# Personalized mathematical oncology: Challenges and opportunities

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)

Jump to: [Introduction] – [Author summary] – [Data Set 79] – [Mathematical Model 96] and [Supporting information]

### 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

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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

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

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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

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

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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

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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

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TL;DR: The BATTLE study is the first completed prospective, adaptively randomized study in heavily pretreated NSCLC patients that mandated tumor profiling with "real-time" biopsies, taking a substantial step toward realizing personalized lung cancer therapy by integrating real-time molecular laboratory findings.

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[4]: /lookup/external-ref?link_type=CLINTRIALGOV&access_num=NCT00411632&atom=%2Fcandisc%2F1%2F1%2F44.atom
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