A Bayesian Sequential Learning Framework to Parameterise Continuum Models of Melanoma Invasion into Human Skin

Abstract

We present a novel framework to parameterise a mathematical model of cell invasion that describes how a population of melanoma cells invades into human skin tissue. Using simple experimental data extracted from complex experimental images, we estimate three model parameters: (i) the melanoma cell proliferation rate, \(\lambda \); (ii) the melanoma cell diffusivity, D; and (iii) \(\delta \), a constant that determines the rate that melanoma cells degrade the skin tissue. The Bayesian sequential learning framework involves a sequence of increasingly sophisticated experimental data from: (i) a spatially uniform cell proliferation assay; (ii) a two-dimensional circular barrier assay; and (iii) a three-dimensional invasion assay. The Bayesian sequential learning approach leads to well-defined parameter estimates. In contrast, taking a naive approach that attempts to estimate all parameters from a single set of images from the same experiment fails to produce meaningful results. Overall, our approach to inference is simple-to-implement, computationally efficient, and well suited for many cell biology phenomena that can be described by low-dimensional continuum models using ordinary differential equations and partial differential equations. We anticipate that this Bayesian sequential learning framework will be relevant in other biological contexts where it is challenging to extract detailed, quantitative biological measurements from experimental images and so we must rely on using relatively simple measurements from complex images.

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

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

References

  1. Anderson ARA, Quaranta V (2008) Integrative mathematical oncology. Nat Rev Cancer 8:227–34

    Google Scholar 

  2. Beaumont MA, Zhang W, Balding DJ (2002) Approximate Bayesian computation in population genetics. Genetics 162:2025–2035

    Google Scholar 

  3. Bowden LG, Maini PK, Moulton DE, Tang JB, Wang XT, Liu PY, Byrne HM (2014) An ordinary differential equation model for full thickness wounds and the effects of diabetes. J Theor Biol 361:87–100

    MATH  Google Scholar 

  4. Browning AP, McCue SW, Simpson MJ (2017) A Bayesian computational approach to explore the optimal duration of a cell proliferation assay. Bull Math Biol 79:1888–1906

    MathSciNet  MATH  Google Scholar 

  5. Browning AP, McCue SW, Binny RN, Plank MJ, Shah ET, Simpson MJ (2018) Inferring parameters for a lattice-free model of cell migration and proliferation using experimental data. J Theor Biol 437:251–260

    MathSciNet  MATH  Google Scholar 

  6. Cai AQ, Landman KA, Hughes BD (2007) Multi-scale modeling of a wound-healing cell migration assay. J Theor Biol 245:576–594

    MathSciNet  Google Scholar 

  7. Carey TE, Takahashi T, Resnick LA, Oettgen HF, Old LJ (1976) Cell surface antigens of human malignant melanoma: mixed hemadsorption assays for humoral immunity to cultured autologous melanoma cells. PNAS 73:3278–3282

    Google Scholar 

  8. Collis J, Connor AJ, Paczkowski M, Kannan P, Pitt-Francis J, Byrne HM, Hubbard ME (2017) Bayesian calibration, validation and uncertainty quantification for predictive modelling of tumour growth: a tutorial. J R Soc Interface 15:20180318

    MATH  Google Scholar 

  9. Daly AD, Gavaghan D, Cooper J, Tavener S (2018) Inference-based assessment of parameter identifiability in nonlinear biological models. Bull Math Biol 79:939–974

    Google Scholar 

  10. Fasano A, Herrero MA, Rodrigo MR (2009) Slow and fast invasion waves in a model of acid-mediated tumour growth. Math Biosci 220:45–56

    MathSciNet  MATH  Google Scholar 

  11. Fearnhead P, Prangle D (2012) Constructing summary statistics for approximate Bayesian computation: semi-automatic approximate Bayesian computation. J R Stat Soc B 74:419–474

    MathSciNet  MATH  Google Scholar 

  12. Gatenby RA, Gawlinski ET (1996) A reaction–diffusion model of cancer invasion. Cancer Res 56:5745–5753

    Google Scholar 

  13. Gelman A, Carlin JB, Stern HS, Rubin DB (2004) Bayesian data analysis. CRC Press, Boca Raton

    Google Scholar 

  14. Haridas P, Penington CJ, McGovern JA, McElwain DLS, Simpson MJ (2017a) Quantifying rates of cell migration and cell proliferation in co-culture barrier assays reveals how skin and melanoma cells interact during melanoma spreading and invasion. J Theor Biol 423:13–25

    Google Scholar 

  15. Haridas P, McGovern JA, McElwain DLS, Simpson MJ (2017b) Quantitative comparison of the spreading and invasion of radial growth phase and metastatic melanoma cells in a three-dimensional human skin equivalent model. PeerJ 5:e3754

    Google Scholar 

  16. Haridas P, Browning AP, McGovern JA, McElwain DLS, Simpson MJ (2018) Three-dimensional experiments and individual based simulations show that cell proliferation drives melanoma nest formation in human skin tissue. BMC Syst Biol 12:34

    Google Scholar 

  17. Illowsky B, Dean S (2015) Introductory statistics. OpenStax College, Houston

    Google Scholar 

  18. Jain HV, Richardson A, Meyer-Hermann M, Byrne HM (2014) Exploiting the synergy between Carboplatin and ABT737 in the treatment of ovarian carcinomas. PLoS ONE 9:e81582

    Google Scholar 

  19. Johnston ST, Simpson MJ, McElwain DLS, Binder BJ, Ross JV (2014) Interpreting scratch assays using pair density dynamics and approximate Bayesian computation. Open Biol 4:140097

    Google Scholar 

  20. Landman KA, Pettet GJ (1998) Modelling the action of proteinase and inhibitor in tissue invasion. Math Biosci 154:23–37

    MATH  Google Scholar 

  21. Landman KA, Pettet GJ, Newgreen NF (2003) Chemotactic cellular migration: smooth and discontinuous travelling wave solutions. SIAM J Appl Math 63:1666–1681

    MathSciNet  MATH  Google Scholar 

  22. Landman KA, Simpson MJ, Slater JL, Newgreen DF (2005) Diffusive and chemotactic cellular migration: smooth and discontinuous traveling wave solutions. SIAM J Appl Math 65:1420–1442

    MathSciNet  MATH  Google Scholar 

  23. Landman KA, Simpson MJ, Pettet GJ (2008) Tactically-driven nonmonotone travelling waves. Physica D 237:678–691

    MathSciNet  MATH  Google Scholar 

  24. Maclaren OJ, Parker A, Pin C, Carding SR, Watson AJM, Fletcher AG, Byrne HM, Maini PK (2017) A hierarchical Bayesian model for understanding the spatiotemporal dynamics of the intestinal epithelium. PLoS Comput Biol 13:1005688

    Google Scholar 

  25. Maini PK, McElwain DLS, Leavesley DI (2004a) Traveling wave model to interpret a wound-healing cell migration assay for human peritoneal mesothelial cells. Tissue Eng 10:475–482

    Google Scholar 

  26. Maini PK, McElwain DLS, Leavesley D (2004b) Travelling waves in a wound healing assay. Appl Math Lett 17:575–580

    MathSciNet  MATH  Google Scholar 

  27. Marchant BP, Norbury J, Sherratt JA (2001) Travelling wave solutions to a haptotaxis-dominated model of malignant invasion. Nonlinearity 14:1653–1671

    MathSciNet  MATH  Google Scholar 

  28. Marchant BP, Norbury J (2002) Discontinuous travelling wave solutions for certain hyperbolic systems. IMA J Appl Math 67:201–224

    MathSciNet  MATH  Google Scholar 

  29. Marchant BP, Norbury J, Byrne HM (2006) Biphasic behaviour in malignant invasion. Math Med Biol 3:173–196

    MATH  Google Scholar 

  30. Massey SC, Assanah MC, Lopez KA, Canoll P, Swanson KR (2012) Glial progenitor cell recruitment drives aggressive glioma growth: mathematical and experimental modelling. J R Soc Interface 9:1757–1766

    Google Scholar 

  31. Mathworks (2018a) Image processing toolbox. https://au.mathworks.com/products/image.html. Accessed Aug 2018

  32. Mathworks (2018b) Gridded and scattered data interpolation, data gridding, piecewise polynomials. https://au.mathworks.com/help/matlab/interpolation.html. Accessed Aug 2018

  33. Mathworks (2018c) Kernel smoothing function estimate for univariate and bivariate data. http://www.mathworks.com/help/stats/ksdensity.html. Accessed Aug 2018

  34. Perumpanani A, Sherratt JA, Norbury J, Byrne HM (1999) A two parameter family of travelling waves with a singular barrier arising from the modelling of extracellular matrix mediated cellular invasion. Physica D 126:145–159

    MATH  Google Scholar 

  35. Perumpanani AJ, Marchant BP, Norbury J (2000) Travelling shock waves arising in a model of malignant invasion. SIAM J Appl Math 60:463–476

    MathSciNet  MATH  Google Scholar 

  36. Sarapata EA, de Pillis LG (2014) A comparison and catalog of intrinsic tumor growth models. Bull Math Biol 76:2010–2024

    MathSciNet  MATH  Google Scholar 

  37. Sengers BG, Please CP, Oreffo ROC (2007) Experimental characterization and computational modelling of two-dimensional cell spreading for skeletal regeneration. J R Soc Interface 4:1107–1117

    Google Scholar 

  38. Sisson SA, Fan Y, Tanaka MM (2007) Sequential Monte Carlo without likelihoods. PNAS 104:1760–1765

    MathSciNet  MATH  Google Scholar 

  39. Simpson MJ, Landman KA (2007) Nonmonotone chemotactic invasion: high-resolution simulations, phase plane analysis and new benchmark problems. J Comput Phys 225:6–12

    MathSciNet  MATH  Google Scholar 

  40. Simpson MJ, McInerney S, Carr EJ, Cuttle L (2017) Quantifying the efficacy of first aid treatments for burn injuries using mathematical modelling and in vivo porcine experiments. Sci Rep 7:10925

    Google Scholar 

  41. Smallbone K, Gavaghan DJ, Gatenby RA, Maini PK (2005) The role of acidity in solid tumour growth and invasion. J Theor Biol 235:476–484

    MathSciNet  Google Scholar 

  42. Swanson KR, Rockne RC, Claridge J, Chaplain MA, Alvord EC, Anderson ARA (2011) Quantifying the role of angiogenesis in malignant progression of gliomas: in silico modeling integrates imaging and histology. Cancer Res 71:7366–7375

    Google Scholar 

  43. Toni T, Welch D, Strelkowa N, Ipsen A, Stumpf MPH (2009) Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. J R Soc Interface 6:187–202

    Google Scholar 

  44. Treloar KK, Simpson MJ (2013) Sensitivity of edge detection methods for quantifying cell migration assays. PLoS ONE 8:e67389

    Google Scholar 

  45. Treloar KK, Simpson MJ, Haridas P, Manton KJ, Leavesley DI, McElwain DLS, Baker RE (2013) Multiple types of data are required to identify the mechanisms influencing the spatial expansion of melanoma cell colonies. BMC Syst Biol 7:137

    Google Scholar 

  46. Tremel A, Tirtaatmadja N, Hughes BD, Stevens GW, Landman KA, OConnor AJ (2009) Cell migration and proliferation during monolayer formation and wound healing. Chem Eng Sci 64:247–253

    Google Scholar 

  47. Vittadello ST, McCue SW, Gunasingh G, Haass NK, Simpson MJ (2018) Mathematical models for cell migration wit real-time cell cycle dynamics. Biophys J 114:1241–1253

    Google Scholar 

  48. Warne DJ, Baker RE, Simpson MJ (2017) Optimal quantification of contact inhibition in cell populations. Biophys J 113:1920–1924

    Google Scholar 

  49. Warne DJ, Baker RE, Simpson MJ (2018) Multilevel rejection sampling for approximate Bayesian computation. Comput Stat Data Anal 124:71–86

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by the Australian Research Council (DP170100474). We appreciate helpful comments from David Warne, and two anonymous referees.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Matthew J. Simpson.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 350 KB)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Browning, A.P., Haridas, P. & Simpson, M.J. A Bayesian Sequential Learning Framework to Parameterise Continuum Models of Melanoma Invasion into Human Skin. Bull Math Biol 81, 676–698 (2019). https://doi.org/10.1007/s11538-018-0532-1

Download citation

Keywords

  • Cell invasion
  • Melanoma
  • Invasion assay
  • Cancer
  • Bayesian inference