We present a competition model of cancer tumor growth that includes both the immune system response and drug therapy. This is a fourpopulation model that includes tumor cells, host cells, immune cells, and drug interaction. We analyze the stability of the drug-free equilibria with respect to the immune response in order to look for target basins of attraction. One of our goals was to simulate qualitatively the asynchronous tumor-drug interaction known as "Jeff's phenomenon. " The model we develop is successful in generating this asynchronous response behavior. Our other goal was to identify treatment protocols that could improve standard pulsed chemotherapy regimens. Using optimal control theory with constraints and numerical simulations, we obtain new therapy protocols that we then compare with traditional pulsed periodic treatment. The optimal control generated therapies produce larger oscillations in the tumor population over time. However, by the end of the treatment period, total tumor size is smaller than that achieved through traditional pulsed therapy, and the normal cell population suffers nearly no oscillations.
|
18
|
Dynamic Optimization: The Calculus of Variations and
– Kamien, Schwartz
- 1991
|
|
6
|
Optimal control for a cancer chemotherapy problem with general growth and loss functions
– Murray
- 1990
|
|
4
|
Differential equations: A modeling perspective
– Borrelli, Coleman
- 1996
|
|
4
|
Mathematical model of cancer chemotherapy. Periodic schedules of phase-specific cytotoxic-agent administration increasing the selectivity of therapy
– Dibrov, Zhabotinsky, et al.
- 1985
|
|
4
|
The growth law of primary breast cancer as inferred from mammography screening trials data
– Hart, Shochat, et al.
- 1998
|
|
4
|
Modelling the macrophage invasion of tumours: effects on growth and composition
– Owen, Sherratt
- 1998
|
|
4
|
Growth Kinetics of Tumors
– Steel
- 1977
|
|
4
|
Evaluation of some mathematical models for tumor growth
– VAIDYA, J
- 1982
|
|
3
|
The dynamics of growth factor-modified immune response to cancer growth: one dimensional models
– Adam
- 1993
|
|
3
|
Modeling immunotherapy of the tumorimmune interaction
– Kirschner, JC
|
|
3
|
A mathematical model of cancer chemotherapy with an optimal selection of parameters
– Martin, Fisher, et al.
- 1990
|
|
3
|
Optimal control problems arising in cell-cycle-specific cancer chemotherapy
– Swierniak, Polanski, et al.
- 1996
|
|
2
|
Mathematical Models in Cell Biology and Cancer
– Eisen
- 1979
|
|
2
|
Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis
– Kuznetsov, Makalkin, et al.
- 1994
|
|
2
|
Optimal control drug scheduling of cancer chemotherapy, Automatica
– Martin
- 1992
|
|
2
|
Optimal control analysis of a cancer chemotherapy problem
– Swan
- 1987
|
|
2
|
Some control problems for simplest differential models of proliferation cycle
– Swierniak
- 1994
|
|
2
|
Dormancy of mammary carcinoma after mastectomy
– Karrison, Ferguson, et al.
- 1999
|
|
1
|
A simple mathematical model and alternative paradigm for certain chemotherapeutic regimens
– Adam, Panetta
- 1995
|
|
1
|
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(7
– Arnerv, Emdin, et al.
- 1992
|
|
1
|
Some stochastic models of cancer metastases
– Bartoszy'nski, Jones, et al.
- 1985
|
|
1
|
The asymptotic behavior of a probabilistic model for tumor growth
– Bramson, Griffeath
- 1980
|
|
1
|
A stochastic model for the evolution of cancerous tumors, Revista Mexicana de F'isica
– Moreno, A
- 1996
|
|
1
|
A stochastic model for the origin and treatment of tumors containing drug-resistant cells
– Coldman, Goldie
- 1986
|
|
1
|
Interactions between macrophages and Tlymphocytes: tumor sneaking through intrinsic to helper T cell dynamics
– Boer, Hogeweg
- 1986
|
|
1
|
A stochastic model of mutant growth due to mutation in tumors, based on stem cell considerations
– Duc
- 1985
|
|
1
|
Cancer dormancy. VII. a regulatory role for CD8+ T cells and IFN-gamma in establishing and maintaining the tumor-dormant state
– Farrar, Katz, et al.
- 1999
|
|
1
|
Immunologic approach to cancer therapy
– Gray, Jr, et al.
- 1975
|
|
1
|
Cell kinetic modelling and the chemotherapy of cancer
– Knolle
- 1988
|
|
1
|
The gross rates of growth of human mammary carcinoma
– Kusama, Spratt, et al.
- 1972
|
|
1
|
Bifurcation-analysis of mathematicalmodel of interactions between cytotoxic lymphocytes and tumor-cells -- effect of immunological amplification of tumor-growth and its connection with other phenomena of oncoimmunology
– Kuznetsov, Makalkin
- 1992
|
|
1
|
1991a, Characterization of a strongly immunogenic mouse leukemia line (DL811) undergoing spontaneous cure with tumor dormancy, Cancer Letters
– Matsuzawa, Kaneko, et al.
|
|
1
|
1991b, Survival of leukemic cells in a dormant state following cyclophosphamide-induced cure of strongly immunogenic mouse leukemia (DL811
– Matsuzawa, Takeda, et al.
|
|
1
|
Host response in tumor growth and progression, Invasion and Metastasis
– Michelson, Leith
- 1996
|
|
1
|
Tumorcell vaccination induces tumor dormancy in a murine model of B-cell leukemia/lymphoma (BCL1
– Morecki, Pugatsch, et al.
- 1996
|
|
1
|
EblacZ tumor dormancy in bone marrow and lymph nodes: active control of proliferating tumor cells by CD8+ immune T cells
– Muller, Gounari, et al.
- 1998
|
|
1
|
A mathematical model of periodically pulsed chemotherapy: Tumor recurrence and metastasis in a competitive environment
– Panetta
- 1996
|
|
1
|
Stimulatory effects of immune-reactions upon the growths of untransplanted tumors
– Prehn
- 1994
|
|
1
|
Conversations. Physician with Pomona Valley Hospital
– Rieker
- 1999
|
|
1
|
Two-stage stochastic model for carcinogenesis with timedependent parameters
– Serio
- 1984
|
|
1
|
Using computer simulations for evaluating the efficacy of breast cancer chemotherapy protocols
– Shochat, Hart, et al.
- 1999
|
|
1
|
Immune mechanisms and tumor dormancy
– Stewart
- 1996
|
|
1
|
Optimal control applications in the chemotherapy of multiple myeloma
– Swan
- 1985
|
|
1
|
Irregularity in scheduling of cancer chemotherapy
– Swierniak, Polanski
- 1994
|
|
1
|
Measurement and management of carcinoma of the breast
– Thomlinson
- 1982
|
|
1
|
Estimate of tumor growth time for reast cancer local recurrences: rapid growth after wakeup ?, Breast Cancer Research and Treatment
– Demicheli, Terenziani, et al.
- 1998
|
|
1
|
Cancer dormancy: Role of cyclin-dependent kinase inhibitors in induction of cell cycle arrest mediated via membrane igm
– Marches, Scheuerman, et al.
- 1998
|
|
1
|
Immune mechanism leads to dormancy
– Stewart
- 1996
|
|
