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Modeling of Longitudinal Tumor Size Data in Clinical Oncology Studies of Drugs in Combination N. Frances 1 , L. Claret 2 , F. Schaedeli Stark 3 , R. Bruno 2 , A. Iliadis 1 1 School of Pharmacy, University Méditerranée, Marseille, France;

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slide1

Modeling of Longitudinal Tumor Size Data

in Clinical Oncology Studies of Drugs in Combination

N. Frances1, L. Claret2, F. Schaedeli Stark3, R. Bruno2, A. Iliadis1

1School of Pharmacy, University Méditerranée, Marseille, France;

2Pharsight Corp., Mountain View, USA; 3Hoffman-La Roche, Basel, Switzerland

#1355

« Posterior Predictive Check » on

( at the first visit )

Actual data

(drug combination)

Intra- and inter-patient variability

NONMEM

Posthoc estimated

parameters n=222

Random drawn

parameters n=1000

Reference design

Reference design

Data presentation

  • Retrospective analysis :
    • 2 drugs in metastatic breast cancer
      • Docetaxel (D)
      • Capecitabine (C)
    • PD-data : observed tumor burden
      • sum of the longest diameter of metastatic sites measured (dependent variable in NONMEM)
  • 3 studies [1, 2, 3] :
    • # 14697 : phase II data on C ( )
    • # 15542 : phase II data on C ( )
    • # 14999 : phase III data on C+D ( ) vs. D ( )
  • Data already treated by a different growth model [4, 5]

Objectives

  • Elaborate the best model (parsimonious principle) fitting the longitudinal tumor size data on :
      • Single agent data
      • Combination data
  • Is this model able to describe the contribution of each drug in the combination data ?
  • Can a drug interaction term be estimated ?
  • What is the minimum number of patients in a study to obtain a good enough estimation of the model parameters ?
      • e.g. in a prospective Phase Ib or Phase II study

« Observed » ratio

(from actual data)

ABSTRACT

Introduction : The analysis of tumor size measurements, obtained in clinical studies involving combination chemotherapy, remains an open modeling problem. We used retrospective clinical data in metastatic breast cancer in order to investigate whether the contribution to the anti-tumor effect of each compound in a combination setting can be estimated 1) from single agent data and combination data with or without single agent data, and 2) from datasets with a limited number of patients .

Methods : Data concerning tumor size measurements and treatments characteristics were available for docetaxel (D, n=223), capecitabine (C, n=168) [1, 2] given as single agents and their combination (D+C, n=222) [3]. The developed model is an extension of already presented disturbed growth models [4, 5] and it is based on the following hypotheses: 1) Tumor growth is exponential or Gompertz; 2) K-PD model describes administration protocols; 3) Resistance is materialized by exponential decline of cell-kill rate; 4) Drugs are combined either in a linear, or Emax, or Weibull model involving a drug interaction term. Population analyses were performed using NONMEM Version 6 within a MATLAB environment. The models were validated using posterior predictive checks.

Results : In the developed models, over-parameterization was the most frequent problem. K-PD models involve only one parameter expressing the dynamics of drug amounts in the cell-kill rate formulation. This parameter was obtained for D and C from the single agent studies and was fixed in the analysis using the combination data only. When using the combination data only, the contribution of each drug to the anti-tumor effect was accurately estimated and the estimates were consistent with those obtained using single-agent data. The effect of the 2 drugs was found to be additive with no drug interaction term. Situation #2 is still under investigation.

Conclusion : Using combination data, the tumor size dynamic model parameters were successfully estimated. Further investigations are in progress for assessing the minimum required extent and type of clinical data for evaluating drug combinations in oncology. This model will be part of a modeling framework to simulate expected clinical response of new compounds and to support end-of-phase II decisions and design of phase III studies [6].

Predicted ratio

pdf(ratio)

KL

KEC

KED

R

n0

0.008

0.008

0.008

0.008

100

KL

40

50

60

Patient ID n° 30

Patient ID n° 74

Patient ID n° 85

0.002

0.002

0.002

0.002

80

0

50

35

60

40

50

0.0012

0.0012

0.0012

0.0012

30

40

30

KEC

20

20

25

0.0004

0.0004

0.0004

0.0004

0

0

10

20

30

0

10

20

30

40

0

5

10

15

80

2

2

2

2

30

Patient ID n° 91

120

Patient ID n° 88

Patient ID n° 87

100

KED

70

25

100

0.5

0.5

0.5

0.5

Tumor

20

60

0

size

15

80

50

(mm)

10

0.12

0.12

0.12

0.12

40

60

5

100

R

50

0

5

10

15

20

25

0

10

20

30

40

50

0

10

20

30

40

50

0.04

0.04

0.04

0.04

0

80

120

Patient ID n° 122

Patient ID n° 147

Patient ID n° 153

50

100

  • Choose a model
    • Proliferation Gompertz vs. exponential
    • Resistance term
    • Dynamical dose model
    • Estimation of the initial condition
    • Residual variability
    • Define bounds in the parameters

60

300

300

300

300

100

40

50

n0

100

100

100

100

40

30

80

0

0.002

0.008

0.0004

0.0012

0.5

2

0.04

0.12

100

200

300

20

20

60

10

0

5

10

0

10

20

30

40

0

10

20

30

40

Figure 1. Dispersion plots and histograms

Time (weeks)

Figure 2. Individual fits

  • For 9 patients from the phase III combination study (C+D), observed tumor size data (o) and model predictions vs. time :
    • population ( - - -),
    • individual ( )

Final model

Non usable models

  • Define the KB values in the dose model
    • Studies on single agent data
      • Phase II-C (14697+15542)
      • Phase III-D (14999)
    • Validation on the combination (C+D) data

Remove the variability

added by individual designs

  • Model explanations
  • : “Effective dose” for C and D respectively
  • : Administration protocols for C and D respectively
  • : Tumor size
  • : K-PD elimination constants (already evaluated on single agent data)
  • Estimated parameters :
    • : Proliferation parameter (max tumor size : mm, fixed)
    • : Resistance parameter, common to both drugs
    • : Constant cell kill rate (efficacy parameter), distinct for each drug
    • : Initial tumor size
  • Modeling tests : Phase III-(C+D) 14999
    • Covariance matrix of estimates
    • Parameter variability (Omega)
    • Residual error (Sigma)

Numerical results

Validated models

Fixed effects :

Random effects are log-normal distributed :

Residual error is proportional :

Objective function : (>100 models tested)

Figure 3a. Flow chart for Posterior Predictive Check

5

5

Design ID n° 172

Design ID n° 487

4

4

3

3

pdf

2

2

  • Comparison between retained models
    • Objective function
    • Residual error

1

1

0

0

0

0.5

1

1.5

0

0.5

1

1.5

5

5

Design ID n° 425

Design ID n° 333

4

4

KL

KEC

n0

300

800

0.08

3

3

700

pdf

250

2

2

600

0.07

200

500

1

1

Final model

150

400

0

0

0.06

0

0.5

1

1.5

0

0.5

1

1.5

300

100

Tumor size ratio

Tumor size ratio

200

50

Figure 3b. Posterior Predictive Check

100

Probability density functions of the ratio :

0.05

0

0

0

0.005

0.01

0

0.001

0.002

0.003

0.04

KED

R

  • For 4 typical designs, predicted ratio from :
    • posthoc estimated parameters ( , ) and
    • randomly drawn parameters ( , )

5

25

0.03

4

20

3

15

0.02

  • Ultimate validation
    • Dispersion plots and histograms (Figure 1)
    • Individual fits (Figure 2)
    • “Posterior Predictive Check” (Figure 3a, b)
    • Minimum number of patients (Figure 4)

2

10

0.01

1

5

  • Conclusion
  • The model is built from Phase III data :
    • two drugs (D+C) in combination,
    • resistance parameter common to both drugs and acting by increasing proliferation term (selection of resistant cells by the treatment),
    • interaction term not estimated (assumes additive effects).
  • This model can be used to predict therapy efficacy in a future clinical trial [6] :
    • using Bayesian approach,
    • a minimum number of patient seems to be necessary, but small sample sizes typical to those in early clinical studies (e.g. 50 patients) may be enough,
    • instead of K-PD, a PK-PD model would supply consistent information.

0

0

0

0

0.2

0.4

0

0.05

0.1

0.15

40

60

80

Figure 4. Minimum number of patients

(Probability density functions of model parameters)

  • Samples were obtained from 100 random permutations of the 222 patients data in the phase III combination study.
    • 50-patients tailed samples () and
    • 70-patients tailed samples ().
  • Covariance matrix obtained : 26/100 with 50 patients and 38/100 with 70 patients.

Flow chart for model development

References:

[1] Blum JL, Jones SE, Buzdar AU, et al: Multicenter Phase II Study of Capecitabine in Paclitaxel-Refractory Metastatic Breast Cancer. J. Clin. Oncol. 17: 485-493, 1999.

[2] Blum JL, Dieras V, Mucci Lo Russo P, et al: Multicenter, phase II study of capecitabine in taxane pretreated metastatic breast carcinoma patients, Cancer 92:1759-1768, 2001.

[3] O’Shaughnessy J, Miles D, Vukelja S et al. Superior survival with capecitabine plus docetaxel combination therapy in anthracycline-pretreated patients with advanced breast cancer: Phase III trial results. J. Clin. Oncol. 12: 2812-2823, 2002.

[4] Iliadis A, Barbolosi D: Optimizing drug regimens in cancer chemotherapy by an efficacy-toxicity mathematical model. Comput. Biomed. Res. 33:211-226, 2000.

[5] Claret L, Girard P, Zuideveld KP, et al: A longitudinal model for tumor size measurements in clinical oncology studies. PAGE 15 (abstract 1004), 2006a [www.page-meeting.org/?abstract=1004].

[6] Claret L, Girard P, O'Shaughnessy J et al: Model-based predictions of expected anti-tumor response and survival in phase III studies based on phase II data of an investigational agent. Proc. Am. Soc. Clin. Oncol, 24, 307s (abstract 6025), 2006b.