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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
1 Introduction and Background
2 Methods
3 Results
Elvira Erhardt, Politecnico di Torino 3/28
v ⇥ y = z
Elvira Erhardt, Politecnico di Torino 4/28
v ⇥ y = z
v = 14r + 10
y = 3n � 2
) v ⇥ y is a convolution.
Elvira Erhardt, Politecnico di Torino 5/28
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 .
Elvira Erhardt, Politecnico di Torino 6/28
v ⇥ y = z
bla
bla
Convolution and deconvolution methods arefrequently used to establish an...
Elvira Erhardt, Politecnico di Torino 7/28
v ⇥ y = z
[absorption function]⇥ [disposition function]
= [blood concentration-time curve]
Convolution and deconvolution methods arefrequently used to establish an...
Elvira Erhardt, Politecnico di Torino 8/28
v ⇥ y = z
[absorption function]⇥ [disposition function]
= [blood concentration-time curve]
Convolution and deconvolution methods arefrequently used to establish an...
Elvira Erhardt, Politecnico di Torino 9/28
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.
Elvira Erhardt, Politecnico di Torino 10/28
Data sets and general idea
Elvira Erhardt, Politecnico di Torino 11/28
Data sets and general idea
Elvira Erhardt, Politecnico di Torino 12/28
Data sets and general idea
Elvira Erhardt, Politecnico di Torino 13/28
Outline
1 Introduction and Background
2 Methods
3 Results
Elvira Erhardt, Politecnico di Torino 14/28
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
Elvira Erhardt, Politecnico di Torino 15/28
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
Elvira Erhardt, Politecnico di Torino 16/28
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.
Elvira Erhardt, Politecnico di Torino 17/28
Outline
1 Introduction and Background
2 Methods
3 Results
Elvira Erhardt, Politecnico di Torino 18/28
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 - -
Elvira Erhardt, Politecnico di Torino 19/28
Model 1: in vitro absorption/permeation
Estimation using nonlinear mixed e↵ects models:
Elvira Erhardt, Politecnico di Torino 20/28
Model 2: in vivo immediate release
Estimation using nonlinear mixed e↵ects models:
Elvira Erhardt, Politecnico di Torino 21/28
Model 3: individual predictions
Elvira Erhardt, Politecnico di Torino 22/28
Model 3: Population predictions
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0.0
0.5
1.0
1.5
2.0
0 25 50 75 100 125Time [h]
Plas
ma
conc
entra
tion
[ng/
ml]
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0.00
0.25
0.50
0.75
1.00
1.25
0 15 30 45 60Time [h]
Plas
ma
conc
entra
tion
[ng/
ml]
Elvira Erhardt, Politecnico di Torino 23/28
Model 3: Area under the curve (AUC)
Comparing our model (top row) to the model without uncertaintypropagation (bottom row).
Elvira Erhardt, Politecnico di Torino 24/28
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
Elvira Erhardt, Politecnico di Torino 25/28
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.
Elvira Erhardt, Politecnico di Torino 26/28
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.
Elvira Erhardt, Politecnico di Torino 27/28
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.
Elvira Erhardt, Politecnico di Torino 28/28
Backup I
Elvira Erhardt, Politecnico di Torino 2/12
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)
Elvira Erhardt, Politecnico di Torino 3/12
Backup III
CR data
Elvira Erhardt, Politecnico di Torino 4/12
Backup IV
Elvira Erhardt, Politecnico di Torino 5/12
Backup V
Individual predictions
Elvira Erhardt, Politecnico di Torino 6/12
Backup VI
Visual predictive check population predictions - 95th
Elvira Erhardt, Politecnico di Torino 7/12
Backup VII
Visual predictive check population predictions - 75th
Elvira Erhardt, Politecnico di Torino 8/12
Backup VIII
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●
●
1.0 1.5 2.0 2.5
0.5
1.0
1.5
2.0
2.5
C_max observations (72h)
C_m
ax p
redi
ctio
ns (7
2h)
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●
●
●
●
●
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●●
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●●
●
0.4 0.6 0.8 1.0 1.2
0.1
0.2
0.5
1.0
C_max observations (24h)
C_m
ax p
redi
ctio
ns (2
4h)
1.0 1.5 2.0 2.5
0.5
1.0
1.5
2.0
2.5
C_max observations (72h)
C_m
ax p
redi
ctio
ns (7
2h)
0.4 0.6 0.8 1.0 1.2
0.1
0.2
0.5
1.0
C_max observations (24h)
C_m
ax p
redi
ctio
ns (2
4h)
Elvira Erhardt, Politecnico di Torino 9/12
Backup IX
Gelman-Rubin statistic (R̂) and trace plot
1.00 1.01 1.02R̂
R̂ > 1.1R̂ ≤ 1.1R̂ ≤ 1.05
Elvira Erhardt, Politecnico di Torino 10/12
Backup X
Pairs plot
Elvira Erhardt, Politecnico di Torino 11/12
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
Elvira Erhardt, Politecnico di Torino 12/12