HAL Id: hal-02424592
https://hal.archives-ouvertes.fr/hal-02424592
Submitted on 27 Dec 2019
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Combining Population Modeling and Bayesian Inference for Tumor Growth Prediction
Cristina Vaghi
To cite this version:
Cristina Vaghi. Combining Population Modeling and Bayesian Inference for Tumor Growth Prediction.
SMB 2019 annual meeting, Jul 2019, Montreal, Canada. �hal-02424592�
Combining Population Modeling and Bayesian Inference for Tumor Growth Prediction
Cristina Vaghi
Ph.D. Candidate at Inria Sud-Ouest-Bordeaux, MONC team
SMB 2019
Montreal, July 22nd 2019
Motivations
•
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 [1].•
Prediction of the time from cancer initiation would have important clinical implications, such as the determination of invisible metastasis at diagnosis.[1] Brunton GF, Wheldon TE. (1978) Characteristic Species Dependent Growth Patterns of Mammalian Neoplasms. Cell Tissue Kinet. !2
Objectives
!3
•
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.Outline
!4
•
Material and methods‣ Preclinical data
‣ Tumor growth models
‣ Nonlinear mixed effects modeling: population approach
‣ Bayesian inference
•
Results‣ Population analysis
‣ The reduced Gompertz model
‣ Backward prediction of the initiation time of the tumor
•
ConclusionsMaterial and methods
!5
Data
•
Preclinical data•
Longitudinal measurements of the primary tumor•
Three data sets:‣ Breast cancer (volume data): 66 individuals (J. Ebos et al., Roswell Cancer Park)
‣ Breast cancer (fluorescence data): 8 individuals (A. Rodallec et al., SMARTc team)
‣ Lung cancer (volume data): 20 individuals (S. Benzekry, C. Lamont, Center of Cancer and System Biology)
Material and methods
Tumor growth models
!6
Exponential Logistic Gompertz
dV
dt = αi − βi log (
V
Vinj ) V V(tI) = VI
dV
dt = αiV (1− V
Ki ) V(tI) = VI
dV
dt = αiV V(tI) = VI
Benzekry S. et al., Classical Mathematical Models for Description and Prediction of Experimental Tumor Growth. PLoS Comput Biol. (2014)
Structural model
Model for the individual parameters
Error model
yji = f(tij; ✓i) + eij
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yji = f(tij; ✓i) + eij
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✓i = µ (⌘i)
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⌘i ⇠ N (0, ⌦)
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eij = ( 1 + 2f(tij; ✓i))"ij
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of 188 observations.
2.2 Tumor growth models.
We denote by t0 and V0 the initial condition of the equation. At time of injection (tinj = 0), we assumed that all animal tumor volumes within a group have the same
volume Vinj (taken to be equal to the number of injected converted in the appropriate unit) and denote by ↵ the specific growth rate (↵ = V1 dVdt ) at this time and volume.
We considered the Exponential, Logistic and Gompertz models [10]. The first two are respectively defined by:
V (t) = V0 exp(↵(t t0)) and V (t) = V0K
V0 + (K V0)e ↵(t t0) . (2) In the logistic equation, K is a carrying capacity parameter.
The Gompertz model is characterized by an exponential decrease of the specific growth rate with rate . The differential form thus writes:
8>
<
>: dV
dt =
✓
↵ log
✓ V Vinj
◆◆ V, V (t0) = V0.
(3)
Note here that the injected volume Vinj appears in the differential equation defining V . This is natural from our assumption that ↵ is the specific growth rate at V = Vinj.
2.3 Population approach.
Let N be the total number of subjects within the population and Y i = {y1i, ..., yni i} the vector of longitudinal measurements of the animal i, where yji is the observation of subject i at time tij for i = 1, ..., N and j = 1, ..., ni (ni is the total number of
measurements of individual i). We assumed the following statistical model
yji = f(tij; ✓i) + eij, j = 1, ..., ni, i = 1, ..., N, (4) where f(tij; ✓i) is the evaluation of the tumor growth model at time tij, ✓i 2 Rp is the vector of the parameters relative to the individual i and eij the residual error model, to be defined later. Parameter set ✓i depends on the fixed effects µ, identical within the population, and on the random effects ⌘i, that are specific for each animal. They follow a normal distribution ⌘i ⇠ N(0, !) with mean zero and variance matrix !. The
relation that identifies the individual parameter ✓i is given by ✓i = µ exp(⌘i).
We considered a combined residual error model eij, defined as
eij = 1 + 2f(tij; ✓i)"ij, where "ij ⇠ N(0, 1) are the residual errors and = [ 1, 2] is the vector with the residual error model parameters.
In order to compute the population parameters, we maximized a population
likelihood, obtained by pooling together all the data. Usually, this likelihood cannot be computed explicitely for nonlinear mixed-effect models. The optimization procedure can be implemented using the stochastic approximation expectation minimization algorithm (SAEM) [11], implemented in Monolix.
The initial condition of the equations defined in the previous section was considered at the time of injection, i.e. t0 = tinj and V0 = Vinj.
From now on we denote by = {µ, !, } the set of the population parameters
containing the fixed effects µ and the random effects ! of the parameters and the error model parameters .
February 12, 2019 3/14
Material and methods
Nonlinear mixed effects modeling
!7
•
Statistical framework to study longitudinal data•
The goal is to understand the intra-subject process of tumor growth and its variability across the individualsLavielle, M. (2014). Mixed effects models for the population approach: models, tasks, methods and tools. Chapman & Hall/CRC biostatistics series. Taylor & Francis, Boca Raton.
Monolix Version 2018R2. Lixoft SAS (2018)
Exponential Logistic Gompertz
A
B
C
D
1
Material and methods
p(✓j|yj ; )
| {z }
posterior distribution
= p(✓j; )
| {z }
prior
distribution
p(yj |✓j; )
| {z }
likelihood
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Bayesian inference
!8
Algorithm:
•
Draw a realization from the posterior distribution•
Compute !a
pred(l)i= f
−1( V
inj; θ
(l)i)
, !l = 1,...,L
Gelman A. (2014) Bayesian Data Analysis. Third edition ed. Chapman & Hall/CRC texts in statistical science. Boca Raton: CRC Press.
Carpenter B, Gelman A, Hoffman MD, Lee D, Goodrich B, Betancourt M, et al. (2017) Stan: A Probabilistic Programming Language. J Stat Softw.
err!i
= a
predi− a
ia
ik-fold cross validation
!a tn-2
Results
!9
Population analysis
Exponential Logistic Gompertz AB
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
Exponential Logistic Gompertz
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
Results
!10
Population analysis
Exponential Logistic Gompertz
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
Exponential Logistic Gompertz
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
Results
!
α
i= kβ
iThe reduced Gompertz model
dV
dt = kβi − βi log (
V
Vinj ) V V(tI) = VI
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
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
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
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
!11
Results
!12
The reduced Gompertz model: biological interpretation
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 Bayesian 19.6 (2.99) 18.2 (2.38)
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 Bayesian 13.5 (3.5) 45.4 (4.43)
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
Model Parameter Unit Fixed effects CV (%) R.S.E. (%)
Gompertz α day−1 0.573 34.73 2.56
β day−1 0.0705 391.49 3.61
σ - [19.1, 0.12] [18.3, 7.36]
Reduced Gompertz β day−1 0.0725 180.69 1.91
k - 7.98 0 0.363
σ - [13.9, 0.183] [22.3, 5.17]
Logistic α day−1 0.324 42.90 1.88
K mm3 1332 0.02 4.39
σ - [57.2, 0.136] [9.8, 8.74]
Exponential α day−1 0.229 34.98 1.35
σ - [283, 0.254] [6.06, 14.3]
Table 2: Fixed effects (typical values) of the parameters of the different models. CV = Coefficient of Variation, expressed in percentage and estimated as the standard deviation of the parameter divided by the fixed effect and multiplied by 100. σ is vector of the residual error model parameters. Last column shows the relative standard errors (R.S.E.) of the estimates.
Ki = Vinje α
i βi
≃ Vinjek ≃ 2900 mm3, ∀i.
1
Constant maximal tumor size within a tumor type in a given species
Breast cancer Lung cancer
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 Bayesian 19.6 (2.99) 18.2 (2.38)
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 Bayesian 13.5 (3.5) 45.4 (4.43)
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
Model Parameter Unit Fixed effects CV (%) R.S.E. (%)
Gompertz α day−1 0.573 34.73 2.56
β day−1 0.0705 391.49 3.61
σ - [19.1, 0.12] [18.3, 7.36]
Reduced Gompertz β day−1 0.0725 180.69 1.91
k - 7.98 0 0.363
σ - [13.9, 0.183] [22.3, 5.17]
Logistic α day−1 0.324 42.90 1.88
K mm3 1332 0.02 4.39
σ - [57.2, 0.136] [9.8, 8.74]
Exponential α day−1 0.229 34.98 1.35
σ - [283, 0.254] [6.06, 14.3]
Table 2: Fixed effects (typical values) of the parameters of the different models. CV = Coefficient of Variation, expressed in percentage and estimated as the standard deviation of the parameter divided by the fixed effect and multiplied by 100. σ is vector of the residual error model parameters. Last column shows the relative standard errors (R.S.E.) of the estimates.
Ki = Vinje α
i βi
≃ Vinjek ≃ 2900 mm3, ∀i.
1
Results
!13
Population analysis
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 Bayesian 19.6 (2.99) 18.2 (2.38)
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 Bayesian 13.5 (3.5) 45.4 (4.43)
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
Model Parameter Unit Fixed effects CV (%) R.S.E. (%)
Gompertz α day−1 0.573 34.73 2.56
β day−1 0.0705 391.49 3.61
σ - [19.1, 0.12] [18.3, 7.36]
Reduced Gompertz β day−1 0.0725 180.69 1.91
k - 7.98 0 0.363
σ - [13.9, 0.183] [22.3, 5.17]
Logistic α day−1 0.324 42.90 1.88
K mm3 1332 0.02 4.39
σ - [57.2, 0.136] [9.8, 8.74]
Exponential α day−1 0.229 34.98 1.35
σ - [283, 0.254] [6.06, 14.3]
Table 2: Fixed effects (typical values) of the parameters of the different models. CV = Coefficient of Variation, expressed in percentage and estimated as the standard deviation of the parameter divided by the fixed effect and multiplied by 100. σ is vector of the residual error model parameters. Last column shows the relative standard errors (R.S.E.) of the estimates.
1
Model -2LL AIC BIC
Gompertz 7128 7142 7157
Reduced Gompertz 7259 7269 7280
Logistic 7584 7596 7609
Exponential 8652 8660 8669
Table 3: Models ranked in ascending order of AIC (Akaike information criterion). Other statistical indices are the log- likelihood estimate (-2LL) and the Bayesian information criterion (BIC).
2
Results
!14
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
Exponential Logistic Gompertz
A
B
C
Figure 2. Individual fits from population analysis. Three representative examples of individual fits computed with the population approach relative to the Exponential (left), the Logistic (center) and the Gompertz (right) models.
9
Exponential Logistic Gompertz
A
B
C
Figure 2. Individual fits from population analysis. Three representative examples of individual fits computed with the population approach relative to the Exponential (left), the Logistic (center) and the Gompertz (right) models.
9
Exponential Logistic Gompertz
A
B
C
Figure 2. Individual fits from population analysis. Three representative examples of individual fits computed with the population approach relative to the Exponential (left), the Logistic (center) and the Gompertz (right) models.
9
Population analysis: example of individual fits
Exponential Logistic
Gompertz Reduced Gompertz
Results
!15
Backward predictions: posterior distribution
Gompertz
Reduced Gompertz
0 tn-2
Results
!16
Backward predictions: posterior distribution
N = 3 N = 4 A N = 5 B N = 6 C
Gompertz (LM)
Reduced Gompertz
(LM)
Gompertz (Bayesian inference)
Reduced Gompertz (Bayesian inference)
Figure 4. Backward predictions computed with likelihood maximization and with Bayesian infer- ence. 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
Results
!17
Backward predictions
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 infer- ence. 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
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 infer- ence. 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
Results
!18
•
Reduced Gompertz >> Gompertz•
Bayesian inference >> likelihood maximization (LM)Backward predictions
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 Bayesian 19.6 (2.99) 18.2 (2.38)
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 Bayesian 13.5 (3.5) 45.4 (4.43)
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
Conclusions
!19
•
The Gompertz model described well tumor growth kinetics while the Exponential and the Logistic showed inferior predictive power.•
We proposed a novel reduced Gompertz model with only one individual parameter.•
The combination of nonlinear mixed effects modelling and Bayesian inference allowed to have reliable predictions of individual tumor age.•
The method proposed herein remains to be extended to clinical data: it would yield important epidemiological insights and could also be informative in routine clinical practice for prediction of metastatic extent.Thank you for your attention!
Preprint available!