PKPD model of multiple follicular development during controlled

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

PK-PD model of multiple follicular development during controlled ovarian stimulation application of Markovian elements PAGE meeting 2008 Anthe Zandvliet, Anton de Haan, Pieta IJzerman-Boon, Rik de Greef, Thomas Kerbusch

Controlled ovarian stimulation Diagnosis • Subfertility – reduced chance of conception Treatment • Gonadotropins

Controlled ovarian stimulation Diagnosis • Subfertility – reduced chance of conception Treatment • Gonadotropins to induce multiple follicular development – Recombinant FSH – Corifollitropin alfa PAGE Meeting – Stuck in modelling 20 -Jun-2008 2

Controlled ovarian stimulation Clinical trials corifollitropin alfa • Phase I, III • n =

Controlled ovarian stimulation Clinical trials corifollitropin alfa • Phase I, III • n = 495 Pharmacokinetics • 3 compartment model • Empirical Bayes estimates used in PK-PD model PAGE Meeting – Stuck in modelling 20 -Jun-2008 3

Ultrasound scan measurements • • • Count data Categorical ordinal Repeated measurements Dependent measurements

Ultrasound scan measurements • • • Count data Categorical ordinal Repeated measurements Dependent measurements Follicles not individually tracked Table. Total follicle count (left and right ovary) of a representative subject. 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 PAGE Meeting – Stuck in modelling 20 -Jun-2008 4

Transit compartment model ≤ 1 mm 16 mm 15 mm 14 mm 17+ mm

Transit compartment model ≤ 1 mm 16 mm 15 mm 14 mm 17+ mm 2 mm 3 mm 13 mm k tr: follicular growth k out: follicular decline 4 mm 12 mm 5 mm 11 mm 6 mm 7 mm 8 mm 9 mm 10 mm PAGE Meeting – Stuck in modelling 20 -Jun-2008 5

Poisson model ≤ 1 mm 2 mm 16 mm 15 mm 14 mm 17+mm

Poisson model ≤ 1 mm 2 mm 16 mm 15 mm 14 mm 17+mm 3 mm = 1. 3 13 mm 4 mm 12 mm 5 mm 6 mm 7 mm 8 mm 9 mm 11 mm 10 mm PAGE Meeting – Stuck in modelling 20 -Jun-2008 6

Multinomial model P P ≤ 1 mm P P 2 mm P 3 mm

Multinomial model P P ≤ 1 mm P P 2 mm P 3 mm P 4 mm 16 mm P 15 mm P 14 mm 17+mm P =P ≤ 1 mm +P 2 mm +…+P 16 mm +P 17 mm +P out =1 n =50 P 13 mm P 12 mm P 5 mm P 6 mm P 7 mm P 8 mm P 9 mm P P 11 mm 10 mm PAGE Meeting – Stuck in modelling 20 -Jun-2008 7

Multinomial model P P ≤ 1 mm P P 2 mm P 3 mm

Multinomial model P P ≤ 1 mm P P 2 mm P 3 mm P 4 mm P 15 mm P 14 mm 17+mm P likelihood P (n 11 -14 mm = k 1 , n 15 -16 mm = k 2 , n 17+ mm = k 3) = 13 mm P 11 -14 mmk 1 *P 15 -16 mmk 2 *P 17+ mmk 3 *Pother(50 - k 1 -k 2 -k 3) * P 5 mm 16 mm P 6 mm 50! k 1!* k 2!* k 3!*(50 - k 1 -k 2 -k 3)! P 7 mm P 8 mm P 9 mm P 12 mm P P 11 mm 10 mm PAGE Meeting – Stuck in modelling 20 -Jun-2008 8

Follicles 11 -14 mm 100 Relative frequency (%) 80 observed model predicted 60 Day

Follicles 11 -14 mm 100 Relative frequency (%) 80 observed model predicted 60 Day 3 40 20 0 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 PAGE Meeting – Stuck in modelling 20 -Jun-2008 9

Follicles 11 -14 mm 100 Relative frequency (%) 80 observed model predicted 60 Day

Follicles 11 -14 mm 100 Relative frequency (%) 80 observed model predicted 60 Day 5 40 20 0 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 PAGE Meeting – Stuck in modelling 20 -Jun-2008 10

Follicles 11 -14 mm 100 Relative frequency (%) 80 observed model predicted 60 Day

Follicles 11 -14 mm 100 Relative frequency (%) 80 observed model predicted 60 Day 8 40 20 0 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 PAGE Meeting – Stuck in modelling 20 -Jun-2008 11

Follicles 15 -16 mm 100 Relative frequency (%) 80 observed model predicted 60 Day

Follicles 15 -16 mm 100 Relative frequency (%) 80 observed model predicted 60 Day 3 40 20 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Number of follicles 15 -16 mm PAGE Meeting – Stuck in modelling 20 -Jun-2008 12

Follicles 15 -16 mm 100 Relative frequency (%) 80 observed model predicted 60 Day

Follicles 15 -16 mm 100 Relative frequency (%) 80 observed model predicted 60 Day 5 40 20 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Number of follicles 15 -16 mm PAGE Meeting – Stuck in modelling 20 -Jun-2008 13

Follicles 15 -16 mm 100 Relative frequency (%) 80 observed model predicted 60 Day

Follicles 15 -16 mm 100 Relative frequency (%) 80 observed model predicted 60 Day 8 40 20 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Number of follicles 15 -16 mm PAGE Meeting – Stuck in modelling 20 -Jun-2008 14

Follicles 17+ mm 100 Relative frequency (%) 80 observed model predicted 60 Day 3

Follicles 17+ mm 100 Relative frequency (%) 80 observed model predicted 60 Day 3 40 20 0 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 PAGE Meeting – Stuck in modelling 20 -Jun-2008 15

Follicles 17+ mm 100 Relative frequency (%) 80 observed model predicted 60 Day 5

Follicles 17+ mm 100 Relative frequency (%) 80 observed model predicted 60 Day 5 40 20 0 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 PAGE Meeting – Stuck in modelling 20 -Jun-2008 16

Follicles 17+ mm 100 Relative frequency (%) 80 observed model predicted 60 Day 8

Follicles 17+ mm 100 Relative frequency (%) 80 observed model predicted 60 Day 8 40 20 0 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 PAGE Meeting – Stuck in modelling 20 -Jun-2008 17

Follicles 11 -14 mm (representative subject) Number of follicles 11 -14 mm 18 Observed

Follicles 11 -14 mm (representative subject) Number of follicles 11 -14 mm 18 Observed and predicted follicle counts. 16 P 75 14 P 50 12 10 P 25 8 6 4 2 0 0 1 2 3 4 Time (days) 5 PAGE Meeting – Stuck in modelling 6 20 -Jun-2008 7 18

Simulation without Markovian features - Independent measurements. - Simulated values highly variable. - Simulated

Simulation without Markovian features - Independent measurements. - Simulated values highly variable. - Simulated profile physiologically not plausible. PAGE Meeting – Stuck in modelling 20 -Jun-2008 19

Physiologically plausible profile Number of follicles 11 -14 mm 18 16 14 12 10

Physiologically plausible profile Number of follicles 11 -14 mm 18 16 14 12 10 8 6 4 2 0 0 1 2 3 4 Time (days) 5 PAGE Meeting – Stuck in modelling 6 20 -Jun-2008 7 20

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

Markovian features • Model should ‘remember’ the size of follicles at previous time point. • Attempts to implement Markovian elements in NONMEM: unsuccessful. PAGE Meeting – Stuck in modelling 20 -Jun-2008 21

Markovian features: implementation in SAS • Empirical Bayes estimation of PK-PD parameters in NONMEM

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 – Pgrow – Punchanged = 1 - exp(-0. 1*kout) = 1 - exp(-0. 1*ktr) = 1 – Pdecline – Pgrow • Markov simulation for individual follicles in SAS – 50 growth courses of individual follicles are simulated for each subject PAGE Meeting – Stuck in modelling 20 -Jun-2008 22

Simulation with Markovian features 3 examples of simulated profiles in SAS PAGE Meeting –

Simulation with Markovian features 3 examples of simulated profiles in SAS PAGE Meeting – Stuck in modelling 20 -Jun-2008 23

Conclusion • A transit compartment multinomial Markov model seems suitable to describe follicular growth

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. PAGE Meeting – Stuck in modelling 20 -Jun-2008 24

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

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? PAGE Meeting – Stuck in modelling 20 -Jun-2008 25

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