Heterogeneity in biological populations, from cancer to ecological systems, is ubiquitous. Despite this knowledge, current mathematical models in population biology often do not account for inter-individual heterogeneity. In systems such as cancer, this means assuming cellular homogeneity and deterministic phenotypes, despite the fact that heterogeneity is thought to play a role in therapy resistance. Glioblastoma Multiforme (GBM) is an aggressive and fatal form of brain cancer notoriously difficult to predict and treat due to its heterogeneous nature. In this talk, I will discuss several approaches I have developed towards incorporating and estimating cellular heterogeneity in partial differential equation (PDE) models of GBM growth.

1. Modeling and estimating biological
heterogeneity in spatiotemporal data
Erica Rutter
Department of Mathematics
Center for Research in Scientific Computation
North Carolina State University
Funding: NSF Math Biology (DMS-1514929), US EPA STAR (RD-835165), DOE GAANN (P200A120120)
October 23, 2018
SAMSI E&O: Undergraduate Workshop

2. Biological Heterogeneity
Source: Banks, H. T. & Davis, J. L. A
comparison of approximation methods for
the estimation of probability distributions
on parameters. Appl. Numer. Math. 57,
753–777, (2007).
𝜕𝑢(𝑡, 𝑙)
𝜕𝑡
+
𝜕𝑢(𝑡, 𝑙)
𝜕𝑙
= −μ𝑢(𝑡, 𝑙)

3. Partial Differential Equations in
Biological Systems
Source: Falconer, Heather MacNeill, et al.
“Diffusion I | Chemistry.” Visionlearning,
Visionlearning, Inc., 11 Feb. 2017,
www.visionlearning.com/en/library/Chemi
stry/1/Diffusion-I/216.

4. Partial Differential Equations in
Biological Systems
Source: Falconer, Heather MacNeill, et al.
“Diffusion I | Chemistry.” Visionlearning,
Visionlearning, Inc., 11 Feb. 2017,
www.visionlearning.com/en/library/Chemi
stry/1/Diffusion-I/216.
Movement of particles from high concentration -> low
concentration

5. Partial Differential Equations in
Biological Systems
Source: Falconer, Heather MacNeill, et al.
“Diffusion I | Chemistry.” Visionlearning,
Visionlearning, Inc., 11 Feb. 2017,
www.visionlearning.com/en/library/Chemi
stry/1/Diffusion-I/216.
Movement of particles from high concentration -> low
concentration
Flux = −𝐷
𝜕𝑢
𝜕𝑥

6. Partial Differential Equations in
Biological Systems
Source: Falconer, Heather MacNeill, et al.
“Diffusion I | Chemistry.” Visionlearning,
Visionlearning, Inc., 11 Feb. 2017,
www.visionlearning.com/en/library/Chemi
stry/1/Diffusion-I/216.
Movement of particles from high concentration -> low
concentration
Flux = −𝐷
𝜕𝑢
𝜕𝑥
Using Laws of Conservation:
𝜕𝑢
𝜕𝑡
+
𝜕
𝜕𝑥
Flux = 0

7. Partial Differential Equations in
Biological Systems
Source: Falconer, Heather MacNeill, et al.
“Diffusion I | Chemistry.” Visionlearning,
Visionlearning, Inc., 11 Feb. 2017,
www.visionlearning.com/en/library/Chemi
stry/1/Diffusion-I/216.
Movement of particles from high concentration -> low
concentration
Flux = −𝐷
𝜕𝑢
𝜕𝑥
Using Laws of Conservation:
𝜕𝑢
𝜕𝑡
+
𝜕
𝜕𝑥
Flux = 0
Which becomes
𝜕𝑢
𝜕𝑡
= 𝐷
𝜕2 𝑢
𝜕𝑥2

8. Partial Differential Equations in
Biological Systems
𝜕𝑢
𝜕𝑡
= 0.1
𝜕2 𝑢
𝜕𝑥2
𝑢 𝑥, 0 = 𝑒
−
𝑥2
0.01
𝜕𝑢
𝜕𝑥
−1, 𝑡 = 0 =
𝜕𝑢
𝜕𝑥
(1, 𝑡)
Source: Falconer, Heather MacNeill, et al.
“Diffusion I | Chemistry.” Visionlearning,
Visionlearning, Inc., 11 Feb. 2017,
www.visionlearning.com/en/library/Chemi
stry/1/Diffusion-I/216.

9. Partial Differential Equations in
Biological Systems
Source: Epanchin-Niell, Rebecca. “Examining the Benefits
of Invasive Species Prevention: The Role of Invader
Temporal Characteristics.” Resources for the Future, 31
Aug. 2017, www.rff.org/blog/2015/examining-benefits-
invasive-species-prevention-role-invader-temporal-
characteristics.

10. Partial Differential Equations in
Biological Systems
𝜕𝑢
𝜕𝑡
= 0.01
𝜕2 𝑢
𝜕𝑥2
+ 0.2𝑢(1 − 𝑢)
𝑢 𝑥, 0 = 𝑒
−
𝑥2
0.01
𝑢 −10, 𝑡 = 0 = 𝑢(10, 𝑡)
Source: Epanchin-Niell, Rebecca. “Examining the Benefits
of Invasive Species Prevention: The Role of Invader
Temporal Characteristics.” Resources for the Future, 31
Aug. 2017, www.rff.org/blog/2015/examining-benefits-
invasive-species-prevention-role-invader-temporal-
characteristics.

11. Glioblastoma
Multiforme Models

12. Glioblastoma Multiforme (GBM)
Sagittal cross-section of human brain with
GBM
𝑑𝑢
𝑑𝑡
= 𝑓 𝑢, 𝑡, 𝑥,
𝑑𝑢
𝑑𝑥
,
𝑑2 𝑢
𝑑𝑥2 , 𝑃
Cancers often modeled by
partial differential
equations, because they can
incorporate
• Spatial structures
• Diffusion
• Taxis

13. Reaction-Diffusion Equation
𝜕𝑢
𝜕𝑡
= 𝐷
𝜕2 𝑢
𝜕𝑥
diffusion
+ 𝜌𝑢(1 − 𝑢)
logistic growth
Extensively used to model biological systems from
cancer to spread of rabies among animals.

14. Experimental Data
MR images from day 25 for all mice
• Mice were imaged using MR 5
times (day 11, 15, 18, 22, 25)
• Mice were euthanized on day
26 brains harvested for
histology
• Why large differences in final
tumor size? Time Since Implantation (days)
10 15 20 25
Visbletumorvolume(mm3
)
0
10
20
30
40
50
60
70
Mouse 1
Mouse 2
Mouse 3

15. Parameter Estimation
Hypotheses
a) Fit 𝐷 and 𝜌 over the full time course
b) Update 𝐷 and 𝜌 over each time step
c) Fit 𝐷 and 𝜌 to each time point, using new MRI as initial
condition (short-term predictions)
Seek to minimize:
𝐸(𝐷, 𝜌) =
1
𝑛
𝑘=1
𝑛
1 −
data(𝑡 𝑘) sim(𝑡 𝑘, 𝐷, 𝜌)
data(𝑡 𝑘) sim(𝑡 𝑘, 𝐷, 𝜌)
𝜕𝑢
𝜕𝑡
= 𝐷
𝜕2 𝑢
𝜕𝑥
diffusion
+ 𝜌𝑢(1 − 𝑢)
logistic growth

16. Results
Representative simulation results for Mouse 1 at day 25 following implantation under (a) Hypothesis 1, (b) Hypothesis 2,
and (c) Hypothesis 3. Red represents simulated enhancing tumor volume and green is the segmented regions of
enhancement in the laboratory tumor.
Rutter et al. Mathematical Analysis of Glioma Growth in a Murine Model. Scientific Reports, 2017

17. Confidence in Parameter Estimates
Parameters are non-identifiable!
Inability to relate MR image intensities
with cell densities represents a major
hurdle in parameterizing mathematical
models to in vivo data
Rutter et al. Mathematical Analysis of Glioma Growth in a Murine Model. Scientific Reports, 2017
Mouse 1 Mouse 2 Mouse 3

18. Current Road Blocks in GBM Modeling
1. Assumptions of cellular homogeneity are unreliable
and overly optimistic
2. Non-identifiability of the simplest in vivo models

19. Modeling heterogeneity
in in vitro models of
GBM growth

20. Importance of heterogeneity
Source: Saunders, Nicholas A., et al. "Role of intratumoural heterogeneity in
cancer drug resistance: molecular and clinical perspectives." EMBO molecular
medicine 4.8 (2012): 675-684.

21. In vitro Data of Glioma Growth
Source: Stein et al. "A mathematical model of glioblastoma tumor spheroid invasion in a three-dimensional in vitro
experiment." Biophysical Journal 92.1 (2007): 356-365.

22. Model 1: Separate Phenotypes
Stepien, Rutter, and Kuang. Traveling Waves of a Go-or-Grow Model of Glioma Growth. SIAM Journal of Applied Mathematics, 2018
𝑃 𝑀
𝜕𝑃
𝜕𝑡
= 𝑔 1 −
𝑇
𝑇max
𝑃
proliferation
𝑀

23. Model 1: Separate Phenotypes
𝑃 𝑀
𝜕𝑃
𝜕𝑡
= 𝑔 1 −
𝑇
𝑇max
𝑃
proliferation
− 𝜀𝑘
𝑇 𝑛
𝑇 𝑛 + 𝐾 𝑀
𝑛 𝑃
transition from
proliferating to
migrating
+ 𝑘
𝐾 𝑃
𝑛
𝑇 𝑛 + 𝐾 𝑃
𝑛 𝑀
transition from
migrating to
proliferating
𝜕𝑀
𝜕𝑡
= 𝐷
𝜕2 𝑀
𝜕𝑥2
diffusion
+ 𝜀𝑘
𝑇 𝑛
𝑇 𝑛 + 𝐾 𝑀
𝑛 𝑃
transition from
proliferating to
migrating
− 𝑘
𝐾 𝑃
𝑛
𝑇 𝑛 + 𝐾 𝑃
𝑛 𝑀
transition from
migrating to
proliferating
− 𝜇𝑀
cell
death
𝑀
Stepien, Rutter, and Kuang. Traveling Waves of a Go-or-Grow Model of Glioma Growth. SIAM Journal of Applied Mathematics, 2018

24. Model 1: Separate Phenotypes
𝑃 𝑀
𝜕𝑃
𝜕𝑡
= 𝑔 1 −
𝑇
𝑇max
𝑃
proliferation
− 𝜀𝑘
𝑇 𝑛
𝑇 𝑛 + 𝐾 𝑀
𝑛 𝑃
transition from
proliferating to
migrating
+ 𝑘
𝐾 𝑃
𝑛
𝑇 𝑛 + 𝐾 𝑃
𝑛 𝑀
transition from
migrating to
proliferating
𝜕𝑀
𝜕𝑡
= 𝐷
𝜕2 𝑀
𝜕𝑥2
diffusion
+ 𝜀𝑘
𝑇 𝑛
𝑇 𝑛 + 𝐾 𝑀
𝑛 𝑃
transition from
proliferating to
migrating
− 𝑘
𝐾 𝑃
𝑛
𝑇 𝑛 + 𝐾 𝑃
𝑛 𝑀
transition from
migrating to
proliferating
− 𝜇𝑀
cell
death
𝑀
Stepien, Rutter, and Kuang. Traveling Waves of a Go-or-Grow Model of Glioma Growth. SIAM Journal of Applied Mathematics, 2018

25. Model 1: Separate Phenotypes
𝑃 𝑀
𝜕𝑃
𝜕𝑡
= 𝑔 1 −
𝑇
𝑇max
𝑃
proliferation
− 𝜀𝑘
𝑇 𝑛
𝑇 𝑛 + 𝐾 𝑀
𝑛 𝑃
transition from
proliferating to
migrating
+ 𝑘
𝐾 𝑃
𝑛
𝑇 𝑛 + 𝐾 𝑃
𝑛 𝑀
transition from
migrating to
proliferating
𝜕𝑀
𝜕𝑡
= 𝐷
𝜕2 𝑀
𝜕𝑥2
diffusion
+ 𝜀𝑘
𝑇 𝑛
𝑇 𝑛 + 𝐾 𝑀
𝑛 𝑃
transition from
proliferating to
migrating
− 𝑘
𝐾 𝑃
𝑛
𝑇 𝑛 + 𝐾 𝑃
𝑛 𝑀
transition from
migrating to
proliferating
− 𝜇𝑀
cell
death
𝑀
Stepien, Rutter, and Kuang. Traveling Waves of a Go-or-Grow Model of Glioma Growth. SIAM Journal of Applied Mathematics, 2018

26. Model 1: Separate Phenotypes
𝑃 𝑀
𝜕𝑃
𝜕𝑡
= 𝑔 1 −
𝑇
𝑇max
𝑃
proliferation
− 𝜀𝑘
𝑇 𝑛
𝑇 𝑛 + 𝐾 𝑀
𝑛 𝑃
transition from
proliferating to
migrating
+ 𝑘
𝐾 𝑃
𝑛
𝑇 𝑛 + 𝐾 𝑃
𝑛 𝑀
transition from
migrating to
proliferating
𝜕𝑀
𝜕𝑡
= 𝐷
𝜕2 𝑀
𝜕𝑥2
diffusion
+ 𝜀𝑘
𝑇 𝑛
𝑇 𝑛 + 𝐾 𝑀
𝑛 𝑃
transition from
proliferating to
migrating
− 𝑘
𝐾 𝑃
𝑛
𝑇 𝑛 + 𝐾 𝑃
𝑛 𝑀
transition from
migrating to
proliferating
− 𝜇𝑀
cell
death
Stepien, Rutter, and Kuang. Traveling Waves of a Go-or-Grow Model of Glioma Growth. SIAM Journal of Applied Mathematics, 2018

27. Numerical Simulations

28. Numerical Results
1 2 3 4 5 6 7
0
0.05
0.1
0.15
time(days)
invasiveradius(cm)
Invasive Radius comparison

29. Model 2: Density-Dependent Diffusion
𝜕𝑢
𝜕𝑡
= 𝛻 ∙ 𝐷
𝑢
𝑢max
𝛻𝑢
Density−dependent
diffusion
+ 𝑔𝑢 1 −
𝑢
𝑢max
logistic growth
− sgn(𝑥)𝜈𝛻 ∙ 𝑢
taxis
Where
𝐷
𝑢
𝑢max
= 𝐷1 −
𝐷2
𝑢
𝑢max
𝑛
𝑎 𝑛 +
𝑢
𝑢max
𝑛
Stepien, Rutter, and Kuang. A Data-Motivated Density-Dependent Diffusion Model of in vitro Glioblastoma Growth. Mathematical Biosciences
and Engineering, 2015

30. Numerical Solutions

31. Numerical Results
Model Type Error
Reaction-Diffusion 0.7459
Go-or-Grow 0.2781
Density-Dependent Diffusion 0.1989
1 2 3 4 5 6 7
0
0.05
0.1
0.15
0.2
time(days)
invasiveradius(cm)
Invasive Radius comparison

32. Model 3: Random Differential
Equations
Dispersion and bifurcation features are not accounted
for when using static parameters
Source: Banks, H. T. & Davis, J. L. A
comparison of approximation methods for
the estimation of probability distributions
on parameters. Appl. Numer. Math. 57,
753–777, (2007).

33. Model 3: Random Differential
Equations
Consider the diffusion (𝑫) and growth (𝝆) as random
variables defined on a compact set Ω = Ω 𝑫 × Ω 𝝆
Model
𝜕𝑢(𝑡,𝑥,𝑫,𝝆)
𝑑𝑡
= 𝛻 ∙ 𝑫𝛻𝑢 𝑡, 𝑥, 𝑫, 𝝆 +
𝝆𝑢(𝑡, 𝑥, 𝑫, 𝝆)(1 − 𝑢(𝑡, 𝑥, 𝑫, 𝝆))
Observation
𝑢 𝑡, 𝑥 = 𝔼 𝑢 𝑡, 𝑥,∙,∙ , 𝑃
= Ω
𝑢 𝑡, 𝑥, 𝑫, 𝝆 𝑑𝑃(𝑫, 𝝆)

34. Prohorov Metric Framework (PMF)
Idea: Using data, determine the approximate
distributions of 𝑫 and 𝝆, without any underlying
assumptions about the pdf/cdf
𝑃 = argmin
𝑃 𝑀(Ω)
𝑖,𝑗
data 𝑡𝑗, 𝑥𝑖 −
Ω
𝑢 𝑡𝑗, 𝑥𝑖, 𝑫, 𝝆 𝑑𝑃(𝑫, 𝝆)
2

35. Performing the Inverse Problem:
Delta Functions
Example: we have M=11
nodes, equispaced over
[0,2]
We are solving for the
𝜔 𝑘, the discrete weights
𝑃 = argmin
𝑃 𝑀(Ω)
𝑖,𝑗
data 𝑡𝑗, 𝑥𝑖 −
𝑘=1
𝑀
𝑢 𝑡𝑗, 𝑥𝑖, 𝐷, 𝜌 𝑘 𝜔 𝑘
2
0 0.5 1 1.5 2
0
0.5
1
1.5
2
r
Probability
0
0.1
0.2
0.3
0.4
Actual
Estimated
𝑘=1
𝑀
𝜔 𝑘 = 1

36. Performing the Inverse Problem:
Spline Functions
Example: we have M=11
nodes, equispaced over
[0,2]
We are solving for the 𝑎 𝑘
𝑃 = argmin
𝑃 𝑀(Ω)
𝑖,𝑗
data 𝑡𝑗, 𝑥𝑖 −
𝑘=1
𝑀
𝑎 𝑘
Ω 𝝆
𝑢 𝑡𝑗, 𝑥𝑖, 𝐷, 𝝆 𝑠 𝑘 𝝆 𝑑𝝆
2
0 0.5 1 1.5 2
0
0.5
1
1.5
2
r
Probability
Actual
Estimated
𝑘=1
𝑀
𝑎 𝑘
Ω 𝝆
𝑠 𝑘 𝝆 𝑑𝝆 = 1

37. Representative Results
𝝆 normally distributed and 𝑫 bigaussian
Goal: Recover parameter distributions
data 𝑡𝑗, 𝑥𝑖 =sim 𝑡𝑗, 𝑥𝑖 + 𝜀sim 𝑡𝑗, 𝑥𝑖
𝜀~0.05𝑁(0,1)
Rutter, Banks and Flores. Estimating Intratumoral Heterogeneity from Spatiotemporal Data. Journal of Mathematical Biology. In Press

38. Resulting pdf Estimates
Rutter, Banks and Flores. Estimating Intratumoral Heterogeneity from Spatiotemporal Data. Journal of Mathematical Biology. In Press

39. Resulting cdf Estimates
Rutter, Banks and Flores. Estimating Intratumoral Heterogeneity from Spatiotemporal Data. Journal of Mathematical Biology. In Press

40. Treatment Prediction Assuming
Heterogeneity
Assuming a log-kill hypothesis, we add the term:
−𝑟
𝜌
𝜌
𝑢(𝑡, 𝑥)
0 1 2 3 4 5
0
0.5
1
r = 0.1
0 1 2 3 4 5
0
0.5
1
r = 0.4
0 1 2 3 4 5
0
0.5
1
r = 0.7
0 1 2 3 4 5
0
0.5
1
time (days)
TumorBurden
r = 1.0
True Solution
RD Model
RDE Model

41. Current Road Blocks in GBM Modeling
1. Assumptions of cellular homogeneity are unreliable
and overly optimistic
2. Non-identifiability of the simplest in vivo models

42. Addressing Non-
Identifiability Issues with
Machine Learning

43. Introduction to Machine Learning
Convolutional
Neural
Network
(CNN)
Cat
Dog
Training phase

44. Introduction to Neural Networks

45. Introduction to Neural Networks

46. Introduction to Neural Networks
A naïve approach to image classification using neural
networks

47. Introduction to Convolutions

48. Introduction to Convolutional Neural
Networks for Image Classification
Input 128 x 128
32 5x5 filters
2x2 max
pooling
Size:
32x64x64
2x2 max
pooling
Size:
64x32x32
64 5x5
filters
Fully Connected layer 1024
neurons
Output
to
classes
(0-9)

49. Introduction to Convolutional Neural
Networks for Image Segmentation

50. Problems with current techniques
Predicted masks, however, may be patchy, leading to
inaccurate cell counts (i.e., may count two cells when
there is actually only one)

51. Intro to Convolutional Neural Networks
Rutter, Lagergren and Flores. Automated Object Tracing for Biomedical Image Segmentation Using a Deep Convolutional Neural Network.
MICCAI, 2018.
Reimagine as a tracing task, as a human would segment
an image

52. Tracing Algorithm in Action
Rutter, Lagergren and Flores. Automated Object Tracing for Biomedical Image Segmentation Using a Deep Convolutional Neural Network.
MICCAI, 2018.

53. Future Work
1. Validate the tracer network on MR images of brains
to segment tumor
2. Use the tracer to count cells in histopathological
images of brain tumors
3. Determine relationship (if existing) between MR
images and cell density.
4. Validate heterogeneous models of GBM growth in
vivo

54. References
1. Erica M. Rutter et al. Mathematical Analysis of Glioma Growth in a Murine Model. Scientific Reports, 7(2508), 2017.
2. Tracy L. Stepien, Erica M. Rutter and Yang Kuang. Traveling Waves of a Go-or-grow Model of Glioma Growth. SIAM
Journal of Applied Mathematics, 78(3): 1778-1801, 2018.
3. Tracy L. Stepien, Erica M. Rutter, and Yang Kuang. A data-motivated density-dependent diffusion model of in vitro
glioblastoma growth. Mathematical Biosciences and Engineering, 12(6):1157-1172, 2015.
4. Erica M. Rutter. A Mathematical Journey of Cancer Growth. PhD Thesis, Arizona State University, 2016.
5. Erica M. Rutter, H. T. Banks, and Kevin B. Flores. Estimating Intratumoral Heterogeneity from Spatiotemporal Data.
Journal of Mathematical Biology. In Press.
6. Erica M. Rutter, H. T. Banks, Gerald A. LeBlanc, and Kevin B. Flores. Continuous Structured Population Models for
Daphnia magna. Bulletin of Mathematical Biology, 79(11):2627–2648.
7. Erica M. Rutter, John Lagergren, and Kevin B. Flores. Automated Object Tracing for Biomedical Image Segmentation
Using a Deep Convolutional Neural Network. In: International Conference on Medical Image Computing and
Computer Assisted Intervention (MICCAI), 686–694. Springer, Cham, 2018.
8. Erica M. Rutter et al. Detection of Bladder Contractions from the Activity of the External Urethral Sphincter in Rats
Using Sparse Regression. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 26(8):1636—1644,
2018.
9. Kiesler, Eva. “What Is Tumor Heterogeneity?” What Is Tumor Heterogeneity, Memorial Sloan Kettering, 2 July 2014,
www.mskcc.org/blog/what-tumor-heterogeneity.
10. Stein, Andrew M., et al. "A mathematical model of glioblastoma tumor spheroid invasion in a three-dimensional in
vitro experiment." Biophysical Journal 92.1 (2007): 356-365.
11. Saunders, Nicholas A., et al. "Role of intratumoural heterogeneity in cancer drug resistance: molecular and clinical
perspectives." EMBO molecular medicine 4.8 (2012): 675-684.
12. Havaei, Mohammad, et al. "Brain tumor segmentation with deep neural networks." Medical image analysis 35
(2017): 18-31.

55. Acknowledgements
Mathematical Collaborators
• Arizona State University
• Yang Kuang
• Eric Kostelich
• Tracy Stepien (now at
University of Arizona)
• North Carolina State University
• Kevin Flores
• H. T. Banks
• Franz Hamilton
• John Lagergren
(graduate student)
• Scott Baldwin
(graduate student)
• Elizabeth Collins
(undergraduate)
• Graedon Martin
(undergraduate)
Biological Collaborators
• Arizona State University
• David Frakes
• Jonathan Placensia
• Barrow Neurological Institute
• Mark Preul
• Adrienne Scheck
• Eric Woolf
• Gregory Turner
• Qingwei Liu
• North Carolina State University
• Gerald LeBlanc
• Duke
• Warren Grill
• UNC-Chapel Hill
• Jim Bear

56. Thank you for your attention!
Questions?