Population modeling of tumor growth curves, the reduced Gompertz model and prediction of the age of a tumor

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HAL Id: hal-02424578

https://hal.archives-ouvertes.fr/hal-02424578

Submitted on 27 Dec 2019

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Population modeling of tumor growth curves, the reduced Gompertz model and prediction of the age of a

tumor

Cristina Vaghi, Anne Rodallec, Raphaelle Fanciullino, Joseph Ciccolini, Jonathan Mochel, Michalis Mastri, John Ebos, Clair Poignard, Sébastien

Benzekry

To cite this version:

Cristina Vaghi, Anne Rodallec, Raphaelle Fanciullino, Joseph Ciccolini, Jonathan Mochel, et al..

Population modeling of tumor growth curves, the reduced Gompertz model and prediction of the age of a tumor. PAGE 2019 Twenty-eighth meeting, Jun 2019, Stockholm, Sweden. �hal-02424578�

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BACKGROUND

Population modeling of tumor growth curves, the reduced Gompertz model and prediction of the age of a tumor

1

MONC team, Inria Sud-Ouest Bordeaux, France

2

Institut de Mathématiques de Bordeaux, France

3

SMARTc, Center for Research on Cancer of Marseille, France

4

Iowa State University, Department of Biomedical Sciences, Ames, USA

5

Roswell Park Comprehensive Cancer Center, Buffalo, NY, USA

C. Vaghi

^1,2,*

, A. Rodallec

³

, R. Fanciullino

³

, J. Ciccolini

³

, J. Mochel

⁴

, M. Mastri

⁵

, J. ML Ebos

⁵

, C. Poignard

^1,2

, S. Benzekry

^1,2

OBJECTIVES

•

Test the descriptive power of different tumor growth models within a population.

•

Study the correlation between the parameters of the Gompertz model within a population and define a novel, simplified model: the reduced Gompertz model.

•

Use the estimated population parameters to perform individual predictions of tumor initiation using Bayesian inference.

MATERIAL AND METHODS

RESULTS

Preclinical data

The experimental data comprised three data sets:

•

Breast data measured by volume (66 animals).

•

Breast data measured by fluorescence (8 animals)

•

Lung tumor measured by volume (20 animals)

Tumor growth models Prediction of individual tumor age

The tumor age of the individual i was determined taking into account only the last three measurements. It was found according to , where was the volume of injected cells and was estimated by:

•

Likelihood maximization (LM)

•

Bayesian inference : the population parameter distribution was considered as prior information. k-fold cross validation was performed (leave-one-out strategy).

Nonlinear mixed effects modeling

yⁱ_j

!"#$

observation of individual i

at time tⁱ_j

= f(tⁱ_j; θⁱ)

! "# $

structural model

+ eⁱ_j

!"#$

error model

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ϵⁱ_j ⇠ N (0, 1)

<latexit sha1_base64="kEB0M2PVuqXVUkJV1uMPWkKiKfc=">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</latexit><latexit sha1_base64="kEB0M2PVuqXVUkJV1uMPWkKiKfc=">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</latexit><latexit sha1_base64="kEB0M2PVuqXVUkJV1uMPWkKiKfc=">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</latexit><latexit sha1_base64="kEB0M2PVuqXVUkJV1uMPWkKiKfc=">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</latexit>

θⁱ = μexp(ηⁱ)

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eⁱ_j = (σ₁ + σ₂f(tⁱ_j; θⁱ))ϵⁱ_j

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f ⇣

aˆⁱ, _θˆⁱ⌘

= V_inj

<latexit sha1_base64="VOLdE4qYcgwyW/n2yS6et9wa3gI=">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</latexit><latexit sha1_base64="VOLdE4qYcgwyW/n2yS6et9wa3gI=">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</latexit><latexit sha1_base64="VOLdE4qYcgwyW/n2yS6et9wa3gI=">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</latexit><latexit sha1_base64="VOLdE4qYcgwyW/n2yS6et9wa3gI=">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</latexit>

f ⇣

aˆⁱ, _θˆⁱ⌘

= V_inj

θˆ<latexit sha1_base64="Y7Y8mESzb31Ksl8YM7rF6I9irrg=">AAACjHicbVFbSxtBFJ5sL17a2tg+SmFpKFiQsCtCBRHEFvWhDwrGCNk0nJ09SYbMziwzZ8Uw7JO/pq/tr/HfOBtDMbEHhvn4vnM/aSGFpSi6bwQvXr56vbK6tv7m7buN983ND1dWl4Zjh2upzXUKFqVQ2CFBEq8Lg5CnErvp5Hutd2/QWKHVJU0L7OcwUmIoOJCnBs1PyRjIJamWmZ3m/nMJjZGgqn45UQ2aragdzSx8DuI5aLG5nQ82G90k07zMURGXYG0vjgrqOzAkuMRqPSktFsAnMMKehwpytH03m6MKv3gmC4fa+KconLFPIxzktu7Se+ZAY7us1eT/tF5Jw/2+E6ooCRV/LDQsZUg6rJcSZsIgJzn1ALgRvteQj8EAJ7+6hSqz3AXyhUncbakE1xkusZJuyYAnLVIOQtVTuTOUN+gr/ON9wlrY/iFGguzOT38ZtXNqECdfnzj7U8TLi38OrnbbcdSOL/ZaR8fzo6yyLfaZbbOYfWNH7Iydsw7j7I79Zn/Y32Aj2AsOgsNH16Axj/nIFiw4eQDJnMwL</latexit><latexit sha1_base64="Y7Y8mESzb31Ksl8YM7rF6I9irrg=">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</latexit><latexit sha1_base64="Y7Y8mESzb31Ksl8YM7rF6I9irrg=">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</latexit><latexit sha1_base64="Y7Y8mESzb31Ksl8YM7rF6I9irrg=">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</latexit> ⁱ

f ⇣

aˆⁱ, _θˆⁱ⌘

= V_inj

fixed effects, random effects combined error model

θ<latexit sha1_base64="fXsyKpgZwVvRADSS7tiPsSI6Yp8=">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</latexit><latexit sha1_base64="fXsyKpgZwVvRADSS7tiPsSI6Yp8=">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</latexit><latexit sha1_base64="fXsyKpgZwVvRADSS7tiPsSI6Yp8=">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</latexit><latexit sha1_base64="fXsyKpgZwVvRADSS7tiPsSI6Yp8=">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</latexit> ⁱ = μexp(ηⁱ)

ηⁱ ⇠ N (0, ω)

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⎧⎨

⎩

dV

dt = αV V(t_I) = V_I

⎧⎨

⎩

dV

dt = αV

$

1 −

$V K

%%

V(t_I) = V_I

⎧⎨

⎩

dV

dt =

$

α − βlog

$ V V_inj

%%

V V(t_I) = V_I

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Exponential Logistic Gompertz

Breast data measured by volume.

1

2

3

CONCLUSIONS AND PERSPECTIVES

Population analysis

•

The optimal fits of the Exponential and Logistic models were unable to give appropriate description of the data

•

The Gompertz model demonstrated excellent goodness-of-fit in all the experimental systems that we investigated

•

Correlation between the individual parameters of the Gompertz model was observed in our analysis

Prediction of individual tumor age

•

The Gompertz model described well tumor growth kinetics.

•

The combination of nonlinear mixed effects modelling and Bayesian inference allowed to have reliable predictions of individual tumor age.

•

The reduced Gompertz model fitted well the data and improved predictions in terms of accuracy and precision.

•

The method proposed herein remains to be extended to clinical data. Personalized estimations of the age of a given patient’s tumor would yield important epidemiological

insights and could also be informative in routine clinical practice.

•

Thanks to its simplicity, the reduced Gompertz model showed superior predictive power.

•

Drastic improvements were observed when leveraging population priors using Bayesian inference, both in terms of accuracy and precision.

The reduced Gompertz model

We considered , where could be a characteristic constant of tumor growth within a given animal model.

Fixed effects

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Two examples of individual tumor age predictions (breast, volume).

Accuracy and precision of methods for prediction of the age of experimental tumors of the three data sets.

Reported are the means and standard errors (in parenthesis)

VPCs and scatterplot of the individual weighted residuals (IWRES) of the diﬀerent tumor growth models (breast, volume).

Correlation of the Gompertz parameters

References

1. Benzekry S. et al., Classical Mathematical Models for Description and Prediction of Experimental Tumor Growth. PLoS Comput Biol. (2014)

2. Monolix Version 2018R2. Lixoft SAS (2018)

3. Carpenter B. et al., Stan: A Probabilistic Programming Language. J Stat Softw. (2017)

Animal 1

•

Few studies have modeled tumor growth kinetics with a population approach and across tumor types.

•

The Gompertz model is a widely accepted model of tumor growth. Several studies have reported a strong correlation between the two parameters of the model.

•

Prediction of the time from cancer initiation would have important clinical implications, such as the determination of invisible metastasis at diagnosis.

* cristina.vaghi@inria.fr

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⎧⎨

⎩

dV dt =

$

kβⁱ − βⁱlog

$ V V_inj

%%

V V(t_I) = V_I

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A

B

C

D

Figure 1. Population analysis of experimental tumor growth kinetics. A) Visual predictive checks assess goodness-of-fit for both structural dynamics and inter-animal variability by reporting model-predicted percentiles (together with confidence prediction intervals (P.I) in comparison to empirical ones. B) Prediction distributions.

C) Individual weighted residuals (IWRES) with respect to time. D) Observations vs predictions Left: Exponential, Center: Logistic, Right: Gompertz models.

8

A

B

C

D

Figure 1. Population analysis of experimental tumor growth kinetics. A) Visual predictive checks assess goodness-of-fit for both structural dynamics and inter-animal variability by reporting model-predicted percentiles (together with confidence prediction intervals (P.I) in comparison to empirical ones. B) Prediction distributions.

C) Individual weighted residuals (IWRES) with respect to time. D) Observations vs predictions Left: Exponential, Center: Logistic, Right: Gompertz models.

8

A B

C D

E

Figure 3. Correlation of the Gompertz parameters and diagnostic plots of the reduced Gompertz model from population analysis. Correlation between the individual parameters of the Gompertz model (A) and results of the population analysis of the reduced Gompertz model: visual predictive check (B), examples of individual fits (C) and scatter plots of the residuals (D).

12

A B

C D

E

12

A B

C D

E

12 VPC and scatterplot of the individual weighted residuals (IWRES) of the reduced Gompertz model (breast, volume).

Cell line Model Method Accuracy (%) Precision (days)

Breast, volume Reduced Gompertz Bayesian 12.1 (1.02) 15.2 (0.503)

Reduced Gompertz LM 74.1 (11.6) 186 (52.8)

Gompertz Bayesian 19.6 (1.77) 40.1 (1.94)

Gompertz LM 205 (55.4) -

Lung, volume Reduced Gompertz Bayesian 9.4 (1.57) 7.34 (0.33)

Reduced Gompertz LM 66.1 (31) 81.6 (71.7)

Gompertz LM 175 (69.6) -

Breast, ﬂuorescence Reduced Gompertz Bayesian 12.3 (2.9) 23.6 (5.15)

Reduced Gompertz LM 91.7 (21.1) 368 (223)

Gompertz LM 236 (150) -

Table 1: Accuracy and precision of methods for prediction of the age of experimental tumors of the three cell lines.

Accuracy was deﬁned as the absolute value of the relative error (in percent). Precision was deﬁned as the width of the 95% prediction interval (in days). Reported are the means and standard errors (in parenthesis). LM = likelihood maximization

1

A B C

Gompertz (LM)

Reduced Gompertz

(LM)

Gompertz (Bayesian inference)

Reduced Gompertz (Bayesian inference)

Figure 4. Backward predictions computed with likelihood maximization and with Bayesian inference. Three examples of backward predictions of individuals A, B and C computed with likelihood maximization (LM) and Bayesian inference: Gompertz model with likelihood maximization (first row); reduced Gompertz with likelihood maximization (second row); Gompertz with Bayesian inference (third row) and reduced Gompertz with Bayesian inference (fourth row). Only the last three points are considered to estimate the parameters. The grey area is the 90% prediction interval (P.I) and the dotted blue line is the median of the posterior predictive distribution. The red line is the predicted initiation time and the black vertical line the actual initiation time.

13

Animal 2