Bayesian Knowledge Integration for an in Vitroâ•ﬁ-in Vivo ...“Basic Pharmacokinetics and...

Bayesian Knowledge Integration for anin Vitro-in Vivo Correlation Model

Elvira Erhardt1, Moreno Ursino2, Tom Jacobs3,Jeike Biewenga3, Mauro Gasparini1

1Politecnico di Torino, IT 2INSERM Paris, FR3Janssen Pharmaceutica, BE

IDEAS Summer School III, 23-27 September 2018

Basel, Switzerland

Outline

1 Introduction and Background

2 Methods

3 Results

Elvira Erhardt, Politecnico di Torino 2/28

Outline


2 Methods

3 Results


v ⇥ y = z


v ⇥ y = z

v = 14r + 10

y = 3n � 2

) v ⇥ y is a convolution.


v ⇥ y = z

v = 14r + 10

y = 3n � 2

) v ⇥ y is a convolution.

) Knowing z and y , a deconvolution can beperformed to find v .


v ⇥ y = z

bla

bla

Convolution and deconvolution methods arefrequently used to establish an...


v ⇥ y = z

[absorption function]⇥ [disposition function]

= [blood concentration-time curve]



Background

In Vitro-In Vivo Correlation (IVIVC)

Defined by the U.S. Food and Drug Administration (FDA) as“predictive mathematical model describing the relationship betweenan in vitro property of a dosage form and an in vivo response”.

Tool to predict entire in vivo drug concentration–time coursebased on in vitro drug release profiles! supports biowaivers ! saves resources.

Aim:Establish a new method of IVIVCmodelling, overcoming the disadvantages ofcurrent IVIVC methodology (averageddata, artificial certainty, deconvolution).Case study: transdermal patch.


Data sets and general idea


Outline


2 Methods

3 Results


Model 1: in vitro absorption/permeation

Cumulative amount of drug passed through the skin

PDFW

(t; h, s) =h

s

⇣t

s

⌘h�1

exp

⇢�⇣t

s

⌘h

�

CDFW

(t; h, s) = 1� exp

⇢�⇣t

s

⌘h

�.

CDFW

Weibull distribution with

time t � 0, shape h > 0, scale s > 0.

y

ijp

= D

p

·fip

·CDFW

(tijp

; hip

, sip

)+✏ijp

✏ijp

⇠ N (0,�2✏ )

D Dose (p = 1, 2); f fraction of dose delivered.) To estimate: h

ip

, sip

, fip

(i = 1, . . . ,N skin portions).

) Frequentist nonlinear mixed e↵ects model


Model 2: in vivo immediate release

Three-compartment-model with intravenous infusion

da1(tijp

)

dt

= k21a2(tijp

) + k31a3(tijp

)�⇥k12a1(t

ijp

) + k13a1(tijp

) + k

e

a1(tijp

)⇤,

da2(tijp

)

dt

= k12a1(tijp

)�⇥k21a2(t

ijp

)⇤,

da3(tijp

)

dt

= k13a1(tijp

)�⇥k31a3(t

ijp

)⇤,

C1(tijp

) =a1(t

ijp

)

V1,i.

) To estimate:V1,i , ke , k12, k21, k13, k31(i = 1, . . . ,N individuals).

) Frequentist nonlinear mixed e↵ects model


Model 3: combining to in vivo controlled release

I

n

(tijp

) = Dp

· Fip

· Bi

· hipsip

·✓tijp

sip

◆h

ip

�1

· exp(✓

� tijp

sip

◆h

ip

)

da1(tijp)dt

= I

n

(tijp

) + k21a2(tijp) + k31a3(tijp)� [k12a1(tijp) + k13a1(tijp) + ke

a1(tijp)]

da2(tijp)dt

= k12a1(tijp))� [k21a2(tijp))]

da3(tijp)dt

= k13a1(tijp))� [k31a3(tijp))]

C1(tijp) =a1(tijp)V1,i

.

I

n

(tijp

): input function, ac

: amount of drug in compartment c.

At time t = 0: a1(0) = a2(0) = a3(0) = 0.

D: Dose; F : fraction of dose delivered; B: fraction of dose delivered which is

actually absorbed into the systemic circulation.

) Bayesian hierarchical 2-stage model

) natural integration and transfer of parameter knowledge,incl. corresponding uncertainty, from submodels to CR model.


Outline


2 Methods

3 Results


Estimates of all model parameters

mean (SD)

Parameter Frequentist Bayesianfixed random fixed R̂ random R̂

permeation controlled release

B - - 0.34 (0.09) 1.01 0.26 (0.15) 1.03f1 0.83 (0.02) (0.16) - - - -f2 0.91 (0.05) (0.07) - - - -log(h1) 0.52 (0.02) (0.15) 0.52 (0.02) 1.00

0.08 (0.02) 1.00log(h2) 0.61 (0.05) (0.05) 0.66 (0.03) 1.00log(s1) 3.84 (0.03) (0.16) 3.34 (0.13) 1.00

0.42 (0.06) 1.00log(s2) 3.82 (0.07) (0.11) 2.72 (0.12) 1.01

immediate release

log(V ) 10.47 (0.11) (0.38) 10.50 (0.11) 1.00 0.17 (0.09) 1.00log(k

e

) 0.39 (0.28) - -1.00 (0.19) 1.01 - -log(k12) 0.15 (0.45) - 0.32 (0.42) 1.00 - -log(k21) -2.16 (1.05) - -2.12 (0.28) 1.00 - -log(k13) 1.82 (0.15) - 2.13 (0.15) 1.00 - -log(k31) 0.61 (0.20) - 0.35 (0.17) 1.00 - -� - - 0.18 (0.01) 1.00 - -


Model 1: in vitro absorption/permeation

Estimation using nonlinear mixed e↵ects models:


Model 2: in vivo immediate release

Estimation using nonlinear mixed e↵ects models:


Model 3: individual predictions


Model 3: Population predictions

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ml]

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ma

conc

entra

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[ng/

ml]


Model 3: Area under the curve (AUC)

Comparing our model (top row) to the model without uncertaintypropagation (bottom row).


Model 3: Density of the AUC percent prediction errors

%PE = 100 ⇤|AUC

observations

� AUC

predictions

|AUC

observations

555555555555555555 252525252525252525252525252525252525 505050505050505050505050505050505050 757575757575757575757575757575757575 959595959595959595959595959595959595

0.00

0.05

0.10

0 5 10 15 20%PE AUC (72h)

dens

ity

555555555555555555 252525252525252525252525252525252525 505050505050505050505050505050505050 757575757575757575757575757575757575 959595959595959595959595959595959595

0.00

0.01

0.02

0.03

0 25 50 75 100%PE AUC (24h)

dens

ity

555555555555555555 252525252525252525252525252525252525

0.00

0.05

0.10

0 5 10 15 20%PE AUC (72h)

dens

ity

555555555555555555

252525252525252525252525252525252525

505050505050505050505050505050505050

757575757575757575757575757575757575

959595959595959595959595959595959595

0.00

0.01

0.02

0.03

0 25 50 75 100%PE AUC (24h)

dens

ity

555555555555555555 252525252525252525252525252525252525 505050505050505050505050505050505050 757575757575757575757575757575757575 959595959595959595959595959595959595

0.00

0.05

0.10

0 5 10 15 20%PE AUC (72h)

dens

ity

555555555555555555 252525252525252525252525252525252525 505050505050505050505050505050505050 757575757575757575757575757575757575 959595959595959595959595959595959595

0.00

0.01

0.02

0.03

0 25 50 75 100%PE AUC (24h)

dens

ity

555555555555555555 252525252525252525252525252525252525

0.00

0.05

0.10

0 5 10 15 20%PE AUC (72h)

dens

ity

555555555555555555

252525252525252525252525252525252525

505050505050505050505050505050505050

757575757575757575757575757575757575

959595959595959595959595959595959595

0.00

0.01

0.02

0.03

0 25 50 75 100%PE AUC (24h)

dens

ity


Conclusions

Developed combined in vitro-in vivo model providessatisfactory estimation of the transdermal patch’s PKpopulation data.

Innovation consists of shared parameter space merging thetwo frequentist submodels into a system of ODEs, estimatedin Bayesian way.

Bayesian framework allows natural integration and transferof knowledge between di↵erent sources of information, whileaccounting for parameter uncertainty.

) Flexible approach yielding results for broad range of datasituations.

) Extension of current IVIVC methodology where frequentistone-stage or two-stage approaches are the standard.


Bibliography

Gelman, A., J. Carlin, H. Stern, D. Dunson, A. Vehtari, and D. Rubin (2013).

“Bayesian Data Analysis, Third Edition”. Taylor & Francis.

Jacobs, T., S. Rossenu, A. Dunne, G. Molenberghs, R. Straetemans, and L. Bijnens

(2008). “Combined Models for Data from In Vitro–In Vivo Correlation Experiments”.

Journal of Biopharmaceutical Statistics 18.6., pages 1197-1211.

O’Hara, T., S. Hayes, J. Davis, J. Devane, T. Smart, and A. Dunne (2001). “In

Vivo–In Vitro Correlation (IVIVC) Modeling Incorporating a Convolution Step”.

Journal of Pharmacokinetics and Pharmacodynamics 28.3, pages 277-298

Davidian, M. and D.M. Giltinan (1995). “Nonlinear Models for Repeated

Measurement Data”. Taylor & Francis.

Pinheiro J., Bates D. (2000). “Mixed-E↵ects Models in S and S-PLUS.” Statistics

and Computing. Springer New York.

Piotrovskii, V.K. (1987). “The use of Weibull distribution to describe the in vivo

absorption kinetics”. J Pharm Biopharm; 15, pages 681-686.

Rosenbaum, S.E. (2011). “Basic Pharmacokinetics and Pharmacodynamics: An

Integrated Textbook and Computer Simulations”. Wiley.


Thank you for your attention!

This project has received funding from theEuropean Unions Horizon 2020 research andinnovation programme under the MarieSklodowska-Curie grant agreement No 633567.


Backup I


Backup II

0 2 4 6 8

Time [h]

Plas

ma

conc

entra

tion

[ng/

ml]

(log−

scal

e)

0.05

0.2

0.5

1

2

51−comp (AIC: 568.16)2−comp (AIC: 374.19)3−comp (AIC: 373.21)


Backup III

CR data


Backup IV


Backup V

Individual predictions


Backup VI

Visual predictive check population predictions - 95th


Backup VII

Visual predictive check population predictions - 75th


Backup VIII

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C_max observations (72h)

C_m

ax p

redi

ctio

ns (7

2h)

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0.4 0.6 0.8 1.0 1.2

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C_m

ax p

redi

ctio

ns (2

4h)

1.0 1.5 2.0 2.5

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ax p

redi

ctio

ns (7

2h)

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0.1

0.2

0.5

1.0


C_m

ax p

redi

ctio

ns (2

4h)


Backup IX

Gelman-Rubin statistic (R̂) and trace plot

1.00 1.01 1.02R̂

R̂ > 1.1R̂ ≤ 1.1R̂ ≤ 1.05


Backup X

Pairs plot


Backup XI

Posterior densities overlaid

mu_B_orig

0.2 0.4 0.6 0.8

Chain1234

eta_B[1]

−2 0 2

Chain1234

eta_B[9]

−2 0 2 4

Chain1234

sigma_B

0.5 1.0 1.5 2.0 2.5

Chain1234

mu_ls1

3.0 3.3 3.6 3.9

Chain1234

mu_ls2

2.4 2.7 3.0 3.3 3.6

Chain1234

sigma_ls

0.2 0.4 0.6

Chain1234

eta_ls[1]

−1.5−1.0−0.5 0.0

Chain1234

eta_ls[9]

0 1 2

Chain1234

mu_lh1

0.45 0.50 0.55

Chain1234

mu_lh2

0.550.600.650.700.750.80

Chain1234

eta_lh[1]

−2 −1 0 1 2

Chain1234

eta_lh[9]

−3 −2 −1 0 1 2

Chain1234

sigma_lh

0.04 0.08 0.12 0.16

Chain1234

mu_lv

10.210.410.610.811.0

Chain1234

eta_lv[1]

−3 −2 −1 0 1 2

Chain1234

eta_lv[9]

−2 0 2

Chain1234

sigma_lv

0.1 0.2 0.3 0.4

Chain1234

lke

−1.6 −1.2 −0.8 −0.4

Chain1234

lk12

0 1 2

Chain1234

lk21

−2.5−2.0−1.5−1.0

Chain1234

lk13

1.61.82.02.22.42.6

Chain1234

lk31

0.000.250.500.751.00

Chain1234

sigma

0.5 1.0 1.5 2.0

Chain1234


Bayesian Knowledge Integration for an in Vitroâ•ﬁ-in Vivo ...“Basic Pharmacokinetics and...

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Transcript of Bayesian Knowledge Integration for an in Vitroâ•ﬁ-in Vivo ...“Basic Pharmacokinetics and...