Skip to main content

Advertisement

SpringerLink
Log in

Investigation of solid tumor progression with account of proliferation/migration dichotomy via Darwinian mathematical model

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

A new continuous spatially-distributed model of solid tumor growth and progression is presented. The model explicitly accounts for mutations/epimutations of tumor cells which take place upon their division. The tumor grows in normal tissue and its progression is driven only by competition between populations of malignant cells for limited nutrient supply. Two reasons for the motion of tumor cells in space are taken into consideration, i.e., their intrinsic motility and convective fluxes, which arise due to proliferation of tumor cells. The model is applied to investigation of solid tumor progression under phenotypic alterations that inversely affect cell proliferation rate and cell motility by increasing the value of one of the parameters at the expense of another.It is demonstrated that the crucial feature that gives evolutionary advantage to a cell population is the speed of its intergrowth into surrounding normal tissue. Of note, increase in tumor intergrowth speed in not always associated with increase in motility of tumor cells. Depending on the parameters of functions, that describe phenotypic alterations, tumor cellular composition may evolve towards: (1) maximization of cell proliferation rate, (2) maximization of cell motility, (3) non-extremum values of cell proliferation rate and motility. Scenarios are found, where after initial tendency for maximization of cell proliferation rate, the direction of tumor progression sharply switches to maximization of cell motility, which is accompanied by decrease in total speed of tumor growth.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Alfonso J, Köhn-Luque A, Stylianopoulos T, Feuerhake F, Deutsch A, Hatzikirou H (2016) Why one-size-fits-all vaso-modulatory interventions fail to control glioma invasion: in silico insights. Sci Rep UK 6:37283

    Google Scholar 

  • Andersen M, Sajid Z, Pedersen R, Gudmand-Hoeyer J, Ellervik C, Skov V, Kjær L, Pallisgaard N, Kruse T, Thomassen M et al (2017) Mathematical modelling as a proof of concept for MPNS as a human inflammation model for cancer development. PLoS ONE 12(8):e0183620

    Google Scholar 

  • Anderson A, Weaver A, Cummings P, Quaranta V (2006) Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment. Cell 127(5):905–915

    Google Scholar 

  • Araujo R, McElwain D (2004) New insights into vascular collapse and growth dynamics in solid tumors. J Theor Biol 228(3):335–346

    MathSciNet  Google Scholar 

  • Association AD et al (2004) Screening for type 2 diabetes. Diabetes Care 27(suppl 1):s11–s14

    Google Scholar 

  • Baker P, Mottram R (1973) Metabolism of exercising and resting human skeletal muscle, in the post-prandial and fasting states. Clin Sci 44(5):479–491

    Google Scholar 

  • Bielas J, Loeb K, Rubin B, True L, Loeb L (2006) Human cancers express a mutator phenotype. Proc Natl Acad Sci 103(48):18238–18242

    Google Scholar 

  • Boris J, Book D (1973) Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works. J Comput Phys 11(1):38–69

    MATH  Google Scholar 

  • Boucher Y, Baxter L, Jain R (1990) Interstitial pressure gradients in tissue-isolated and subcutaneous tumors: implications for therapy. Cancer Res 50(15):4478–4484

    Google Scholar 

  • Bouchnita A, Belmaati F, Aboulaich R, Koury M, Volpert V (2017) A hybrid computation model to describe the progression of multiple myeloma and its intra-clonal heterogeneity. Computation 5(1):16

    Google Scholar 

  • Byrne H, Drasdo D (2009) Individual-based and continuum models of growing cell populations: a comparison. J Math Biol 58(4–5):657

    MathSciNet  MATH  Google Scholar 

  • Casciari J, Sotirchos S, Sutherland R (1992) Mathematical modelling of microenvironment and growth in EMT6/Ro multicellular tumour spheroids. Cell Prolif 25(1):1–22

    Google Scholar 

  • Chisholm R, Lorenzi T, Lorz A, Larsen A, Almeida L, Escargueil A, Clairambault J (2015) Emergence of drug tolerance in cancer cell populations: an evolutionary outcome of selection, non-genetic instability and stress-induced adaptation. Cancer Res 75(6):930–939

    Google Scholar 

  • Chmielecki J, Foo J, Oxnard G, Hutchinson K, Ohashi K, Somwar R, Wang L, Amato K, Arcila M, Sos M et al (2011) Optimization of dosing for EGFR-mutant non-small cell lung cancer with evolutionary cancer modeling. Sci Transl Med 3(90):90ra59–90ra59

    Google Scholar 

  • Citron M, Berry D, Cirrincione C, Hudis C, Winer E, Gradishar W, Davidson N, Martino S, Livingston R, Ingle J et al (2003) Randomized trial of dose-dense versus conventionally scheduled and sequential versus concurrent combination chemotherapy as postoperative adjuvant treatment of node-positive primary breast cancer: first report of intergroup Trial C9741/Cancer and Leukemia Group B Trial 9741. J Clin Oncol 21(8):1431–1439

    Google Scholar 

  • Esteller M (2008) Epigenetics in cancer. New Engl J Med 358(11):1148–1159

    Google Scholar 

  • Eymontt M, Gwinup G, Kruger F, Maynard D, Hamwi G (1965) Cushing’syndrome with hypoglycemia caused by adrenocortical carcinoma. J Clin Endocrinol Metab 25(1):46–52

    Google Scholar 

  • Fisher R (1937) The wave of advance of advantageous genes. Ann Eugen 7(4):355–369

    MATH  Google Scholar 

  • Freyer J, Sutherland R (1985) A reduction in the in situ rates of oxygen and glucose consumption of cells in EMT6/Ro spheroids during growth. J Cell Physiol 124(3):516–524

    Google Scholar 

  • Gatenby R, Gawlinski E, Gmitro A, Kaylor B, Gillies R (2006) Acid-mediated tumor invasion: a multidisciplinary study. Cancer Res 66(10):5216–5223

    Google Scholar 

  • Gerlee P, Anderson A (2008) A hybrid cellular automaton model of clonal evolution in cancer: the emergence of the glycolytic phenotype. J Theor Biol 250(4):705–722

    MathSciNet  MATH  Google Scholar 

  • Giese A, Loo M, Tran N, Haskett D, Coons S, Berens M (1996) Dichotomy of astrocytoma migration and proliferation. Int J Cancer 67(2):275–282

    Google Scholar 

  • Gottesman M (2002) Mechanisms of cancer drug resistance. Annu Rev Med 53(1):615–627

    Google Scholar 

  • Greaves M, Maley C (2012) Clonal evolution in cancer. Nature 481(7381):306

    Google Scholar 

  • Hadjiandreou M, Mitsis G (2014) Mathematical modeling of tumor growth, drug-resistance, toxicity, and optimal therapy design. IEEE T Bio-Med Eng 61(2):415–425

    Google Scholar 

  • Hanahan D, Weinberg R (2000) The hallmarks of cancer. Cell 100(1):57–70

    Google Scholar 

  • Hanahan D, Weinberg R (2011) Hallmarks of cancer: the next generation. Cell 144(5):646–674

    Google Scholar 

  • Hart D, Shochat E, Agur Z (1998) The growth law of primary breast cancer as inferred from mammography screening trials data. Br J Cancer 78(3):382

    Google Scholar 

  • Holash J, Maisonpierre P, Compton D, Boland P, Alexander C, Zagzag D, Yancopoulos G, Wiegand S (1999) Vessel cooption, regression, and growth in tumors mediated by angiopoietins and VEGF. Science 284(5422):1994–1998

    Google Scholar 

  • Iwasa Y, Nowak M, Michor F (2006) Evolution of resistance during clonal expansion. Genetics 172(4):2557–2566

    Google Scholar 

  • Izuishi K, Kato K, Ogura T, Kinoshita T, Esumi H (2000) Remarkable tolerance of tumor cells to nutrient deprivation: possible new biochemical target for cancer therapy. Cancer Res 60(21):6201–6207

    Google Scholar 

  • Jiao Y, Torquato S (2011) Emergent behaviors from a cellular automaton model for invasive tumor growth in heterogeneous microenvironments. PLoS Comput Biol 7(12):e1002314

    Google Scholar 

  • Kathagen-Buhmann A, Schulte A, Weller J, Holz M, Herold-Mende C, Glass R, Lamszus K (2016) Glycolysis and the pentose phosphate pathway are differentially associated with the dichotomous regulation of glioblastoma cell migration versus proliferation. Neuro Oncol 18(9):1219–1229

    Google Scholar 

  • Kolobov A, Kuznetsov M (2015) Investigation of the effects of angiogenesis on tumor growth using a mathematical model. Biophysics 60(3):449–456

    Google Scholar 

  • Kolobov A, Polezhaev A, Solyanik G (2000) The role of cell motility in metastatic cell dominance phenomenon: analysis by a mathematical model. Comput Math Methods Med 3(1):63–77

    MATH  Google Scholar 

  • Kuznetsov M, Kolobov A (2017) Mathematical modelling of chemotherapy combined with bevacizumab. Russ J Numer Anal Model 32(5):293–304

    MathSciNet  MATH  Google Scholar 

  • Kuznetsov M, Kolobov A (2018) Transient alleviation of tumor hypoxia during first days of antiangiogenic therapy as a result of therapy-induced alterations in nutrient supply and tumor metabolism-analysis by mathematical modeling. J Theor Biol 451:86–100

    MathSciNet  MATH  Google Scholar 

  • Kuznetsov M, Gorodnova N, Simakov S, Kolobov A (2016) Multiscale modeling of angiogenic tumor growth, progression, and therapy. Biophysics 61(6):1042–1051

    Google Scholar 

  • Kuznetsov M, Gubernov V, Kolobov A (2018) Analysis of anticancer efficiency of combined fractionated radiotherapy and antiangiogenic therapy via mathematical modelling. Russ J Numer Anal Model 33(4):225–242

    MathSciNet  MATH  Google Scholar 

  • Ledzewicz U, Naghnaeian M, Schättler H (2012) Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics. J Math Biol 64(3):557–577

    MathSciNet  MATH  Google Scholar 

  • Levick JR (2013) An introduction to cardiovascular physiology. Butterworth-Heinemann, Oxford

    Google Scholar 

  • Lorenzi T, Chisholm R, Clairambault J (2016) Tracking the evolution of cancer cell populations through the mathematical lens of phenotype-structured equations. Biol Direct 11(1):43

    Google Scholar 

  • Lorenzi T, Venkataraman C, Lorz A, Chaplain M (2018) The role of spatial variations of abiotic factors in mediating intratumour phenotypic heterogeneity. J Theor Biol 451:101–110

    MathSciNet  MATH  Google Scholar 

  • Lorz A, Lorenzi T, Hochberg M, Clairambault J, Perthame B (2013) Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies. ESAIM Math Model Numer Anal 47(2):377–399

    MathSciNet  MATH  Google Scholar 

  • Lorz A, Lorenzi T, Clairambault J, Escargueil A, Perthame B (2015) Modeling the effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors. Bull Math Biol 77(1):1–22

    MathSciNet  MATH  Google Scholar 

  • Louis D, Ohgaki H, Wiestler O, Cavenee W, Burger P, Jouvet A, Scheithauer B, Kleihues P (2007) The 2007 who classification of tumours of the central nervous system. Acta Neuropathol 114(2):97–109

    Google Scholar 

  • Macklin P, McDougall S, Anderson A, Chaplain M, Cristini V, Lowengrub J (2009) Multiscale modelling and nonlinear simulation of vascular tumour growth. J Math Biol 58(4–5):765–798

    MathSciNet  MATH  Google Scholar 

  • Moreno-Sánchez R, Rodríguez-Enríquez S, Marín-Hernández A, Saavedra E (2007) Energy metabolism in tumor cells. FEBS J 274(6):1393–1418

    Google Scholar 

  • Owen M, Alarcón T, Maini P, Byrne H (2009) Angiogenesis and vascular remodelling in normal and cancerous tissues. J Math Biol 58(4–5):689

    MathSciNet  MATH  Google Scholar 

  • Patra K, Hay N (2014) The pentose phosphate pathway and cancer. Trends Biochem Sci 39(8):347–354

    Google Scholar 

  • Phan L, Yeung S, Lee M (2014) Cancer metabolic reprogramming: importance, main features, and potentials for precise targeted anti-cancer therapies. Cancer Biol Med 11(1):1

    Google Scholar 

  • Press WH (2007) Numerical recipes. The art of scientific computing, 3rd edn. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Robertson K (2001) Dna methylation, methyltransferases, and cancer. Oncogene 20(24):3139

    Google Scholar 

  • Rockne R, Alvord E, Rockhill J, Swanson K (2009) A mathematical model for brain tumor response to radiation therapy. J Math Biol 58(4–5):561

    MathSciNet  MATH  Google Scholar 

  • Shiraishi T, Verdone J, Huang J, Kahlert U, Hernandez J, Torga G, Zarif J, Epstein T, Gatenby R, McCartney A et al (2015) Glycolysis is the primary bioenergetic pathway for cell motility and cytoskeletal remodeling in human prostate and breast cancer cells. Oncotarget 6(1):130

    Google Scholar 

  • Skehan P (1986) On the normality of growth dynamics of neoplasms in vivo: a data base analysis. Growth 50(4):496–515

    Google Scholar 

  • Sonveaux P, Végran F, Schroeder T, Wergin M, Verrax J, Rabbani Z, De Saedeleer C, Kennedy K, Diepart C, Jordan B et al (2008) Targeting lactate-fueled respiration selectively kills hypoxic tumor cells in mice. J Clin Investig 118(12):3930

    Google Scholar 

  • Stamatelos S, Kim E, Pathak A, Popel A (2014) A bioimage informatics based reconstruction of breast tumor microvasculature with computational blood flow predictions. Microvasc Res 91:8–21

    Google Scholar 

  • Stiehl T, Lutz C, Marciniak-Czochra A (2016) Emergence of heterogeneity in acute leukemias. Biol Direct 11(1):51

    Google Scholar 

  • Strong L (1958) Genetic concept for the origin of cancer: historical review. Ann NY Acad Sci 71(6):810–838

    Google Scholar 

  • Sudhakar A (2009) History of cancer, ancient and modern treatment methods. J Cancer Sci Ther 1(2):1

    Google Scholar 

  • Swanson K, Alvord E Jr, Murray J (2000) A quantitative model for differential motility of gliomas in grey and white matter. Cell Prolif 33(5):317–329

    Google Scholar 

  • Theodorescu D, Cornil I, Sheehan C, Man S, Kerbel R (1991) Dominance of metastatically competent cells in primary murine breast neoplasms is necessary for distant metastatic spread. Int J Cancer 47(1):118–123

    Google Scholar 

  • Tuchin V, Bashkatov A, Genina E, Sinichkin Y, Lakodina N (2001) In vivo investigation of the immersion-liquid-induced human skin clearing dynamics. Tech Phys Lett 27(6):489–490

    Google Scholar 

  • Vander Heiden M, Cantley L, Thompson C (2009) Understanding the warburg effect: the metabolic requirements of cell proliferation. Science 324(5930):1029–1033

    Google Scholar 

  • Velicescu M, Weisenberger D, Gonzales F, Tsai Y, Nguyen C, Jones P (2002) Cell division is required for de novo methylation of CpG islands in bladder cancer cells. Cancer Res 62(8):2378–2384

    Google Scholar 

Download references

Acknowledgements

The reported study was funded by RFBR according to the research Projects Nos. 16-01-00709, 17-01-00070 and 19-01-00768. Numerical simulations have been prepared with the support of the “RUDN University Program 5-100”.

Author information

Authors and Affiliations

  1. Division of Theoretical Physics, P.N. Lebedev Physical Institute of the Russian Academy of Sciences, 53 Leninskii Prospekt, Moscow, Russia, 119991

    Maxim Kuznetsov & Andrey Kolobov

  2. Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, Russia, 117198

    Andrey Kolobov

Authors
  1. Maxim Kuznetsov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrey Kolobov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maxim Kuznetsov.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 8503 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kuznetsov, M., Kolobov, A. Investigation of solid tumor progression with account of proliferation/migration dichotomy via Darwinian mathematical model. J. Math. Biol. 80, 601–626 (2020). https://doi.org/10.1007/s00285-019-01434-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-019-01434-4

Keywords

Mathematics Subject Classification

Search

Navigation