This presentation begins with a survey of the principles of predictive computational science: the discipline concerned with assessing the predictability of mathematical and computational models of events that occur in the physical universe in the presence of uncertainties. We then focus on key aspects of predictability: modeling error and how to estimate it, and model selection. The idea of optimal control of modeling error in which a sequence of models is generated so as control error relative to a high-fidelity ground truth model is discussed, as well as the related problem in which, given noisy data , we wish to select, calibrate, and validate a model’s ability to predict quantities of interest with a preset level of accuracy. As applications, we consider the analysis of random heterogeneous media and the construction and selection of mathematical models of tumor growth. We discuss OPAL, the Occam Plausibility Algorithm, as a framework for systematic model selection and validation. Examples of applications of these methodologies are given.

1. Principles of Predictive Computational Science:
Predictive Models of Random Heterogenous
Materials and Tumor Growth
J. Tinsley Oden
Institute for Computational Engineering and Sciences
The University of Texas at Austin
Collaborators: Danial Faghihi (ICES), Laura Scarabosio (TUM),
Barbara Wohlmuth (TUM), Ernesto Lima (ICES)
MUMS: Model Uncertainty: Mathematical and Statistical – SAMSI
Duke University
August 20-24, 2018
ODEN MUMS August 20-24, 2018 1 / 102

2. Collaborators:
Danial Faghihi (ICES)
Barbara Wohlmuth (TUM)
Laura Scarabosio (TUM)
Ernesto Lima (ICES)
Supported by
• DOE-DESC009286 MMICC
• ICES: IGSSE
ODEN MUMS August 20-24, 2018 2 / 102

3. Outline
1. Foundations of predictive computational science
2. Predictive models of random materials
3. Model Selection and Occam Plausibility Algorithm (OPAL)
4. Concluding Comments
ODEN MUMS August 20-24, 2018 3 / 102

4. Foundations of predictive computational science
1. Foundations of predictive computational science
1.1. Definitions
1.2. Sources of Uncertainty
1.3. The Imperfect Paths to Knowledge
1.4. Fundamental Principles
ODEN MUMS August 20-24, 2018 4 / 102

5. Foundations of predictive computational science
1. Foundations of predictive computational science
1.1 Definitions
V V & UQ ⇒ Predictive Computational Science
The scientific discipline concerned with assessing the predictability of
mathematical and computational models of events in the physical
universe in the presence of uncertainty.
ODEN MUMS August 20-24, 2018 5 / 102

6. Foundations of predictive computational science
1. Foundations of predictive computational science
1.1 Definitions
V V & UQ ⇒ Predictive Computational Science
The scientific discipline concerned with assessing the predictability of
mathematical and computational models of events in the physical
universe in the presence of uncertainty.
Prediction: Forecast, prophesy,
prognosis, prognostication, · · · guess
ODEN MUMS August 20-24, 2018 5 / 102

7. Foundations of predictive computational science
1. Foundations of predictive computational science
1.1 Definitions
V V & UQ ⇒ Predictive Computational Science
The scientific discipline concerned with assessing the predictability of
mathematical and computational models of events in the physical
universe in the presence of uncertainty.
Prediction: Forecast, prophesy,
prognosis, prognostication, · · · guess
Science: The systematic acquisition
of knowledge through observations
(data) and theory (mathematical and
computational models)
ODEN MUMS August 20-24, 2018 5 / 102

8. Foundations of predictive computational science
1. Foundations of predictive computational science
1.1 Definitions
V V & UQ ⇒ Predictive Computational Science
The scientific discipline concerned with assessing the predictability of
mathematical and computational models of events in the physical
universe in the presence of uncertainty.
Prediction: Forecast, prophesy,
prognosis, prognostication, · · · guess
Science: The systematic acquisition
of knowledge through observations
(data) and theory (mathematical and
computational models)
Uncertain: absence of complete
knowledge, indeterminate
ODEN MUMS August 20-24, 2018 5 / 102

9. Foundations of predictive computational science
1. Foundations of predictive computational science
1.1 Definitions
V V & UQ ⇒ Predictive Computational Science
The scientific discipline concerned with assessing the predictability of
mathematical and computational models of events in the physical
universe in the presence of uncertainty.
Prediction: Forecast, prophesy,
prognosis, prognostication, · · · guess
Science: The systematic acquisition
of knowledge through observations
(data) and theory (mathematical and
computational models)
Uncertain: absence of complete
knowledge, indeterminate
How do you “quantify uncertainty”?
ODEN MUMS August 20-24, 2018 5 / 102

10. Foundations of predictive computational science
1.2. Sources of Uncertainty in Predictive Science
1. System of Reasoning
(Epistemology)
2. Observational Data
3. Model Selection
4. Model Parameters
5. Discretization Error
ODEN MUMS August 20-24, 2018 6 / 102

11. Foundations of predictive computational science
1.2. Sources of Uncertainty in Predictive Science
1. System of Reasoning
(Epistemology)
2. Observational Data
3. Model Selection
4. Model Parameters
5. Discretization Error
P
Systemic Uncertainty
• Probability Theory
• Dempster–Schafer
• Possibility Theory
• Fuzzy Sets
...
ODEN MUMS August 20-24, 2018 6 / 102

12. Foundations of predictive computational science
1.2. Sources of Uncertainty in Predictive Science
1. System of Reasoning
(Epistemology)
2. Observational Data
3. Model Selection
4. Model Parameters
5. Discretization Error
P
Systemic Uncertainty
• Probability Theory
• Dempster–Schafer
• Possibility Theory
• Fuzzy Sets
...
Probability Theory:
But which one
The Logic of Science: Cox-
Jaynes Theory of Probability
• Every natural extension of
Aristotelian logic with
uncertainties is Bayesian
ODEN MUMS August 20-24, 2018 6 / 102

13. Foundations of predictive computational science
1.2. Sources of Uncertainty in Predictive Science
1. System of Reasoning
(Epistemology)
2. Observational Data
3. Model Selection
4. Model Parameters
5. Discretization Error
Y
y-uncertainty
y = {y(xi)}
y = f(g, ε)
truth
exp. noise
gi + εi = yi
εi ∼ pµ
Noise
Model
ODEN MUMS August 20-24, 2018 6 / 102

14. Foundations of predictive computational science
1.2. Sources of Uncertainty in Predictive Science
1. System of Reasoning
(Epistemology)
2. Observational Data
3. Model Selection
4. Model Parameters
5. Discretization Error
M
Model Uncertainty and
Model (in)adequacy
“All models are wrong but
some are useful” (Box (1978))
gi = di(θ) − ηi(θ)
gi = yi − εi
yi − di(θ) = εi − ηi(θ)
discrepancy model (GP)
or model validation
Pnoise+model(εi − ηi) =
Pnoise+model(yi − di(θ)) =
πlikelihood(yi|θ)
ODEN MUMS August 20-24, 2018 6 / 102

15. Foundations of predictive computational science
1.2. Sources of Uncertainty in Predictive Science
1. System of Reasoning
(Epistemology)
2. Observational Data
3. Model Selection
4. Model Parameters
5. Discretization Error
θ
θ-uncertainty
The Product Rule
P(A|B)P(B) = P(B|A)P(A)
⇓
π(θ|y) =
π(y|θ)π(θ)
π(y)
θ
θ-sensitivity
Y (θ) = output
STi = 1 −
Vθ∼i[Eθi
(Y |θi)]
V (Y )
A. Saltelli et. al. 2004, 2008.
M. D. Morris 1991.
I. M. Sobol 2006.
ODEN MUMS August 20-24, 2018 6 / 102

16. Foundations of predictive computational science
1.2. Sources of Uncertainty in Predictive Science
1. System of Reasoning
(Epistemology)
2. Observational Data
3. Model Selection
4. Model Parameters
5. Discretization Error
h-uncertainty
goal-oriented
a-posterior
error estimator
ODEN MUMS August 20-24, 2018 6 / 102

17. Foundations of predictive computational science
1.2. Sources of Uncertainty in Predictive Science
1. System of Reasoning
(Epistemology)
2. Observational Data
3. Model Selection
4. Model Parameter
5. Discretization Error
Logical Probability-Cox-Jaynes
Aristotelian Logic + Uncertainty ⇔ Bayesian
y ∈ Y : yi = fi(gi, εi) ⇐ yi + εi =
gi, εi ∼ Pi( i)
gi = di(θ) − ηi(θ) ⇒ yi − di(θ) = εi − ηi(θ)
P(A|B)P(B) = P(B|A)P(A) ⇒
π(θ, y) ∝ π(y|θ)π(θ)
A posteriori error estimation
ODEN MUMS August 20-24, 2018 7 / 102

18. Foundations of predictive computational science
1.3. The Imperfect Paths to Knowledge
THE UNIVERSE
OF PHYSICAL
REALITIES
THEORY /
MATHEMATICAL
MODELS
OBSERVATIONS
COMPUTATIONAL
MODELS
KNOWLEDGE
PREDICTION
The Three Pillars of Science
VERIFICATION
Discretization
Errors
Modeling
Errors
Observational
Errors
VALIDATION
JTO, Moser, Ghattas – SIAM News, 2010
ODEN MUMS August 20-24, 2018 8 / 102

19. Foundations of predictive computational science
1.4. Fundamental Principles in V, V, & UQ
1. Jaynes’ Principle:
“The essence of honesty and objectivity demands that we take into account all
the evidence we have, not just some arbitrary chosen subset of it”.
E. T. Jaynes, Probability Theory, The Logic of Science, 2003
2. Model Scenarios, Ultra Cross-Validation, The Prediction Pyramid:
P(A|B)P(B) = P(B|A)P(A)
⇓
π(θ|y) =
π(y|θ)π(θ)
π(y)
⇓
Ai(θ, S; ui(θ, S)) = 0
Reality
ODEN MUMS August 20-24, 2018 9 / 102

20. Foundations of predictive computational science
1.4. Fundamental Principles in V, V, & UQ
1. Jaynes’ Principle:
“The essence of honesty and objectivity demands that we take into account all
the evidence we have, not just some arbitrary chosen subset of it”.
E. T. Jaynes, Probability Theory, The Logic of Science, 2003
2. Model Scenarios, Ultra Cross-Validation, The Prediction Pyramid:
P(A|B)P(B) = P(B|A)P(A)
⇓
π(θ|y) =
π(y|θ)π(θ)
π(y)
⇓
Ai(θ, S; ui(θ, S)) = 0
Reality
Bayes rule in terms of probability densities of
model parameters θ and observational data y.
ODEN MUMS August 20-24, 2018 9 / 102

21. Foundations of predictive computational science
1.4. Fundamental Principles in V, V, & UQ
1. Jaynes’ Principle:
“The essence of honesty and objectivity demands that we take into account all
the evidence we have, not just some arbitrary chosen subset of it”.
E. T. Jaynes, Probability Theory, The Logic of Science, 2003
2. Model Scenarios, Ultra Cross-Validation, The Prediction Pyramid:
P(A|B)P(B) = P(B|A)P(A)
⇓
π(θ|y) =
π(y|θ)π(θ)
π(y)
⇓
Ai(θ, S; ui(θ, S)) = 0
Reality
Bayes rule in terms of probability densities of
model parameters θ and observational data y.
The forward problem for scenario S =
{Sc, Sv, Sp}.
ODEN MUMS August 20-24, 2018 9 / 102

22. Foundations of predictive computational science
Bayesian Model Calibration, Validation, and Prediction
Reality
Prior
π(θ)
Calibration (Sc, yc) ("Training data")
π(θ|yc) =
π(yc|θ)π(θ)
π(yc)
Validation (Sv, yv) ("Testing data")
π(θ|yv, yc) =
π(yv|θ, yc)π(θ|yc)
π(yv, yc)
Validation Forward Problem
A(θ, Sv; u(θ, Sv)) = 0
||d(u(θ, Sv)) − yv|| ≤ γtol
Prediction (Sp, QoI)
A(θ, Sp; u(θ, Sp)) = 0
Q(u(θ, Sp)) ∼ π(Q) = π(Q|θ, Sv, Sc, γtol)
JTO, Moser, Ghattas, 2010; JTO, Babuska, Faghihi, 2017
ODEN MUMS August 20-24, 2018 10 / 102

23. Foundations of predictive computational science
3. The central goal of a computer prediction is to evaluate the Quantity of
Interest (QoI), not simply to compute the solution of the stochastic forward
problem. [Q : Ω × U → R]
ODEN MUMS August 20-24, 2018 11 / 102

24. Foundations of predictive computational science
3. The central goal of a computer prediction is to evaluate the Quantity of
Interest (QoI), not simply to compute the solution of the stochastic forward
problem. [Q : Ω × U → R]
The QoI is NOT data; it is never observed: It is an extrapolation
outside the calibration and validation observational data.
ODEN MUMS August 20-24, 2018 11 / 102

25. Foundations of predictive computational science
3. The central goal of a computer prediction is to evaluate the Quantity of
Interest (QoI), not simply to compute the solution of the stochastic forward
problem. [Q : Ω × U → R]
The QoI is NOT data; it is never observed: It is an extrapolation
outside the calibration and validation observational data.
4. Jeffrey’s Principle:
“ Parameters in a law that make no contribution to the results of any observation
can be eliminated mathematically, leaving the observations to be described only
in terms of the relevant parameters. ”
(provided 0 ∈ Θ)
Sir Harold Jeffrey, Theory of Probability,
Oxford, 1936-1961
ODEN MUMS August 20-24, 2018 11 / 102

26. Foundations of predictive computational science
3. The central goal of a computer prediction is to evaluate the Quantity of
Interest (QoI), not simply to compute the solution of the stochastic forward
problem. [Q : Ω × U → R]
The QoI is NOT data; it is never observed: It is an extrapolation
outside the calibration and validation observational data.
4. Jeffrey’s Principle:
“ Parameters in a law that make no contribution to the results of any observation
can be eliminated mathematically, leaving the observations to be described only
in terms of the relevant parameters. ”
(provided 0 ∈ Θ)
Sir Harold Jeffrey, Theory of Probability,
Oxford, 1936-1961
5. A model can never be validated as a perfect portrayal of the truth; it can only
be deemed ‘not invalid’, contingent on its agreement with available observational
data for (subjective) choices of metrics and tolerances.
ODEN MUMS August 20-24, 2018 11 / 102

27. Foundations of predictive computational science
3. The central goal of a computer prediction is to evaluate the Quantity of
Interest (QoI), not simply to compute the solution of the stochastic forward
problem. [Q : Ω × U → R]
The QoI is NOT data; it is never observed: It is an extrapolation
outside the calibration and validation observational data.
4. Jeffrey’s Principle:
“ Parameters in a law that make no contribution to the results of any observation
can be eliminated mathematically, leaving the observations to be described only
in terms of the relevant parameters. ”
(provided 0 ∈ Θ)
Sir Harold Jeffrey, Theory of Probability,
Oxford, 1936-1961
5. A model can never be validated as a perfect portrayal of the truth; it can only
be deemed ‘not invalid’, contingent on its agreement with available observational
data for (subjective) choices of metrics and tolerances.
6. The CENTRAL PROBLEM of predictive computational science (UQ in CSE) is
the selection of a valid model and model parameters for a given QoI.
ODEN MUMS August 20-24, 2018 11 / 102

28. Foundations of predictive computational science
3. The central goal of a computer prediction is to evaluate the Quantity of
Interest (QoI), not simply to compute the solution of the stochastic forward
problem. [Q : Ω × U → R]
The QoI is NOT data; it is never observed: It is an extrapolation
outside the calibration and validation observational data.
4. Jeffrey’s Principle:
“ Parameters in a law that make no contribution to the results of any observation
can be eliminated mathematically, leaving the observations to be described only
in terms of the relevant parameters. ”
(provided 0 ∈ Θ)
Sir Harold Jeffrey, Theory of Probability,
Oxford, 1936-1961
5. A model can never be validated as a perfect portrayal of the truth; it can only
be deemed ‘not invalid’, contingent on its agreement with available observational
data for (subjective) choices of metrics and tolerances.
6. The CENTRAL PROBLEM of predictive computational science (UQ in CSE) is
the selection of a valid model and model parameters for a given QoI.
The design of appropriate validation experiment.
ODEN MUMS August 20-24, 2018 11 / 102

29. Predictive Models of Random Materials
2. Predictive Models of Random Materials
2.1. Mathematical Models and Modeling Error
2.2. Strategy 1 – Optimal Control of Modeling Error
2.3. An Old Example
2.4. Random Two-Phase Heterogeneous Media
2.5. Selection of the Most Plausible Models
2.6. Accelerated MLMC
2.7. The Inverse Problem: Machine Learning Methods
ODEN MUMS August 20-24, 2018 12 / 102

30. Predictive Models of Random Materials
2. Predictive Models of Random Materials
Returning to the Central Problem: M and θ Uncertainty
I. Model Adaptivity and Control:
• Model Reduction through
Surrogate Models
• Control (estimated) error in QoIs
⇒
II. Model Selection:
• Selection in the Bayesian VVUQ
Framework
• Select most plausible valid model
⇒
ODEN MUMS August 20-24, 2018 13 / 102

31. Predictive Models of Random Materials
2. Predictive Models of Random Materials
Returning to the Central Problem: M and θ Uncertainty
I. Model Adaptivity and Control:
• Model Reduction through
Surrogate Models
• Control (estimated) error in QoIs
⇒
Adaptive Control of
Goal-Oriented Esti-
mates of Error in QoIs
II. Model Selection:
• Selection in the Bayesian VVUQ
Framework
• Select most plausible valid model
⇒
OPAL: the Occam
Plausibility Algorithm
ODEN MUMS August 20-24, 2018 13 / 102

32. Predictive Models of Random Materials
2.1. Mathematical Models and Modeling Error
Mathematical Model: a mathematical description of a system – here a physical or engineered
system – representing mathematical abstractions of the functioning or behavior of the system,
characterized by mathematical constructions (e.g. equations, inequalities, etc.).
ODEN MUMS August 20-24, 2018 14 / 102

33. Predictive Models of Random Materials
Modeling Error
1 - “Ground Truth” is defined by a High-Fidelity Base Model but: The forward problem
A(θ, S, u(ω, S)) = 0 is intractable. Replace it by a surrogate model of lower dimension
and complexity
A0(θ0, S; u0(ω, S)) = 0
Q(u0(ω, Sp)) = surrogate QoI
The Strategy: Develop methods to choose surrogates so that the error
ε(u) = Q(u(ω, Sp)) − Q(u0(ω, Sp))
can be estimated and controlled1
.
2 - Error in the Choice of Model: Given a possibly large set M = M1, M2, · · · , Mm of
models, and (noisy) observational data y ∈ Y ⊂ Rn
, find the models(s) that best fit the
data for any parameters θk ∈ Θk, k = 1, 2, · · · , m. Then establish if the best-fit models is
valid through additional validation tests in Sv
2
.
1
JTO, Prudhomme 2001, JTO, Vemaganti 2000, 2001, JTO, Prudhomme, Romkes,
Bauman, 2000
2
JTO, Babuska, Faghihi 2018, JTO, Acta Numerica 2018.
ODEN MUMS August 20-24, 2018 15 / 102

34. Predictive Models of Random Materials
2.2. Strategy I: Optimal Control of Models
A : U → V
Au = F in V : The Forward Problem
V Au, v V := B(u; v) = V F, v V = F(v) ∀v ∈ V
Q : U → R = The Quantity of Interest
Q(u) = inf Q(w)
{w ∈ U : B(w; v) = F(v) ∀v ∈ V }
B(u; v) = F(v) ∀v ∈ V
B (u; w, v) = Q (u; v) ∀v ∈ V
ODEN MUMS August 20-24, 2018 16 / 102

35. Predictive Models of Random Materials
Theorem: If B(·; ·), Q(·) ∈ C3
(U) and (u, w) ∈ U × U is the solution pair of the
forward and adjoint problems, and if (u0, w0) is an arbitrary pair drawn from
U × U, then
ε = Q(u) − Q(u0) = R(u0, w) + ∆
where R(u0; ·, ·) is the residual functional
R(u0; v) = F(v) − B(u0, v)
and ∆ is a remainder: ∆ = O( u − u0
2
, w − w0
2
).
Strategy I:
1. Construct a Surrogate System:
B0(u0, v) = F(v)
B0(u0; w0, v) = Q (u0; v) ∀v ∈ V
such that
∆(u0, w0) ≈ 0
2. Pick u0, w0 so that w0 ≈ w and ε ≤ γtol
JTO, Prudhomme, JCP, 2002
ODEN MUMS August 20-24, 2018 17 / 102

36. Multiscale Modeling
Modeling of Polymeric Materials for Nanomanufacturing
Step-and-Flash
Imprint Lithography (SFIL)
∗Bailey, Johnson, Resnick, Eckerdt, and Willson, J. Photopolymer Sci. Tech. (2002)
∗Resnick, Sreenivasan, and Willson, Materials Today (2005)
J.T. Oden von Neumann Symposium 2011 July 4–7, 2011 27 / 44
Predictive Models of Random Materials
ODEN MUMS August 20-24, 2018 18 / 102

37. Multiscale Modeling
Multi-Algorithm Coupling
Kinetic Monte Carlo (AA) Coarse-Graining (CG) RVE Averaging
UCG =
bond
kb(r − r0)2
+
angle
(θ − θ0)2
≈ −kBT log[P(r)/r2]
−kBT log[P(θ)/ sin θ]
∗Izvekov and Voth, J. Phys. Chem. B (2005)
∗Shinoda, Devane, and Klein, Mol. Sim. (2007)
J.T. Oden von Neumann Symposium 2011 July 4–7, 2011 28 / 44
Predictive Models of Random Materials
ODEN MUMS August 20-24, 2018 19 / 102

38. Multiscale Modeling
Molecular Statics Model
Polymerization
Densification
Two-step approach:
1) Polymerization by Monte-Carlo Algorithm.
2) Densification by energy minimization:
u = argmin
v
N
i=1
Ni
k=1
Eik(vi, vk) − fi · vi
N
i=1
Ni
k=1
∂Eik
∂ui
· vi
B(u;v)
=
N
i=1
fi · vi
F (v)
where Eik(ui, uk) is the potential energy between particle
i and neighbor k. Use TAO & PETSc to solve problem.
∗P.T. Bauman, Ph.D. Dissertation, The University of Texas at Austin (2008).
∗Bauman, Oden, and Prudhomme, CMAME, 198, 799–818 (2009).
J.T. Oden von Neumann Symposium 2011 July 4–7, 2011 30 / 44
Predictive Models of Random Materials
ODEN MUMS August 20-24, 2018 20 / 102

39. Multiscale Modeling
Multiscale Modeling
1. Molecular statics model (lattice)
2. Continuum model by homogenization
3. Coupling method by Arlequin
4. Adaptation of interface region
+ ⇒
J.T. Oden von Neumann Symposium 2011 July 4–7, 2011 29 / 44
Predictive Models of Random Materials
ODEN MUMS August 20-24, 2018 21 / 102

40. Multiscale Modeling
Construction of Continuum Model
VIRTUAL EXPERIMENTS ON RVE’s:
1) Choose strain energy density function∗
W = W(I1, I2, I3)
2) Exploit isotropy and homogeneity
3) Use molecular RVE to fit parameters
FITTING PROCEDURE:
1) Relax RVE
2) Deform RVE
3) Measure E, V0, Ii
4) Compute W = E/V0
5) Fit W for model parameters
6) Repeat over several samples
∗e.g. Mooney-Rivlin: W = α(I1 − 3) + β(I2 − 3) + γ(J − 1)2 − (2α + 4β)lnJ
J.T. Oden von Neumann Symposium 2011 July 4–7, 2011 31 / 44
Predictive Models of Random Materials
ODEN MUMS August 20-24, 2018 22 / 102

41. Multiscale Modeling
Atomic-to-Continuum Coupling Method
00000000000000000000000
0000000000000000000000000000000000000000000000
11111111111111111111111
1111111111111111111111111111111111111111111111
00
0000
00
11
1111
11 Ωc
domain
Overlap
f
Continuum model
Ωd
Particle model
Ωo
0
1
ααc d
ΩdoΩ
x
Constant
LinearCubic
Ωc
THE ARLEQUIN FRAMEWORK:
1) Partition of energies.
2) Weight coefficients may be
chosen constant, linear,
cubic in overlap region.
3) Coupling through Lagrange
multipliers.
4) Resulting mixed problem is
well-posed.
∗Ben Dhia, Comptes Rendus de l’Acad´emie des Sciences (1998)
∗Xiao and Belytschko, IJNME (2004)
∗Bauman, Ben Dhia, Elkhodja, Oden, and Prudhomme, CM (2008)
J.T. Oden von Neumann Symposium 2011 July 4–7, 2011 32 / 44
Predictive Models of Random Materials
ODEN MUMS August 20-24, 2018 23 / 102

42. Multiscale Modeling
Atomic-to-Continuum Coupling Method
1) The particle model is replaced
with a continuum model far from
the zone of interest and coupled
to the lattice model via the
Arlequin framework∗.
⇓
Sequence of surrogate models
2) Computational study of solution
convergence with respect to QoI.
J.T. Oden von Neumann Symposium 2011 July 4–7, 2011 33 / 44
Predictive Models of Random Materials
ODEN MUMS August 20-24, 2018 24 / 102

43. Multiscale Modeling
Example: Regular Lattice
E ≈ Ru(w0; p) ≈ Ru(w0; p0)
Step Er (%) η0 η
0 30.4 0.95 1.08
1 25.4 0.95 1.08
2 23.4 0.94 1.11
3 20.2 0.93 1.13
4 18.0 0.92 1.16
5 16.2 0.89 1.18
6 14.8 0.88 1.20
7 14.2 0.88 1.20
8 10.8 0.87 1.21
9 6.5 0.81 1.31
Relative error: Er = |E|/|Q(u)|
Eff. index: η0 = |R(u0, p0)|/|E|
Eff. index: η = |R(u0, p)|/|E|
Dual solution p for uz at center of top face.
G = continuum, R = particle, Y = overlap
J.T. Oden von Neumann Symposium 2011 July 4–7, 2011 34 / 44
Predictive Models of Random Materials
ODEN MUMS August 20-24, 2018 25 / 102

44. Multiscale Modeling
Example: Polymer Lattice
Dual solution p for uz
at center of top face.
Error contributions in QoI
Sequence of adapted surrogate models
∗Bauman, Oden, and Prudhomme, CMAME, 2009.
J.T. Oden von Neumann Symposium 2011 July 4–7, 2011 35 / 44
Predictive Models of Random Materials
ODEN MUMS August 20-24, 2018 26 / 102

45. Predictive Models of Random Materials
2.4. Random Two-Phase Heterogeneous Media
ODEN MUMS August 20-24, 2018 27 / 102

46. Predictive Models of Random Materials
Error in Surrogate Models: Goal-Oriented A Posteriori
Estimates of Modeling Error ∗
The Base (high-fidelity) Model
− · E(ω, x) u(ω, x) = f(ω, x)
ω ∈ Ω , x ∈ D ⊂ Rd
⇓
B(u, v) = F(v) ∀v ∈ H
B∗
(v, w) = Q(v) ∀v ∈ H
The Surrogate Model
− · E0(ω, x) u0(ω, x) = f(ω, x)
ω ∈ Ω , x ∈ D ⊂ Rd
⇓
B0(u0, v) = F(v) ∀v ∈ H
B∗
0 (v, w0) = Q(v) ∀v ∈ H
∗
JTO/Vemaganti, JCP, 2000; Vemaganti, JTO, CMAME 2001 ; JTO, Prudhomme, JCP, 2000;
JTO, Prudhomme, Baumann, Romkes, SIAM J. Sc. Comp., 2006
ODEN MUMS August 20-24, 2018 28 / 102

47. Predictive Models of Random Materials
Error in Surrogate Models: Goal-Oriented A Posteriori
Estimates of Modeling Error ∗
The Base (high-fidelity) Model
− · E(ω, x) u(ω, x) = f(ω, x)
ω ∈ Ω , x ∈ D ⊂ Rd
⇓
B(u, v) = F(v) ∀v ∈ H
B∗
(v, w) = Q(v) ∀v ∈ H
E
D
E u : vdx
H = L2
P Θ ⊗ H1
ΓD
(D)
The Surrogate Model
− · E0(ω, x) u0(ω, x) = f(ω, x)
ω ∈ Ω , x ∈ D ⊂ Rd
⇓
B0(u0, v) = F(v) ∀v ∈ H
B∗
0 (v, w0) = Q(v) ∀v ∈ H
E
D
E0 u : vdx
E
D
f · vdx +
Γ
t · vds
∗
JTO/Vemaganti, JCP, 2000; Vemaganti, JTO, CMAME 2001 ; JTO, Prudhomme, JCP, 2000;
JTO, Prudhomme, Baumann, Romkes, SIAM J. Sc. Comp., 2006
ODEN MUMS August 20-24, 2018 28 / 102

48. Predictive Models of Random Materials
Error in Surrogate Models: Goal-Oriented A Posteriori
Estimates of Modeling Error ∗
The Base (high-fidelity) Model
− · E(ω, x) u(ω, x) = f(ω, x)
ω ∈ Ω , x ∈ D ⊂ Rd
⇓
B(u, v) = F(v) ∀v ∈ H
B∗
(v, w) = Q(v) ∀v ∈ H
E
D
E u : vdx
H = L2
P Θ ⊗ H1
ΓD
(D)
The Surrogate Model
− · E0(ω, x) u0(ω, x) = f(ω, x)
ω ∈ Ω , x ∈ D ⊂ Rd
⇓
B0(u0, v) = F(v) ∀v ∈ H
B∗
0 (v, w0) = Q(v) ∀v ∈ H
E
D
E0 u : vdx
E
D
f · vdx +
Γ
t · vds
E · =
Ω
(·)dP(ω)
∗
JTO/Vemaganti, JCP, 2000; Vemaganti, JTO, CMAME 2001 ; JTO, Prudhomme, JCP, 2000;
JTO, Prudhomme, Baumann, Romkes, SIAM J. Sc. Comp., 2006
ODEN MUMS August 20-24, 2018 28 / 102

49. Predictive Models of Random Materials
Theorem
Under the assumptions laid down thus far, the error e0 = Q(u) − Q(u0) in the
quantity of interest predicted by the surrogate model satisfies the two-sided
bounds,
ηlow ≤ Q(e0) ≤ ηupp
where ηlow and ηupp are computable using only properties of the surrogate and
A:
ηlow =
1
4
(η+
low)2
−
1
4
(η−
upp)2
+ Ru0 (w0)
ηupp =
1
4
(η+
upp)2
−
1
4
(η−
low)2
+ Ru0 (w0)
(η±
upp)2
= s2
I0u0
2
± 2B(I0u0, I0w0)B + s−2
I0w0
2
B, s ∈ R+
η±
low = Rsu0+s−1w0
(u0 + θ±
w0)/ u0 ± θ±
w + 0 B
ODEN MUMS August 20-24, 2018 29 / 102

50. Predictive Models of Random Materials
Theorem
Under the assumptions laid down thus far, the error e0 = Q(u) − Q(u0) in the
quantity of interest predicted by the surrogate model satisfies the two-sided
bounds,
ηlow ≤ Q(e0) ≤ ηupp
where ηlow and ηupp are computable using only properties of the surrogate and
A:
ηlow =
1
4
(η+
low)2
−
1
4
(η−
upp)2
+ Ru0 (w0)
ηupp =
1
4
(η+
upp)2
−
1
4
(η−
low)2
+ Ru0 (w0)
(η±
upp)2
= s2
I0u0
2
± 2B(I0u0, I0w0)B + s−2
I0w0
2
B, s ∈ R+
η±
low = Rsu0+s−1w0
(u0 + θ±
w0)/ u0 ± θ±
w + 0 B
E
D
I0E0 u0 : ·w0dx
ODEN MUMS August 20-24, 2018 29 / 102

51. Predictive Models of Random Materials
Theorem
Under the assumptions laid down thus far, the error e0 = Q(u) − Q(u0) in the
quantity of interest predicted by the surrogate model satisfies the two-sided
bounds,
ηlow ≤ Q(e0) ≤ ηupp
where ηlow and ηupp are computable using only properties of the surrogate and
A:
ηlow =
1
4
(η+
low)2
−
1
4
(η−
upp)2
+ Ru0 (w0)
ηupp =
1
4
(η+
upp)2
−
1
4
(η−
low)2
+ Ru0 (w0)
(η±
upp)2
= s2
I0u0
2
± 2B(I0u0, I0w0)B + s−2
I0w0
2
B, s ∈ R+
η±
low = Rsu0+s−1w0
(u0 + θ±
w0)/ u0 ± θ±
w + 0 B
E
D
I0E0 u0 : ·w0dx
= I − E−1
E0
ODEN MUMS August 20-24, 2018 29 / 102

52. Predictive Models of Random Materials
An Adaptive Modeling Algorithm for Materials
ηest =
N
j=1 η
(l)
est,j :
≥ αηmax , Dj → Dω
< αηmax , Dj → D/Dω
ODEN MUMS August 20-24, 2018 30 / 102

53. Predictive Models of Random Materials
Approximate Error Estimator
Given a nested sequence of solutions of the surrogate models
(u
(k)
0 , w
(k)
0 ) ∈ H × H, k = 1, 2, · · · , N such that
E |Q(u) − Q(u(k)
)| ≥ E |Q(u) − Q(u(k+1)
)|
and
Q(u
(N)
0 ) = Q(u), w
(N)
0 − w H = 0,
then ∀ε ∈ R+
there is a model k = k∗
≤ N
and a constant C = C(κ, κ0, u
(k∗
)
0 ) such that
|E[Q(u) − Q(u
(k∗
)
0 )]| ≤ η
(k∗
)
est + Cε
where
η
(k∗
)
est = E
D
(E − E0) u
(k∗
)
0 · w
(k∗
)
0 dx
ODEN MUMS August 20-24, 2018 31 / 102

54. Predictive Models of Random Materials
Accelerated MC with Surrogate Model
Figure: The geometry and one realization of the two phase material under elastic
deformation.
ODEN MUMS August 20-24, 2018 32 / 102

55. Predictive Models of Random Materials
Accelerated MC with Surrogate Model
Surrogate Models:
l = 1 l = 2 l = 3 l = 4
l = 5 l = 6 l = 7 l = 8
ODEN MUMS August 20-24, 2018 33 / 102

56. Predictive Models of Random Materials
Accelerated MC with Surrogate Model
Selecting the surrogate model given a tolerance γtol = 0.07
E[Q(u)] − E[Q(u0)] ≈ E[ηest] < γtol
surrogate model E[ηest]
l = 1 2.6334
l = 2 0.8857
l = 3 0.5910
l = 4 0.2868
l = 5 0.1966
l = 6 0.1402
l = 7 0.0892
l = 8 0.0460 < γtol
Results of MC simulation using the the base and surrogate models
QoI cost = realizations × dof
Surrogate (l = 8) E[Q(u
(l=8)
0 )] = 33.81040 ≈ 160 × 60532
Base E[Q(u)] = 33.75003 ≈ 160 × 105202
ODEN MUMS August 20-24, 2018 34 / 102

57. Predictive Models of Random Materials
2.5. Strategy II: Selection of Most Plausible Models
If training (or testing) data y ∈ Y is available.
M = set of parametric model classes = {M1, M2, . . . , Mm}
Each M has its own likelihood and parameters θj
Bayes’ rule in expanded form:
π(θj|y, Mj, M) =
π(y|θj, Mj, M)π(θj|Mj, M)
π(y|Mj, M)
, 1 ≤ j ≤ m
where π(y|Mj, M) = π(y|θj, Mj, M)π(θj|Mj, M) dθj
Now apply Bayes’ Rule to the evidence1
:
ρj = π(Mj|y, M) =
π(y|Mj, M)π(Mj|M)
π(y, M)
= model plausibility
m
j=1
ρj = 1
1
H. Jeffreys, 1961; Beck and Yuen, 2004;
E. E. Prudencio & H. Cheung, 2012; Farrel, JTO, Faghihi, 2015.
ODEN MUMS August 20-24, 2018 35 / 102

58. Predictive Models of Random Materials
2.6. Accelerated MLMC with Adaptive Modeling
(L. Scarabosio, B. Wohlmuth)
MLMC Philosophy: To accelerate MC by taking most samples from low accuracy
models (low cost) and very few at high accuracy model (high cost).
Approximate computational expensive QoI = E[Q] with
E[Q] ≈ E[QL] =
L
l=1
E[
Y l
Q(ul
0) − Q(ul−1
0 )]
≈ EL
[QL] =
L
l=1
1
Ml
Ml
j=1
Y l
j
E[Q] − EL
[QL] ≤ |E[Q − QL]| + E[QL] − EL
[QL]
requiring that
|E[Q − QL]| ≤ TOLmodel
E E[QL] − EL
[QL]
2
≤ TOL2
sampling
ODEN MUMS August 20-24, 2018 36 / 102

59. Predictive Models of Random Materials
Determining Sequence of Surrogate Models
Selecting surrogate models associated with each level guided by the
goal-oriented estimation of the modeling error:
1. Choosing possible models:
• Construct sequence of models {Mk}K
k=1 such that MK is the most
accurate among all surrogate models and
E Q(u) − Q(u
(K)
0 ) ≈ E η
(K)
est ≤ TOLmodel
2. Choose the MLMC levels:
• Select {Ml}L
l=1 ⊆ {Mk}K
k=1 that minimizes the computational cost, with
ML = MK
• Estimated work of MLMC estimator
E[W] = TOL−2
sampling
L
l=1
VlWl
where V l
= V[Y l
] and Wl
is the estimated cost of one realization of Y l
ODEN MUMS August 20-24, 2018 37 / 102

60. Predictive Models of Random Materials
Optimal Number of Samples per Level
classical MLMC
Determine number of samples per level: {Ml}L
l=1 to minimize total work,
subject to the constraint
E E[QL] − EL
[QL]
2
≤ TOL2
sampling
Ml = λ V l
W l with λ = 1
T OL2
sampling
L
l=1
√
V lWl
• V l
= V ar[Q(ul
0) − Q(ul−1
0 )]
• Wl
= average cost of computing Q(ul
0) − Q(ul−1
0 ) = sample average for the
number of degree of freedoms at level l
ODEN MUMS August 20-24, 2018 38 / 102

61. Predictive Models of Random Materials
Heat Conduction in Random Two Phase Materials
1m
1m
(0.4,0.4)
Figure: The geometry and one realization of the two phase material.
QoI: q(u) = 1
˜Aq D
χ(x)(−uy)(x) dx where uy is the derivative in y-direction.
ODEN MUMS August 20-24, 2018 39 / 102

62. Predictive Models of Random Materials
MLMC Selected Surrogate Models
Tolerance samples l = 1 samples l = 2 samples l = 3
0.05 7027.50 558.60 200 (80)
0.025 52269.20 4954.10 829.10
0.0125 111694.30 10367.40 1721.70
Table: Number of samples per level for the three-level MLMC.
ODEN MUMS August 20-24, 2018 40 / 102

63. Predictive Models of Random Materials
MLMC Speedup Comparing to MC
Figure: Convergence plot for the two-level and three level Monte Carlo comparing
with the estimated cost for the plain Monte Carlo on the fine scale model.
ODEN MUMS August 20-24, 2018 41 / 102

64. Predictive Models of Random Materials
Elastic Deformation in Random Two Phase Materials
Figure: The geometry and one realization of the two phase material for the elastic
deformation problem.
QoI: q(u) = 1
˜Aq D
χ(x)(−u2)(x) dx where u2 is the displacement in y-direction.
ODEN MUMS August 20-24, 2018 42 / 102

65. Predictive Models of Random Materials
Elasticity: MLMC Selected Surrogate Models
Figure: Selected models for two-level MLMC
ODEN MUMS August 20-24, 2018 43 / 102

66. Predictive Models of Random Materials
Elasticity: MLMC Speedup Comparing to MC
Figure: Convergence plot for the two-level Monte Carlo comparing with the
estimated cost for the plain Monte Carlo on the fine scale model.
ODEN MUMS August 20-24, 2018 44 / 102

67. Predictive Models of Random Materials
2.7. The Inverse Problem: Machine Learning Methods
Stochastic model: − κ(ω, x) u(ω, x) = f(ω, x)
+b.c. ω ∈ Ω, x ∈ D
Synthetic Models of Ω for two-phase random media:
• m(x) ∼= m(ω, x) = parameter field
(u(ω, x) = u(m(x)))
Assume linear parameter-to-observation map G:
G(m) = {G (u(ω, xj))}
Nsenario
j=1
dobs = G(m) + noise with
noise ∼ N(0, Γnoise)
sensors
𝑢 = 𝑔&
𝑢 = 𝑔%
dobs
• Prior
πprior(m) ∼ N(mpr, Cpr)
•
πpost(m) = π(m|dobs) ∝ πlike(dobs|m)πpr(m)
= exp −
1
2
dobs − G(m) 2
Γ−1
noise
−
1
2
m − mpr
2
C−1
pr
ODEN MUMS August 20-24, 2018 45 / 102

68. Predictive Models of Random Materials
Whipple-Matern Covariance
(k2
I − ∆)α/2
m(x) = σW
m(x) =
Rα
Cpr(x, x )dW(x ) = Gaussian Random Field w. Matern Covariance
C(x, x ) =
1
Γ(ν)2ν−1
2ν
k
|x − x |
ν
× Kν
√
2ν
k
|x − x |
= (k2
I − ∆)α/2
−1
κ(m) =
κI + κM
2
+
κI − κM
2
tanh
m − τ
h
size and density of the inclusions are controlled by τ = cut-off threshold,
1/k = correlation length
P. Whipple 1963, Matern, B. 1986, Simpson, D., Lindgren, F., and Rue, H. 2010,
2015, Alexanderian, A. Petra, N., Stadler, G. and Ghattas, O. 2016, Villa, U.,
Petra, N., and Ghattas, O. 2016.
ODEN MUMS August 20-24, 2018 46 / 102

69. Predictive Models of Random Materials
True parameters m(x) and state κ(m(x)):
κ(m(x)) obtained from samples drawn from the posterior distribution
πpost(m):
ODEN MUMS August 20-24, 2018 47 / 102

70. Predictive Models of Random Materials
Surrogate Models for Bayesian Inversion
M ∼ A(m, Sp; u(m, x)) = 0
Q(u) = E[q(u)] ∼ q(m) ∼ π(Q|m, M)
M0 ∼ A0(m0, Sp; u0(m, x)) = 0
Q(u0) = E[q(u0)] ∼ q(m0) ∼ π(Q|m0, M0)
Variational Bayes Method: Data ∼ Y(∈ Sv)
π(y|M) = π(y|m, M)πprior(m|M)dm
= evidence of model M
ln π(y|M) ≥ π(m0) ln
π(y, m|M)
π(m0)
dm
= DKL(π(m0) π(y|m, M))
= H(π(m0)) + m0 ln π(y|m, M)dm
π(m∗
0) = inf E(m0, y, M)
ODEN MUMS August 20-24, 2018 48 / 102

71. Model Selection and OPAL
3. Model Selection and OPAL: the Occam Plausibility
Algorithm
3.1. Bayesian Plausibility
3.2. OPAL: the Occam Plausibility Algorithm
3.3. An OPAL Scheme for Predictive Models of Tumor Growth
ODEN MUMS August 20-24, 2018 49 / 102

72. Model Selection and OPAL
3. Model Selection and OPAL: the Occam Plausibility
Algorithm
Instead of adapting a model to meet error con-
straints, select a model from a class of models
that best fits data – in the presence of uncertain-
ties.
• Bayesian Model Posterior Plausibilities
• OPAL – the Occam Plausibility Algorithm
ODEN MUMS August 20-24, 2018 50 / 102

73. Model Selection and OPAL
3.1. Bayesian Model Plausibilities
M = set of parametric model classes = {P1, P2, . . . , Pm}
Each P has its own likelihood and parameters θj
Bayes’ rule in expanded form:
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; Beck and Yuen, 2004;
E. E. Prudencio & H. Cheung, 2012; Farrel, JTO, Faghihi, 2015.
ODEN MUMS August 20-24, 2018 51 / 102

74. Model Selection and OPAL
3.2. OPAL: the Occam Plausibility Algorithm
Occam’s Razor
Among competing theories that lead to the same
prediction, the one that relies on the fewest
assumptions is the best.
When choosing among a set of models:
M = {P1(θ1), P2(θ2), · · · , Pm(θm)}
The simplest valid model is the best choice.
o simple ⇒ number of relevant parameters
o valid ⇒ passes Bayesian validation test
How do we choose a model that adheres to this principle?
ODEN MUMS August 20-24, 2018 52 / 102

75. Model Selection and OPAL
3.2. OPAL: the Occam Plausibility Algorithm
START
Identify a set of possible models
M = {P1(θ1), . . . , Pm(θm)}
SENSITIVITY ANALYSIS
Fix parameters for which model
output is insensitive, or eliminate
models with such parameters
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∗
ITERATIVE OCCAM STEP
Choose models in
next Occam category
PLAUSIBILITY STEP
Compute plausibilities and
identify most plausible model P∗
j
Does P∗
j have the most
parameters in ¯M?
Identify a new set
of possible models
VALIDATION STEP
Submit P∗
j to validation test
Is P∗
j valid?
Use validated params
to predict QoI
Can additional calibration
data be accessed?
No Yes No
No
Yes
Yes
K. Farrell, JTO, D. Faghihi, 2015
ODEN MUMS August 20-24, 2018 53 / 102

76. Model Selection and OPAL
3.3. An OPAL Scheme for Predictive Models of Tumor
Growth
• Continuum Mixture
Theory (Balance
Laws of Physics)
• Hallmarks of
Cancer
@⇢↵ ↵
@t
+ r · (⇢↵v↵ ↵) =
⇢↵S↵ ⇢↵r · J↵
J↵ = M↵( )rµ↵
divT↵ + ˆp↵ = ⇢↵
dv↵
dt
1  ↵  N = no. of species
x
Mass density
of ⍺th species
Volume fraction
of ⍺th species
Species
velocity
Mass flux Partial
stress
Chemical
potential
Source
control
Represents Many Model Class M
S↵ = prof T (1 T )
apot
µ↵ =
c T ) Reaction Di↵ussion
( T ) + "|r T | · · · ) Phase Field
Represents Many Model Class M
S↵ = prof T (1 T )
apot
µ↵ =
c T ) Reaction Di↵ussion
( T ) + "|r T | · · · ) Phase Field
Represents Many Model Class M
S↵ = prof T (1 T )
apot
µ↵ =
c T ) Reaction Di↵ussion
( T ) + "|r T | · · · ) Phase Field
Represents Many Model Class M
S↵ = prof T (1 T )
apot
µ↵ =
c T ) Reaction Di↵ussion
( T ) + "|r T | · · · ) Phase Field
Represents Many Model Class M
S↵ = prof T (1 T )
apot
µ↵ =
c T ) Reaction Di↵ussion
( T ) + "|r T | · · · ) Phase Field
ODEN MUMS August 20-24, 2018 54 / 102

77. Model Selection and OPAL
Most Complete Phase-Field Model
Tumor: φT = φP + φH + φN
Hypoxic, φH
Proliferative (φP )
Necrotic, φN
Nutrient, φσ
φMDE
φECM Mechanical, u
φα = φα(x, t), x ∈ D, t > 0
α
φα = 1
ODEN MUMS August 20-24, 2018 55 / 102

78. Model Selection and OPAL
∂φT
∂t
= · MT (φT , φECM ) µ + λgrowφσφT (1 − φT ) − λapopφT
µ = Ψ (φT ) − 2
T ∆φT + λ · u − χ0φσ
∂φH
∂t
= · M(φH, φECM ) µ + λP HH (σP H − φσ) φP
− λHP H (φσ − σHP ) φH − λHN H (σHN − φσ) φH
∂φN
∂t
= · M(φN , φECM ) µ+λHN H (σHN − φσ) φH
0 = · G u +
G
1 − 2ν
( · u) + λ φT
∂φσ
∂t
= · Mσδ−1
σ φσ − · Mσχ0 φT − λconsφT φσ
dφECM
dt
= −λECM
decay (φMDE)φECM + λECM
rem φT (1 − φECM )
∂φMDE
∂t
= · MMDE φMDE + λMDE
prod (1 − φMDE)φT − λMDE
decay φMDE
− λECM
decay φECM φMDE
ODEN MUMS August 20-24, 2018 56 / 102

79. Model Selection and OPAL
Extracellular
matrix
Proliferative
tumor cell
Hypoxic
tumor cell
Healthy
cell
Endothelial
cell
Fibroblast
Macrophage
VEGF
VEGF receptor
Necrotic
tumor cell
ODEN MUMS August 20-24, 2018 57 / 102

80. Model Selection and OPAL
Data: Murine Subject-CMRI
Murine model implanted by tumor cells
Magnetic resonance
imaging (MRI)
Source: Huang, Jun, et al.
(2015) Molecular medicine
reports
ODEN MUMS August 20-24, 2018 58 / 102

81. Model Selection and OPAL
Glioma Data With 40Gy Radiotherapy
• MRI at days: 10, 12, 14, 16.5, 18.5, 20.5, 22;
• Carrying capacity: 40761 cells;
• Voxel dimensions (dx×dy×dz): 250 × 250 × 1000 µm.
• Number of voxels: 41 × 61 × 16
10 12 14 16.5 18.5 20.5 22
Calibration
Validation
Prediction
Tumor area evolution at slice=11.
Hormuth II, Weis, Barnes, Miga, Rericha, Quaranta, and Yankeelov, 2015.
ODEN MUMS August 20-24, 2018 59 / 102

82. Model Selection and OPAL
Calibration
6
8
10
12
14
16
18
20
22
24
10 12 14 16 18 20 22
TumorArea(mm2)
Time (day)
Calibration
Likelihood Function
ln (π(yc|θ)) =
N
2
ln(2π) +
N
i=1
− ln(σi) −
1
2
yci − di(θ)
σi
2
• N: number of data points (days) used for calibration;
• di: tumor area/volume model prediction at day i;
• σ2
i : variance of the data (estimated);
ODEN MUMS August 20-24, 2018 60 / 102

83. Model Selection and OPAL
Glioma Data With 40 Gy Radiotherapy
• MRI at days: 10, 12, 14, 16.5, 18.5, 20.5, 22;
6
8
10
12
14
16
18
20
22
24
10 12 14 16 18 20 22
TumorArea(mm2)
Time (day)
Calibration
π(θ|yc)
Invalid
Tumor area evolution at slice=11.
Lima, E., JTO, Wohlmuth, B., Shahmoradi, A., Hormuth II, D., Yankeelov, T.E., CMAME 2017.
ODEN MUMS August 20-24, 2018 61 / 102

84. Model Selection and OPAL
Glioma Data With 40 Gy Radiotherapy
• MRI at days: 10, 12, 14, 16.5, 18.5, 20.5, 22;
6
8
10
12
14
16
18
20
22
24
10 12 14 16 18 20 22
TumorArea(mm2)
Time (day)
Calibration
π(θ|yc) π(θ|yv, yc)
Validation Prediction
Tumor area evolution at slice=11.
Lima, E., JTO, Wohlmuth, B., Shahmoradi, A., Hormuth II, D., Yankeelov, T.E., CMAME 2017.
ODEN MUMS August 20-24, 2018 61 / 102

85. Model Selection and OPAL
Model Derivation
• The tumor cell volume fraction must satisfy its own mass balance given
by
∂φT
∂t
= · MT (φT ) µ + λg
T φT (1 − φT ) −R(Ttreat, ˜Dt)φT
where the chemical potential is given by
µ(φT ) = DφT
E,
and DφT
(·) is the Gâteaux derivative with respect to φT .
• The choice of the total free energy E leads to different model classes:
PF ⇒ E = Ω
Ψ(φT ) +
2
T
2 | φT |2
+ W(φT , E(u)) dx;
RD ⇒ E = Ω
c
2 φ2
T dx;
MD ⇒ E = Ω
c
2 φ2
T + W(φT , E(u)) dx.
ODEN MUMS August 20-24, 2018 62 / 102

86. Model Selection and OPAL
Occam Categories
• PF# = phase-field model
• RD# = reaction-diffusion model without mechanical coupling
• MD# = reaction-diffusion model with mechanical coupling
Model
Variables Parameters (θ)
# θ
φT µ u MT M∗
T c λg
T
¯ET T G ν λ γ γg
RD01 2
PF01 4
RD02 6
RD03 6
MD01 6
RD04 7
MD02 7
MD03 7
PF02 7
MD04 8
PF03 8
PF04 8
PF05 9
E. A. B. F. Lima, JTO, D. A. Hormuth, T. E. Yankeelov, R. C. Almeida, 2016
ODEN MUMS August 20-24, 2018 63 / 102

87. Model Selection and OPAL
Radiotherapy Models
(T01) Memory model
R(Ttreat, ˜Dt) =
λk( ˜Dt)
1+λr(t−Ttreat) , for t ≥ Ttreat,
0, for t < Ttreat,
(T02) Partial memory model
R(Ttreat, ˜Dt) =
αD, for t = Ttreat,
0, for t = Ttreat,
¯λg
T =
λg
T Rg
, for t ≥ Ttreat,
λg
T , for t < Ttreat,
(T03) No memory model
R(Ttreat, ˜Dt) =
κ(1 − exp(−α ˜Dt − β ˜D2
t )), for t = Ttreat,
0, for t = Ttreat,
ODEN MUMS August 20-24, 2018 64 / 102

88. Model Selection and OPAL
Radiotherapy Models
Model
Parameters (θ)
#θ
λr λk α β κ αD Rg
T01 2
T02 2
T03 3
• Ttreat = 14.5: treatment day;
• λr: recovery from the treatment;
• λk: death rate of tumor cells;
• αD: death by radiation;
• α: linear cell kill;
• β: quadratic cell kill;
• κ: positive parameter;
• Rg
: tumor growth reduction;
ODEN MUMS August 20-24, 2018 65 / 102

89. Model Selection and OPAL
Combining the 13 models × 3 treatment types
Model
Variables Parameters (θ)
#θ
Occam
CategoryφT µ u MT M∗
T c λg
T
¯ET T E ν λ γ γg λr λk α β κ αD Rg
RD01T01 4 1
RD01T02 4 1
RD01T03 5 2
PF01T01 6 3
PF01T02 6 3
PF01T03 7 4
RD02T01 8 5
RD02T02 8 5
RD03T01 8 5
RD03T02 8 5
MD01T01 8 5
MD01T02 8 5
RD02T03 9 6
RD03T03 9 6
MD01T03 9 6
RD04T01 9 6
RD04T02 9 6
MD02T01 9 6
MD02T02 9 6
MD03T01 9 6
ODEN MUMS August 20-24, 2018 66 / 102

90. Model Selection and OPAL
Combining the 13 models × 3 treatment types
Model
Variables Parameters (θ)
#θ
Occam
CategoryφT µ u MT M∗
T c λg
T
¯ET T E ν λ γ γg λr λk α β κ αD Rg
MD03T02 9 6
PF02T01 9 6
PF02T02 9 6
RD04T03 10 7
MD02T03 10 7
MD03T03 10 7
PF02T03 10 7
MD04T01 10 7
MD04T02 10 7
PF03T01 10 7
PF03T02 10 7
PF04T01 10 7
PF04T02 10 7
MD04T03 11 8
PF03T03 11 8
PF04T03 11 8
PF05T01 11 8
PF05T02 11 8
PF05T03 12 9
ODEN MUMS August 20-24, 2018 67 / 102

91. Model Selection and OPAL
Results - 40Gy
γ18.5
tol = 1.98mm2
γ20.5
tol = 2.33mm2
γ22.0
tol = 2.47mm2
Model
Occam
Category
Calibration Validation Prediction
Plausibility Plausibility d18.5 ( mm2) d20.5 ( mm2) d22 ( mm2)
RD01T01 1 0.99 1.00 1.13 0.98 1.11
RD01T02 1 0.01 0.00 3.83 6.81 1.70
RD01T03 2 n/a n/a 3.74 6.58 1.64
PF01T01 3 1.00 1.00 1.31 1.72 1.06
PF01T02 3 0.00 0.00 2.55 2.81 1.23
PF01T03 4 n/a n/a 6.67 12.40 3.45
RD02T01 5 0.15 0.71 1.08 1.07 1.12
RD02T02 5 0.00 0.00 1.64 1.08 1.13
RD03T01 5 0.20 0.23 1.16 1.14 0.99
RD03T02 5 0.01 0.00 4.11 7.26 1.94
MD01T01 5 0.63 0.06 1.43 1.49 1.07
MD01T02 5 0.01 0.00 1.97 1.84 1.14
ODEN MUMS August 20-24, 2018 68 / 102

92. Model Selection and OPAL
Results - 40Gy
γ18.5
tol = 1.98mm2
γ20.5
tol = 2.33mm2
γ22.0
tol = 2.47mm2
Model
Occam
Category
Calibration Validation Prediction
Plausibility Plausibility d18.5 ( mm2) d20.5 ( mm2) d22 ( mm2)
RD02T03 6 0.01 0.00 3.69 6.46 1.62
RD03T03 6 0.01 0.00 4.45 8.31 2.03
MD01T03 6 0.05 0.00 4.18 7.69 1.85
RD04T01 6 0.06 0.15 0.92 2.22 0.91
RD04T02 6 0.00 0.00 2.16 2.36 1.15
MD02T01 6 0.05 0.02 1.69 2.08 1.20
MD02T02 6 0.00 0.00 2.14 2.43 1.16
MD03T01 6 0.04 0.53 1.29 1.45 1.07
MD03T02 6 0.00 0.00 2.62 3.73 1.21
PF02T01 6 0.74 0.30 1.15 1.72 1.07
PF02T02 6 0.03 0.00 1.82 1.22 1.11
ODEN MUMS August 20-24, 2018 69 / 102

93. Model Selection and OPAL
Results - 40Gy
γ18.5
tol = 1.98mm2
γ20.5
tol = 2.33mm2
γ22.0
tol = 2.47mm2
Model
Occam
Category
Calibration Validation Prediction
Plausibility Plausibility d18.5 ( mm2) d20.5 ( mm2) d22 ( mm2)
RD04T03 7 0.01 0.00 4.03 7.12 1.87
MD02T03 7 0.02 0.00 7.99 15.30 4.26
MD03T03 7 0.02 0.00 4.58 8.88 2.05
PF02T03 7 0.05 0.00 7.64 14.73 4.08
MD04T01 7 0.04 0.41 1.51 1.75 1.12
MD04T02 7 0.01 0.00 1.98 2.03 1.16
PF03T01 7 0.41 0.26 1.30 1.71 1.09
PF03T02 7 0.00 0.00 1.78 1.17 1.13
PF04T01 7 0.43 0.33 1.26 1.75 1.09
PF04T02 7 0.00 0.00 1.85 1.28 1.11
MD04T03 8 0.05 0.00 6.96 15.16 3.47
PF03T03 8 0.05 0.00 5.60 11.76 2.59
PF04T03 8 0.03 0.00 7.12 13.53 3.74
PF05T01 8 0.83 1.00 1.25 1.88 1.07
PF05T02 8 0.03 0.00 1.72 1.14 1.14
PF05T03 9 n/a n/a 7.39 14.12 3.92
ODEN MUMS August 20-24, 2018 70 / 102

94. Model Selection and OPAL
RD01 - 3D With 40 Gy Radiotherapy
Tumor at day t = 11, treatment at day t = 14.5.
ODEN MUMS August 20-24, 2018 71 / 102

95. Model Selection and OPAL
RD01 - 3D With 40 Gy Radiotherapy
Tumor at day t = 14, treatment at day t = 14.5.
ODEN MUMS August 20-24, 2018 72 / 102

96. Model Selection and OPAL
RD01 - 3D With 40 Gy Radiotherapy
Tumor at day t = 17, treatment at day t = 14.5.
ODEN MUMS August 20-24, 2018 73 / 102

97. Model Selection and OPAL
4. Concluding Comments
1. The Central Problem of Predictive Computational Science is the selection of
valid model and model parameters for a given QoI.
2. Bayesian setting provides a natural framework for handling all sources of
uncertainty in predictive modeling.
3. Goal-Oriented estimates of error in surrogate model approximations of QoIs
provide a basis for analyzing random heterogeneous media with reduced
models involving considerably fewer dofs than the high fidelity base model.
4. The adaptive modeling algorithm provides a framework for developing new
model-based Multilevel Monte Carlo methods that can often provide
considerable savings in cost over the traditional Monte Carlo method.
5. Predictive multiscale models of random heterogeneous materials and tumor
growth can be developed if sufficient relative data are available.
ODEN MUMS August 20-24, 2018 74 / 102

98. Model Selection and OPAL
General
Oden, J. T. (2018). Adaptive multiscale predictive modelling. Acta Numerica, 27, 353-450.
Goal Oriented Error Estimation
Oden, J. T., Prudhomme, S. (2002). Estimation of modeling error in computational mechanics.
Journal of Computational Physics, 182(2), 496-515.
Bauman, P. T., Oden, J. T., Prudhomme, S. (2009). Adaptive multiscale modeling of polymeric
materials with Arlequin coupling and Goals algorithms. Computer Methods in Applied
Mechanics and Engineering, 198(5-8), 799-818.
Oden, J. T., Prudhomme, S., Romkes, A.,Bauman, P. T. (2006). Multiscale modeling of physical
phenomena: Adaptive control of models. SIAM Journal on Scientific Computing, 28(6),
2359-2389.
Oden, J. T., Vemaganti, K. S. (2000). Estimation of local modeling error and goal-oriented
adaptive modeling of heterogeneous materials: I. Error estimates and adaptive algorithms.
Journal of Computational Physics, 164(1), 22-47.
Vemaganti, K. S., Oden, J. T. (2001). Estimation of local modeling error and goal-oriented
adaptive modeling of heterogeneous materials: Part II: a computational environment for
adaptive modeling of heterogeneous elastic solids. Computer Methods in Applied Mechanics
and Engineering, 190(46-47), 6089-6124.
ODEN MUMS August 20-24, 2018 75 / 102

99. Model Selection and OPAL
Tumor Growth Papers
Hawkins-Daarud, A., van der Zee, K. G., Tinsley Oden, J. (2012). Numerical simulation of a
thermodynamically consistent four?species tumor growth model. International journal for
numerical methods in biomedical engineering, 28(1), 3-24.
Oden, J. T., Hawkins, A., Prudhomme, S. (2010). General diffuse-interface theories and an
approach to predictive tumor growth modeling. Mathematical Models and Methods in Applied
Sciences, 20(03), 477-517.
Oden, J. T., Lima, E. A., Almeida, R. C., Feng, Y., Rylander, M. N., Fuentes, D., Faghihi, D.,
Rahman, M.M., DeWitt, M., Gadde, M., Zhou, J. C. (2016). Toward predictive multiscale
modeling of vascular tumor growth. Archives of Computational Methods in Engineering, 23(4),
735-779.
Lima, E. A. B. F., Oden, J. T., Almeida, R. C. (2014). A hybrid ten-species phase-field model of
tumor growth. Mathematical Models and Methods in Applied Sciences, 24(13), 2569-2599.
Lima, E. A. B. F., Oden, J. T., Hormuth, D. A., Yankeelov, T. E., Almeida, R. C. (2016). Selection,
calibration, and validation of models of tumor growth. Mathematical Models and Methods in
Applied Sciences, 26(12), 2341-2368.
Lima, E. A. B. F., Oden, J. T., Wohlmuth, B., Shahmoradi, A., Hormuth II, D. A., Yankeelov, T. E.,
Scarabosiob, L., Horger, T. (2017). Selection and validation of predictive models of radiation
effects on tumor growth based on noninvasive imaging data. Computer methods in applied
mechanics and engineering, 327, 277-305.
ODEN MUMS August 20-24, 2018 76 / 102

100. Model Selection and OPAL
Multiscale Models of Cancer
ODEN MUMS August 20-24, 2018 77 / 102

101. Model Selection and OPAL
Definition of Agent-Based Model
Agent-based model (ABM) is a method of
computational modeling that simulates inter-
actions and actions of autonomous agents
tumor and healthy cells
with the purpose of viewing their effects on
the system as a whole
tumor environment
.
ODEN MUMS August 20-24, 2018 78 / 102

102. Model Selection and OPAL
Avascular Tumor Growth Model
Avascular Tumor Growth Model
Nutrient
P D E
Tumor Cells
Quiescent, Proliferative,
Hypoxic, Apoptotic
and Necrotic
A B M
Healthy Cells
Homeostasis
A B M
Consumption
Consumption
Tissue Scale
Cell Scale
Ernesto A. B. F. Lima Avascular Tumor Growth Model February 14, 2018 7 / 45
ODEN MUMS August 20-24, 2018 79 / 102

103. Model Selection and OPAL
Cell Representation
• Cell geometry is a perturbation of a spherical
(circular) core.
• Two neighbor cells may deform, which is cap-
tured by cytoplasm overlap.
• Cell nucleus is incompressible.
• Each cell has phenotypic and physical prop-
erties:
Phenotypic properties
cell state: Si
calcification: Ci
proliferation: αP,i
apoptosis: αA,i
Physical properties
cell radius: Ri
nucleus radio: Ri
N
action radius: Ri
A
position: xi
velocity: vi
Ri
Ri
N
Ri
A
xi
ODEN MUMS August 20-24, 2018 80 / 102

104. Model Selection and OPAL
Cell Forcesell Forces
Cell-cell adhesion and repulsion:
Fij
cca = cccar'(xj xi ; Ri
A + Rj
A),
Fij
ccr = cccr r (xj xi ; Ri
N + Rj
N, Ri + Rj ).
E↵ects of the compressive stress:
Fi
ct = cctK(V , t)r'(di
; Ri
A),
Fi
rct = crtcK(V , t)r (di
; Ri
N, Ri ).
Drag force: Fi
drag = ⌫vi .
K(V , t): a real function in [0, 1] depending on the tumor volume V at time t;
di
: the distance between the center of the cell and the domain boundary;
ccca, cccr , cct, crtc and ⌫: scale parameters;
' and : interaction potentials for adhesion and repulsion, respectively.
Ernesto A. B. F. Lima Avascular Tumor Growth Model February 14, 2018 9 /
• Cell-cell adhesion and repulsion:
Fij
cca = −ccca ϕ(xj − xi; Ri
A + Rj
A)
Fij
ccr = −cccr ϕ(xj − xi; Ri
N + Rj
N , Ri + Rj)
• Cell-cell adhesion and repulsion:
Fij
ct = −cctK(V, t) ϕ(di
; Ri
A)
Fij
rct = −crctK(V, t) ϕ(di
; Ri
N , Ri)
Macklin, P., Kim, J., Tomaiuolo, G., Edgerton, M. E., Cristini, V. Comp. Biology, 2009.
Rocha, H. L., Almeida, R. C., Lima, E. A. B. F., Resende, A. C. M., Oden, J. T., Yankeelov,
T. E., M3MS, 2018.
ODEN MUMS August 20-24, 2018 81 / 102

105. Model Selection and OPAL
Balance of Forces ⇒ cell movement
alance of forces ) cell movement
Balance of forces on cell i:
mi ˙vi =
cell-cell
z }| {
N(t)
X
j=1
j6=i
(Fij
cca + Fij
ccr ) +
cell-microenvironment
z }| {
(Fi
drag + Fi
ct + Fi
rct)
ODEN MUMS August 20-24, 2018 82 / 102

106. Model Selection and OPAL
Tumor Cell Transition
Tumor Cell Transition
P P
S and G2
⌧P ⌧G1
Q
G1⌧G1↵P( )
H
< H
< H
N
⌧N
A
↵A
Removed
⌧A
P
PTumor Cells:
P: Proliferative
Q: Quiescent
H: Hypoxic
N: Necrotic
A: Apoptotic
Nutrient:
⌧N, ⌧A, ⌧P and ⌧G1 : time for necrosis, apoptosis, cell cycle, and G1 phase, respectively;
H: hypoxic threshold;
↵P and ↵A: proliferation and apoptosis rate, respectively;
Red Arrows: stochastic process and Black Arrows: deterministic process.
Ernesto A. B. F. Lima Avascular Tumor Growth Model February 14, 2018 11 /
τs = time to phase change
αP , αA ∼ proliferation and apoptosis rate
ODEN MUMS August 20-24, 2018 83 / 102

107. Model Selection and OPAL
Stochastic Transition
Considering small time intervals [t, t + ∆t]
• The transition from quiescent (Q) to apoptotic (A) is given by
P(A|Q) = 1 − exp(αA∆t),
∆t is the time interval.
• The transition from quiescent (Q) to proliferative (P) is given by
P(P|Q) = 1 − exp(αP (σ)∆t),
αP (σ) = ¯αP
σ − σH
1 − σH
1 −
Nt
out
Nmax
out
,
Nt
out: is the number of cells outside the domain;
Nmax
out : is maximum number of cells that can leave the domain.
ODEN MUMS August 20-24, 2018 84 / 102

108. Model Selection and OPAL
Numerical Experiments
Tumor cells: 5 Nutrient
Numerical Experiments
Healthy
cells
Quiescent
tumor
cells
Proliferative
tumor
cells
Apoptotic
tumor
cells
Necrotic
tumor
cells
Figure: Day 0
Ernesto A. B. F. Lima Avascular Tumor Growth Model February 14, 2018 15 / 45
ODEN MUMS August 20-24, 2018 85 / 102

109. Model Selection and OPAL
Numerical Experiments
Tumor cells: 171 Nutrient
Numerical Experiments
Healthy
cells
Quiescent
tumor
cells
Proliferative
tumor
cells
Apoptotic
tumor
cells
Necrotic
tumor
cells
Figure: Day 8.33
Ernesto A. B. F. Lima Avascular Tumor Growth Model February 14, 2018 15 / 45
ODEN MUMS August 20-24, 2018 86 / 102

110. Model Selection and OPAL
Numerical Experiments
Tumor cells: 861 Nutrient
Numerical Experiments
Healthy
cells
Quiescent
tumor
cells
Proliferative
tumor
cells
Apoptotic
tumor
cells
Necrotic
tumor
cells
Figure: Day 16.67
Ernesto A. B. F. Lima Avascular Tumor Growth Model February 14, 2018 15 / 45
ODEN MUMS August 20-24, 2018 87 / 102

111. Model Selection and OPAL
IncuCyte Live Cell Analysis
Figure: Image of the whole well
with the mask for cell confluence
(area of the well 0.32cm2
.)
Live/death cell confluence time series
• 3 initial confluences (low, medium and
high);
• 4 glucose levels (1, 2, 5 and 10 mM);
• 4 replicas of each;
• 48 wells.
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0 50 100 150 200 250 300 350 400
Confluence
Time (hours)
10 mM glucose
The line represents the mean of 4 replicas.
ODEN MUMS August 20-24, 2018 88 / 102

112. Model Selection and OPAL
Avascular Model
Tumor Cell Transition
P P
S and G2
⌧P ⌧G1
Q
G1⌧G1↵P( )
D
↵D( )
Removed
⌧D
P
P
Tumor Cells:
P: Proliferative
Q: Quiescent
D: Dying
Nutrient:
⌧N, ⌧D, ⌧P and ⌧G1 : time for necrosis, to die, cell cycle, and G1 phase, respectively;
↵P and ↵D: proliferation and death rate, respectively.
Ernesto A. B. F. Lima Avascular Tumor Growth Model February 14, 2018 20 / 45
ODEN MUMS August 20-24, 2018 89 / 102

113. Model Selection and OPAL
Stochastic Transition
Considering small time intervals [t, t + ∆t]
• The transition from quiescent (Q) to dying (D) is given by
P(D|Q) = 1 − exp(αD(σ)∆t), αD(σ) = ¯αD (1 − σ) ,
∆t is the time interval.
• The transition from quiescent (Q) to proliferative (P) is given by
P(P|Q) = 1 − exp(αP (σ)∆t), αP (σ) = ¯αP σ.
ODEN MUMS August 20-24, 2018 90 / 102

114. Model Selection and OPAL
Numerical Experiments
Numerical Experiments
Figure: Agent-based model (left) and nutrient (right) evolution.
Ernesto A. B. F. Lima Avascular Tumor Growth Model February 14, 2018 22 / 45
ODEN MUMS August 20-24, 2018 91 / 102

115. Model Selection and OPAL
Numerical Experiments
Numerical Experiments
Figure: Agent-based model (left) and nutrient (right) evolution.
Ernesto A. B. F. Lima Avascular Tumor Growth Model February 14, 2018 22 / 45
ODEN MUMS August 20-24, 2018 92 / 102

116. Model Selection and OPAL
Stochastic Calibration
Stochastic Calibration
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0 50 100 150 200 250 300 350 400
Confluence
Time (hours)
10 mM glucose
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 20 40 60 80 100 120 140 160 180 200
Confluence
Time Steps
Generated Confluence
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0 2 4 6 8 10 12 14 16 18 20
Confluence
Time Steps
Generated Confluence
Figure: Mean of 4 in vitro experiments (top) and 10 model simulations (bottom).
Ernesto A. B. F. Lima Avascular Tumor Growth Model February 14, 2018 23 / 45
ODEN MUMS August 20-24, 2018 93 / 102

117. Model Selection and OPAL
Stochastic Calibration: Likelihood Function
For both calibration and validation we assume:
Additive Noise :
Ωreal + ηdata = y,
Ωreal + ηmodel = d(θ).
• ηdata ∼ N(0N×1, σ2
dataIN×N ) → accounting for the data uncertainty;
• ηmodel ∼ N(0N×1, σ2
modelIN×N ) → accounting for the model inadequacy.
Considering the parameter σ such as σ2
= σ2
data + σ2
model:
Likelihood Function
π(ˆy|θ) =
Nt
i=1
1
2πσ2
i
e
−
(ˆyi − ˆdi(θ))2
2σ2
i ,
ˆdi(θ) is the mean of Mm realizations of the stochastic model;
ˆyi is the mean of 4 data points.
ODEN MUMS August 20-24, 2018 94 / 102

118. Model Selection and OPAL
Vascular Tumor Growth Model
Vascular Tumor Growth Model
Nutrient
P D E
VEGF
P D E
Tumor Cells
Quiescent, Proliferative,
Hypoxic, Apoptotic
and Necrotic
A B M
Endothelial Cells
Quiescent, Tip, Stalk
A B M
Consumption
Release Release
Consumption
Tissue Scale
Cell Scale
Ernesto A. B. F. Lima Vascular Tumor Growth Model February 14, 2018 33 / 45
ODEN MUMS August 20-24, 2018 95 / 102

119. Model Selection and OPAL
Tumor Cell Transition
Tumor Cell Transition
P P
S and G2
⌧P ⌧G1
Q
G1⌧G1↵P( )
H
< H
< H
N
⌧N
A
↵A
Removed
⌧A
P
PTumor Cells:
P: Proliferative
Q: Quiescent
H: Hypoxic
N: Necrotic
A: Apoptotic
Nutrient:
⌧N, ⌧A, ⌧P and ⌧G1 : time for necrosis, apoptosis, cell cycle, and G1 phase, respectively;
H: hypoxic threshold;
↵P and ↵A: proliferation and apoptosis rate, respectively;
Red Arrows: stochastic process and Black Arrows: deterministic process.
Ernesto A. B. F. Lima Avascular Tumor Growth Model February 14, 2018 11 / 45
ODEN MUMS August 20-24, 2018 96 / 102

120. Model Selection and OPAL
Endothelial Cell Transition
Endothelial Cell Transition
S S
⌧M > ⌧grow
⌧C
TQE
vegf < tip
dtip < D dtip < D
S
S
Endothelial Cells:
QE : Quiescent
T : Tip
S: Stalk
VEGF: vegf
⌧M, ⌧grow , and ⌧C : time to mature, grow, and endothelial cell cycle, respectively;
tip: VEGF threshold;
dtip: distance from the closest tip cell;
D: minimum distance required to become stalk cell.
Ernesto A. B. F. Lima Vascular Tumor Growth Model February 14, 2018 34 / 45
ODEN MUMS August 20-24, 2018 97 / 102

121. Model Selection and OPAL
Lumen Formation
Lumen Formation
QE
+
QE
+
QE
+
QE
+
S
+
S
+ T
Frep
Frep
Frep
Frep Frep Frep
QE
+
QE
+
QE
+
QE
+
S
+
S
+ T
Frep
Frep Frep
Frep
Frep Frep
Lumen repulsion:
Fij
rep = ceer (xj xi ; Ri
N + Rj
N, Ri + Rj ).
Ernesto A. B. F. Lima Vascular Tumor Growth Model February 14, 2018 35 / 45
ODEN MUMS August 20-24, 2018 98 / 102

122. Model Selection and OPAL
Tip Cell Movement
Tip Cell Movement
QE
QE
QE
QE
S
S
T Fvegf
VEGF Force:
Fi
vegf = cvegf r vegf .
Ernesto A. B. F. Lima Vascular Tumor Growth Model February 14, 2018 36 / 45
ODEN MUMS August 20-24, 2018 99 / 102

123. Model Selection and OPAL
Numerical Simulation
Figure: Agent-based model (left), nutrient (middle) and VEGF (right) evolution.
ODEN MUMS August 20-24, 2018 100 / 102

124. Model Selection and OPAL
Numerical Simulation
Figure: Agent-based model (left), nutrient (middle) and VEGF (right) evolution.
ODEN MUMS August 20-24, 2018 100 / 102

125. Model Selection and OPAL
Numerical Simulation
Figure: Agent-based model (left), nutrient (middle) and VEGF (right) evolution.
ODEN MUMS August 20-24, 2018 100 / 102

126. Model Selection and OPAL
Numerical Simulation
Figure: Agent-based model (left), nutrient (middle) and VEGF (right) evolution.
ODEN MUMS August 20-24, 2018 100 / 102

127. Model Selection and OPAL
Numerical Simulation
Figure: Agent-based model (left), nutrient (middle) and VEGF (right) evolution.
ODEN MUMS August 20-24, 2018 100 / 102

128. Model Selection and OPAL
Unbreakable Vessel
Figure: Agent-based model evolution.
ODEN MUMS August 20-24, 2018 101 / 102

129. Model Selection and OPAL
Unbreakable Vessel
Figure: Agent-based model evolution.
ODEN MUMS August 20-24, 2018 101 / 102

130. Model Selection and OPAL
Unbreakable Vessel
Figure: Agent-based model evolution.
ODEN MUMS August 20-24, 2018 101 / 102

131. Model Selection and OPAL
Unbreakable Vessel
Figure: Agent-based model evolution.
ODEN MUMS August 20-24, 2018 101 / 102

132. Model Selection and OPAL
Breakable Vessel
Figure: Agent-based model evolution.
ODEN MUMS August 20-24, 2018 102 / 102

133. Model Selection and OPAL
Breakable Vessel
Figure: Agent-based model evolution.
ODEN MUMS August 20-24, 2018 102 / 102

134. Model Selection and OPAL
Breakable Vessel
Figure: Agent-based model evolution.
ODEN MUMS August 20-24, 2018 102 / 102

135. Model Selection and OPAL
Breakable Vessel
Figure: Agent-based model evolution.
ODEN MUMS August 20-24, 2018 102 / 102