Mathematical modeling, analysis and simulation are set to play crucial roles in explaining tumor behavior, and the uncontrolled growth of cancer cells over multiple time and spatial scales. This book, the first to integrate state-of-the-art numerical techniques with experimental data, provides an in-depth assessment of tumor cell modeling at multiple scales. The first part of the text presents a detailed biological background with an examination of single-phase and multi-phase continuum tumor modeling, discrete cell modeling, and hybrid continuum-discrete modeling. In the final two chapters, the authors guide the reader through problem-based illustrations and case studies of brain and breast cancer, to demonstrate the future potential of modeling in cancer research. This book has wide interdisciplinary appeal and is a valuable resource for mathematical biologists, biomedical engineers and clinical cancer research communities wishing to understand this emerging field.
Author(s): Vittorio Cristini, John Lowengrub
Edition: 1
Year: 2010
Language: English
Pages: 298
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 9
List of contributors......Page 14
Preface......Page 17
Acknowledgements......Page 19
Vectors, matrices, and tensors......Page 20
Probability and statistics......Page 21
I Theory......Page 23
1 Introduction......Page 25
2.1.1 Tissue microarchitecture and maintenance......Page 30
2.1.2 Cellular adhesion and cell sorting......Page 32
2.1.4 Cell cycle, proliferation, and apoptosis......Page 33
2.1.5 Genetics, gene expression, and cell signaling......Page 35
2.1.6 Cell motility......Page 38
2.2.1 Carcinogenesis......Page 39
2.2.2 Avascular solid-tumor growth......Page 40
2.2.3 Interaction with the microenvironment......Page 41
2.2.4 Vascular growth and metastasis......Page 42
2.3 Concluding remarks......Page 45
3.1.1 Background......Page 46
3.2.1 Model without necrosis......Page 48
3.2.2 Growth in heterogeneous tissue......Page 51
3.2.3 Effect of stress......Page 53
3.3.1 Background......Page 55
3.3.2 Modeling of angiogenesis......Page 57
3.3.3 Basic model......Page 58
3.3.4 Coupling of tumor growth with angiogenesis......Page 60
3.4 Conclusion......Page 61
4.1 Regimes of growth......Page 63
4.2 Linear analysis......Page 65
4.3 Nonlinear results......Page 71
4.4 Model calibration from experimental data......Page 75
4.5 Tumor growth with chemotaxis......Page 79
4.6 Tumor growth with necrosis......Page 81
4.7 Tumor growth in heterogeneous microenvironments......Page 84
4.8 Vascular tumor growth......Page 86
4.9 Conclusion......Page 87
5.1 Background......Page 89
5.2 General conservation equations......Page 91
5.3 A solid--liquid biphasic model......Page 92
5.4 Liquid--liquid mixture model I......Page 93
5.5 Liquid--liquid mixture model II......Page 95
5.6 A special case of the generalized Darcy's law liquid--liquid mixture model......Page 96
5.7 Fluxes and velocities......Page 97
5.8 Mass-exchange terms......Page 98
5.10 Mutation of tumor-cell species......Page 99
5.11.1 Morphological instability as an invasive mechanism......Page 101
5.11.2 Chemotaxis as an invasive mechanism......Page 102
5.12.1 Background......Page 104
5.12.2 Biased circular random walk......Page 106
5.13 Nonlinear results: vascularized tumor growth......Page 107
6.1 A brief review of discrete modeling in cancer biology......Page 110
6.1.2 Lattice-free models......Page 111
6.1.3 Comparison with continuum methods......Page 112
6.1.4 Some discrete modeling examples......Page 113
6.2 An agent-based cell modeling framework......Page 118
6.2.1 A brief review of exponential random variables and Poisson processes......Page 119
6.2.3 Cell states......Page 121
6.2.4 Forces acting on the cells......Page 125
6.3 Subcellular modeling......Page 131
6.4 Dynamic coupling with the microenvironment......Page 132
6.5 A brief analysis of the volume-averaged model behavior......Page 135
6.6 Numerical examples from breast cancer......Page 136
6.6.1 Baseline calibrated run......Page 137
6.6.2 Impact of hypoxic survival time......Page 140
6.6.3 Impact of the cell lysis time......Page 141
6.6.5 Impact of cell motility......Page 142
6.7 Conclusions......Page 144
7.1 Background......Page 145
7.2.1 Conservation laws in continuum models......Page 149
7.2.2 Conservation laws in discrete models......Page 150
7.2.3 ``Inertialess'' assumption......Page 151
7.2.4 Linking discrete and continuum variables......Page 152
7.3.1 The continuum-to-discrete (C2D) conversion......Page 153
7.3.2 The discrete-to-continuum (D2C) conversion......Page 156
7.3.3 Momentum exchange through continuum--discrete interactions......Page 158
7.4 A hybrid model of multicellular tumor spheroids (MCTSs)......Page 161
7.4.1 The continuum and the discrete components......Page 162
7.4.2 Continuum--discrete coupling mechanism......Page 164
7.4.3 Results......Page 165
7.5 A hybrid model of vascularized tumor growth......Page 166
7.5.1 The continuum and the discrete components......Page 167
7.5.2 The C2D conversion process......Page 168
7.5.3 The D2C conversion process......Page 170
7.5.4 Momentum exchange through continuum--discrete interactions......Page 171
7.5.6 Discussion......Page 172
8 Numerical schemes......Page 175
8.1 Review of the multiphase mixture model......Page 176
8.2 Uniform mesh discretization......Page 178
8.2.1 The discrete gradient operator d......Page 179
8.2.2 Treatment of the advection terms......Page 180
8.2.3 Discretization of the boundary conditions......Page 183
8.2.5 Treatment of the source terms......Page 184
8.3 Multigrid mesh hierarchy and block-structured adaptive mesh......Page 186
8.3.1 Block-structured adaptive-mesh refinement......Page 187
8.3.2 Initialization of the refined meshes......Page 190
8.4.1 Operators and sources......Page 193
8.4.2 The MLAT--FAS V-cycle method......Page 195
8.4.3 Smoothing procedures......Page 197
8.4.4 Restriction and interpolation......Page 201
8.4.5 Criteria for mesh refinement......Page 203
II Applications......Page 205
9.1.1 Avascular growth......Page 207
9.1.2 Vascularized tumors......Page 209
9.2.1 Avascular growth......Page 210
9.2.2 Vascularized tumors......Page 214
9.2.3 Clinical observations......Page 218
9.3 Modeling outlook......Page 226
10.1 Introduction......Page 228
10.1.1 Biology of breast-duct epithelium......Page 229
10.1.2 Biology of DCIS......Page 231
10.2.2 Duct geometry......Page 232
10.2.3 Intraductal oxygen diffusion......Page 234
10.2.4 Numerical method......Page 235
10.3.1 Estimating ``universal'' parameters......Page 236
10.3.2 Calibrating patient-specific parameters......Page 237
10.3.3 Sample patient calibration and verification......Page 240
10.4 Case studies......Page 241
10.4.1 Estimating difficult physical parameters......Page 242
10.4.2 Generating and testing hypotheses......Page 244
10.4.3 Calibrating multiscale modeling frameworks: preliminary results......Page 248
10.5 Concluding remarks......Page 254
11 Conclusion......Page 257
References......Page 260
Index......Page 297
