要了解腫瘤的演化和制訂適當的療程，環境條件如何影響固態腫瘤型態的知識就變得相當重要。在此研究中，我們透過反應-擴散模型去模擬腫瘤的演化過程，並考慮細胞的增生率是養分濃度的函數。我們的模擬結果顯示當環境養分低於一個臨界值時，腫瘤表面就會出現型態不穩定性，並最後造成腫瘤分裂成數個小腫瘤，而腫瘤透過分裂可以增加細胞與養分接觸的表面積，藉此提高在養分缺乏的環境中的生存機率。為了研究此機制，我們透過線性穩定性分析去求出微擾的本徵值和本徵方程式，且此分析結果和模擬的量測結果高度相符。另外，所有理論和模擬數據都顯示提高養分濃度、越強的細胞-細胞吸引力和越大的養分對細胞的擴散係數比例都會抑制腫瘤的不穩定性。在相圖中也顯示腫瘤會因為不同的養分濃度和增生率係數而有不同的生長行為。以上結果都指出環境條件會對腫瘤形態學有重要影響。

The morphology of solid tumors driven by environments is crucial to understanding the progression of tumors and assessing proper treatments. In this study, we propose the reaction-diffusion model accounting for the nutrient-dependent growth rate of tumor cells. The simulation results show that the surface instability would happen as the nutrient level of surroundings is lower than a threshold value and eventually lead to the break-up of a solid tumor. By doing so, it pronouncedly increases the contact surface between cells and nutrients, which forms a strategy for cells to raise their survival probability in a nutrient-deficient environment. To investigate its mechanism, we present a linear stability analysis around the unperturbed planar tumor surface. The eigenfunction and its growth rate are solved analytically, and the dispersion relation is in quantitative agreement with simulations. In addition, both simulation and theoretical results demonstrate that the instability can be inhibited with higher nutrient level, stronger cell-to-cell adhesion and the larger diffusion coefficient ratio of nutrients to cells. The phase diagram shows four distinct behaviors of tumor growth associated with different nutrient concentrations and growth rate of cells. These results indicate that an environmental condition plays an important role in controlling tumor morphology.

[1] Clément Chatelain, Pasquale Ciarletta, and Martine Ben Amar. Morphological changes in early melanoma development: Influence of nutrients, growth inhibitors and cell-adhesion mechanisms. Journal of Theoretical Biology, 290:46–59, 2011.

[2] Hermann B Frieboes, Xiaoming Zheng, Chung-Ho Sun, Bruce Tromberg, Robert Gatenby, and Vittorio Cristini. An integrated computational/experimental model of tumor invasion. Cancer research, 66(3):1597—1604, February 2006.

[3] Adi Karsch-Bluman, Ariel Feiglin, Eliran Arbib, Tal Stern, Hila Shoval, Ouri Schwob, Michael Berger, and Ofra Benny. Tissue necrosis and its role in cancer progression. Oncogene, 38(11):1920–1935, Mar 2019.

[4] Joseph J Casciari, Stratis V Sotirchos, and Robert M Sutherland. Variations in tumor cell growth rates and metabolism with oxygen concentration, glucose concentration, and extracellular ph. Journal of cellular physiology, 151(2):386–394, 1992.

[5] Clément Chatelain, Pasquale Ciarletta, and Martine Ben Amar. Morphological changes in early melanoma development: influence of nutrients, growth inhibitors and cell-adhesion mechanisms. Journal of theoretical biology, 290:46–59, 2011.

[6] Mark AJ Chaplain, Steven R McDougall, and ARA Anderson. Mathematical modeling of tumor-induced angiogenesis. Annu. Rev. Biomed. Eng., 8:233–257, 2006.

[7] James D Murray. Mathematical biology II: spatial models and biomedical applications, volume 3. Springer New York, 2001.

[8] Mark AJ Chaplain, Steven R McDougall, and ARA Anderson. Mathematical modeling of tumor-induced angiogenesis. Annu. Rev. Biomed. Eng., 8:233–257, 2006.

[9] Robert M. Sutherland and Ralph E. Durand. Hypoxic cells in an in vitro tumour model. International Journal of Radiation Biology and Related Studies in Physics, Chemistry and Medicine, 23(3):235–246, 1973.

[10] Hsiang-Ying Chen, Yi-Teng Hsiao, Shu-Chen Liu, Tien Hsu, Wei-Yen Woon, and I Lin. Enhancing cancer cell collective motion and speeding up confluent endothelial dynamics through cancer cell invasion and aggregation. Physical review letters, 121(1):018101, 2018.

[11] Sarah Kessel, Scott Cribbes, Surekha Bonasu, William Rice, Jean Qiu, and Leo Li-Ying Chan. Real-time viability and apoptosis kinetic detection method of 3d multicellular tumor spheroids using the celigo image cytometer. Cytometry Part A, 91(9):883–892, 2017.

[12] Yoshiki Kuramoto. Chemical oscillations, waves, and turbulence. Courier Corporation, 2003.

[13] Michael P Brenner, Leonid S Levitov, and Elena O Budrene. Physical mechanisms for chemotactic pattern formation by bacteria. Biophysical journal, 74(4):1677–1693, 1998.

[14] Liyang Xiong, Yuansheng Cao, Robert Cooper, Wouter-Jan Rappel, Jeff Hasty, and Lev Tsimring. Flower-like patterns in multi-species bacterial colonies. Elife, 9:e48885, 2020.

[15] Lisa Fayed and J Seong. Differences between a malignant and benign tumor, 2020.

[16] Chao-Shiang Li, Guo-Shu Huang, Hong-Da Wu, Wei-Tsung Chen, Li-Sun Shih, Jiunn-Ming Lii, Shyi-Jye Duh, Ran-Chou Chen, Hsing-Yang Tu, and Wing P Chan. Differentiation of soft tissue benign and malignant peripheral nerve sheath tumors with magnetic resonance imaging. Clinical imaging, 32(2):121–127, 2008.

[17] Harald Kittler, Pascale Guitera, Elisabeth Riedl, Michelle Avramidis, Ligia Teban, Manfred Fiebiger, Rickard A Weger, Markus Dawid, and Scott Menzies. Identification of clinically featureless incipient melanoma using sequential dermoscopy imaging. Archives of dermatology, 142(9):1113–1119, 2006.

[18] RH Thomlinson and LH Gray. The histological structure of some human lung cancers and the possible implications for radiotherapy. British journal of cancer, 9(4):539, 1955.

[19] AC Burton. Rate of growth of solid tumours as a problem of diffusion. Growth, 30(2):157—176, June 1966.

[20] H.P. Greenspan. On the growth and stability of cell cultures and solid tumors. Journal of Theoretical Biology, 56(1):229–242, 1976.

[21] Robert M. Sutherland, John A. McCredie, and W. Rodger Inch. Growth of Multicell Spheroids in Tissue Culture as a Model of Nodular Carcinomas2. JNCI: Journal of the National Cancer Institute, 46(1):113–120, 01 1971.

[22] Vittorio Cristini, John Lowengrub, and Qing Nie. Nonlinear simulation of tumor growth. Journal of Mathematical Biology, 46(3):191–224, Mar 2003.

[23] X. Zheng, S. M. Wise, and V. Cristini. Nonlinear simulation of tumor necrosis, neo-vascularization and tissue invasion via an adaptive finite-element/level-set method. Bulletin of Mathematical Biology, 67(2):211, Mar 2005.

[24] John S Lowengrub, Hermann B Frieboes, Fang Jin, Yao-Li Chuang, Xian-grong Li, Paul Macklin, Steven M Wise, and Vittorio Cristini. Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity, 23(1):R1, 2009.

[25] Steven M Wise, John S Lowengrub, Hermann B Frieboes, and Vittorio Cristini. Three-dimensional multispecies nonlinear tumor growth—i: model and numerical method. Journal of theoretical biology, 253(3):524–543, 2008.

[26] M Ben Amar, C Chatelain, and Pasquale Ciarletta. Contour instabilities in early tumor growth models. Physical review letters, 106(14):148101, 2011.

[27] Thibaut Balois and Martine Ben Amar. Morphology of melanocytic lesions in situ. Scientific Reports, 4(1):3622, Jan 2014.

[28] Jonathan A Sherratt and Martin A Nowak. Oncogenes, anti-oncogenes and the immune response to cancer: a mathematical model. Proceedings of the Royal Society of London. Series B: Biological Sciences, 248(1323):261–271, 1992.

[29] Robert A. Gatenby and Edward T. Gawlinski. A reaction-diffusion model of cancer invasion. Cancer Research, 56(24):5745–5753, 1996.

[30] Christian Märkl, Gülnihal Meral, Christina Surulescu, et al. Mathematical analysis and numerical simulations for a system modeling acid-mediated tumor cell invasion. International Journal of Analysis, 2013:1–15, 2013.

[31] Ahmed M Fouad. Dynamical stability analysis of tumor growth and invasion: A reaction-diffusion model. arXiv preprint arXiv:1905.11341, 2019.

[32] Thierry Gallay and Corrado Mascia. Propagation fronts in a simplified model of tumor growth with degenerate cross-dependent self-diffusivity. arXiv preprint arXiv:2103.07775, 2021.

[33] Alexander RA Anderson, Mark AJ Chaplain, E Luke Newman, Robert JC Steele, and Alastair M Thompson. Mathematical modelling of tumour invasion and metastasis. Computational and mathematical methods in medicine, 2(2):129–154, 2000.

[34] Dumitru Trucu, Ping Lin, Mark AJ Chaplain, and Yangfan Wang. A multi-scale moving boundary model arising in cancer invasion. Multiscale Modeling & Simulation, 11(1):309–335, 2013.

[35] Peter Romeo Nyarko and Martin Anokye. Mathematical modeling and numerical simulation of a multiscale cancer invasion of host tissue. AIMS Mathematics, 5(4):3111–3124, 2020.

[36] Kristin R Swanson, Ellsworth C Alvord Jr, and JD Murray. A quantitative model for differential motility of gliomas in grey and white matter. Cell proliferation, 33(5):317–329, 2000.

[37] Kristin R Swanson, Carly Bridge, JD Murray, and Ellsworth C Alvord Jr. Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion. Journal of the neurological sciences, 216(1):1–10, 2003.

[38] Yong Chen, Hengtong Wang, Jiangang Zhang, Ke Chen, and Yumin Li. Simulation of avascular tumor growth by agent-based game model involving phenotype-phenotype interactions. Scientific Reports, 5(1):17992, Dec 2015.

[39] Vittorio Cristini, Hermann B. Frieboes, Robert Gatenby, Sergio Caserta, Mauro Ferrari, and John Sinek. Morphologic instability and cancer invasion. Clinical Cancer Research, 11(19):6772–6779, 2005.

[40] Vittorio Cristini, Xiangrong Li, John S. Lowengrub, and Steven M. Wise. Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching. Journal of Mathematical Biology, 58(4):723, Sep 2008.

[41] Judith Müller and Wim van Saarloos. Morphological instability and dynamics
of fronts in bacterial growth models with nonlinear diffusion. Phys. Rev. E,
65:061111, Jun 2002.

[42] Thomas Risler and Markus Basan. Morphological instabilities of stratified epithelia: a mechanical instability in tumour formation. New Journal of Physics, 15(6):065011, jun 2013.

[43] J. H. Merkin and I. Z. Kiss. Dispersion curves in the diffusional instability of autocatalytic reaction fronts. Phys. Rev. E, 72:026219, Aug 2005.

[44] Rui Peng, Junping Shi, and Mingxin Wang. On stationary patterns of a reaction–diffusion model with autocatalysis and saturation law. Nonlinearity, 21(7):1471–1488, may 2008.

[45] Rui Peng, Feng qi Yi, and Xiao qiang Zhao. Spatiotemporal patterns in a reaction–diffusion model with the degn–harrison reaction scheme. Journal of Differential Equations, 254(6):2465–2498, 2013.

[46] DLS McElwain and LE Morris. Apoptosis as a volume loss mechanism in mathematical models of solid tumor growth. Mathematical Biosciences, 39(1-2):147–157, 1978.

[47] Christopher S Potten, James Wilson, et al. Apoptosis: the life and death of cells. Cambridge University Press, 2004.

[48] S. C. Ferreira, M. L. Martins, and M. J. Vilela. Reaction-diffusion model for the growth of avascular tumor. Phys. Rev. E, 65:021907, Jan 2002.

[49] Hana LP Harpold, Ellsworth C Alvord Jr, and Kristin R Swanson. The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology & Experimental Neurology, 66(1):1–9, 2007.

[50] Adam A Wheeler, William J Boettinger, and Geoffrey B McFadden. Phase-field model for isothermal phase transitions in binary alloys. Physical Review A, 45(10):7424, 1992.

[51] Seong Gyoon Kim, Won Tae Kim, and Toshio Suzuki. Phase-field model for binary alloys. Physical review e, 60(6):7186, 1999.

[52] John FR Kerr, Andrew H Wyllie, and Alastair R Currie. Apoptosis: a basic biological phenomenon with wideranging implications in tissue kinetics. British journal of cancer, 26(4):239–257, 1972.

[53] Sébastien Benzekry, Clare Lamont, Afshin Beheshti, Amanda Tracz, John M. L. Ebos, Lynn Hlatky, and Philip Hahnfeldt. Classical mathematical models for description and prediction of experimental tumor growth. PLOS Computational Biology, 10(8):1–19, 08 2014.

[54] Hope Murphy, Hana Jaafari, and Hana M. Dobrovolny. Differences in predictions of ode models of tumor growth: a cautionary example. BMC Cancer, 16(1):163, Feb 2016.

[55] Philippe R Spalart, Robert D Moser, and Michael M Rogers. Spectral methods for the navier-stokes equations with one infinite and two periodic directions. Journal of Computational Physics, 96(2):297–324, 1991.

[56] Alfonso Bueno-Orovio, David Kay, and Kevin Burrage. Fourier spectral meth-
ods for fractional-in-space reaction-diffusion equations. BIT Numerical Math-
ematics, 54(4):937–954, Dec 2014.

[57] Agota Tóth, Dezső Horváth, and Wim van Saarloos. Lateral instabilities of cubic autocatalytic reaction fronts in a constant electric field. The Journal of Chemical Physics, 111(24):10964–10968, 1999.

[58] V. Gubernov, G. N. Mercer, H. S. Sidhu, and R. O. Weber. Numerical methods for the travelling wave solutions in reaction-diffusion equations. In K. Burrage and Roger B. Sidje, editors, Proc. of 10th Computational Techniques and Applications Conference CTAC-2001, volume 44, pages C271–C289, April 2003. [Online] http://anziamj.austms.org.au/V44/CTAC2001/Gube [April 1, 2003].

[59] J. Yang, A. D’Onofrio, S. Kalliadasis, and A. De Wit. Rayleigh–taylor instability of reaction-diffusion acidity fronts. The Journal of Chemical Physics, 117(20):9395–9408, 2002.

[60] S. Kalliadasis, J. Yang, and A. De Wit. Fingering instabilities of exothermic reaction-diffusion fronts in porous media. Physics of Fluids, 16(5):1395–1409, 2004.

[61] Vincent TY Ng, Benny YM Fung, and Tim K Lee. Determining the asymmetry of skin lesion with fuzzy borders. Computers in biology and medicine, 35(2):103–120, 2005.

[62] Paul Macklin and John Lowengrub. Nonlinear simulation of the effect of microenvironment on tumor growth. Journal of Theoretical Biology, 245(4):677–704, 2007.

[63] Gregg L Semenza, Mary K Nejfelt, Suzie M Chi, and Stylianos E Antonarakis. Hypoxia-inducible nuclear factors bind to an enhancer element located 3’to the human erythropoietin gene. Proceedings of the National Academy of Sciences, 88(13):5680–5684, 1991.

[64] Guang L Wang, Bing-Hua Jiang, Elizabeth A Rue, and Gregg L Semenza. Hypoxia-inducible factor 1 is a basic-helix-loop-helix-pas heterodimer regulated by cellular o2 tension. Proceedings of the national academy of sciences, 92(12):5510–5514, 1995.

[65] Patrick H Maxwell, Michael S Wiesener, Gin-Wen Chang, Steven C Clifford, Emma C Vaux, Matthew E Cockman, Charles C Wykoff, Christopher W Pugh, Eamonn R Maher, and Peter J Ratcliffe. The tumour suppressor protein vhl targets hypoxia-inducible factors for oxygen-dependent proteolysis. Nature, 399(6733):271–275, 1999.

[66] Mircea Ivan, Keiichi Kondo, Haifeng Yang, William Kim, Jennifer Valiando, Michael Ohh, Adrian Salic, John M Asara, William S Lane, and William G Kaelin Jr. Hifα targeted for vhl-mediated destruction by proline hydroxylation: implications for o2 sensing. Science, 292(5516):464–468, 2001.

[67] Panu Jaakkola, David R Mole, Ya-Min Tian, Michael I Wilson, Janine Gielbert, Simon J Gaskell, Alexander von Kriegsheim, Holger F Hebestreit, Mridul Mukherji, Christopher J Schofield, et al. Targeting of hif-α to the von hippel-lindau ubiquitylation complex by o2-regulated prolyl hydroxylation. Science, 292(5516):468–472, 2001.

[68] Miguel Martín-Landrove. Reaction-diffusion models for glioma tumor growth. arXiv preprint arXiv:1707.09409, 2017.

[69] Shuichi Miyamoto, Kenji Atsuyama, Keisuke Ekino, and Takashi Shin. Estimating the diffusion coefficients of sugars using diffusion experiments in agar-gel and computer simulations. Chemical and Pharmaceutical Bulletin, 66(6):632–636, 2018.

[70] Gao-Feng Xiong and Ren Xu. Function of cancer cell-derived extracellular matrix in tumor progression. Journal of Cancer Metastasis and Treatment, 2:357–364, 2016.

[71] Alexander RA Anderson. A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. Mathematical medicine and biology: a journal of the IMA, 22(2):163–186, 2005.