A simple immune system model of effector cells and regulator cells is coupled to a compartment model tuned to neuroblastoma xenograft data. Radiation is modeled as a raised death rate on a fixed day that reduces tumor size by transferring a fraction of cells into a compartment of “doomed” cells whose presence can be detected by effector cells, producing an immune response. The model predicts that the time required to regrow the tumor to its initial size is positively correlated with the ratio of effector to regulator cells naturally present in the patient. Individual immune cell distribution is likely to affect the patient’s response to radiation therapy.

1. D.Wallace
Dartmouth College

2. ¡ Joint work with Maya Srinivasan and Heiko
Enderling
¡ And also Shannon Fee, JadeYen, Alice Hsu,
Lawrence Abu-Hammour,Yixuan He
¡ Based on a suite of models developed in
collaboration with: Xinyue Guo, Paula Chen,
Michelle Chen, Milan Huynh, Evan Rheingold,
Ann Dunham,Olivia Prosper, Alisa DeStefano,
Sophia Jiang, Celeste Rodriguez, Molly
Carpenter, Rachel Chang

3. ¡ Why does the response to radiation therapy
differ among patients?
¡ The model we are using and why
¡ Prior results from this model
¡ What the model says about the immune
system-tumor interaction.
¡ What the model says about tumor regrowth
after radiation
¡ Caveats and future directions
¡ Implications for personalized medicine

4. Heuvers et al. BMC Cancer 2012, 12:580
http://www.biomedcentral.com/1471-2407/12/580
N2 neutrophils
RegulatoryT cells
M2 macrophages
And others
Tumor progression
Natural killer cells
NKT cells
N1 neutrophils
CD4+ helperT cells
CD8+ cytotoxic cells
M1 Macrophages
And others
Tumor regression

5. ¡ Hypotheses:
¡ The response is mediated by the immune system
interacting with cells damaged by radiation
¡ Assumptions:
¡ We need only represent 2 general classes of immune
cells: ”effector” and “regulator” cells, corresponding
to the two classes in the PCA
¡ The time it takes for a tumor to regrow after radiation
therapy is a measure of the effectiveness of therapy
¡ We can test this hypothesis in silica through a model

6. ¡ Is built as a series of nested models with
features that can be turned on and off
¡ The sub-models represent various kinds of
experiments that can be done with a cell line
¡ This allows us to use multiple published
studies of different sorts to parametrize the
model
¡ And it allows us to test its qualitative
behavior in various situations against
experiment

7. ¡ a
Small amounts
of very
expensive
data!
Almost
too much
data!
Problem driven Theory driven
Image
processing
Ecological
models
Social
networks

8. Monolayer
Spheroid
Xenograft Patient
In vitro,
No hypoxia
No vasculature
No immune
system
In vitro,
Hypoxia
No vasculature
No immune
system
In vivo
Hypoxia
Vasculature
No immune
system
In vivo
Hypoxia
Vasculature
Immune system

9. The smallest submodel is a linear model for unrestrained monolayer growth
Tumor
cells
G1 S G2
2𝑐# 𝐺#
𝑐% 𝐵𝐺% 𝑐' 𝑆
d𝐺#
The linear model can perfectly
match:
1. Observed doubling time
2. Observed natural death rate
3. Observed percentages in each
part of the cell cycle
4. Completely determines all
parameters you see here
5. The model and the monolayer
both grow exponentially

10. 𝑐% 1 − 𝐵 𝐺%
Q
N
𝑐+ 𝐹𝐺%
𝐶𝑄
𝑒𝐻𝑄
𝑚𝑁
𝑐34 𝑄
A
𝑘𝐴
The nonlinear model for spheroid growth includes the interior hypoxic region (Q), a
necrotic core (N) and the action of TNF-alpha (A) to trigger apoptosis of
proliferating cells.
d𝐺%
Tumor
cells
G1 S G2
2𝑐# 𝐺#
𝑐% 𝐵𝐺%
𝑐+ 𝐹𝑆
d𝑆
𝑐' 𝑆
𝑐# 𝐺#
𝑐+ 𝐹𝐺#
d𝐺#
Spheroids are known
to cease growth with a
thin layer of
proliferating cells
balanced byTNF-alpha
mediated death.
The model does this,
whether nutrient
availability is bounded
or unlimited but in
proportion to surface
area.
In addition the model
can approximate
growth patterns seen
in the data.
Carlsson et al Int. J. Cancer: 31,
523-533 (1983)

11. Differences inTherapeutic Indexes of
Combination Metronomic
Chemotherapy and an Anti-VEGFR- Antibody in
Multidrugresistant
Human Breast Cancer Xenografts
Giannoula Klement, Ping Huang, Barbara Mayer,
Shane K. Green, Shan Man, Peter Bohlen,
Daniel Hicklin, and Robert S. Kerbel
Clinical cancer research. 2002 Jan 1;8(1):221-32.
A model for spheroid versus monolayer response
of SK-N-SH neuroblastoma cells to
treatment with 15-deoxy-PGJ2.
Wallace DI, Dunham A, Chen PX, Chen M, Huynh
M, Rheingold E, Prosper O.
Computational and mathematical methods in
medicine.
2016;2016.
Prior results from this model, part 1
In both model and experiment, monolayer and spheroid cultures exhibit different
responses to treatment, with monolayer culture giving the more pronounced effect.

12. Q
N
R
A
V
The model we use, continued.
The xenograft model for immuno-compromised mouse include production of
theVEGF signal (R) that induces vascular growth (V), produced by hypoxic cells
(Q) and also proliferating cells (G1,S,G2) triggered byTNF-alpha (A).
Tumor
cells
Shared
vasculature
Shared signals
G1 S G2
2𝑐# 𝐺# * Most parameters are determined by
monolayer and spheroid data.
* Maximum vasculature growth rate comes
from data on healthy tissue.
* Many transitions are nonlinear rate
bounded functional responses
Xenografts exhibit
an initial phase of
rapid growth,
followed by a
constant growth
rate.
The model does
this also.

13. Control 1 is Ackerman
control data starting at
day 3 of tumor
implantation.
Control 2 is Ackerman
control data starting at
size 18 cubic
micrometers in size.
Control 3 is Segerstrom
control data.
Solid black line is model
control.
Blue diamonds are
treatment data from
Segerstrom
Green line is regression
fit to blue diamonds.
Dotted black line is
model fit to treatment
regrssion line.
Prior results from this model, part 2
Yixuan He, Anita Kodali, and Dorothy I.Wallace.
"Predictive Modeling of Neuroblastoma Growth
Dynamics in Xenograft Model After Bevacizumab
Anti-VEGFTherapy."
Bulletin of Mathematical Biology (2018): 1-23.

14. Q
N
R
A
V
𝑘 𝑣8 + 𝑉 𝐴
The model we use, continued.
Model with immune response including cells damaged by radiation or immune response
(D), “effector” immune cells that destroy tumor cells (E), and “regulator” cells that
suppress the immune response (T). Immune cell recruitment and death parameters
estimated from HIV literature.
D E
T
Tumor
cells
Shared
vasculature
Damaged cells
and immune
response
Shared signals
𝑐; 𝑇 𝑣8 + 𝑉
G1 S G2
𝑟; 𝑣8 + 𝑉
Recruitment,
clearance and
proliferation of
immune cells
mediated by
vasculature.
Tumor cells
damaged by
radiation and/or
immune
response.

15. What the model says about the immune system-tumor interaction.
What follows are all preliminary results.
Same in silica tumor
A. without immune response and
B. with immune response.
No radiation therapy is included in B.This is the control run.We see a slight
reduction in tumor growth rate.
A. B.

16. What the model says about tumor regrowth after radiation
Radiation is modeled as an extra linear relative rate of death over some fixed period,
starting at day 35. Here is the result with default parameters.Tumor growth resumes
after radiation.

17. For the same numerical experiment, this shows the ratio of effector cells (E) to
regulator cells (T) over time.
A. The immune system reaches homeostasis.
B. After radiation, the effector cells increase.
C. This creates a response of regulator cells.
D. Finally the immune system returns to homeostasis.
A B C D

18. What the model says about tumor regrowth after radiation, continued.
Hypothesis (due to Heiko Enderling): The ratio E/T is correlated with the
success of radiation therapy.
To test the hypothesis we need:
1. A reasonable way to vary E/T at the homeostatic ratio. We do this
by varying appropriate parameters.
2. A way to decide how successful radiation therapy was. We do this
by measuring how long it takes the tumor to regrow to its starting
size.

19. Only 7 parameters determine the immune response.
Of these, only 5 were deemed significant based on a simple sensitivity analysis (below).
Of those 5, we were unable to obtain biological measurements for two of them.
The remaining 3 were varied in a range around the reported estimates, to produce varying
values for the ratio E/T.

20. Hypothesis confirmed in silica:
Tumor regrowth time is positively correlated with E/T ratio.

21. ¡ Why does the response to radiation therapy
differ among patients?
¡ The model we are using and why
¡ Prior results from this model
¡ What the model says about the immune
system-tumor interaction.
¡ What the model says about tumor regrowth
after radiation
¡ Caveats and future directions
¡ Implications for personalized medicine

22. ¡ The difference in time to regrowth was only 2 days at
most. OK but those are mouse days.
¡ Two parameters are mysteries and should be varied both
numerically and in some kind of experiment if possible.
¡ Each parameter should be checked separately to see
independent correlations.
¡ It is thought that many tumors succumb to the immune
response and are never detected. If so, the efficiency of
the response is much larger perhaps than we made it
here.
¡ It remains to check the effect of initial tumor size and
composition on the behavior of the model.

23. ¡ How effective radiation therapy is likely to be may
depend on the state of the person’s immune
system.This can be measured in advance of
treatment. The decision whether to treat can be
based on personal data.
¡ Perhaps it is possible to bolster the immune system
in advance of treatment in order to improve the
likelihood of success. The model suggests that
suppressing the regulatory immune cells and/or
increasing the effector cells would be the right
strategy.

24. ¡ Knowing the makeup of the patient’s immune
system should shed light on their likely response
to radiation therapy.
¡ A patient-specific pre-treatment of the immune
system could enhance the results of therapy.
¡ The effector/regulator ratio may be a good basis
for decisions about radiation.
¡ Thank you!