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MATHEMATICAL MODELING OFPHARMACOKINETIC DATA

DR. SHIVPRAKASHMANAGING DIRECTOR

SYNCHRON RESEARCH SERVICES PVT. LTD.,AHMEDABADINDIA

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PHARMACOKINETICS- DEFINITION ANDSCOPE

Pharmacon =drug and Kinetics =movement

Mathematical description of biological process affectingdrugs and affected by drugs

Important:

• Design and development of new drugs• Reassessment of old drugs• Clinical applications (Clin. Pkinetics)• Evaluating bioavailability and bioequivalence

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SHEMATIC REPRESENTATION OF ADME AND PD OF A MOLECULE

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REASONS FOR MEASURING BLOODLEVELS AND URINARY EXCRETION

1. Is the drug absorbed and to what extent?2. Do other things interfere with absorption of the drug?3. What is the nature of the dose response curve?4. How do levels obtained with different routes of administration

compare?5. How is the drug eliminated and how fast?6. What factors affect rate of elimination?7. Is the pharmacological action due to the parent drug or a metabolite?

8. Is there a correlation between pharmacologic response and pharmacokinetic data?

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WHY DO WE TRY TO FIT DATA WITH EQUATIONSAND DERIVE KINETIC MODELS?

1. To summarize observed data- data exploration2. To increase understanding of the process or processes

involved3. To be able to make predictions4. To compare several drugs with similar pharmacological

actions5. To quantitatively relate biological activity with

pharmacokinetic data

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COMPARTMENT MODELS

A model in pharmacokinetics is a hypotheticalstructure which can be used to characterize with

reproducibility, the behavior and fate of a drug in biological system when given by a certain route ofadministration and in a particular dosage form.

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PHARMACOKINETIC MODELS

Model parametersfit to priori

4-20Physiological

Model parametersfit to data

1-3Compartmental

Curve fitting to

data

0 Non-

compartmental

MathematicalCharacteristics

No. ofCompartments

Type

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PHARMACOKINETIC MODELS NON-COMPARTMENTAL MODELS• Offers little insight into the rate or processes involved in drug

distribution• Imposes a model structure and more restrictive than multi-

compartmental modelCOMPARTMENT MODELS

• Catenary and mamillary modelsPHYSIOLOGICAL MODELS• Organ blood flow, volume, partition coefficient, and binding affinity

for a particular drug• Assumes that a fraction of the drug in the body will be extracted form

the blood on each pass• Blood flow is a determental of drug clearance

E QCL .=

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PHYSIOLOGICAL MODELS

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ESTIMATION OF PARAMETERS

1. The statistical problem2. Numerical methods3. Computer implementation

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NUMERICAL GRAMMAR (Daniel & Wood, 1971)

1. Use all relevant data2. Have reasonable parsimony in the number of unknown parameters3. Take into account the error in the data4. Provide some measure for the precision of estimates for unknown

model parameters5. Be able to locate systematic deviations in data form the chosen

model equations6. Provide some measure of how well the model will predict future

experimental frames

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WAGNERs SIX POINT CHECK LIST FOR DERIVINGMODELS

1. State the assumption involved2. Excert caution particularly in making physiological and biochemical inferences

3. Ask yourself whether models could equally as well

explain the data4. Check whether your model agrees with known

physiology, pharmacology etc.,5. Check not only for closeness of fit, but also trends in

areas of poor fit6. Check whether your model provides accurate predictions

particularly if the system is perturbed

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GENERAL MODELING STRATEGIES

1. Initial model selection• Start with a plot• How many linear segments are involved?2. Obtaining initial estimates• Curve stripping method• Computer programs – PCNONLIN, KINETICA etc.,3. Selection of minimization algorithm

• Gauss-Newton• Marquardt• Nelder-Mead Simplex Optimization

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GENERAL MODELINGSTRATEGIES…continued

4. Assessing the goodness of fit• AIC• Correlation between observed and predicted values

• Analysis of residuals

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START WITH A PLOT

1. Plot on Log scale2. See for the number ofsegments

3. Straight line in theelimination segment- 1compartment (mono exp)

4. One segment – 2compartment (bi-exp)

5. Two segments – 3compartments (Tri-exp)…..

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RATE PROCESSESFIRST ORDER PROCESS

ZERO ORDER PROCESS

CAPACITY LIMITED PROCESS

t k C C

t k C C

C k dt

dc

.ln

).exp(.

.

0

0

−=

−=

−=

t k C C

k dt

dC

.00

0

−=

−=

)1()(maxmax)1()(

)()1(

max

..1

lnln

.

+

+

−+=

−

−

+=−

nnm

nn

nn

m

C C V V

K

C C

t t

C K C V

dt dC

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CAPACITY LIMITED PROCESS

1. Plot Con Vs Time on logscale at increasing doses

2. Low dose may look like firstorder plot (indicated non-saturation)

3. At higher doses the plot looklike dome shaped (due tosaturation)

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HOW COMPLICATED MODEL CAN BE FIT TO DATA?

NP=2.EX+PE+2.TS+NL NP=No. of parametersEX=No. of exponentialsPE=No. of elimination or excretory pathwaysTS=No. of tissue spaces or binding proteins

NL=No. of visible non-linear features

Ref.: Jusko, Applied Pharmacokinetics: Principles of TDM, 2 nd Edn.1986

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OBTAINING INITIAL ESTIMATES

1. RANDOM SEARCH METHODS2. STRIPPING OR PEELING METHODS3. LINEARIZATION METHODS

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CURVE STRIPPING METHOD

t T Be AeC .. β α −− +=

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∑

∑

∑2i

i

i x

x

x

n=

1

0

β

β

∑

∑ii

i

y x

y

LINEARIZATION METHODS

NORMAL EQUATIONS TO SOLVE LEAST SQUARES

C B A =− .1

A C B

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COMPUTER PROGRAMS

A large number of computer programs are available to perform thefunctions of mathematical modeling. Some are:

1. ADAPT II – Fortran based by D’Argenio and Schumitzky2. BOOMER – Fortran Based by WA Bourne3. MK MODEL - by Holford4. MULTI – Basic based by Yamaoka et al.5. WINNONLIN – Industry standard popular program, Windows

based.6. KINETICA – Industry based popular program, Windows based

7. NONMEM – Fortran based by Beal and Sheiner 8. QUICKCALC – Basic based by Shivprakash- 14 models9. MATHEMATICA – Most advanced programming capabilities, can

be used for alpha-numerical programming

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CONFUSION BETWEEN MATHS ANDBIOLOGY PROCESS

DO NOT GET CONFUSED BETWEEN NON-LINEARPHARMACOKINETICS AND NON-LINEAR MODELS

Non=Linear Pharmacokinetics is saturation kinetics in which PK parametersare not linear with increasing dose (do not exhibit dose linearity)

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NON-LINEAR PHARMACOKINETICS

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LINEAR AND NON-LINEAR MODELS

3.....................................ln

2................................)exp(

1............

221

221

22110

ε θ θ

ε θ θ

ε β β β β

++=

++=

+++++=

t Y

t Y

Z Z Z Y nn

Read: Draper and Smith: Applied Regression analysis

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OPTIMIZATION- CRITERIA FOR BEST FIT

These criteria are achieved by minimizing the following quantities:

))((log)(

)(

)(

)(

2

1

2

1

2

1

ie

i

iin

i

ii

n

ii

n

iii

C Var C Var

C C ELS

C C W WLS

C C OLS

∧

∧

∧

=

∧

=

=

∧

+−

=

−=

−=

∑

∑

∑

Read on Weighting Schemes in PK: Peck CC et al. Drug Met. Rev., 1984

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OPTIMIZATION TECHNIQUESSTEEPEST DESCENT METHOD

Seems to be most efficient. Problems arise when translatedinto computer method.Large steps- method gets lostSmall steps- too long

Quite slow at minimum

GUASS-NEWTON METHOD

Linear least squares method. Linearisation by first orderTaylors expansion. It may lost with poor initial estimates.

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OPTIMIZATION TECHNIQUEScontinued…

MARQUARDT METHOD

Minimization of is used as a criterion of best fit.

SIMPLEX METHOD

∑=

−=

n

i i

ii xY Y

1

2

2 )(σ

χ

2 χ

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SIMPLEX OPTIMIZATION

Contour plot of residual sum ofsquare surface

Ref. :Nelder and Mead,Computing Journal, 1965

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MODEL SEECTION CRITERIA1. THE AKAIKE INFORMATION CRITERIA (AIC)

Minimum AIC value is the best representation of model.

Warning: Improper weight could select wrong equation.

2. F-TEST

df=(No. of experimental data points)-(No. of parameters)

∑=

∧

−=

+=

n

iiiie

e

C C w R p R N AIC

1

2)(2.ln.

21

2

2

21 .df df

df WSS

WSS WSS F

−

−= )( 21 df df

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MODEL SELECTION CRITERIONcontinue…3. STANDARD DEVIATION OF ESTIMATED

PARAMETER

N=No. of data points

P= No. of parameters estimated N-P= degrees of freedomC ii=i th diagonal element of the variance-covariancematrix of the elements

∑ ∑

∑∧

−=

−=

=

22

22

2

)(

.

Y Y dev P N

dev s

C s DS ii

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MODEL SELECTION continue…4. COEFFICIENT OF DETERMINATION

5. MODEL SELECTION CRITERION (MSC)

For equal weight

∑ ∑

∑

=

=

∧

−=

−−=

n

i y

n

i y

i

N

Y Y

Y Y wr

S

S

1

222

12

22

)(

)(1

N

P

Y Y w

Y Y w MSC n

iii

n

iii

.2)(

)(ln

1

2

1

2 _

−

−

−

=

∑

∑

=

∧

=

N

P

dev MSC S y

.2ln 2

2

−=

∑

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MODEL SELECTION continue…

6. ANALYSIS OFRESIDUALS

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MODEL SELECTION continue..

6. ANALYSIS OFRESIDUALS

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THANK YOU