pk-pd model of multiple follicular development during controlled ovarian stimulation

PAGE meeting 2008

Anthe Zandvliet, Anton de Haan, Pieta IJzerman-Boon, Rik de Greef, Thomas Kerbusch

PK-PD model of multiple follicular development during controlled ovarian stimulation

application of Markovian elements

20-Jun-2008PAGE Meeting – Stuck in modelling 2

Controlled ovarian stimulation

AuthorFunctionDate

12-Mar-2008 |M&S corifollitropin alfa - ACoP | 2


Diagnosis:

• Subfertility – reduced chance of conception– Tubal factor, pelvic pathology, endometriosis, unknown

– Male factor

Treatment:

• Gonadotropins (e.g. recFSH) to induce multifollicular growth for– In vitro fertilization (IVF)

– Intra-cytoplasmic sperm injection (ICSI)

hCGrecFSH

Multifollicular growth Oocyte retrieval

IVF/ICSI

Embryo transfer

Diagnosis • Subfertility – reduced chance of conception

Treatment• Gonadotropins to induce multiple follicular development

– Recombinant FSH– Corifollitropin alfa



AuthorFunctionDate

12-Mar-2008 |M&S corifollitropin alfa - ACoP | 2


Diagnosis:

• Subfertility – reduced chance of conception– Tubal factor, pelvic pathology, endometriosis, unknown

– Male factor

Treatment:

• Gonadotropins (e.g. recFSH) to induce multifollicular growth for– In vitro fertilization (IVF)

– Intra-cytoplasmic sperm injection (ICSI)

hCGrecFSH

Multifollicular growth Oocyte retrieval

IVF/ICSI

Embryo transfer

Clinical trials corifollitropin alfa

• Phase I, II, III• n = 495

Pharmacokinetics

• 3 compartment model• Empirical Bayes estimates used in PK-PD model


Ultrasound scan measurements

2-4 mm 5-7 mm 8-10 mm 11-14 mm 15-16 mm 17+ mm

Day 1 0 3 2 0 0 0

Day 3 - - - 1 0 0

Day 5 - - - 6 1 0

Day 6 - - - 5 2 0

Day 7 - - - 1 4 3

• Count data• Categorical ordinal• Repeated measurements• Dependent measurements• Follicles not individually tracked

Table. Total follicle count (left and right ovary) of a representative subject.


Transit compartment model

≤1 mm

2 mm

3 mm

4 mm

5 mm

6 mm

7 mm 8 mm9 mm

10 mm

11 mm

12 mm

13 mm

14 mm

15 mm16 mm

17+ mm

k out: follicular decline

k tr: follicular growthCorifollitropin alfa concentration

k tr (follicular growth)

Corifollitropin alfa concentration

k out (follicular decline)


Poisson model

2mm

3mm

4mm

5mm

6mm

7mm

8mm

9mm

10mm

11mm

12mm

13mm

14mm

15mm

16mm

17+mm

≤1mm

= 1.3

1 2 3 4 5 6 7

P


Multinomial model

P2mm

P3mm

P4mm

P5mm

P6mm P

7mmP

8mm

P9mm

P10mm

P11mm

P12mm

P13mm

P14mm

P15mm

P16mm

P17+mm

P≤1mm

P =P ≤1mm +P 2mm +…+P 16mm +P 17mm +P out =1

n =50


Multinomial model

P2mm

P3mm

P4mm

P5mm

P6mm P

7mmP

8mm

P9mm

P10mm

P11mm

P12mm

P13mm

P14mm

P15mm

P16mm

P17+mm

P≤1mm

likelihood

P (n11-14mm = k1 , n15-16 mm = k2 , n17+ mm = k3) =

50! k1!* k2!* k3!*(50- k1-k2-k3)!

P11-14 mmk1 *P15-16 mm

k2 *P17+ mmk3 *Pother

(50- k1-k2-k3) *


Follicles 11-14 mm

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Number of follicles 11-14 mm

Rel

ativ

e fr

equ

ency

(%

) observed

model predicted

Day 3


0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


Rel

ativ

e fr

equ

ency

(%

) observed

model predicted

Day 5

Follicles 11-14 mm


0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


Rel

ativ

e fr

equ

ency

(%

) observed

model predicted

Day 8

Follicles 11-14 mm


Follicles 15-16 mm

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


Rel

ativ

e fr

equ

ency

(%

) observed

model predicted

Day 3


0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


Rel

ativ

e fr

equ

ency

(%

) observed

model predicted

Day 5

Follicles 15-16 mm


0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


Rel

ativ

e fr

equ

ency

(%

) observed

model predicted

Day 8

Follicles 15-16 mm


Follicles 17+ mm

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Number of follicles 17+ mm

Rel

ativ

e fr

equ

ency

(%

) observed

model predicted

Day 3


0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


Rel

ativ

e fr

equ

ency

(%

) observed

model predicted

Day 5

Follicles 17+ mm


0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


Rel

ativ

e fr

equ

ency

(%

) observed

model predicted

Day 8

Follicles 17+ mm


Follicles 11-14 mm (representative subject)

P25

P75

P50

0

2

4

6

8

10

12

14

16

18

0 1 2 3 4 5 6 7Time (days)

Nu

mb

er o

f fo

llic

les

11-1

4 m

m Observed and predicted follicle counts.


02468

1012141618

0 1 2 3 4 5 6 7

Time (days)

Num

ber

of f

ollic

les

11-1

4 m

m

- Independent measurements.- Simulated values highly variable.- Simulated profile physiologically not plausible.

Simulation without Markovian features


Physiologically plausible profile

0

2

4

6

8

10

12

14

16

18

0 1 2 3 4 5 6 7Time (days)

Nu

mb

er o

f fo

llic

les

11-1

4 m

m


Markovian features

• Model should ‘remember’ the size of follicles at previous time point.

• Attempts to implement Markovian elements in NONMEM: unsuccessful.


Markovian features: implementation in SAS

• Empirical Bayes estimation of PK-PD parameters in NONMEM

• Calculation of transition rates for each 0.1-hour interval:

– Pdecline = 1- exp(-0.1*kout)

– Pgrow = 1- exp(-0.1*ktr)

– Punchanged = 1 – Pdecline – Pgrow

• Markov simulation for individual follicles in SAS

– 50 growth courses of individual follicles are simulated for each subject


0

5

10

15

20

0 1 2 3 4 5 6 7

Time (days)

Num

ber

of f

ollic

les

11-1

4 m

m

0

5

10

15

20

0 1 2 3 4 5 6 7

Time (days)

Num

ber

of f

ollic

les

11-1

4 m

m

0

5

10

15

20

0 1 2 3 4 5 6 7

Time (days)

Num

ber

of f

ollic

les

11-1

4 m

m

Simulation with Markovian features

3 examples of simulated profiles in SAS


Conclusion

• A transit compartment multinomial Markov model seems suitable

to describe follicular growth during treatment with corifollitropin alfa.

• The transit compartment multinomial model required ordinary differential equation calculation in NONMEM.

• Markovian features were implemented for simulation purposes in SAS.


Discussion

• How to apply Markovian elements in NONMEM?– Poisson model – multinomial model

• Models for count data with less dispersion?

• Is the work-around acceptable?– Estimation in NONMEM (empirical Bayes estimates of PK and PD parameters)– Simulation in SAS (Markov simulation of 50 follicles for each subject)

• Other examples of repeated dependent categorical count data?

• Diagnostic plots? Diagnostic methods?

pk-pd model of multiple follicular development during controlled ovarian stimulation

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