Erica Rutter presents non-parametric techniques for estimating tumor heterogeneity from data. She describes using a Prohorov metric framework to determine the approximate distributions of diffusion (D) and growth (ρ) parameters from data, without assumptions about their distributions. She creates synthetic data from a known ρ distribution and solves the inverse problem to estimate ρ, comparing solutions using delta functions and spline functions with varying numbers of nodes. The Akaike Information Criteria is used to select the optimal number of nodes. Representative results show the estimated ρ distribution matching the true distribution well.

1. Non-Parametric Techniques for
Estimating Tumor Heterogeneity
Erica Rutter
Department of Mathematics
Center for Research in Scientific Computation
North Carolina State University
Funding: NSF Math Biology (DMS-1514929)
May 20, 2019
SAMSI Precision Medicine Transition Workshop

2. 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).

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
+,(.,0,𝑫,𝝆)
2.
= 𝛻 4 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,4,4 , 𝑃
= ∫ 𝑢 𝑡, 𝑥, 𝑫, 𝝆 𝑑𝑃(𝑫, 𝝆)I

4. Prohorov Metric Framework (PMF)
Idea: Using data, determine the approximate
distributions of 𝑫 and 𝝆, without any underlying
assumptions about the pdf/cdf
𝑃J = argmin
MN(I)
O data 𝑡P, 𝑥Q − R 𝑢 𝑡P, 𝑥Q, 𝑫, 𝝆 𝑑𝑃(𝑫, 𝝆)
I
S
Q,P

5. Prohorov Metric Framework
Theory
1. Since Ω = Ω 𝑫×Ω 𝝆 is a compact set, 𝑃 Ω is a
compact metric space
2. The minimizer is continuous in 𝑃
⟹ There exists a (not necessarily unique) minimizer
Theorem
There exists a (not necessarily unique) minimizer 𝑃J (Banks, Hu, Thompson, 2015)
𝑃J = argmin
M∈MN(I)
O data 𝑡P, 𝑥Q − R 𝑢 𝑡P, 𝑥Q, 𝑫, 𝝆 𝑑𝑃(𝑫, 𝝆)
I
S
Q,P

6. Creating Synthetic Data
We finely mesh over the parameter 𝝆 ∈ [0,2] and create
our desired pdf
0 0.5 1 1.5 2
0
0.5
1
1.5
weight

7. Creating Synthetic Data
0 0.5 1 1.5 2
0
0.5
1
1.5
weight
𝜕𝑢(𝑡, 𝑥, 𝜌])
𝑑𝑡
= 𝛻 4 𝐷 𝛻 𝑢 𝑡, 𝑥, 𝜌] + 𝜌] 𝑢 𝑡, 𝑥, 𝜌] 1 − 𝑢 𝑡, 𝑥, 𝜌]
0
20
1
0.5
u(t,x,k
)
t
10
x
1
0
0 -1

8. Creating Synthetic Data
𝜕𝑢(𝑡, 𝑥, 𝜌])
𝑑𝑡
= 𝛻 4 𝐷 𝛻 𝑢 𝑡, 𝑥, 𝜌] + 𝜌] 𝑢 𝑡, 𝑥, 𝜌] 1 − 𝑢 𝑡, 𝑥, 𝜌]
0 0.5 1 1.5 2
0
0.5
1
1.5
weight
0
20
1
0.5
u(t,x,k
)
t
10
x
1
0
0 -1

9. Creating Synthetic Data
𝜕𝑢(𝑡, 𝑥, 𝜌])
𝑑𝑡
= 𝛻 4 𝐷 𝛻 𝑢 𝑡, 𝑥, 𝜌] + 𝜌] 𝑢 𝑡, 𝑥, 𝜌] 1 − 𝑢 𝑡, 𝑥, 𝜌]
0 0.5 1 1.5 2
0
0.5
1
1.5
weight
0
20
1
0.5
u(t,x,k
)
t
10
x
1
0
0 -1

10. Creating Synthetic Data
0
20
1
0.5
u(t,x,k
)
t
10
x
1
0
0 -1
0
20
1
0.5
u(t,x,k
)
t
10
x
1
0
0 -1
0
20
1
0.5
u(t,x,k
)
t
10
x
1
0
0 -1
0
20
1
0.5
u(t,x)
t
10
x
1
0
0 -1
𝑢 𝑡, 𝑥 = O 𝑢 𝑡, 𝑥, 𝜌] 𝜔]
`
]ab

11. Creating Synthetic Data
0
20
1
0.5
u(t,x)
t
10
x
1
0
0 -1
0
20
0.5
1
u(t,x)
t
10
1
x
0
0 -1
data 𝑡P, 𝑥Q =sim 𝑡P, 𝑥Q + 𝜀sim 𝑡P, 𝑥Q
𝜀~0.05𝑁(0,1)
sim data

12. Performing the Inverse Problem:
Delta Functions
Assume there are 𝑀 nodes equispaced over Ω 𝝆 such
that 𝝆i
= Δkl
, 𝑘 = 1, … , 𝑀
where 𝜔] ≥ 0 represent a discrete probability
density function. Thus, we require
𝑃J = argmin
MN(I)
O data 𝑡P, 𝑥Q − O 𝑢 𝑡P, 𝑥Q, 𝐷, 𝜌] 𝜔]
i
]ab
S
Q,P
O 𝜔]
i
]ab
= 1

13. Performing the Inverse Problem:
Delta Functions
Example: we have M=11
nodes, equispaced over
[0,2] and we precompute
𝑢(𝑡, 𝑥, 𝐷, 𝜌])
We are solving for the 𝜔],
the discrete weights
𝑃J = argmin
MN(I)
O data 𝑡P, 𝑥Q − O 𝑢 𝑡P, 𝑥Q, 𝐷, 𝜌] 𝜔]
i
]ab
S
Q,P
0 0.5 1 1.5 2
0
0.5
1
1.5
2
ρ
Probability
0
0.1
0.2
0.3
0.4
Actual
Estimated

14. Performing the Inverse Problem:
Spline Functions
Assume 𝑀 nodes equispaced over Ω 𝝆 such that
𝝆i
= 𝑠](𝝆), 𝑘 = 1, … , 𝑀 , where 𝑠] are hat
functions
where 𝑝] = 𝑎] 𝑠] 𝝆 ≥ 0 represent a probability
density function. Thus, we require
𝑃J = argmin
MN(I)
O data 𝑡P, 𝑥Q − O 𝑎] R 𝑢 𝑡P, 𝑥Q, 𝐷, 𝝆 𝑠] 𝝆 𝑑𝝆
I 𝝆
i
]ab
S
Q,P
O 𝑎] R 𝑠] 𝝆 𝑑𝝆
I 𝝆
i
]ab
= 1

15. Performing the Inverse Problem:
Spline Functions
Example: we have M=11
nodes, equispaced over
[0,2]
We are solving for the 𝑎]
𝑃J = argmin
MN(I)
O data 𝑡P, 𝑥Q − O 𝑎] R 𝑢 𝑡P, 𝑥Q, 𝐷, 𝝆 𝑠] 𝝆 𝑑𝝆
I 𝝆
i
]ab
S
Q,P
0 0.5 1 1.5 2
0
0.5
1
1.5
2
ρ
Probability
Actual
Estimated

16. How to choose M: the optimal number
of nodes?
Increasing M: the number of nodes
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
Probability
Exact
Discrete
0 5 10 15 20
time
0
100
200
300
400
population
Exact
Discrete
RSS = 875.8139

17. How to choose M: the optimal number
of nodes?
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
Probability
Exact
Discrete
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
Probability
Exact
Discrete
Increasing M: the number of nodes
0 5 10 15 20
time
0
100
200
300
400
population
Exact
Discrete
0 5 10 15 20
time
0
100
200
300
400
population
RSS = 3.6191RSS = 875.8139

18. How to choose M: the optimal number
of nodes?
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
Probability
Exact
Discrete
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
Probability
Exact
Discrete
Increasing M: the number of nodes
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
Probability
Exact
Discrete
0 5 10 15 20
time
0
100
200
300
400
population
Exact
Discrete
0 5 10 15 20
time
0
100
200
300
400
population
0 5 10 15 20
time
0
100
200
300
400
population
2.5656e+03RSS = 3.6191RSS = 875.8139

19. How to choose M: the optimal number
of nodes?
Akaike Information Criteria (AIC) as a model
comparison test in the context of least-squares
𝐴𝐼𝐶 = 𝑁 ln
RSS
𝑁
+ 𝑁 1 − ln 2𝜋 + 2(𝑀 + 1)
𝑁: number of data points
RSS: error between data and solution 𝑢(𝑡, 𝑥)
𝑀: number of parameters being fit (our 𝑀 nodes)

20. How to choose M: the optimal number
of nodes?
10 20 30 40 50
−2.5
−2
−1.5
−1
−0.5x 10
5
Number of Nodes
AICScore
Spline
twopoint
point
normal
lognormal
bigaussian
10 20 30 40 50
−2.5
−2
−1.5
−1
−0.5x 10
5
Number of Nodes
AICScore
Discrete

21. Representative Results: finding 𝝆
0 0.5 1 1.5 2
0
0.25
0.5
0.75
1
Probability
0
0.25
0.5
0.75
1Exact
Discrete
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Probability
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Exact
Spline

22. Performing the Inverse Problem:
Delta Functions
Assume there are 𝑀k nodes equispaced over Ω 𝝆 such
that 𝝆i
= Δkl
, 𝑘 = 1, … , 𝑀k and that there are 𝑀z nodes
equispaced over Ω 𝑫 such that 𝑫i
= Δz{
, 𝑙 = 1, … , 𝑀z
where 𝜔], 𝜔} ≥ 0 represent a discrete probability
density function. Thus, we require
𝑃J = argmin
MN(I)
O data 𝑡P, 𝑥Q − O O 𝑢 𝑡P, 𝑥Q, 𝐷} , 𝜌] 𝜔] 𝜔}
i~
]ab
i•
}ab
S
Q,P
O 𝜔]
i~
]ab
= 1 O 𝜔}
i•
}ab
= 1

23. Performing the Inverse Problem:
Spline Functions
Assume 𝑀knodes equispaced over Ω 𝝆 such that 𝝆i~ =
𝑠](𝝆), 𝑘 = 1, … , 𝑀k , and 𝑀znodes equispaced over
Ω 𝑫 such that 𝑫i• = 𝑠}(𝑫), 𝑙 = 1, … , 𝑀z , where
𝑠], 𝑠} are hat functions
where 𝑝] = 𝑏] 𝑠] 𝝆 ≥ 0 , 𝑝} = 𝑎} 𝑠} 𝑫 ≥ 0
represent probability density functions.
𝑃J = argmin
MN(I)
O data 𝑡P, 𝑥Q − O 𝑎} R O 𝑏] R 𝑢 𝑡P, 𝑥Q, 𝑫, 𝝆 𝑠} 𝑫 𝑑𝑫
I 𝝆
i~
]ab
𝑠] 𝝆 𝑑𝝆
I 𝑫
i•
}ab
S
Q,P
O 𝑏] R 𝑠] 𝝆 𝑑𝝆
I 𝝆
i~
]ab
= 1 O 𝑎} R 𝑠} 𝑫 𝑑𝑫
I 𝑫
i•
}ab
= 1

24. Selection of M

25. Representative Results
𝝆 normally distributed and 𝑫 bigaussian
Goal: Recover parameter distributions
data 𝑡P, 𝑥Q =sim 𝑡P, 𝑥Q + 𝜀sim 𝑡P, 𝑥Q
𝜀~0.05𝑁(0,1)
Rutter, Banks and Flores. Estimating Intratumoral Heterogeneity from Spatiotemporal Data. Under review
−1
0
1 0
10
200
0.5
1
1.5
Time (t)
Distance (x)
CellDensity(u(t,x))
Exact
Discrete
Spline

26. Resulting pdf Estimates
Rutter, Banks and Flores. Estimating Intratumoral Heterogeneity from Spatiotemporal Data. Under review
0 1 2
x 10
−4
0
1
2
3
x 10
4
D
Probability
0
0.1
0.2
0.3
0 1 2
0
0.5
1
1.5
ρ
0
0.1
0.2
0.3Exact
Spline
Discrete

27. Resulting cdf Estimates
Rutter, Banks and Flores. Estimating Intratumoral Heterogeneity from Spatiotemporal Data. Under review
0 1 2
x 10
−4
0
0.2
0.4
0.6
0.8
1
D
CumulativeDistribution
Exact
Discrete
Spline
0 1 2
0
0.2
0.4
0.6
0.8
1
ρ

28. Modeling Inter-Individual
Heterogeneity via PMF
• Growth rates in size-structured population models
of shrimp
• Diffusion rates in transdermal alcohol
measurements
• If we can model inter-individual heterogeneity in
GBM, we may be able to glean insights on
treatment success for subpopulations

29. Conclusions
• We can recover parameter distributions from
noisy spatiotemporal data from a variety of pdfs
for various PDEs
• We need to extend this framework to consider
inter-individual heterogeneity
• How much data is necessary?
• What are the uncertainties in the estimates?

30. References
1. Banks, H. T., et al. "Experimental design and estimation of growth
rate distributions in size-structured shrimp populations." Inverse
Problems 25.9 (2009): 095003.
2. Banks, H. T., and W. Clayton Thompson. "Existence and consistency of
a nonparametric estimator of probability measures in the prohorov
metric framework." International Journal of Pure and Applied
Mathematics 103.4 (2015).
3. Erica M. Rutter, H. T. Banks, and Kevin B. Flores. Estimating
Intratumoral Heterogeneity from Spatiotemporal Data. Journal of
Mathematical Biology, 77(6-7):1999–2022, 2018 .
4. Banks, H. T., et al. "The Prohorov metric framework and aggregate
data inverse problems for random PDEs." Commun. Appl. Anal 22
(2018): 415-46.
5. Sirlanci, Melike, et al. "Estimating the distribution of random
parameters in a diffusion equation forward model for a transdermal
alcohol biosensor." Automatica 106 (2019): 101-109.

31. Thank you for your attention!
Questions?