if the decline in the drugconcentrations is cur!ilinear on the semilog scale, the two-compartment pharmacokinetic modelwill be appropriate in this case" 3n the contrary, if the drug concentrations after a single IV bolusdose on the semilog scale decline at three distinct rates, this pharmacokinetic beha!ior may bedescribed by the three-compartment pharmacokinetic model"In addition to the number of the compartments, other model components such as the inputfunction that depends on the route of drug administration and the output function that describesthe drug elimination process should be included" Drug input into the systemic circulation, whichis usually part of the central compartment, can be instantaneous such as in the case of IV bolus

administration, first order as in oral administration, or ero order as in constant rate IV infusionand drugs absorbed by ero-order process, while drug elimination can follow first-order process,Hichaelis-Henten process, or a combination of the two" $he compartment where drugelimination occurs is usually the central compartment because most of the eliminating organsare highly perfused organs, unless there are e!idences that drug elimination takes place inorgans that are part of the peripheral compartments" 3nce all the model components areincluded, the model is defined mathematically"

17.18.1 M-T6EM-TIC-5 "ESC!IPTIO O$ T6E MO"E5 -" ESTIM-TIO O$ T6EMO"E5 P-!-METE!S

6ased on the constructed compartmental model, a set of differential e#uations are usually

utilied to describe the rate of change of the dependent !ariable that is usually the drug amountor concentration with respect to time, which is the independent !ariable" $hese e#uationsusually include the pharmacokinetic parameters that control the drug pharmacokinetic beha!iorin addition to constants like the administered dose" 'or each compartment, the rate of change of the dependent !ariable is the difference between the rate of drug entering the compartment andthe rate of drug lea!ing the compartment" $he differential e#uations describe the rate of changeof the dependent !ariable with time at a finite period of time" Integration of these differentiale#uations gi!es the integrated e#uations which can be used to calculate the dependent !ariableat any time" $he integrated e#uations contain the pharmacokinetic model parameters that ha!e

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to be estimated to allow prediction of the drug concentration at any time" $he pharmacokineticparameters are estimated from the drug concentration data obtained from pharmacokineticstudies" &ince most pharmacokinetic studies in!ol!e determination of the plasma drugconcentrations, the obser!ed plasma drug concentrations and their corresponding time !aluesare fitted to the integrated e#uation that describes the drug concentration–time profile in thecentral compartment to estimate the pharmacokinetic model parameters"

55555

17.18.2 $ITTI% T6E E@PE!IMET-5 "-T- TO T6E MO"E5 E>#-TIO

'itting of the e*perimentally obser!ed data to the model e#uation to estimate thepharmacokinetic parameters utilies nonlinear regression analysis, which is usually performedwith the aid of specialied statistical software" $he basic principle for estimation of thepharmacokinetic parameters in!ol!es selecting the best !alues for the pharmacokineticparameters, which will minimie the sum of the s#uared differences between the e*perimentallyobser!ed drug concentrations and the model predicted drug concentrations .)/" $he modelpredicted concentrations are determined by substituting the estimated pharmacokineticparameters to the model e#uation and calculating the drug concentration at different time points"$he process is not simple because the data analysis program has to check all possible

combinations of the parameter !alues to find the combination of the parameter !alues that willminimie the sum of the s#uared error for all data points" &o most programs re#uire the input ofan initial estimate for each parameter to be used as the starting point for the parameterestimation process" Different programs utilie different algorithms to search for the bestestimates for the pharmacokinetic parameters"'or e*ample, the pharmacokinetic parameters in the e#uation for the two-compartmentpharmacokinetic model for drugs administered by IV bolus dose are A, 6, @, and , E#uation()"(" $he dose and the e*perimentally determined drug concentrations at different time pointsare used to estimate the pharmacokinetic parameters" $he best estimates for these parametersare the !alues that should minimie the sum of the s#uared differences between all theobser!ed and predicted concentrations, whereas the pharmacokinetic parameters in thee#uation for the three-compartment pharmacokinetic model for drugs administered by a single

IV bolus dose are A, 6, 0, @, , and S, E#uation ()"5)" $hese parameters are estimated byselecting the best !alues for the si* parameters that when used together minimies the sum ofthe s#uared differences between all the obser!ed and predicted concentrations" +hen thepharmacokinetic e*periment is repeated se!eral times, the data for each indi!idual e*perimentare fitted to the model e#uation to estimate the parameters for that particular e*periment" $hefitting is repeated for each data set and the parameters for the different e*periments can besummaried using descripti!e statistics"

Estimation of the pharmacokinetic parameters depends primarily on the drug concentrations inthe e*perimentally obtained samples" &o the #uality of the obtained data is important inimpro!ing the accuracy in pharmacokinetic parameter estimation" $he #uality of data isdependent on the number of samples, the period of time o!er which the samples were obtained,

and the accuracy of the analytical method used for determination of the drug concentration inthe samples" Accurate estimation of the three-compartment pharmacokinetic parameters cannotbe obtained if only few samples were obtained after a single IV drug administration" In general,complicated models ha!e more parameters that re#uire larger number of samples for theiraccurate estimation" Also, samples obtained o!er a short period of time cannot be used toaccurately estimate the parameters of a drug that follows two-compartment pharmacokineticmodel with long elimination half-life e!en if large number of samples were obtained after drug

administration" Enough samples should be obtained 555557during each phase of the plasma

drug concentration–time profile to obtain accurate estimation of all model parameters" $here is

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no strict rule for the number and the timing of samples that should be obtained inpharmacokinetic e*periments because fre#uent sampling can be obtained in some e*perimentalsettings, while samples can be !ery limited in other settings" %owe!er, the minimum number ofsamples re#uired to obtain reasonable estimates for the pharmacokinetic parameters should notbe less than three samples for each phase of the drug profile, and samples should be spreado!er the entire drug concentration–time profile" In addition to the number and the timing of the

obtained samples, the analytical procedures used for the determination of the drugconcentration in the obtained samples ha!e to be accurate and precise to minimie the error inthe e*perimental data" &o, fewer number of samples obtained at the proper time and analyedusing accurate analytical method can be used to obtain estimates for the pharmacokineticparameters that are better than the parameter estimates obtained from larger number ofsamples obtained at inappropriate time schedule and analyed using inaccurate method"

After drug administration, the drug concentration declines e*ponentially and usually there is bigdifference in the concentrations measured shortly after drug administration and during theterminal elimination phase specially after IV bolus doses" &ince the obTecti!e of the fittingprocedures is to obtain the parameter estimates that will minimie the sum of the s#uarederrors, the samples with the high concentrations usually ha!e larger influence in the estimation

of the parameters" $his is because absolute error of ( mgB9 represents F relati!e error if thedrug concentration is 8? mgB9 and represents ??F relati!e error if the drug concentration is ?"8mgB9" &o the fitting program will usually fit the high concentration !alues better than the lowconcentration !alues when all the data points are treated e#ually, that is, weighted e#ually" Inthis case, fitting of the measured concentrations to the model e#uation during the distributionphase will be much better compared to fitting the concentrations during the elimination phase"

+eighting of the pharmacokinetic data is used during the fitting procedures to gi!e the differentdata points different weights to compensate for the difference in their magnitude .)/" Differentweighting schemes can be used for weighting the pharmacokinetic data" $he reciprocal of the!ariance at each data point has been used as a weighting function to make all the data pointsha!e appro*imately the same influence while estimating the pharmacokinetic parameters" &o if

the pharmacokinetic e*periment is repeated se!eral times, the drug concentrations measured atspecific time point in all of these e*periments are used to calculate the !ariance for theconcentrations obtained at this time" $he !alue for (B!ariance is used as the weight for the drugconcentrations at this particular time" $he same is done for each time point to determine theweight for each data point" +hen the !ariance for each data point cannot be determined, otherweighting functions can be used such as (B1predicted concentration2, (B1obser!edconcentration2, or (B1obser!ed concentration2, that is, (BU, (By, or (By, respecti!ely" $heseweighting functions gi!e less weight to the high drug concentration and higher weight to thelower drug concentrations, which compensate for the difference in the absolute !alues of theconcentrations" &o when the appropriate weighting function is used, all the drug concentrationsshould ha!e similar influence in the estimation of the pharmacokinetic parameters"

()"(5"5 EVA9KA$I3 3' $%E %AGHA03MIE$I0 H3DE9

$he primary goal of modeling in pharmacokinetics is to choose the best model that can describethe drug pharmacokinetic beha!ior in the body and to estimate the model parameters withacceptable accuracy" 'or e*ample, in the pre!ious discussion of the two-compartmentpharmacokinetic model, it was mentioned that after IV bolus administration the drug isdistributed rapidly to the tissues of the central compartment and slowly to the tissues of theperipheral compartment" $his does not mean that the drug distribution to the different tissues is

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either rapid or slow" In reality, the drug may be distributed to the different tissues at differentrates but the drug distribution can be appro*imated by two rates and the two-compartmentmodel can appro*imate the o!erall beha!ior of the drug in the body" &o the obser!ed plasmadrug concentrations should fit with reasonable accuracy to the e#uation for the two-compartment pharmacokinetic model" Ksing the three-compartment pharmacokinetic modelmay impro!e fitting the obser!ed data to the model e#uation and usually decreases the sum of

the s#uared de!iations between the obser!ed and the predicted concentrations" %owe!er, theuse of more complicated models is not necessarily better because increasing the number ofcompartments increases the number of pharmacokinetic parameters as in E#uations()"( and ()"5" +hen the same number of data points is used to estimate larger number ofpharmacokinetic parameters, the accuracy of the parameter estimates is compromised" &o it isnecessary to e!aluate the goodness of fit of the e*perimental data to the pharmacokineticmodel"

$here is no single diagnostic procedure or statistical test that can be used to determine the!alidity of the model to describe the obser!ed data" %owe!er, there are se!eral statistical andgraphical methods that are generally used to e!aluate how well the model fits the data set .)/"Host data analysis programs pro!ide information regarding the !ariability for each of the

estimated parameters such as the standard error or the coefficient of !ariation" 0oefficient of!ariation ?F for any parameter indicates uncertainty of this parameter estimate, whichusually results from o!erparameteriation of the model or insufficient data" Also, the Akaikeinformation criteria are used to determine if going to more comple* model impro!es the fitwithout compromising the accuracy in model parameter estimation"

&e!eral graphical methods can be used to e!aluate the goodness of fit including a scattered plotof the obser!ed concentrations around the model predicted drug concentration–time cur!e todetermine how well the model fit the data" &mall differences between the obser!ed and modelpredicted !alues and random distribution of the obser!ed !alues around the predicted !aluesindicate good fit, as in 'igure ()"(A" $he e*istence of trend in the data points that is presentedby a series of consecuti!e data points abo!e or below the predicted profile is an indication of

inappropriate fit as in'igure ()"(6" Also, the obser!ed !ersus the model predicted drugconcentration !alues is a !ery useful diagnostic plot to e!aluate the goodness of fit" A goodmodel fit can be concluded when the !alues are gathered uniformly and closely around the linewith slope e#ual to one as in 'igure ()"(0, while the presence of trend in the data points is anindication of inappropriate fit as in 'igure ()"(D" Also, se!eral residuals plots can be utilied inthe model e!aluation" $he residuals are the difference between the obser!ed drug concentrationand the model predicted drug concentration, which 55855:is a measure of the error at each

data point" 'or e*ample, the residual !ersus predicted concentration plot to e*amine the error!alue at different drug concentrations and determine if the model fit one end of the cur!e betterthan the other" Also, the residual !ersus time plot to e!aluate if the model accurately accountsfor all the different phases in the drug concentration–time profile" 'urthermore, the plot of theweighed residuals !ersus predicted !alues and weighed residuals !ersus time are useful to

e!aluate the model fit as in 'igure ()"(5A –D" +hen the model fits the obser!ed data properly,the magnitude of the weighted residuals in the weighted residuals !ersus predicted !alue plotand the weighted residuals !ersus time plot, should be small with appro*imately uniformmagnitude, and randomly distributed around the ero residual line, as in 'igure ()"(5Aand 0"+hile the e*istence of trend or inconsistent weighted residual magnitude indicates improper fit,as in 'igure ()"(6 and D"

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$I%#!E 17.12 E9amples o' the dia&nostic &raphs 'or compartmental pharmacokineticdata anal)sis. The scattered plot o' the o(sered concentration and the model predicteddru& plasma pro'ile. !andom distri(ution o' the data around the predicted pro'ileindicates &ood data 'ittin& ,-/0 *hile the presence o' a trend that appears as series o'o(serations a(oe or (elo* the model predicted alues indicates inappropriate data'ittin& ,B/. The o(sered ersus predicted plasma dru& concentration. #ni'orm and

random distri(ution o' data around the line *ith slope o' 1 indicates &ood data 'ittin& ,C/0*hile the presence o' trend indicates unaccepta(le data 'ittin& ,"/.

55:55)

$I%#!E 17.18 E9amples o' the residual plot used as a dia&nostic test 'or ealuatin&cure 'ittin&. ,- and B/ is the *ei&hted residuals ersus predicted concentrations0 and ,Cand "/ is the *ei&hted residuals ersus time. Small0 uni'orm0 and randoml) distri(uted*ei&hted residuals indicate &ood 'ittin& o' data ,- and C/0 *hile the presence o' trend orune4ual *ei&hed residuals su&&est improper data 'ittin& ,B and "/.


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GA0$I0E G369EH&

• 17.1 A drug that follows a two-compartment pharmacokinetic model is gi!en to a patient

by rapid IV inTection" +ould the drug concentration in each tissue be the same after the druge#uilibrates between the plasma and all the tissues in the bodyN E*plain"• 17.2 A drug follows two-compartment pharmacokinetic model" If a single IV bolus dose is

gi!en, what is the cause of the initial rapid decline in the plasma concentration 1@-phase2N+hat is the cause of the slower decline in the plasma concentration 1-phase2N

• 17.8 A drug that follows two-compartment pharmacokinetic model was gi!en as a single

IV bolus dose of 8": mgBkg" $he e#uation• that describes the plasma concentration–time data is

o a" 0alculate the plasma concentration after ?"8, 5, and ( h of drug

administration"o b" +hat will be the e#uation that describes the plasma concentration–time profile

if the dose gi!en was ((" mgBkgN• 17.; After administration of a single IV bolus dose of )8 mg of a drug to a healthy

!olunteer, the pharmacokinetics of this drug followed the two-compartment model" $hefollowing parameters were obtained


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• ;

o a" 0alculate the following parameters;t(B@, t(B, k(, k(, k(?, Vc, Vd, Vdss, AK0,and 09$"o b" +hat will be the amount of drug remaining in the body after hN

• 17.

17 multicompartment pharmacokinetic models.docx

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