1

NONCOMPARTMENTAL ANALYSIS

Deficiencies of compartmental analysis:1. Lack of meaningful physiological basis for derived

parameters.2. Lack of rigorous criteria to determine # of

compartments necessary to describe disposition.3. Lack of ability to elucidate organ specific

elimination.4. Inability to relate derived parameters to quantifiable

physiological parameters.5. Inability to predict impact of pathophysiology.6. Inability to provide insight into mechanism of drug-

drug and drug-nutrient interactions.7. Highly sensitive to sampling frequency.

2

GENERAL PRINCIPLES OF STATISTICAL MOMENTS

MOMENT: A mathematical description of a discrete distribution.

STATISTICAL MOMENTS:•Utilized in chemical engineering to describe flow data•First applied to biological systems by Perl and Samuel in 1969 to describe the kinetics of cholesterol

3

Examples of Statistical Moment UsageIn statistics

M0N

M1 NX

X i (mean)

M2 N

XX i

2

2 (variance)

M3

2/32

3

1

NXX i

(skewness)

M4

22

4

2

NXX i

(kurtosis)

4

In statistics, the mean is a measure of a sample mean and is actually an estimate of the true population mean. In pharmacokinetics, we can calculate the moment of the theoretical probability density function (i.e., the solution of a differential equation describing the plasma concentration time data),or we can calculate moments from measured plasma concentration-time data. These curves are referred to as sample moments and are estimates of the true curves.

5

Assume a theoretical relationship of C(t) as a function of time. The non-normalized moments, Sr , about the origin are calculated as:

0

),...2,1,0( )( mrdttCtS rr

6

Kinetic parameter

0

)( dttC AUCArea under the curve

0

)( dtttC AUMC Area under the moment curve

7

dtCtAUMC

0

0

CdtAUC

8

ADVANTAGES • widely used to estimate the important pharmacokinetic

parameters.• Ease of derivation of pharmacokinetic parameters by simple

algebraic equations.• The same mathematical treatment can be applied to almost any

drug or metabolite provided they follow first order kinetics.• A detailed description of drug disposition is not required.

DISADVANTAGE

• Limited information regarding the plasma drug concentration – time profile,

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From: Rowland M, Tozer TN. Clinical Pharmacokinetics – Concepts and Applications, 3rd edition, Williams and Wilkins, 1995, p. 487.

10

Kinetic parameterFirst moment:

AUCAUMC

dttC

dtttC

0

0

)(

)( MRTMean residence time

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AREA DETERMINATION

A. Integration of Specific Function•Must elucidate the specific function•Influenced by the quality of the fit

2

2

1

1 :example CCAUCCAUC

i

i

22

221

12 :example

CCAUMCCAUMC

i

i

12

B. Numerical Integration1. Linear trapezoidal2. Log trapezoidal

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B. Numerical Integration1. Linear trapezoidal

CC

tt

12

12

Concentration

Time

))(( 2112212

1CCttArea t

t

))((

...))(())((

1121

233221

122121

0

nnnn

t

ttCC

ttCCttCCArea n

14

B. Numerical Integration1. Linear trapezoidal

Advantages: Simple (can calculate by hand)

Disadvantages:•Assumes straight line btwn data points•If curve is steep, error may be large•Under or over estimate depends on whether curve is ascending of descending

15

16

B. Numerical Integration1. Linear trapezoidal2. Log trapezoidal

21

1221

lnln))((2

1 CCttCC

Area t

t

17

B. Numerical Integration1. Linear trapezoidal2. Log trapezoidal 21

1221

lnln))((2

1 CCttCC

Area t

t

Advantages:•Hand calculator•Very accurate for mono-exponential•Very accurate in late time points where interval btwn points is substantially increased

Disadvantages:•Limited application•May produce large errors on an ascending curve, near the peak, or steeply declining polyexponential curve

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B. Numerical Integration1. Linear trapezoidal2. Log trapezoidal3. Extrapolation to infinity

n

nt z

nt

CCdtAUC

z

nn

z

nt

CtCAUMCn 2

19

E

nt

KC

AUCAUC n

00

E

nn

E

nt

KC

tKC

AUMCAUMC n

200

20

Time (hr) C (mg/L) 0 2.55 1 2.00 3 1.13 5 0.70 7 0.43 10 0.20 18 0.025

AUC Determination

Area (mg-hr/L)-2.2753.131.831.130.9450.900

Total 10.21

AUMC Determination C x t(mg/L)(hr) 0 2.00 3.39 3.50 3.01 2.00 0.45

Area(mg-hr2/L) - 1.00 5.39 6.89 6.51 7.52 9.80 37.11

LhrmgAUMC

LhrmgAUCt

t

/ 11.37

/ 21.10 2

0

0

18

18

21

LhrmgAUChr

LmgLhrmgAUC

KC

AUCAUCE

t

/ 31.10 26.0

/ 025.0/ 21.10

0

10

180018

LhrmgAUMC

hrLmg

hrLhrmg

LhrmgAUMC

KC

KCt

AUMCAUMCEE

t

/ 21.39

26.0/ 025.0

26.0/ 45.0

/ 11.37

20

2112

0

2181818

0018

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CLEARANCE CONCEPTS

ORGANQCa

QCv

elimination

If Cv < Ca, then it is a clearing organ

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Rate In = QCa

Rate Out = QCv

Rate of elimination = QCa – QCv

= Q(Ca – Cv)

24

a

va

CCCQQECL )(

Clearance:The volume of blood from which all of the drug would appear to be removed per unit time.

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Extraction Ratio:Ratio of the rate of xenobiotic elimination and the rate at which xenobiotic enters the organ.

a

va

a

va

CCC

QCCCQE

E

)(Entry of Rate

nEliminatio of Rate

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Relationship between CL & QSince CL = QE, if E~1: CL Q

Perfusion rate-limited clearance

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Total Clearance

Total (systemic) Clearance:

bloodin ion concentratraten Eliminatio

Cdt

dXCLT

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Total ClearanceTotal (systemic) Clearance:

bloodin ion concentratraten Eliminatio

Cdt

dXCLT

0

0

0

0

0

Therefore

and

(Div) eliminatedamt total where,

,0 from gIntegratin

AUCDCL

AUCCdt

dtdtdX

Cdt

dtdtdX

CL

ivT

T

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Additivity of clearanceRate of elimination = Rate of Renal Excretion + Rate of Hepatic Metabolism

Dividing removal rate by incoming concentration:

aaa CCCMetabolism Hepatic of RateExcretion Renal of RatenEliminatio of Rate

Total Clearance = Renal Clearance + Hepatic Clearance

CLT = CLR + CLH

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Exception: sig. pulmonary elimination

From: Rowland M, Tozer TN. Clinical Pharmacokinetics – Concepts and Applications, 3rd edition, Williams and Wilkins, 1995, p. 12.

31

RTRiv

uR fCLCL

DXf

,

100 mg drug administered to a volunteer resultedin 10 mg excreted in urine unchanged:

Riv

RTR

iv

uR

fAUCDfCLCL

mgmg

DXf

1.0 100

10

32

Application of Clearance ConceptsPrediction of the effect of pathophysiological changesA new antibiotic has just been introduced onto the market. Currently, there are no studies examining the effect of renal disease on the pharmacokinetics of this compound. Is dosage adjustment necessary for this drug when used in pts with renal failure? How can we gain some insight into this question? A study in normal volunteers was recently published and the following data was included (mean):

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Application of Clearance ConceptsPrediction of the effect of pathophysiological changes

CLT = 1.2 L/hr Div = 500 mgAmount in urine unchanged = 63 mg

hrLhrLfCLCLmgmg

DXf

RTR

iv

uR

/ 15.0126.0/ 2.1

126.0 500

63

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Mechanisms of altered eliminationVerapamil has been shown to elevate serum digoxin concentrations in patients receiving both drugs concurrently. A study by Pedersen et al (Clin Pharmacol Ther 30:311-316, 1981.) examined this interaction with the following results.: TreatmentDigoxinDig + verapamil

CLT

3.282.17

CLR

2.181.73

CLNR

1.100.44

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STEADY-STATE VOLUME OF DISTRIBUTION

Cf

Cbp

Cf

Cbt

VP VT

36

Cf

Cbp

Cf

Cbt

VP VT

T

fut

P

fup C

Cf

CC

f

CP = Cf + Cbp CT = Cf + Cbt

37

At steady-state:

TPss

TssPSSTTssPPssSS

Pss

SSSS

VCCVVorVCVCA

CAV

Substitute:

up

fPss

ut

fTss f

CC

fC

C and

38

Tutf

upfPSS V

fCfC

VV

Simplifying:

Tut

upPSS V

ff

VV

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Tut

ubBSS V

ffVV

Using blood concentrations:

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Calculation via moment analysis:

2AUCAUMCDV iv

SS

Assumptions:•Linear disposition•Administered and eliminated via sampling site•Instantaneous input

41

If administration via a short term infusion:

AUCTK

AUCAUMCTKVSS 2

)( 20

20

K0 = infusion rate T = infusion duration

42

MEAN RESIDENCE/TRANSIT TIMEAdministration of a small dose may represent a large number of molecules:Dose = 1 mg MW = 300 daltons

# of molecules = (10-3 g/300) x (6.023 x 1023)

~2 x 1018 molecules

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Instantaneous administration of the entire dose will result in xenobiotic molecules spending various amounts of time in the body. Evaluation of the time various molecules spend in the body (residence time) can be characterized in the same manner as any statistical distribution.

Mean residence time: The average time the molecules of a given dose spend in the body.

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A conceptual understanding can be gained from the following example: Assume a child received 20 dimes for his birthday and immediately places them in his piggy bank. Over the next month, he periodically removes 1 or more dimes from the piggy bank to purchase candy. Specifically, 3 days after placing the coins in his bank he removes 5 dimes, on day 10 he removes 4 dimes, on day 21 he removes 6 dimes and on day 30 he removes 5 dimes. At the 30th day after placing the coins in his bank, all of the coins have been removed. Hence, the elimination of the deposited dimes is complete. The MRT of the dimes in the piggy bank is simply the sum of the times that coins spend in the bank divided by the number of dimes placed in the bank.

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2030303030302121212121211010101033333

MRT

20)530()621()410()53(

MRT

daysMRT 55.16

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MRT can be determined for any given number of drug molecules (Ai) that spend a given amount of time (ti) in the body:

timesresidence ofnumber total where

1

nA

tAMRT

total

n

iii

47

The mean rate of drug leaving the body relative to the total amount eliminated can be expressed in terms of concentration:

0

0

)(

)(

dttC

dtttCMRT

AUCAUMCMRT

48

AUCAUMCMRT

This is not a definition of MRT, rather it is a means of calculating MRT when CL is constant.

MRTAUCAUMC

po

po

When calculated in this fashion, it is often said that MRT is a function of the route of administration. However, MRT is independent of the route.

Meant Transit Time (MTT): The average time for xenobiotic molecules to leave a kinetic system after administration.

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Since an iv bolus assumes instantaneous input:

iviv

iv MTTMRTAUCAUMC

timeabsorptionmean

MAT

MATMRTMATMTTMTT

MTTAUCAUMC

ivpo

popo

po

50

CLVMRT ss

If drug declines via monoexponential decline:

10

20

C

C

AUCAUMCMRT

51

SYSTEMIC AVAILABILITY

poiv

ivpo

DAUCDAUC

F