A Bayesian PK/PD model for synergy A case study Fabiola La Gamba, - - PowerPoint PPT Presentation

▶
a bayesian pk pd model for synergy
SMART_READER_LITE
LIVE PREVIEW

A Bayesian PK/PD model for synergy A case study Fabiola La Gamba, - - PowerPoint PPT Presentation

Jul 28, 2023 10 likes •210 views

A Bayesian PK/PD model for synergy A case study Fabiola La Gamba, Tom Jacobs, Christel Faes 23/06/2016 0 Importance of being uncertain Case study To assess the safety resulting from the co-administration of a novel molecule with an

slide-1
SLIDE 1

A Bayesian PK/PD model for synergy

Fabiola La Gamba, Tom Jacobs, Christel Faes 23/06/2016

Importance of being uncertain

A case study

slide-2
SLIDE 2

Case study

1 23/06/2016 A Bayesian PK/PD model for synergy; a case study

(Non-clinical) Biostatistics My PhD  To assess the safety resulting from the co-administration of a novel molecule with an existing, marketed treatment using in-vivo data

Data sets:

  • Historical study: Dose-response longitudinal data where only

the existing treatment is administered (55 rats in total)

  • 11 synergy studies: Both existing and novel treatments are
  • administered. One specific dose combination in each study:
slide-3
SLIDE 3

Case study

2 23/06/2016 A Bayesian PK/PD model for synergy; a case study

(Non-clinical) Biostatistics My PhD  To assess the safety resulting from the co-administration of a novel molecule with an existing, marketed treatment using in-vivo data

Data sets:

  • Historical study: Dose-response longitudinal data where only

the existing treatment is administered (55 rats in total)

  • 11 synergy studies: Both existing and novel treatments are
  • administered. One specific dose combination in each study:
slide-4
SLIDE 4

Case study

3 23/06/2016 A Bayesian PK/PD model for synergy; a case study

(Non-clinical) Biostatistics My PhD  To assess the safety resulting from the co-administration of a novel molecule with an existing, marketed treatment using in-vivo data

Data sets:

  • Historical study: Dose-response longitudinal data where only

the existing treatment is administered (55 rats in total)

  • 11 synergy studies: Both existing and novel treatments are
  • administered. One specific dose combination in each study:

Study 1 2 3 4 5 6 7 8 9 10 11

Existing treatment dose (mpk)

10 2.5 10 0.63 10 0.16 2.5 0.63 0.16 0.04 0.04

Novel treatment dose (mpk)

40 40 10 40 2.5 40 10 10 10 10 40

slide-5
SLIDE 5

Case study

4 23/06/2016 A Bayesian PK/PD model for synergy; a case study

(Non-clinical) Biostatistics My PhD  To assess the safety resulting from the co-administration of a novel molecule with an existing, marketed treatment using in-vivo data

Data sets:

  • Historical study: Dose-response longitudinal data where only

the existing treatment is administered (55 rats in total)

  • 11 synergy studies: Both existing and novel treatments are
  • administered. 20 rats in each study, 5 for each treatment group:

(Non-clinical) Biostatistics My PhD  Vehicle Existing treatment only Novel treatment only Treatments combination

slide-6
SLIDE 6

Case study

5 23/06/2016 A Bayesian PK/PD model for synergy; a case study

(Non-clinical) Biostatistics My PhD  To assess the safety resulting from the co-administration of a novel molecule with an existing, marketed treatment using in-vivo data

Data sets:

  • Historical study: Dose-response longitudinal data where only

the existing treatment is administered (55 rats in total)

  • 11 synergy studies: Both existing and novel treatments are

administered

Study variables:

  • Existing treatment dose
  • Novel treatment dose (only in synergy studies)
  • Continuous safety biomarker, measured at the moment of oral

administration, and after 1, 2, 3, 4 hours

slide-7
SLIDE 7

How does the data look like?

6 23/06/2016 A Bayesian PK/PD model for synergy; a case study

(Non-clinical) Biostatistics My PhD 

Example from study 1 Which model is the most suitable?

slide-8
SLIDE 8

A nice answer: PK-PD model

𝑒𝐵𝑓𝑗𝑢 𝑒𝑢

= −𝑙𝑏𝐵𝑓𝑗𝑢

𝑒𝐷𝑗𝑢 𝑒𝑢 = 𝑙𝑏𝐵𝑓𝑗𝑢 − 𝑙𝑓𝐷𝑗𝑢 𝑒𝑆 𝑗𝑢 𝑒𝑢 = 𝑙𝑗𝑜 1 − 𝐽𝑛𝑏𝑦𝐷𝑗𝑢 𝐽𝐷50+𝐷𝑗𝑢 − 𝑙𝑝𝑣𝑢𝑆

𝑗𝑢

7 23/06/2016 A Bayesian PK/PD model for synergy; a case study

PD part: indirect response (turnover) model

Parameters: 𝑙𝑏 ≥ 0: Absorption constant 𝑙𝑓 ≥ 0: Elimination constant 𝑙𝑗𝑜 ≥ 0: Constant for response production 𝑙𝑝𝑣𝑢 ≥ 0: Constant for response loss 0 ≤ 𝐽𝑛𝑏𝑦 ≤ 1: Maximal inhibition attributed to drug 𝛾: Interaction coefficient

PK Part:

  • ne compartment

model with oral absorption

i=1, …, S (subjects) t=0, …, 4 (hours) A𝑓𝑗𝑢=Existing treatment amount 𝐵𝑜𝑗0=Novel treatment dose 𝑆𝑗𝑢=Response: 𝑆𝑗𝑢~𝑂(𝑆 𝑗𝑢, 𝜏2) 𝐷𝑗𝑢=Plasma concentration of the existing treatment (latent!)

𝑓𝛾𝐵𝑓𝑗0𝐵𝑜𝑗0

With: 𝐵𝑓𝑗𝑢=0 = 𝐵𝑓𝑗0 (𝑓𝑦𝑗𝑡𝑢𝑗𝑜𝑕 𝑢𝑠𝑓𝑏𝑢𝑛𝑓𝑜𝑢 𝑒𝑝𝑡𝑓); 𝐷𝑗𝑢=0 = 0; 𝑆 𝑗𝑢=0 = 𝑙𝑗𝑜/𝑙𝑝𝑣𝑢

slide-9
SLIDE 9

A nice answer: PK-PD model

𝑒𝐵𝑓𝑗𝑢 𝑒𝑢

= −𝑙𝑏𝐵𝑓𝑗𝑢

𝑒𝐷𝑗𝑢 𝑒𝑢 = 𝑙𝑏𝐵𝑓𝑗𝑢 − 𝑙𝑓𝐷𝑗𝑢 𝑒𝑆 𝑗𝑢 𝑒𝑢 = 𝑙𝑗𝑜 1 − 𝐽𝑛𝑏𝑦𝐷𝑗𝑢 𝐽𝐷50+𝐷𝑗𝑢 − 𝑙𝑝𝑣𝑢𝑆

𝑗𝑢

8 23/06/2016 A Bayesian PK/PD model for synergy; a case study

PD part: indirect response (turnover) model

Parameters: 𝑙𝑏 ≥ 0: Absorption constant 𝑙𝑓 ≥ 0: Elimination constant 𝑙𝑗𝑜 ≥ 0: Constant for response production 𝑙𝑝𝑣𝑢 ≥ 0: Constant for response loss 0 ≤ 𝐽𝑛𝑏𝑦 ≤ 1: Maximal inhibition attributed to drug 𝛾: Interaction coefficient

PK Part:

  • ne compartment

model with oral absorption

i=1, …, S (subjects) t=0, …, 4 (hours) A𝑓𝑗𝑢=Existing treatment amount 𝐵𝑜𝑗0=Novel treatment dose 𝑆𝑗𝑢=Response: 𝑆𝑗𝑢~𝑂(𝑆 𝑗𝑢, 𝜏2) 𝐷𝑗𝑢=Plasma concentration of the existing treatment (latent!)

𝑓𝛾𝐵𝑓𝑗0𝐵𝑜𝑗0

With: 𝐵𝑓𝑗𝑢=0 = 𝐵𝑓𝑗0 (𝑓𝑦𝑗𝑡𝑢𝑗𝑜𝑕 𝑢𝑠𝑓𝑏𝑢𝑛𝑓𝑜𝑢 𝑒𝑝𝑡𝑓); 𝐷𝑗𝑢=0 = 0; 𝑆 𝑗𝑢=0 = 𝑙𝑗𝑜/𝑙𝑝𝑣𝑢

slide-10
SLIDE 10

A nice answer: PK-PD model

𝑒𝐵𝑓𝑗𝑢 𝑒𝑢

= −𝑙𝑏𝐵𝑓𝑗𝑢

𝑒𝐷𝑗𝑢 𝑒𝑢 = 𝑙𝑏𝐵𝑓𝑗𝑢 − 𝑙𝑓𝐷𝑗𝑢 𝑒𝑆 𝑗𝑢 𝑒𝑢 = 𝑙𝑗𝑜 1 − 𝐽𝑛𝑏𝑦𝐷𝑗𝑢 𝐽𝐷50+𝐷𝑗𝑢 − 𝑙𝑝𝑣𝑢𝑆

𝑗𝑢

9 23/06/2016 A Bayesian PK/PD model for synergy; a case study

PD part: indirect response (turnover) model

Parameters: 𝑙𝑏 ≥ 0: Absorption constant 𝑙𝑓 ≥ 0: Elimination constant 𝑙𝑗𝑜 ≥ 0: Constant for response production 𝑙𝑝𝑣𝑢 ≥ 0: Constant for response loss 0 ≤ 𝐽𝑛𝑏𝑦 ≤ 1: Maximal inhibition attributed to drug 𝛾: Interaction coefficient

PK Part:

  • ne compartment

model with oral absorption

i=1, …, S (subjects) t=0, …, 4 (hours) A𝑓𝑗𝑢=Existing treatment amount 𝐵𝑜𝑗0=Novel treatment dose 𝑆𝑗𝑢=Response: 𝑆𝑗𝑢~𝑂(𝑆 𝑗𝑢, 𝜏2) 𝐷𝑗𝑢=Plasma concentration of the existing treatment (latent!)

𝑓𝛾𝐵𝑓𝑗0𝐵𝑜𝑗0

With: 𝐵𝑓𝑗𝑢=0 = 𝐵𝑓𝑗0 (𝑓𝑦𝑗𝑡𝑢𝑗𝑜𝑕 𝑢𝑠𝑓𝑏𝑢𝑛𝑓𝑜𝑢 𝑒𝑝𝑡𝑓); 𝐷𝑗𝑢=0 = 0; 𝑆 𝑗𝑢=0 = 𝑙𝑗𝑜/𝑙𝑝𝑣𝑢

slide-11
SLIDE 11

A nice answer: PK-PD model

𝑒𝐵𝑓𝑗𝑢 𝑒𝑢

= −𝑙𝑏𝐵𝑓𝑗𝑢

𝑒𝐷𝑗𝑢 𝑒𝑢 = 𝑙𝑏𝐵𝑓𝑗𝑢 − 𝑙𝑓𝐷𝑗𝑢 𝑒𝑆 𝑗𝑢 𝑒𝑢 = 𝑙𝑗𝑜 1 − 𝐽𝑛𝑏𝑦𝐷𝑗𝑢 𝐽𝐷50+𝐷𝑗𝑢 − 𝑙𝑝𝑣𝑢𝑆

𝑗𝑢

10 23/06/2016 A Bayesian PK/PD model for synergy; a case study

PD part: indirect response (turnover) model

Parameters: 𝑙𝑏 ≥ 0: Absorption constant 𝑙𝑓 ≥ 0: Elimination constant 𝑙𝑗𝑜 ≥ 0: Constant for response production 𝑙𝑝𝑣𝑢 ≥ 0: Constant for response loss 0 ≤ 𝐽𝑛𝑏𝑦 ≤ 1: Maximal inhibition attributed to drug 𝛾: Interaction coefficient

PK Part:

  • ne compartment

model with oral absorption

i=1, …, S (subjects) t=0, …, 4 (hours) A𝑓𝑗𝑢=Existing treatment amount 𝐵𝑜𝑗0=Novel treatment dose 𝑆𝑗𝑢=Response: 𝑆𝑗𝑢~𝑂(𝑆 𝑗𝑢, 𝜏2) 𝐷𝑗𝑢=Plasma concentration of the existing treatment (latent!)

𝑓𝛾𝐵𝑓𝑗0𝐵𝑜𝑗0

With: 𝐵𝑓𝑗𝑢=0 = 𝐵𝑓𝑗0 (𝑓𝑦𝑗𝑡𝑢𝑗𝑜𝑕 𝑢𝑠𝑓𝑏𝑢𝑛𝑓𝑜𝑢 𝑒𝑝𝑡𝑓); 𝐷𝑗𝑢=0 = 0; 𝑆 𝑗𝑢=0 = 𝑙𝑗𝑜/𝑙𝑝𝑣𝑢

slide-12
SLIDE 12

A nice answer: PK-PD model

𝑒𝑆 𝑗𝑢 𝑒𝑢 = 𝑙𝑗𝑜 1 − 𝐽𝑛𝑏𝑦𝐷𝑗𝑢 𝐽𝐷50+𝐷𝑗𝑢 − 𝑙𝑝𝑣𝑢𝑆

𝑗𝑢

11 23/06/2016 A Bayesian PK/PD model for synergy; a case study

PD part: indirect response (turnover) model

A𝑓𝑗0=Existing treatment dose 𝐵𝑜𝑗0=Novel treatment dose 𝛾 =Synergy coefficient, expressing the pharmacodynamic drug-drug interaction

𝑓𝛾𝐵𝑓𝑗0𝐵𝑜𝑗0

Observed data Simulated model

Slower

  • nset, larger

decrease

slide-13
SLIDE 13

A nice answer: PK-PD model

𝑒𝐵𝑓𝑗𝑢 𝑒𝑢

= −𝑙𝑏𝐵𝑓𝑗𝑢

𝑒𝐷𝑗𝑢 𝑒𝑢 = 𝑙𝑏𝐵𝑓𝑗𝑢 − 𝑙𝑓𝐷𝑗𝑢 𝑒𝑆 𝑗𝑢 𝑒𝑢 = 𝑙𝑗𝑜 1 − 𝐽𝑛𝑏𝑦𝐷𝑗𝑢 𝐽𝐷50+𝐷𝑗𝑢 − 𝑙𝑝𝑣𝑢𝑆

𝑗𝑢

12 23/06/2016 A Bayesian PK/PD model for synergy; a case study

PD part: indirect response (turnover) model

How to pool data from different studies?

  • Historical data  Frequentist approach (NONMEM)
  • Study 1-11  Bayesian approach (WinBUGS)

PK Part:

  • ne compartment

model with oral absorption

i=1, …, S (subjects) t=0, …, 4 (hours) A𝑓𝑗𝑢=Existing treatment amount 𝐵𝑜𝑗0=Novel treatment dose 𝑆𝑗𝑢=Response: 𝑆𝑗𝑢~𝑂(𝑆 𝑗𝑢, 𝜏2) 𝐷𝑗𝑢=Plasma concentration of the existing treatment (latent!)

𝑓𝛾𝐵𝑓𝑗0𝐵𝑜𝑗0

With: 𝐵𝑓𝑗𝑢=0 = 𝐵𝑓𝑗0 (𝑓𝑦𝑗𝑡𝑢𝑗𝑜𝑕 𝑢𝑠𝑓𝑏𝑢𝑛𝑓𝑜𝑢 𝑒𝑝𝑡𝑓); 𝐷𝑗𝑢=0 = 0; 𝑆 𝑗𝑢=0 = 𝑙𝑗𝑜/𝑙𝑝𝑣𝑢

The results from the previous study are used in order to build the prior distribution for the following study.

slide-14
SLIDE 14

Priors

  • Study 1: distributions centered at the point estimates

resulting from analysis on historical data; variance obtained by doubling the resulting (squared) standard error

  • 𝑙𝑓, 𝑙𝑗𝑜, 𝑙𝑝𝑣𝑢, 𝑆0 ~ LN
  • All precisions ~ Gamma
  • 𝐽𝑛𝑏𝑦 ~ Beta
  • 𝛾 ~ Unif(-10,10)
  • Studies 2-11: distributions having the same mean and

variance as the posteriors resulting from the analysis on previous study

  • 𝑙𝑓, 𝑙𝑗𝑜, 𝑙𝑝𝑣𝑢, 𝑆0, 𝐽𝑛𝑏𝑦 and all precisions ~ same as above
  • 𝛾 ~ N, truncated at (-10,10)

13 23/06/2016 A Bayesian PK/PD model for synergy; a case study

slide-15
SLIDE 15

Different models performed

  • No random effects
  • Random response at baseline
  • Random 𝑙𝑗𝑜
  • Random 𝑙𝑝𝑣𝑢
  • 2 random effects

(among the above parameters)

14 23/06/2016 A Bayesian PK/PD model for synergy; a case study

slide-16
SLIDE 16

Different models performed

  • No random effects
  • Random response at baseline
  • Random 𝑙𝑗𝑜
  • Random 𝑙𝑝𝑣𝑢
  • 2 random effects

(among the above parameters)

15 23/06/2016 A Bayesian PK/PD model for synergy; a case study

No random effect

Trap Trap

slide-17
SLIDE 17

Results

No random effects Random kout

Posterior mean 2.5% CI 97.5% CI Posterior mean 2.5% CI 97.5% CI

log (𝑙𝑓)

  • 0.411
  • 0.535
  • 0.309
  • 0.326
  • 0.378
  • 0.263

log (𝑙𝑝𝑣𝑢) 0.264 0.078 0.460

  • 0.166
  • 0.283
  • 0.040

𝐽𝑛𝑏𝑦 0.261 0.240 0.281 0.292 0.271 0.312 log(𝑆 0) 3.615 3.614 3.617 3.616 3.615 3.617 𝜐 𝑆 2.742 2.525 2.977 2.860 2.663 3.065 𝛾

  • 2.655
  • 3.374
  • 1.919
  • 2.952
  • 3.578
  • 2.486

𝜐 𝑙𝑝𝑣𝑢

  • 1.959

1.476 2.540 DIC 148.230 144.653

16 23/06/2016 A Bayesian PK/PD model for synergy; a case study

slide-18
SLIDE 18
  • No random effects
  • Random kout

17 A Bayesian PK/PD model for synergy; a case study 23/06/2016

Results (e.g. from study 5)

slide-19
SLIDE 19

Future developments

  • Compare Stan (and DIFFMEM?) with WinBUGS
  • Extension to a categorical response
  • Bayesian optimal design of experiments

18 23/06/2016 A Bayesian PK/PD model for synergy; a case study

slide-20
SLIDE 20

Thank you for your attention!

19 23/06/2016 A Bayesian PK/PD model for synergy; a case study