pharmacokinetic models of transdermal drug delivery … · pharmacokinetic models of transdermal...

Pharmacokinetic models of transdermal drug delivery

Annette L. Bunge

Opportunities and pitfalls

PK models of transdermal drug delivery

http://en.wikipedia.org/wiki/Transdermal_patchTime

Combine three models

transdermaldevice

Model 1Release from device

Model 2Skin permeation

Model 3Systemic distribution/metabolism/elimination

Transdermal device model

Represent the device design including all diffusion barriersPartitioning between all parts of deviceDissolution of undissolved drug(if present)

transdermaldevice

Systemic pharmacokinetic model

Could be a 1 or multi-compartment modelCould be a physically-based PK (PBPK) modelModel parameters generally derived by separate experiment (e.g., plasma C arising from IV delivery)

SC alone or with other layer(s)

Skin model

stratum corneum

viable epidermis

dermis

transdermaldevice

Skin model

Simple or complex description of each layer

“Stirred” compartment

Average C(t)

transdermaldevice

Average C(t)

Skin model

“Stirred” compartmentDiffusion barrier

Dlayer

C(t,x)

transdermaldevice

Dlayer

C(t,x)

Skin model

“Stirred” compartmentDiffusion barrierDiffusion barrier with multiple, multiphasic pathways

D1, D2, K1/2

C1(t,x,y,z?)C2(t,x,y,z?)

transdermaldevice

PK modelling opportunities

Test hypothesesPhysically-based models precisely specify mechanisms Identify mechanisms by comparing model predictions with experimentsDisagreement with experiment proves that the model is wrongAgreement with experiment does NOT prove that the model is correct (other models might alsoagree)

PK modelling opportunities (even more)

Simulate experimentsPhysicochemical parameters are known preciselyTest sensitivity of proposed data analysis; e.g.

Does the data analysis give the correct physicochemical parameters?Will the proposed bioequivalence test metric give the correct result?

PK modelling opportunities (still more)

Answer what-if questions

Answer what-if questions; e.g.What happens if the TDD is changed?

Gupta et al., J Pain Symptom Manage, 7:S17-26 (1992)

Effect of membrane thickness(e.g., manufacturing variation in TDD)

Answer what-if questions; e.g.What happens if the TDD is changed?What if the SC thickness is different?

Effect of SC thickness(e.g., TDD applied to different sites)

Answer what-if questions; e.g.What happens if the TDD is changed?What if the SC thickness is different?What if the clearance rate changes?

Effect of clearance rate

Pitfalls

Results are plausible but wrongUnreasonable model parameters are usedNon-unique solutionsMeaningless extrapolations

Pitfalls: Results are plausible but wrong

Just because the calculated results look OK does not mean they are correct

Pitfalls

Errors in one (or more) mathematical elementsMass balance in and between compartmentsThermodynamic limits(partition coefficients at boundaries and equilibrium)Constitutive equations(describing how rates of transport and reaction vary with drug concentration)

: Results are plausible but wrong

Example TDD model

International J of Pharmaceutics, 71:237-243 (1991)

Drug: ClenbuterolEliminated slowly from the body; ½ life is ~ 33 hTDD device: Polymer matrix containing dissolved drug

Example TDD model

Matrix-type TDD

MembraneCm (x,t)

Initial ConcentrationCm,o < Cm,sat

Example TDD model

MatrixMembrane

Cm(x,t)

Stratum Corneum

MembraneCsc(x,t)

Initial ConcentrationCsc = 0

Example TDD model

MatrixMembrane

Cm(x,t)

SCMembrane

Csc(x,t)

Viable Epidermis/Dermis

StirredCe(t)

Initial ConcentrationCe = 0

Tissue

Stirredmt(t)

Initial Massmt = 0

Example TDD model

MatrixMembrane

Cm(x,t)

SCMembrane

Csc(x,t)

VE/DermStirred

Plasma

Stirredmp(t)

Initial Massmp = 0

Example TDD model

MatrixMembrane

Cm(x,t)

SCMembrane

Csc(x,t)

VE/DermStirred

PlasmaStirredmp(t)

TissueStirred

Example TDD model: Model parameters

MatrixMembrane

Cm(x,t)

SCMembrane

Csc(x,t)

VE/DermStirred

PlasmaStirredmp(t)

TissueStirred

Operational Parameters: TDD area, initial concentration

MatrixMembrane

Cm(x,t)

SCMembrane

Csc(x,t)

VE/DermStirred

PlasmaStirredmp(t)

TissueStirred

Dm Dsc

Km/sc Ksc/e

VDLm Lsc Le

Model parameters: 13

MatrixMembrane

Cm(x,t)

SCMembrane

Csc(x,t)

VE/DermStirred

PlasmaStirredmp(t)

TissueStirred

Dm Dsc

Km/sc Ksc/e

VDLm Lsc Le

Model parameters: 13Independent experiment: 6Regressed to data: 7

MatrixMembrane

Cm(x,t)

SCMembrane

Csc(x,t)

VE/DermStirred

PlasmaStirredmp(t)

TissueStirred

Dm Dsc

Km/sc Ksc/e = 1

VDLm Lsc Le

Model parameters: 13Independent experiment: 6Regressed to data: 7

- Plasma concentration(t)- Mass in urine(t)- Mass in TDD (? at end)

Example TDD model: Model equations

MatrixMembrane

Cm(x,t)

SCMembrane

Csc(x,t)

VE/DermStirred

PlasmaStirredmp(t)

TissueStirred

/ interface

e scm sc pe p ep e

x sc e

dm CA D k m k mdt x =

∂= − + −

MatrixMembrane

Cm(x,t)

SCMembrane

Csc(x,t)

VE/DermStirred

PlasmaStirredmp(t)

TissueStirred

Change in mass in VE/D with time

Transfer infrom SC

Transfer infrom plasma

Transfer outto plasma

= + ̶

// interface

e sce sc e sc pe p ep e

x sc e

dm CK L D k m k mdt x =

∂= + −

/ interface

e scm sc pe p ep e

x sc e

dm CA D k m k mdt x =

∂= − + −

MatrixMembrane

Cm(x,t)

SCMembrane

Csc(x,t)

PlasmaStirredmp(t)

TissueStirred

=In thepaper

VE/DermStirred

MatrixMembrane

Cm(x,t)

SCMembrane

Csc(x,t)

PlasmaStirredmp(t)

TissueStirred

e scm sc pe p ep e

dm Cf A D k m k mdt x = +

∂= − + −

/where = 1/( ) in unknown unitse sc e mf K L A length=

This makes the equation dimensionally inconsistent.

VE/DermStirred

Example TDD model: Model calculations

Göpferich and Lee, Intl J Pharmaceut, 71:237-43 (1991)

Model calculationfrom paper

Plasma concentration

Reported Km/sc = 0.45, f = 0.36

Model calculationfrom paper

Reported Km/sc = 0.45, f = 0.36Km/sc = 1 (in Fig. 5), f = 0.53

Reported Km/sc = 0.45, f = 0.36Km/sc = 1 (in Fig. 5), f = 0.53Km/sc = 1/0.45, f = 1

No difference

Mass eliminated

Difference might be too small to discriminate between calculations

Mass fraction in TDD

All results look reasonable

Mass balance error = 1 –

(Mass in TDD + Mass in all compartments + Mass eliminated) / (Initial mass in TDD)

Mass balance

Large errors inmass balance

when f ≠ 1

Pitfalls

Errors in mass balance equations are revealed by doing an overall mass balance.

Pitfalls

Errors in mass balance equations are revealed by doing an overall mass balance.Mass balances should be done and results reported as “proof” of model correctness.

Pitfalls

Errors in mass balance equations are revealed by doing an overall mass balance.Mass balances should be done and results reported as “proof” of model correctness.

Errors in thermodynamic limits or constitutive equations may not produce mass balance errors (and thus, may not be discovered)

Pitfalls

Numerical errors

Example TDD model: Numerical Errors

Numerical solutions approximate derivatives in the model equationsDerivative approximations may be poor:

If spacing between positions (nodes) at which concentration is calculated are spaced too far apart (in finite-difference methods)If size of volume within which concentration is calculated are too large (in finite-element methods)

Numerical errors may be discovered by errors in the mass balance.

Model ParametersCm,o = 91.9 mg/mLLsc = 15 μmLe = 50 μmLm = 13.6 μmKsc/m = 1000Ksc/e = 25Ke/p = 1kep = 15 h-1

kpe = 1.33 x 10-4 h-1

kpt = 0.2 h-1

ktp = 0.5 h-1

kel = 2.8 h-1

VD = 1130 mLDm = 2 x 10-8 cm2/hDsc = 1 x 10-8 cm2/h

No of nodes in matrix = 6

At 8 h

# nodes in SC = 21

# nodes in TDD = 6# nodes in TDD = 6

# nodes in SC = 6

11, 21

Optimize # of nodes in all compartments

# nodes in SC = 21

# nodes in SC = 6

11, 21

Numerical errors are generally larger at shorter times

# nodes in SC = 21

# nodes in SC = 6

11, 21

Numerical errors are generally larger at shorter timesErrors at shorter times can affect longer time results

Pitfalls:

Results are only as good as the model parameters used in the calculations

: Unreasonable parameter values

From http://www.donwardell.ca/cartoons.php

Pitfalls

Poorly chosen values

Example TDD model: Model Parameter

MatrixMembrane

Cm(x,t)

SCMembrane

Csc(x,t)

VE/DermStirred

PlasmaStirredmp(t)

TissueStirred

kelKsc/e = 1

Example TDD model: Model Parameter

MatrixMembrane

Cm(x,t)

SCMembrane

Csc(x,t)

VE/DermStirred

PlasmaStirredmp(t)

TissueStirred

kelKsc/e = 1

Estimated Koct/w = 100Ksc/e = 1 does NOT seem reasonable

logKsc/e ≈ 0.7 logKoct/w

Ksc/e ≈ 25

Pitfalls

Poorly chosen valuesParameter regression without consideration of connections or limits in the parameter values

Example TDD model: Parameter limits

MatrixMembrane

Cm(x,t)

SCMembrane

Csc(x,t)

VE/DermStirred

PlasmaStirredmp(t)

TissueStirred

kep = 1.47 h-1 and kpe = 12.6 h-1

Mass transfer rate from VE to plasma = kep me ̶ kpe mp

Values for kep and kpe are related by equilibrium limits.

At equilibrium between VE and plasma: Ce = Ke/p Cp

Mass transfer rate from VE to plasma = 0

/e p e mpe ep

K L Ak k

V⎛ ⎞

= ⎜ ⎟⎝ ⎠

kep and kpe are NOT independent!

From Göpferich and Lee, 1991:kep = 1.47 h-1 and kpe = 12.6 h-1

Le = 50 μm, VD = 113 L and Am = 2 cm2

If Ke/p ≈ 1 for kep = 1.47 h-1

kpe = 13 x 10-8 h-1 (NOT 12.6 h-1)

Pitfalls

Poorly chosen valuesParameter regression without consideration of connections or limits in the parameter valuesPoorly chosen or incorrect model parameters for a model compartment:

May not matter if that compartment is not rate controlling, butCould matter if conditions change (e.g., longer times, other parameter values change)Might lead to misinterpretation of the physical situation

Pitfalls

Even simple PK models of TDD have many model parametersCommonly multiple combinations of parameter values provide similar resultsDiscriminating between model parameter choices is especially difficult if experimental data are variable

: Non-unique solutions

Pitfalls

0 2 4 6 8Time (days)

Plasma concentration Mass fraction eliminated

Gopferich & Lee, 1991

Pitfalls

0 2 4 6 8Time (days)

Plasma concentration Mass fraction in TDD

Pitfalls

Multiple sets of model parameters may fit the data but extrapolate to different resultsExtrapolation with model parameters determined by regression to data are particularly unreliableExtrapolations may be done with more confidence if the model parameters have been determined in separate experiments (and fit the experimental data)

: Meaningless extrapolations

0 14 28 42 56Time (days)

Pitfalls

Mass fraction eliminated

0 14 28 42 56Time (days)

Pitfalls

Recommendations for PK models of TDD

Write out all equationsWord descriptions leave room for interpretation

Describe derivation of all model parametersRecognize physical constraints on model parameter values

Not all can be set independentlyIndependent experimental sources for model parameters are preferred over regressionCheck the mass balance and report resultsExtrapolate cautiously

The End

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