[PPT] - Selection, calibration, and validation of models of tumor growth PowerPoint Presentation

SLIDE 1

Tumor Modeling Group

Selection, calibration, and validation of models of tumor growth

Regina C. Almeida

Laboratório Nacional de Computação Científica - LNCC

Chemnitz Symposium on Inverse Problems New Trends in Parameter Identification for Mathematical Model

Oct. 30 to Nov. 03, 2017, RJ

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SLIDE 2

Collaborations:

Ernesto A. B. F. Lima (ICES)

J. Tinsley Oden (ICES)

Thomas E. Yankeelov (ICES) David A. Hormuth II (ICES)

Supported by Grants

DOE-DESC009286 MMICC
National Science Foundation
CNPq 305612/2013-1

CCO

The Center for Computational Oncology R.C. Almeida New Trends in Parameter Identification ...

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SLIDE 3

The Imperfect Paths to Knowledge (Oden & Prudhome, 2010)

THE UNIVERSE OF PHYSICAL REALITIES THEORY / MATHEMATICAL MODELS OBSERVATIONS COMPUTATIONAL MODELS KNOWLEDGE PREDICTION VERIFICATION Discretization Errors Modeling Errors Observational Errors VALIDATION

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SLIDE 4

Goals of this Presentation

Review some concepts on the Foundations of Predictive Science:

the scientific discipline concerned with the predictability of compu- tational models of physical events in the presence of uncertainties∗

Describe OPAL (the Occam Plausibility Algorithm) [selection, cali-

bration, validation, model inadequacy, UQ of QoIs]†

Application: Predictive Modeling of Tumor Growth∗∗

∗JTO, Foundations of Predictive Computational Sciences, ICES, 17-01, Austin, 2017 †K. Farrell, JTO, D. Faghihi, J. Comp. Physics, 2015 ∗∗E. A. B. F. Lima, JTO, D. A. Hormuth, T. E. Yankeelov, R. C. Almeida, M3AS, 2016 R.C. Almeida New Trends in Parameter Identification ...

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Some Definitions

Model (Mathematical Model):

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Some Definitions

Model (Mathematical Model): Quantity of Interest - QoI: The goal of constructing and solving the model Tumor volume/area, RECIST (longest diameter), etc.

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SLIDE 7

A hierarchy of scenarios

Calibration Scenarios Sc Domain of unit tests designed to initialize parameters (e.g., in vitro observation in a pathology lab) Validation Scenarios Sv Domains of subsystem experiments designed to yield observational data that represents the QoI or other ability of the model to predict the QoI (e.g. in vivo data through MRI) Prediction Scenarios Sp The full-system domain, in which the target QoI resides

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SLIDE 8

Sources of Uncertainty in Predictive Science Terenin and Draper (2015)

1 Discretization 2 Observational Data 3 Model Parameters 4 Model Selection

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Sources of Uncertainty in Predictive Science Terenin and Draper (2015)

1 Discretization 2 Observational Data 3 Model Parameters 4 Model Selection

h-sensitivity a posteriori goal-oriented error estimator

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Sources of Uncertainty in Predictive Science Terenin and Draper (2015)

1 Discretization 2 Observational Data 3 Model Parameters 4 Model Selection

Y y-uncertainty y = {y(xi)} y = f(g, ε) truth

exp. noise

gi + εi = yi εi ∼ pµ

Noise

Model

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Sources of Uncertainty in Predictive Science Terenin and Draper (2015)

1 Discretization 2 Observational Data 3 Model Parameters 4 Model Selection

θ θ-uncertainty P(A|B)P(B) = P(B|A)P(A) ⇓ π(θ|y, S) ∝ π(y|θ, S)π(θ|S) θ θ-sensitivity Y (θ) = QoI

A. Saltelli et. al. 2004, 2008
M. D. Morris 1991
I. M. Sobol 2006

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SLIDE 12

Sources of Uncertainty in Predictive Science Terenin and Draper (2015)

1 Discretization 2 Observational Data 3 Model Parameters 4 Model Selection

M Model Uncertainty and Model (in)adequacy “All models are wrong but some are useful” (Box,1978) AIC, BIC Plausibilities . . . Validation gi = di(θ) − ηi(θ) gi = yi − εi yi − di(θ) = εi − ηi(θ) discrepancy model (GP)

r model validation

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SLIDE 13

The Major Issues

Computing/selecting priors π(θ).
Model selection: which model is best?
Parameter sensitivity: which parameters do not influence the QoI?
Computational algorithms: for sampling the posterior and solving the

stochastic forward problem.

Validation tolerances and metrics.
Quantifying uncertainty in the QoI.

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SLIDE 14

Model Selection

M = set of parametric model classes = {P1, P2, . . . , Pm} Each P has its own likelihood and parameters θj Bayes’ rule: π(θj|y, Pj, M) = π(y|θj, Pj, M)π(θj|Pj, M) π(y|Pj, M) , 1 ≤ j ≤ m where π(y|Pj, M) =

π(y|θj, Pj, M)π(θj|Pj, M) dθj

Now apply Bayes’ rule to the evidence: ρj = π(Pj|y, M) = π(y|Pj, M)π(Pj|M) π(y, M) = model plausibility

m

j=1

ρj = 1

H. Jeffreys, 1961; E. E. Prudencio & S. H. Cheung, 2012;

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SLIDE 15

Model Validation

Validation Metrics

Total Variation Metric

d(yv, dv(θi)) = ||yv − dv(θi)||L1 = ∞

−∞

|y(x) − d(θ, x)| dx

L1 error between cumulative distribution functions

d(yv, dv(θi)) = ∞

−∞

x

−∞

y(s) − d(θ, s) ds

dx
DKL (Kullback-Leibler Divergence): Measures the difference in information content between

probability distributions: for pdfs q (data) and p (prediction) DKL(q||p) = ∞

−∞

q(x) log q(x) p(x) dx,

Tolerances d(yv, dv(θi)) =

≤ γtol

model is “valid” (not invalid) > γtol model is invalid γtol = ?

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SLIDE 16

Bayesian Model Calibration, Validation, and Prediction

6 4 2

32

2
4

8 6 4

31

2

2

0.04 0.06 0.02

4

:(3)

Prior π(θ) Calibration (Sc, yc) π(θ|yc) = π(yc|θ)π(θ) π(yc) Validation (Sv, yv) π(θ|yv, yc) = π(yv|θ, yc)π(θ|yc) π(yv, yc) Validation Forward Problem A(θ, Sv; u(θ, Sv)) = ||d(u(θ, Sv)) − yv|| ≤ γtol Prediction (Sp, QoI) A(θ, Sp; u(θ, Sp)) = Q(u(θ, Sp)) ∼ π(Q) = π(Q|θ, Sv, Sc, γtol)

JTO, Moser, Ghattas, 2010; JTO, Babuska, Faghihi, 2017 R.C. Almeida New Trends in Parameter Identification ...

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SLIDE 17

Occam’s Razor

“Non sunt multiplicanda entia sine necessitate”

Entities should not be multiplied beyond necessity When choosing among a set of competing models: The simplest valid model is the best choice.

simple ⇒ number of parameters
valid ⇒ passes Bayesian validation test

How do we choose a model that adheres to this principle?

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The Occam-Plausibility Algorithm - OPAL

START Identify a set of possible models M = {P1(θ1), . . . , Pm(θm)} SENSITIVITY ANALYSIS Eliminate models with parameters to which the model output is insensitive ¯ M = { ¯ P1(¯ θ1), . . . , ¯ Pl(¯ θl)} OCCAM STEP Choose model(s) in the lowest Occam Category M∗ = {P∗

1(θ∗ 1), . . . , P∗ k(θ∗ k)}

CALIBRATION STEP Calibrate all models in M∗ PLAUSIBILITY STEP Compute plausibilities and identify most plausible model P∗

j

VALIDATION STEP Submit P∗

j to validation test

Is P∗

j valid?

yes no Does P∗

j have the most

parameters in ¯ M? yes no ITERATIVE OCCAM STEP Choose models in next Occam category Identify a new set

f possible models

Use validated parameters to predict QoI

K. Farrell, JTO, D. Faghihi, 2015

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SLIDE 19

Tumor Model Framework

Cancer in living tissue Intratumor heterogeneity How does

ne develop

mathematical models of tumor growth? Metastatic carcinoma in bone marrow†

† Silver S, at https://drsusansilverpathologist.wordpress.com

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SLIDE 20

Strategies for Tumor Growth Model Building

I. Conservation Laws of Mechanics and Thermodynamics must hold (balance of mass, mo-

mentum, energy, second law of thermodynamics)

II. Continuum Mixture Theory
III. The Hallmarks of Cancer∗

Metastatic carcinoma in bone marrow†

∗Hanahan D, Weinberg R. Hallmarks of cancer: the next generation. Cell 144(5):646-674, 2011.

†Silver S, at https://drsusansilverpathologist.wordpress.com

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SLIDE 21

Strategies for Tumor Growth Model Building

I. Conservation Laws of Mechanics and Thermodynamics must hold (balance of mass, mo-

mentum, energy, second law of thermodynamics)

II. Continuum Mixture Theory
III. The Hallmarks of Cancer∗

x φT (x, t) + φN(x, t) = 1 φN(x, t) = 1 − φT (x, t) where: φT = tumor cells φN = healthy cells Metastatic carcinoma in bone marrow†

∗Hanahan D, Weinberg R. Hallmarks of cancer: the next generation. Cell 144(5):646-674, 2011.

†Silver S, at https://drsusansilverpathologist.wordpress.com

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SLIDE 22

Glioma Data Hormuth II et al., 2015

MRI at days: 10, 12, 14, 15, 16, 18, 20
Carrying capacity: 40761 cells
Voxel dimensions (dx×dy×dz): 250 × 250 × 1000 µm
Number of voxels: 41 × 61 × 16

10 12 14 15 16 18 20 Tumor area evolution at slice=12 of the Diffusion Weighted Magnetic Resonance Imaging (DW-MRI)

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SLIDE 23

Glioma Data Hormuth II et al., 2015

MRI at days: 10, 12, 14, 15, 16, 18, 20
Carrying capacity: 40761 cells
Voxel dimensions (dx×dy×dz): 250 × 250 × 1000 µm
Number of voxels: 41 × 61 × 16

10 12 14 15 16 18 20 Tumor area evolution at slice=12 of the Diffusion Weighted Magnetic Resonance Imaging (DW-MRI)

How to represent the data? → avascular tumor phase → tumor and healthy cells → with/without mechanical deformation

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SLIDE 24

Model Derivation

The tumor cell volume fraction must satisfy its own balance law given by

∂φT ∂t = ST (φ) − ∇ · (MT (φT )∇ · µ(φT )) where the chemical potential is given by µ(φT ) = DφT E and DφT (·) is the functional derivative with respect to φT

The total free energy E is given by

E = Ech + Eel which results from chemical (Ech) and elastic (Eel) contributions

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SLIDE 25

Free Energy

Chemical:

PF: Phase Field Model

Ech

P F =

Ω
Ψ(φT ) + ǫ2

T

2 |∇φT |2

dx
ǫT is the interaction length
Ψ(φT ) = ¯

Eφ2

T (1 − φT )2

¯

E > 0 is an energy scale

RD: Reaction-Diffusion Model Given by the internal energy

f

the system Ech

RD =

Ω

c 2φ2

T

dx

Elastic:

Eel =

Ω

W(φT , E(u)) dx

W(φT , E(u)) is the strain energy density
u the displacement field

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Phase-field model (PF) ∂φT ∂t = ∇ · MT φ2

T (1 − φT )2 ∇µ + λgrow T

φT (1 − φT ) µ = 2 ¯ EφT

1 − 3φT + 2φ2

T

− ǫ2

T ∆φT

Reaction-diffusion (RD)

∂φT ∂t = ∇ · M ∗

T ∇φT + λgrow T

φT (1 − φT )

Deformation Model

= ∇ · G∇u + ∇ G 1 − 2ν (∇ · u) − λ∇φT

MT : mobility of tumor cells
M ∗

T = cMT ; c > 0

λgrow

T

: the tumor growth rate

ǫT : interaction length
¯

E: energy scale

λ: compositional growth strength
G: is the shear modulus
ν: Poisson’s ratio
Feedback at the mobility: MT = ¯

MT e−γσV M

Feedback at the growth rate: λgrow

T

= ¯ λg

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Phase-field model with deformation (PF) ∂φT ∂t = ∇ · MT φ2

T (1 − φT )2 ∇µ + λgrow T

φT (1 − φT ) µ = 2 ¯ EφT

1 − 3φT + 2φ2

T

− ǫ2

T ∆φT −λ∇ · u

Reaction-diffusion with deformation (MD)

∂φT ∂t = ∇ · M ∗

T ∇φT + λgrow T

φT (1 − φT )−λ∇ · u

Deformation Model

= ∇ · G∇u + ∇ G 1 − 2ν (∇ · u) − λ∇φT

MT : mobility of tumor cells - U(0.2, 1.0)
M∗

T = cMT : c positive constant - U(0.05, 0.35)

λgrow

T

: the tumor growth rate - U(0.2, 1.6)

ǫT : interaction length - U(0.5, 0.7)
¯

E: energy scale - U(0.15, 0.35)

λ: compositional growth strength - U(0.2, 1.0)
E: is the modulus of elasticity - U(1.2, 2.8)
ν: Poisson’s ratio - U(0.2, 0.49)
Feedback at the mobility: MT = ¯

MT e−γσV M , γ ∈ U(80, 320)

Feedback at the growth rate: λgrow

T

= ¯ λg

T e−γgσV M , γg ∈ U(80, 320)

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SLIDE 28

Occam Categories

PF# = phase-field model
RD# = reaction-diffusion model without mechanical deformation coupling
MD# = reaction-diffusion model with mechanical deformation coupling

Variables Parameters #P Occam Model φT µ u MT M∗

T

c λgrow

T

¯ ET ǫT G ν λ γ γg Category RD01

2

1 PF01

4

2 RD02

6

3 RD03

6

3 MD01

6

3 RD04

7

4 MD02

7

4 MD03

7

4 PF02

7

4 MD04

8

5 PF03

8

5 PF04

8

5 PF05

9

6

E. A. B. F. Lima, JTO, D. A. Hormuth, T. E. Yankeelov, R. C. Almeida, 2016

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SLIDE 29

Glioma Data

MRI at days: 10, 12, 14, 15, 16, 18, 20
Carrying capacity: 40761 cells
Voxel dimensions (dx×dy×dz): 250 × 250 × 1000 µm.
Number of voxels: 41 × 61 × 16

10 12 14 15 16 18 20 Calibration Validation Prediction

Tumor area evolution at slice=12

Hormuth II et al., 2015 R.C. Almeida New Trends in Parameter Identification ...

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SLIDE 30

The Occam-Plausibility Algorithm - OPAL

START Identify a set of possible models M = {P1(θ1), . . . , Pm(θm)} SENSITIVITY ANALYSIS Eliminate models with parameters to which the model output is insensitive ¯ M = { ¯ P1(¯ θ1), . . . , ¯ Pl(¯ θl)} OCCAM STEP Choose model(s) in the lowest Occam Category M∗ = {P∗

1(θ∗ 1), . . . , P∗ k(θ∗ k)}

CALIBRATION STEP Calibrate all models in M∗ PLAUSIBILITY STEP Compute plausibilities and identify most plausible model P∗

j

VALIDATION STEP Submit P∗

j to validation test

Is P∗

j valid?

yes no Does P∗

j have the most

parameters in ¯ M? yes no ITERATIVE OCCAM STEP Choose models in next Occam category Identify a new set

f possible models

Use validated parameters to predict QoI

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SLIDE 31

Sensitivity Analysis Moris Method

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SLIDE 32

The Occam-Plausibility Algorithm - OPAL

START Identify a set of possible models M = {P1(θ1), . . . , Pm(θm)} SENSITIVITY ANALYSIS Eliminate models with parameters to which the model output is insensitive ¯ M = { ¯ P1(¯ θ1), . . . , ¯ Pl(¯ θl)} OCCAM STEP Choose model(s) in the lowest Occam Category M∗ = {P∗

1(θ∗ 1), . . . , P∗ k(θ∗ k)}

CALIBRATION STEP Calibrate all models in M∗ PLAUSIBILITY STEP Compute plausibilities and identify most plausible model P∗

j

VALIDATION STEP Submit P∗

j to validation test

Is P∗

j valid?

yes no Does P∗

j have the most

parameters in ¯ M? yes no ITERATIVE OCCAM STEP Choose models in next Occam category Identify a new set

f possible models

Use validated parameters to predict QoI

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SLIDE 33

First Occam’s Category: RD01 - Model Calibration

∂φT ∂t = ∇ · M ∗

T ∇φT + λg T φT (1 − φT )

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SLIDE 34

The Occam-Plausibility Algorithm - OPAL

START Identify a set of possible models M = {P1(θ1), . . . , Pm(θm)} SENSITIVITY ANALYSIS Eliminate models with parameters to which the model output is insensitive ¯ M = { ¯ P1(¯ θ1), . . . , ¯ Pl(¯ θl)} OCCAM STEP Choose model(s) in the lowest Occam Category M∗ = {P∗

1(θ∗ 1), . . . , P∗ k(θ∗ k)}

CALIBRATION STEP Calibrate all models in M∗ PLAUSIBILITY STEP Compute plausibilities and identify most plausible model P∗

j

VALIDATION STEP Submit P∗

j to validation test

Is P∗

j valid?

yes no Does P∗

j have the most

parameters in ¯ M? yes no ITERATIVE OCCAM STEP Choose models in next Occam category Identify a new set

f possible models

Use validated parameters to predict QoI

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SLIDE 35

RD01 - Model Validation

Validation step: the posterior from the calibration step is used as a prior to the validation step

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RD01 Model Validation

The forward problem is solved and the tumor area is computed. Validation criterium: DKL/σ2 ≤ γtol = 0.70; plausability step not needed DKL(π(Data)||π(RD01))/σ2

16 = 0.92 > γtol

DKL(π(Data)||π(RD01))/σ2

18 = 0.70 ≥ γtol

Model validation with chain size = 5000 Model is NOT valid

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SLIDE 37

The Occam-Plausibility Algorithm - OPAL

START Identify a set of possible models M = {P1(θ1), . . . , Pm(θm)} SENSITIVITY ANALYSIS Eliminate models with parameters to which the model output is insensitive ¯ M = { ¯ P1(¯ θ1), . . . , ¯ Pl(¯ θl)} OCCAM STEP Choose model(s) in the lowest Occam Category M∗ = {P∗

1(θ∗ 1), . . . , P∗ k(θ∗ k)}

CALIBRATION STEP Calibrate all models in M∗ PLAUSIBILITY STEP Compute plausibilities and identify most plausible model P∗

j

VALIDATION STEP Submit P∗

j to validation test

Is P∗

j valid?

yes no Does P∗

j have the most

parameters in ¯ M? yes no ITERATIVE OCCAM STEP Choose models in next Occam category Identify a new set

f possible models

Use validated parameters to predict QoI

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SLIDE 38

OPAL Results

DKL/σ2 ≤ γtol = 0.70 Model Occam Category Plausibility DKL /σ2

16

DKL/σ2

18

RD01 1 n/a 0.92 0.70 PF01 2 n/a 0.65 0.74 RD02 3 0.2212 RD03 3 0.1425 MD01 3 0.6362 0.93 0.72 RD04 4 0.0220 MD02 4 0.0556 MD03 4 0.0245 PF02 4 0.8979 0.68 0.76 MD04 5 0.0189 PF03 5 0.3371 PF04 5 0.6440 0.70 0.66 PF05 6 n/a

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SLIDE 39

Model Validation - PF04

32.7344 mm2 (data) 32.4517 mm2 (model)

Model validation at t = 16. Model is NOT invalid

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SLIDE 40

Model Validation - PF04

37.9219 mm2 (data) 41.9598 mm2 (model)

Model validation at t = 18. Model is NOT invalid

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SLIDE 41

Model Prediction - PF04

40.9219 mm2 (data) 44.9074 mm2 (model)

Model prediction at t = 20. The model is able to predict the area of the tumor with 9.75% difference from the mean value.

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SLIDE 42

OPAL Results with AIC and BIC Weights

DKL/σ2 ≤ γtol = 0.70 Model Occam Category AIC w BIC w DKL/σ2

16

DKL/σ2

18

RD01 1 n/a n/a 0.92 0.70 PF01 2 n/a n/a 0.65 0.74 RD02 3 0.41 0.41 0.95 0.72 RD03 3 0.25 0.25 MD01 3 0.34 0.34 RD04 4 0.02 0.02 MD02 4 0.08 0.08 MD03 4 0.05 0.05 PF02 4 0.85 0.85 0.68 0.76 MD04 5 0.02 0.02 PF03 5 0.52 0.52 0.64 0.70 PF04 5 0.46 0.46 PF05 6 n/a n/a

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SLIDE 43

Concluding Comments

We develop 13, thermodynamically consistent, continuum tumor growth models: five of them

using the phase-field modeling approach and the others being reaction-diffusion type models.

The OPAL is able to assist in the selection of the best tumor growth model
For these data and with the tumor area as a QoI, the phase-field model with a deformation

feedback is the simplest valid model.

Bayesian-based Logical Probability provides a general framework for selecting and validating

models in the presence of uncertainties in data, parameters, and output.

The OPAL strategy provides a powerful framework for model selection, parameter estimation,

and model validation in the presence of uncertainties.

Applications of OPAL to PDE models of tumor growth suggests that the approach is

applicable to broad classes of problems in engineering, science and medicine.

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SLIDE 44