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Personalized mathematical oncology: Challenges and opportunities

03 Aug 2021-bioRxiv (Cold Spring Harbor Laboratory)-
Abstract: An outstanding challenge in the clinical care of cancer is moving from a one-size-fits-all approach that relies on population-level statistics towards personalized therapeutic design. Mathematical modeling is a powerful tool in treatment personalization, as it allows for the incorporation of patient-specific data so that treatment can be tailor-designed to the individual. In this work, we employ two fitting methodologies to personalize treatment in a mathematical model of murine cancer immunotherapy. Unexpectedly, we found that the predicted personalized treatment response is sensitive to the fitting methodology utilized. This raises concerns about the ability of mathematical models, even relatively simple ones, to make reliable predictions about individual treatment response. Our analyses shed light onto why it can be challenging to make personalized treatment recommendations from a model, but also suggest ways we can increase our confidence in personalized mathematical predictions. Author summaryAs we enter the era of healthcare where personalized medicine becomes a more common approach to treating cancer patients, harnessing the power of mathematical models will only become more essential. Using a preclinical dataset on cancer immunotherapy, we explore the challenges and limitations that arise when trying to move from a one-size-fits-all approach to treatment design towards personalized therapeutic design. These challenges lead to actionable suggestions on how to ascertain when we have enough data to personalize treatment, or how to determine when we can have confidence that an optimal-for-the-average prediction will have a comparable impact on an individual. We also show how mathematical modeling can suggest what data is needed to increased confidence in personalized predictions.

Summary (2 min read)

Introduction

  • An outstanding challenge in the clinical care of cancer is moving from a one-size-fits-all approach that relies on population-level statistics towards personalized therapeutic design.
  • Herein, the authors work with a mathematical model of murine cancer immunotherapy that has been previously-validated against the average of an experimental dataset.
  • Typically, this would be done by choosing a single fitting methodology, and a single cost function, identifying the individualized best-fit parameters, and extrapolating from there to make personalized treatment recommendations.
  • The authors analyses show the potentially problematic nature of this approach, as predicted personalized treatment response proved to be sensitive to the fitting methodology utilized.
  • The authors also demonstrate how a small amount of the right additional experimental measurements could go a long way to improve consistency in personalized fits.

Author summary

  • As the authors enter the era of healthcare where personalized medicine becomes a more common approach to treating cancer patients, harnessing the power of mathematical models will only become more essential.
  • The conventional approach for developing a cancer treatment protocol relies on 2 measuring average efficacy and toxicity from population-level statistics in randomized 3 clinical trials [1–3].
  • Unlike other treatment 16 modalities that directly attack the tumor, immunotherapy depends on the interplay 17 between two complex systems (the tumor and the immune system), and therefore may 18 exhibit more variability across individuals [18].
  • Only then can modeling be harnessed to answer some of the most 25 pressing questions in precision medicine, including selecting the right drug for the right 26 patient, identifying optimal drug combinations for a patient, and prescribing a 27 treatment schedule that maximizes efficacy while minimizing toxicity.
  • The authors explore the consequences of performing individualized fits using a 63 minimal mathematical model previously-validated against the average of an 64 experimental dataset.

Data Set 79

  • The first protocol uses oncolytic viruses (OVs) that are 81 genetically engineered to lyse and kill cancer cells.
  • In [33] the OVs are 82 immuno-enhanced by inserting transgenes that cause the virus to release 4-1BB ligand 83 (4-1BBL) and interleukin (IL)-12, both of which result in the stimulation of the 84 tumor-targeting T cell population [33].
  • 88 The second protocol utilized by Huang et al. are dendritic cell (DC) injections.
  • DCs 89 are antigen-presenting cells that, when exposed to tumor antigens ex vivo and 90 intratumorally injected, can stimulate a strong adaptive immune response against 91 cancer cells [33].

Mathematical Model 96

  • When all parameters and time-varying terms are positive, this 100 models captures the effects of tumor growth and response to treatment with 101 Ad/4-1BBL/IL-12 and DCs [34].
  • It is well-established that estimating a unique parameter set for a mathematical model 170 can be challenging due to the limited availability of often noisy experimental data [42].
  • The QMC-predicted 243 parameters cover a much larger range of values relative to the average mouse.
  • The predictive discrepancies across fitting methodologies begs the question of whether 336 the parameters the authors are fitting are actually practically identifiable given the available 337 experimental data.
  • Mouse 7 is particularly 434 interesting, as there was 95% agreement across methodologies when using V (30) to 435 measure treatment success or failure, and the optimal for the average of VVVDDD is 436 the only protocol for which treatment response differed (with QMC predicting tumor 437 eradication, and NLME predicting treatment failure).

Supporting information

  • Fig S1. Estimated parameter distributions from Monolix’s implementation of NLME.
  • Best-fit value of number of viruses released by lysed cell α, the cytotoxicity enhancement term due to immunostimulants ckill, T cell decay rate δT , DC decay rate δD, and default cytotoxicity rate of T cells κ0.
  • The best-fit values are shown for each mouse and are presented relative to the best-fit value of the parameter in the average mouse [34].
  • Left shows predictions when parameters are fit using QMC and right shows NLME predictions.
  • Compare to heatmap in Fig. 8 which shows the log of the tumor volume 50 days later.

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From fitting the average to fitting the individual: A
cautionary tale for mathematical modelers
Michael C. Luo
1
, Elpiniki Nikolopoulou
2¤
, Jana L. Gevertz
1*
1 Department of Mathematics and Statistics, The College of New Jersey, Ewing, NJ,
USA
2
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ,
USA
¤Current Address: Nationwide, Columbus, OH, USA
* gevertz@tcnj.edu
Abstract
An outstanding challenge in the clinical care of cancer is moving from a one-size-fits-all
approach that relies on population-level statistics towards personalized therapeutic
design. Mathematical modeling is a powerful tool in treatment personalization, as it
allows for the incorporation of patient-specific data so that treatment can be
tailor-designed to the individual. Herein, we work with a mathematical model of murine
cancer immunotherapy that has been previously-validated against the average of an
experimental dataset. We ask the question: what happens if we try to use this same
model to perform personalized fits, and therefore make individualized treatment
recommendations? Typically, this would be done by choosing a single fitting
methodology, and a single cost function, identifying the individualized best-fit
parameters, and extrapolating from there to make personalized treatment
recommendations. Our analyses show the potentially problematic nature of this
approach, as predicted personalized treatment response proved to be sensitive to the
fitting methodology utilized. We also demonstrate how a small amount of the right
additional experimental measurements could go a long way to improve consistency in
personalized fits. Finally, we show how quantifying the robustness of the average
response could also help improve confidence in personalized treatment recommendations.
Author summary
As we enter the era of healthcare where personalized medicine becomes a more common
approach to treating cancer patients, harnessing the power of mathematical models will
only become more essential. Using a preclinical dataset on cancer immunotherapy, we
explore the challenges and limitations that arise when trying to move from fitting and
making predictions for the population-level average, to fitting and making predictions
for an individual. We find that the standard of approach of picking a single fitting
methodology and a single cost function may end up having limited predictive value
when applied to individual data. We also show how having a small amount of the right
additional experimental data, and establishing the robustness of average treatment
response, can help improve confidence in personalized model predictions.
September 7, 2021 1/26
.CC-BY-NC 4.0 International licenseavailable under a
(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made
The copyright holder for this preprintthis version posted September 9, 2021. ; https://doi.org/10.1101/2021.08.03.454882doi: bioRxiv preprint

Introduction 1
The conventional approach for developing a cancer treatment protocol relies on 2
measuring average efficacy and toxicity from population-level statistics in randomized 3
clinical trials [1–3]. However, it is increasingly recognized that heterogeneity, both 4
between patients and within a patient, is a defining feature of cancer [4, 5]. This 5
inevitably results in a portion of cancer patients being over-treated and suffering 6
toxicity consequences from the standard-of-care dose, and another portion being 7
under-treated and not benefiting from the expected efficacy of the treatment [6]. 8
For these reasons, in the last decade there has been much interest in moving away 9
from this ‘one-size-fits-all’ approach to cancer treatment and towards personalized 10
therapeutic design (also called predictive or precision medicine) [1, 2, 7]. Collecting 11
patient-specific data has the potential to improve treatment response to 12
chemotherapy [6,8
11], radiotherapy [12
14], and targeted molecular therapy [11, 15
17].
13
However, it has been proposed that personalization may hold the most promise when it
14
comes to immunotherapy [18]. Immunotherapy is an umbrella term for methods that 15
increase the potency of the immune response against cancer. Unlike other treatment 16
modalities that directly attack the tumor, immunotherapy depends on the interplay 17
between two complex systems (the tumor and the immune system), and therefore may 18
exhibit more variability across individuals [18]. 19
Mathematical modeling has become a valuable tool for understanding tumor-drug 20
interactions. However, just as clinical care is guided by standardized recommendations,
21
most mathematical models are validated based on population-level statistics from 22
preclinical or clinical studies [19]. To truly realize the potential of mathematical models
23
in the clinic, these models must be individually parameterized using measurable, 24
patient-specific data. Only then can modeling be harnessed to answer some of the most
25
pressing questions in precision medicine, including selecting the right drug for the right
26
patient, identifying optimal drug combinations for a patient, and prescribing a 27
treatment schedule that maximizes efficacy while minimizing toxicity. 28
Efforts to personalize mathematical models have been undertaken to understand 29
glioblastoma treatment response [20, 21], to identify optimal chemotherapeutic and 30
granulocyte colony-stimulating factor combined schedules in metastatic breast 31
cancer [22], to identify optimal maintenance therapy chemotherapeutic dosing for 32
childhood acute lymphoblastic leukemia [9], and to identify optimized doses and dosing
33
schedules of the chemotherapeutic everolimus with the targeted agent sorafenib for solid
34
tumors [23]. Interesting work has also been done in the realm of radiotherapy, where 35
individualized head and neck cancer evolution has been modeled through a dynamic 36
carrying capacity informed by patient response to their last radiation dose [24]. 37
Beyond these examples, most model personalization efforts have focused on prostate
38
cancer, as prostate-specific antigen is a clinically measurable marker of prostate cancer
39
burden [25] that can be used in the parameterization of personalized mathematical 40
models. The work of Hirata and colleagues has focused on the personalization of 41
intermittent androgen suppression therapy using retrospective clinical trial data [26, 27].
42
Other interesting work using clinical trial data has been done by Agur and colleagues, 43
focusing on individualizing a prostate cancer vaccine using retrospective phase 2 clinical
44
trial data [25, 28], as well as androgen deprivation therapy using data from an advanced
45
stage prostate cancer registry [29]. Especially exciting work on personalizing prostate 46
cancer has been undertaken by Gatenby and colleagues, who used a mathematical 47
model to discover patient-specific adaptive protocols for the administration of the 48
chemotherapeutic agent abiraterone acetate [30]. Among the 11 patients in a pilot 49
clinical trial treated with the personalized adaptive therapy, they observed the median 50
time to progression increased to at least 27 months as compared to 16.5 months 51
observed with standard dosing, while also using a cumulative drug amount that was 52
September 7, 2021 2/26
.CC-BY-NC 4.0 International licenseavailable under a
(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made
The copyright holder for this preprintthis version posted September 9, 2021. ; https://doi.org/10.1101/2021.08.03.454882doi: bioRxiv preprint

47% less than the standard dosing [17]. 53
Despite these examples, classically mathematical models are not personalized, but 54
are validated against the average of experimental data. In particular, modelers choose a
55
single fitting methodology, a single cost function to minimize, and find the best-fit 56
parameters to the average of the data. Using the best-fit parameters and the 57
mathematical model, treatment optimization can be performed. Recognizing the 58
limitations of this approach in describing variable treatment response across 59
populations, modelers have begun employing virtual population cohorts [31, 32]. There
60
is much value in this population-level approach to study variability, but it is not 61
equivalent to looking at individualized treatment response. 62
In this work, we explore the consequences of performing individualized fits using a 63
minimal mathematical model previously-validated against the average of an 64
experimental dataset. In Materials and methods, we describe the preclinical data 65
collected by Huang et al. [33] on a mouse model of melanoma treated with two forms of
66
immunotherapy, and our previously-developed mathematical model that has been 67
validated against population-level data from this trial [34]. Individual mouse volumetric
68
time-course data is fit to our dynamical systems model using two different approaches 69
detailed in Materials and methods: the first fits each mouse independent of the other 70
mice in the population, whereas the second constrains the fits to each mouse using 71
population-level statistics. In Results and Discussion, we demonstrate that the 72
treatment response identified for an individual mouse is sensitive to the fitting 73
methodology utilized. We explore the causes of these predictive discrepancies and how 74
robustness of the optimal-for-the-average treatment protocol influences these 75
discrepancies. We conclude with actionable suggestions for how to increase our 76
confidence in mathematical predictions made from personalized fits. 77
Materials and methods 78
Data Set 79
The data in this study considers the impact of two immunotherapeutic protocols on a 80
murine model of melanoma [33]. The first protocol uses oncolytic viruses (OVs) that are
81
genetically engineered to lyse and kill cancer cells. In [33] the OVs are 82
immuno-enhanced by inserting transgenes that cause the virus to release 4-1BB ligand 83
(4-1BBL) and interleukin (IL)-12, both of which result in the stimulation of the 84
tumor-targeting T cell population [33]. The preclinical work of Huang et al. has shown
85
that oncolytic viruses carrying 4-1BBL and IL-12 (which we will call Ad/4-1BBL/IL-12)
86
can cause tumor debulking via virus-induced tumor cell lysis, and immune system 87
stimulation from the local release of the immunostimulants [33]. 88
The second protocol utilized by Huang et al. are dendritic cell (DC) injections. DCs
89
are antigen-presenting cells that, when exposed to tumor antigens ex vivo and 90
intratumorally injected, can stimulate a strong adaptive immune response against 91
cancer cells [33]. Huang et al. showed that combination of Ad/4-1BBL/IL-12 with DC 92
injections results in a stronger antitumor response than either treatment 93
individually [33]. Volumetric trajectories of individual mice treated with 94
Ad/4-1BBL/IL-12, along with the average trajectory, is shown in Fig. 1. 95
September 7, 2021 3/26
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(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made
The copyright holder for this preprintthis version posted September 9, 2021. ; https://doi.org/10.1101/2021.08.03.454882doi: bioRxiv preprint

Fig 1. Individual volumetric trajectories are shown for eight mice treated with
Ad/4-1BBL/IL-12. The average, with standard error bars, is also shown in black [33].
Mathematical Model 96
Our model contains the following five ordinary differential equations:
dU
dt
= rU β
UV
N
(κ
0
+ c
kill
I)
UT
N
, U(0) = U
0
, (1)
dI
dt
= β
UV
N
δ
I
I (κ
0
+ c
kill
I)
IT
N
, I(0) = 0, (2)
dV
dt
= u
V
(t) + αδ
I
I δ
V
V, V (0) = 0, (3)
dT
dt
= c
T
I + χ
D
D δ
T
T, T (0) = 0, (4)
dD
dt
= u
D
(t) δ
D
D, D(0) = 0, (5)
where
U
is the volume of uninfected tumor cells,
I
is the volume of OV-infected tumor
97
cells,
V
is the volume of free OVs,
T
is the volume of tumor-targeting T cells,
D
is the
98
volume of injected dendritic cells, and
N
is the total volume of cells (tumor cells and T
99
cells) at the tumor site. When all parameters and time-varying terms are positive, this
100
models captures the effects of tumor growth and response to treatment with 101
Ad/4-1BBL/IL-12 and DCs [34]. By allowing various parameters and time-varying 102
terms to be identically zero, other treatment protocols tested in Huang et al. [33] can 103
also be described. 104
This model was built in a hierarchical fashion, details of which have been described
105
extensively elsewhere [31, 34
36]. Here, we briefly summarize the full model. Uninfected
106
tumor cells grow exponentially at a rate r, and upon being infected by an OV convert 107
to infected cancer cells at a density-dependent rate
βU V/N
. These uninfected cells get
108
lysed by the virus or other mechanisms at a rate of
δ
I
, thus acting as a source term for
109
the virus by releasing an average of
α
free virions into the tissue space. Viruses decay at
110
a rate of δ
V
. 111
The activation/recruitment of tumor-targeting T cells can happen in two ways: 1) 112
stimulation of cytotoxic T cells due to 4-1BBL or IL-12 (modeled through
I
, at a rate of
113
c
T
, as infected cells are the ones to release 4-1BBL and IL-12), and 2) 114
production/recruitment due to the externally-primed dendritic cells at a rate of χ
D
. 115
September 7, 2021 4/26
.CC-BY-NC 4.0 International licenseavailable under a
(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made
The copyright holder for this preprintthis version posted September 9, 2021. ; https://doi.org/10.1101/2021.08.03.454882doi: bioRxiv preprint

These tumor-targeting T cells indiscriminately kill uninfected and infected tumor cells, 116
with the rate of killing that depends on IL-12 and 4-1BBL production (again, modeled
117
through I in the term (κ
0
+ c
kill
I)), and they can also experience natural death at a 118
rate of δ
T
. The time-dependent terms, u
V
(t) and u
D
(t), represent the source of the 119
drug and are determined by the delivery and dosing schedule of interest. 120
Fitting Methodologies 121
Independently Fitting Individuals 122
Our first attempt at individualized fitting is to find the parameter set that minimizes 123
the L
2
-norm between the model and the individual mouse data: 124
ζ =
n
X
t=0
(V
model
(t) V
data
(t))
2
, (6)
where V
model
(t) = U(t) + I(t) is the volumetric output predicted by our model in eqns. 125
(1)-(5), V
data
(t) represents the volumetric data for an individual mouse, and n is the 126
number of days for which tumor volume is measured in the experiments. 127
To independently fit an individual mouse, parameter space is first quasi-randomly 128
sampled using high-dimensional Sobol’ Low Discrepancy Sequences (LDS). LDS are 129
designed to give rise to quasi-random numbers that sample points in space as uniformly
130
as possible, while also (typically) having faster convergence rates than standard Monte
131
Carlo sampling methods [37]. After the best-fit parameter set has been selected among
132
the 10
6
randomly sampled sets chosen by LDS, the optimal is refined using simulated 133
annealing [38]. Having observed that the landscape of the objective function near the 134
optimal parameter set does not contain local minima, we randomly perturb the 135
LDS-chosen parameter set, and accept any parameter changes that decrease the value of
136
the objective function - making the method equivalent to gradient descent. This random
137
perturbation process is repeated until no significant change in
ζ
can be achieved, which
138
we defined as the relative change in
ζ
for the last five accepted parameter sets being less
139
than 10
6
. We call this final parameter set the optimal parameter set. 140
It is important to note that, by approaching fitting in this way, the parameters for 141
Mouse i depend only the volumetric data for Mouse i; that is, the volumetric data for 142
the other mice are not accounted for. 143
Fitting Individuals with Population-Level Constraints 144
Nonlinear mixed effects (NLME) models incorporate fixed and random effects to 145
generate models to analyze data that are non-independent, multilevel/hierarchical, 146
longitudinal, or correlated [39]. Fixed effects refer to parameters that can generalize 147
across an entire population. Random effects refer to parameters that differ between 148
individuals that are randomly sampled from a population. 149
The mixed effects model we will utilize is of the form: 150
y
ij
= T (t
ij
, ψ
i
) + bT (t
ij
, ψ
i
)
ij
, i = 1, ..., M, j = 1, ..., n
i
, (7)
where
y
ij
is the predicted tumor volume at each day
j
for each individual
i
(that is, at
151
time t
ij
), M = 8 is the number of mice, n
i
= 31 is the number of observations per 152
mouse,
ψ
i
is the parameter vector for the structural model for each individual, and
ij
is
153
a variable describing random noise. Here we made the assumption that the error is a 154
scalar value proportional to our structural model. 155
Typically, NLME models attempt to maximize the likelihood of the parameter set 156
given the available data. There does not exist a general closed-form solution to this 157
September 7, 2021 5/26
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(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made
The copyright holder for this preprintthis version posted September 9, 2021. ; https://doi.org/10.1101/2021.08.03.454882doi: bioRxiv preprint

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The authors ask the question: what happens if they try to use this same model to perform personalized fits, and therefore make individualized treatment recommendations ? Their analyses show the potentially problematic nature of this approach, as predicted personalized treatment response proved to be sensitive to the fitting methodology utilized. The authors also demonstrate how a small amount of the right additional experimental measurements could go a long way to improve consistency in personalized fits. Finally, the authors show how quantifying the robustness of the average response could also help improve confidence in personalized treatment recommendations. 

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