Adaptive therapy is an on-and-off treatment depending on the size of tumor which aims to delay the emergence of resistance in cancer treatment. To delay cancer resistance, adaptive therapy will suppress resistant cells growth by allowing a significant number of sensitive cells to remain alive during treatment breaks to compete with resistant cells. It is important to study the characteristics of cancer or factors which will affect the tumor response to adaptive therapy. In this article, we shall model sensitive cells, resistant cells, drug concentration and nutrient concentration by one-dimensional reaction-diffusion equations. It is assumed that the growth of sensitive cells and resistant cells depends on the nutrient concentration. The sensitive cells and resistant cells will compete for resources and we use the Lotka-Volterra model to describe the competition. We study whether the initial fraction of resistant cells and the nutrient concentration will extend the time to progression for adaptive therapy beyond that of the standard of care called continuous therapy. We consider the source of nutrient to be either at both ends of tumor or only at one end of tumor. For a very rare initial resistant cells population, adaptive therapy extends the time to progression the most for both cases of source of nutrient. Tumor progresses slightly faster when the source of nutrient is at both ends of tumor for both adaptive therapy and continuous therapy while the time gained by adaptive therapy is slightly smaller compared to when the source of nutrient is at one end only.
REFERENCES
1.
R. A.
Gatenby
, A. S.
Silva
, R. J.
Gillies
, and B. R.
Frieden
, “Adaptive Therapy
,” Cancer Research
69
, 4894
–4903
(2009
), https://aacrjournals.org/cancerres/article-pdf/69/11/4894/2610147/4894.pdf.2.
M. M.A,
J.-Y.
Kim
, C.-H.
Pan
, and E.
Kim
, “The impact of the spatial heterogeneity of resistant cells and fibroblasts on treatment response
,” PLOS Computational Biology
18
, 1
–33
(2022
).3.
D.
Longley
and P.
Johnston
, “Molecular mechanisms of drug resistance
,” The Journal of Pathology
205
, 275
–292
(2005
), https://onlinelibrary.wiley.com/doi/pdf/10.1002/path.1706.4.
A. M.
González-Angulo
, F.
Morales-Vásquez
, and G. N.
Hortobagyi
, “Overview of resistance to systemic therapy in patients with breast cancer
,” Advances in experimental medicine and biology
608
, 1
–22
(2007
).5.
P.
Ascierto
, G.
Mcarthur
, B.
Dréno
, V.
Atkinson
, G.
Liszkay
, A. M.
Di Giacomo
, M.
Mandala
, L.
Demidov
, D.
Stroyakovskiy
, L.
Thomas
, L.
De la Cruz Merino
, C.
Dutriaux
, C.
Garbe
, Y.
Yan
, M.
Wongchenko
, I.
Chang
, J.
Hsu
, D.
Koralek
, I.
Rooney
, and J.
Larkin
, “Cobimetinib combined with vemurafenib in advanced BRAF(V600)-mutant melanoma (coBRIM): updated efficacy results from a randomised, double-blind, phase 3 trial
:,” The Lancet Oncology
17
(2016
), .6.
R.
Dummer
, P. A.
Ascierto
, H. J.
Gogas
, A.
Arance
, M.
Mandala
, G.
Liszkay
, C.
Garbe
, D.
Schadendorf
, I.
Krajsova
, R.
Gutzmer
, V. Chiarion
Sileni
, C.
Dutriaux
, J. W. B.
de Groot
, N.
Yamazaki
, C.
Loquai
, L. A.
Moutouh-de Parseval
, M. D.
Pickard
, V.
Sandor
, C.
Robert
, and K. T.
Flaherty
, “Overall survival in patients with BRAF-mutant melanoma receiving encorafenib plus binimetinib versus vemurafenib or encorafenib (COLUMBUS): a multicentre, open-label, randomised, phase 3 trial
,” The Lancet Oncology
19
, 1315
–1327
(2018
).7.
J.
Zhang
, J. J.
Cunningham
, J. S.
Brown
, and R. A.
Gatenby
, “Integrating evolutionary dynamics into treatment of metastatic castrate-resistant prostate cancer
,” Nature Communications
8
(2017
), .8.
A.
Silva
, Y.
Kam
, Z.
Khin
, S.
Minton
, R.
Gillies
, and R.
Gatenby
, “Evolutionary approaches to prolong progression-free survival in breast cancer
,” Cancer research
72
(2012
), .9.
P.
Enriquez-Navas
, Y.
Kam
, T.
Das
, S.
Hassan
, A.
Silva
, P.
Foroutan
, E.
Ruiz
, G.
Martinez
, S.
Minton
, R.
Gillies
, and R.
Gatenby
, “Exploiting evolutionary principles to prolong tumor control in preclinical models of breast cancer
,” Science Translational Medicine
8
, 327ra24
–327ra24
(2016
).10.
“
Adaptive tyrosine kinase inhibitor (TKI) therapy in patients with thyroid cancer
,” https://clinicaltrials.gov/ct2/show/NCT03630120 (2021
), H. Lee Moffitt Cancer Center and Research Institute
(NCT03630120).11.
“
Adaptive BRAF-MEK inhibitor therapy for advanced BRAF mutant melanoma
,” https://clinicaltrials.gov/ct2/show/NCT03543969 (2022
), H. Lee Moffitt Cancer Center and Research Institute
(NCT03543969).12.
J. A.
Gallaher
, P. M.
Enriquez-Navas
, K. A.
Luddy
, R. A.
Gatenby
, and A. R.
Anderson
, “Spatial Heterogeneity and Evolutionary Dynamics Modulate Time to Recurrence in Continuous and Adaptive Cancer Therapies
,” Cancer Research
78
, 2127
–2139
(2018
), https://aacrjournals.org/cancerres/article-pdf/78/8/2127/2777885/2127.pdf.13.
E.
Hansen
and A. F.
Read
, “Modifying adaptive therapy to enhance competitive suppression
,” Cancers
12
(2020
), .14.
E.
Kim
, J. S.
Brown
, Z.
Eroglu
, and A. R.
Anderson
, “Adaptive therapy for metastatic melanoma: Predictions from patient calibrated mathematical models
,” Cancers
13
(2021
), .15.
M. A.
Strobl
, J.
West
, Y.
Viossat
, M.
Damaghi
, M.
Robertson-Tessi
, J. S.
Brown
, R. A.
Gatenby
, P. K.
Maini
, and A. R.
Anderson
, “Turnover Modulates the Need for a Cost of Resistance in Adaptive Therapy
,” Cancer Research
81
, 1135
–1147
(2021
), https://aacrjournals.org/cancerres/article-pdf/81/4/1135/2804690/1135.pdf.16.
M. A. R.
Strobl
, J.
Gallaher
, J.
West
, M.
Robertson-Tessi
, P. K.
Maini
, and A. R. A.
Anderson
, “Spatial structure impacts adaptive therapy by shaping intra-tumoral competition
,” Communications Medicine
2
(2022
), .17.
Y.
Viossat
and R.
Noble
, “A theoretical analysis of tumour containment
,” Nature Ecology Evolution
5
(2021
), .18.
A.
Marusyk
and K.
Polyak
, “Tumor heterogeneity: Causes and consequences
,” Biochim. Biophys. Acta
2010
, 105
–117
(2010
).19.
K.
Bacevic
, R.
Noble
, A.
Soffar
, O. W.
Ammar
, B.
Boszonyik
, S.
Prieto
, C.
Vincent
, M. E.
Hochberg
, L.
Krasinska
, and D.
Fisher
, “Spatial competition constrains resistance to targeted cancer therapy
,” Nature Communications
8
, 1995
(2017
).20.
E.
Eisenhauer
, P.
Therasse
, J.
Bogaerts
, L.
Schwartz
, D.
Sargent
, R.
Ford
, J.
Dancey
, S.
Arbuck
, S.
Gwyther
, M.
Mooney
, L.
Rubinstein
, L.
Shankar
, L.
Dodd
, R.
Kaplan
, D.
Lacombe
, and J.
Verweij
, “New response evaluation criteria in solid tumours: Revised recist guideline (version 1.1
),” European Journal of Cancer
45
, 228
–247
(2009
), response assessment in solid tumours (RECIST): Version 1.1 and supporting papers.21.
C.
Holohan
, S.
Van Schaeybroeck
, D. B.
Longley
, and P. G.
Johnston
, “Cancer drug resistance: an evolving paradigm
,” Nature reviews. Cancer
13
, 714
—726
(2013
).22.
J. M.
Greene
, J. L.
Gevertz
, and E.
Sontag
, “Mathematical approach to differentiate spontaneous and induced evolution to drug resistance during cancer treatment
,” JCO Clinical Cancer Informatics
3
(2019
).23.
C.
Grassberger
, D. M.
McClatchy
, C.
Geng
, S. C.
Kamran
, F. J.
Fintelmann
, Y. E.
Maruvka
, Z.
Piotrowska
, H.
Willers
, L. V.
Sequist
, A. N.
Hata
, and H.
Paganetti
, “Patient-specific tumor growth trajectories determine persistent and resistant cancer cell populations during treatment with targeted therapies
.” Cancer research
(2019
).24.
T. L.
Jackson
and H. M.
Byrne
, “A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy
,” Mathematical Biosciences
164
, 17
–38
(2000
).25.
C.
Arnerlöv
, S. O.
Emdin
, B.
Lundgren
, G.
Roos
, J.
Söderström
, L.
Bjersing
, C.
Norberg
, and K. A.
Angquist
, “Breast carcinoma growth rate described by mammographic doubling time and S-phase fraction. correlations to clinical and histopathologic factors in a screened population
,” Cancer
70
, 1928
–1934
(1992
).26.
G. C.
Cruywagen
, D. E.
Woodward
, P.
Tracqui
, G. T.
Bartoo
, J. D.
Murray
, and E. C.
Alvord
, “The modelling of diffusive tumours
,” Journal of Biological Systems
03
, 937
–945
(1995
), .27.
R.
Gatenby
and E.
Gawlinski
, “A reaction-diffusion model of cancer invasion
,” Cancer research
56
, 5745
–53
(1996
).28.
S. C.
Ferreira
, M. L.
Martins
, and M. J.
Vilela
, “Reaction-diffusion model for the growth of avascular tumor
,” Phys. Rev. E
65
, 021907
(2002
).29.
R.
Gatenby
, P.
Maini
, and E.
Gawlinski
, “Analysis of tumor as an inverse problem provides a novel theoretical framework for understanding tumor biology and therapy
,” Applied Mathematics Letters
15
, 339
–345
(2002
).30.
S. A.
Morris
and D.
Pratt
, “Analysis of the Lotka–Volterra competition equations as a technological substitution model
,” Technological Forecasting and Social Change
70
, 103
–133
(2003
).31.
C.
Neuhauser
and S.
Pacala
, “An explicitly spatial version of the lotka-volterra model with interspecific competition
,” Annals of Applied Probability
9
(1999
), .32.
H. P.
Greenspan
, “Models for the growth of a solid tumor by diffusion
,” Studies in Applied Mathematics
51
, 317
–340
(1972
), https://onlinelibrary.wiley.com/doi/pdf/10.1002/sapm1972514317.33.
R.
Thomlinson
and L.
Gray
, “The histological structure of some human lung cancers and the possible implications for radiotherapy
,” British journal of cancer
9
, 539
—549
(1955
).34.
K. J.
Himmelstein
and K. B.
Bischoff
, “Mathematical representations of cancer chemotherapy effects
,” Journal of Pharmacokinetics and Biopharmaceutics
1
, 51
–68
(1973
).35.
L.
Norton
and R.
Simon
, “Tumor size, sensitivity to therapy, and design of treatment schedules
,” Cancer treatment reports
61
, 1307
—1317
(1977
).36.
J.
West
and P. K.
Newton
, “Chemotherapeutic Dose Scheduling Based on Tumor Growth Rates Provides a Case for Low-Dose Metronomic High-Entropy Therapies
,” Cancer Research
77
, 6717
–6728
(2017
), https://aacrjournals.org/cancerres/article-pdf/77/23/6717/2761043/6717.pdf.37.
L.
Norton
and R.
Simon
, “The Norton-Simon hypothesis revisited
,” Cancer treatment reports
70
, 163
—169
(1986
).38.
J. C.
Panetta
and K. R.
Fister
, “Optimal control applied to competing chemotherapeutic cell-kill strategies
,” SIAM J. Appl. Math.
63
, 1954
–1971
(2003
).39.
N.
Holford
and L.
Sheiner
, “Pharmacokinetic and pharmacodynamic modeling in vivo
,” Critical reviews in bioengineering
5
, 273
—322
(1981
).40.
X.
Sun
, J.
Bao
, and Y.
Shao
, “Mathematical modeling of therapy-induced cancer drug resistance: Connecting cancer mechanisms to population survival rates
,” Scientific Reports
6
(2016
), .41.
S.
Goutelle
, M.
Maurin
, F.
Rougier
, X.
Barbaut
, L.
Bourguignon
, M.
Ducher
, and P.
Maire
, “The Hill equation: a review of its capabilities in pharmacological modelling
,” Fundamental & Clinical Pharmacology
22
, 633
–648
(2008
), https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1472-8206.2008.00633.x.42.
J. P.
Ward
and J. R.
King
, “Mathematical modelling of avascular-tumour growth
,” Mathematical Medicine and Biology: A Journal of the IMA
14
, 39
–69
(1997
), https://academic.oup.com/imammb/article-pdf/14/1/39/1878306/14-1-39.pdf.43.
J. P.
Ward
and J. R.
King
, “Mathematical modelling of avascular-tumour growth II: Modelling growth saturation
,” Mathematical Medicine and Biology: A Journal of the IMA
16
, 171
–211
(1999
), https://academic.oup.com/imammb/article-pdf/16/2/171/1942305/16-2-171.pdf.44.
C.
Breward
, H.
Byrne
, and C.
Lewis
, “The role of cell-cell interaction in a two-phase model for avascular tumour growth
,” Journal of mathematical biology
45
, 125
–52
(2002
).45.
J. J.
Casciari
, S. V.
Sotirchos
, and R. M.
Sutherland
, “Variations in tumor cell growth rates and metabolism with oxygen concentration, glucose concentration, and extracellular pH
,” Journal of Cellular Physiology
151
, 386
–394
(1992
), https://onlinelibrary.wiley.com/doi/pdf/10.1002/jcp.1041510220.46.
L.
Hlatky
, R. K.
Sachs
, and E. L.
Alpen
, “Joint oxygen-glucose deprivation as the cause of necrosis in a tumor analog
,” Journal of Cellular Physiology
134
, 167
–178
(1988
), https://onlinelibrary.wiley.com/doi/pdf/10.1002/jcp.1041340202.47.
C. K. N.
Li
, “The glucose distribution in 9L rat brain multicell tumor spheroids and its effect on cell necrosis
,” Cancer
50
, 2066
–2073
(1982
), https://acsjournals.onlinelibrary.wiley.com/doi/pdf/10.1002/1097-0142%2819821115%2950%3A10%3C2066%3A%3AAID-CNCR2820501017%3E3.0.CO%3B2-X.48.
C. J. W.
Breward
, H. M.
Byrne
, and C. E.
Lewis
, “Modelling the interactions between tumour cells and a blood vessel in a microenvironment within a vascular tumour
,” European Journal of Applied Mathematics
12
, 529
–556
(2001
).49.
R.
Courant
, K.
Friedrichs
, and H.
Lewy
, “Über die partiellen differenzengleichungen der mathematischen physik
,” Mathematische Annalen
100
, 32
–74
.50.
V.
Ruas
, Numerical Methods for Partial Differential Equations: An Introduction
(Wiley
, 2016
).51.
Y.
Saad
, Iterative Methods for Sparse Linear Systems, EngineeringPro collection
(Society for Industrial and Applied Mathematics
, 2003
).52.
W.
Inch
, J.
Credie
, and R.
Sutherland
, “Growth of nodular carcinomas in rodents compared with multi-cell spheroids in tissue culture
,” Growth
34
, 271
–282
(1970
).53.
R. M.
Sutherland
, J. A.
McCredie
, and W. R.
Inch
, “Growth of Multicell Spheroids in Tissue Culture as a Model of Nodular Carcinomas2
,” JNCI: Journal of the National Cancer Institute
46
, 113
–120
(1971
), https://academic.oup.com/jnci/article-pdf/46/1/113/2712819/46-1-113.pdf.54.
J.
Folkman
, “Anti-angiogenesis: new concept for therapy of solid tumors
.” Annals of surgery
175
, 409
(1972
).55.
J.
Folkman
, E.
Merler
, C.
Abernathy
, and G.
Williams
, “Isolation of a tumor factor responsible for angiogenesis
,” The Journal of experimental medicine
133
, 275
—288
(1971
).56.
J. F.
Crow
, “Genetics of insect resistance to chemicals
,” Annual Review of Entomology
2
, 227
–246
(1957
), .57.
R.
Gatenby
and J.
Brown
, “The evolution and ecology of resistance in cancer therapy
,” Cold Spring Harbor Perspectives in Medicine
10
, a040972
(2020
).58.
T.
Lee
, M.
Baines
, S.
Langdon
, and M.
Tindall
, “A moving mesh approach for modelling avascular tumour growth
,” Applied Numerical Mathematics
72
, 99
–114
(2013
).
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