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Convected element method for simulation of angiogenesis

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Abstract

We describe a novel Convected Element Method (CEM) for simulation of formation of functional blood vessels induced by tumor-generated growth factors in a process called angiogenesis. Angiogenesis is typically modeled by a convection-diffusion-reaction equation defined on a continuous domain. A difficulty arises when a continuum approach is used to represent the formation of discrete blood vessel structures. CEM solves this difficulty by using a hybrid continuous/discrete solution method allowing lattice-free tracking of blood vessel tips that trace out paths that subsequently are used to define compact vessel elements. In contrast to more conventional angiogenesis modeling, the new branches form evolving grids that are capable of simulating transport of biological and chemical factors such as nutrition and anti-angiogenic agents. The method is demonstrated on expository vessel growth and tumor response simulations for a selected set of conditions, and include effects of nutrient delivery and inhibition of vessel branching. Initial results show that CEM can predict qualitatively the development of biologically reasonable and fully functional vascular structures. Research is being carried out to generalize the approach which will allow quantitative predictions.

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References

  1. Alarcon T., Byrne H., Maini P. and Panovska J. (2006). Mathematical modelling of angiogenesis and vascular adaptation. In: Patony, R. and McNamara, L. (eds) Studies in Multidisciplinarity, vol 3. Elsevier B.V., Amsterdam

    Google Scholar 

  2. Ambrosi D. and Preziosi L. (2002). On the closure of mass balance models for tumor growth. Math. Models Methods Appl. Sci. 12(5): 737–754

    Article  MATH  MathSciNet  Google Scholar 

  3. Anderson A.R.A. and Chaplain M.A.J. (1998). Continuous and discrete mathematical models of tumour-induced angiogenesis. Bull. Math. Bio. 60: 857–899

    Article  MATH  Google Scholar 

  4. Bauer A.L., Jackson T.L. and Jiang Y. (2007). A cell-based model exhibiting branching and anastomosis during tumor-induced angiogenesis. Biophys. J. 92: 3105–3121

    Article  Google Scholar 

  5. Bellomo N., De Angelis E. and Preziosi L. (2003). Multiscale modeling and mathematical problems related to tumor evolution and medical therapy. J. Theor. Med. 5: 111–136

    Article  MATH  Google Scholar 

  6. Breward C.J.W., Byrne H.M. and Lewis C.E. (2001). Modelling the interactions between tumour cells and a blood vessel in a microenvironment within a vascular tumour.. Eur. J. Appl. Math. 12(5): 529–556

    Article  MATH  MathSciNet  Google Scholar 

  7. Chaplain M.A.J., McDougall S.R. and Anderson A.R.A. (2006). Mathematical modelling of tumor induced angiogenesis. Annu. Rev. Biomed. Eng. 2006(8): 233–257

    Article  Google Scholar 

  8. Chaplain M.A.J. and Preziosi L. (2000). Macroscopic Modeling of the Growth and Development of Tumor Masses, Internal Report Nr. 27. Politecnico di Torino, Italy

  9. Chorin A.J. (1973). Numerical study of slightly viscous flow. J. Fluid Mech. 57: 785–796

    Article  MathSciNet  Google Scholar 

  10. Cristini V., Lowengrub J. and Nie Q. (2003). Nonlinear simulation of tumor growth. J. Math. Biol. 46: 191–224

    Article  MATH  MathSciNet  Google Scholar 

  11. Davis B. (1990). Reinforced Random Walks. Probab. Theory Related Fields 84: 203–229

    Article  MATH  MathSciNet  Google Scholar 

  12. DeAngelis E. and Preziosi L. (2000). Advection-diffusion models for solid tumour evolution in-vivo and related free boundary problem. Math. Mod. Meth. App. Sci. 10(3): 379–407

    Article  MathSciNet  Google Scholar 

  13. Einstein, A.: Investigations on the Theory of the Brownian Movement. Dover Publications, Inc., New York (1956)

    MATH  Google Scholar 

  14. Holmes M.J. and Sleeman B.D. (2000). A mathematical model of tumor angiogenesis incorporating cellular traction and viscoelastic effects. J. Theor. Biol. 202: 95–112

    Article  Google Scholar 

  15. Jain R.K. (1988). Determinants of tumor blood flow, a review. Cancer Res. 44: 2641–2656

    Google Scholar 

  16. Less J.R., Skalak T.C., Sevick E.M. and Jain R.K. (1991). Microvascular architecture in mammary carcinoma: branching patterns and vessel dimensions.. Microcirculation 4(1): 25–33

    Article  Google Scholar 

  17. Less J.R., Posner M.C., Skalak T.C., Wolmark N. and Jain R.K. (1997). Rapid communication geometric resistance and microvascular network architecture of human colorectal carcinoma. Microcirculation 4(1): 25–33

    Article  Google Scholar 

  18. Levine H.A., Sleeman B.D. and Nilsen-Hamilton M. (2000). A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis, I: the role of protease inhibitors in preventing angiogenesis. Math. Biosci. 168: 77–115

    Article  MATH  MathSciNet  Google Scholar 

  19. Levine H.A., Palmuk S., Sleeman B.D. and Nielsen-Hamilton M. (2001). Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma. Bull. Math. Biol. 63: 195–238

    Article  Google Scholar 

  20. Levine H.A., Sleeman B.D. and Nielsen-Hamilton M. (2001). Mathematical modeling of the onset of capillary formation initiating angiogenesis. J. Math. Biol. 42: 195–238

    Article  MATH  MathSciNet  Google Scholar 

  21. Mantzaris N., Webb S. and Othmer H.G. (2004). Mathematical modeling of tumor-induced angiogenesis. J. Math. Biol. 49: 111–187

    Article  MATH  MathSciNet  Google Scholar 

  22. McDougall S.R., Anderson A.R.A., Chaplain M.A.J. and Sherratt J.A. (2002). Mathematical modelling of flow through vascular networks: implications for tumour-induced angiogenesis and chemotherapy strategies. Bull. Math. Biol. 64: 673–702

    Article  Google Scholar 

  23. NCI: http://press2.nci.nih.gov/sciencebehind/angiogenesis (2003)

  24. Orme M.E. and Chaplain M.A.J. (1996). A mathematical model of the first steps of tumour-related angiogenesis: capillary sprout formation and secondary branching. IMA J. Math. Appl. Med. Biol. 13: 73–98

    Article  MATH  Google Scholar 

  25. Othmer H.G. and Stevens A. (1997). Aggregation, blowup and collapse: the ABC’s of taxis in reinforced random walks. SIAM J. Appl. Math. 57(4): 1044–1081

    Article  MATH  MathSciNet  Google Scholar 

  26. Papetti M. and Herman I.M. (2002). Mechanisms of normal and tumor-derived angiogenesis. Am. J. Physiol. Cell Physiol. 282: C947–C970

    Google Scholar 

  27. Patankar S.V. (1980). Numerical Heat Transfer and Fluid Flow. McGraw-Hill, NY

    MATH  Google Scholar 

  28. Pindera M.-Z. and Talbot L. (1988). Some fluid dynamic considerations in the modeling of flames. Combust. Flame 73: 111–125

    Article  Google Scholar 

  29. Pindera, M.Z., Ding, H., Chen, Z., Przekwas, A.J.: Simulation of Angiogenesis-Tumor Dynamics and Treatment. NIH Report 8492, Grant # R43 CA 97827-01, August (2003)

  30. Pindera, M.Z., Bayyuk, S.B., Upadhya, V., Przekwas, A.J.: A Computational Framework for Modeling One-Dimensional, Sub-Grid Components and Phenomena in Multi-Dimensional Microsystems. Proceedings of the Design, Test, Integration and Packaging of MEMS/MOEMS, SPIE, Paris France, vol. 4109, pp. 272–279 (2000)

  31. Preziosi L. (2003). Cancer Modeling and Simulation. CRC Press, Boca Raton

    Google Scholar 

  32. Pries A.R., Secomb T.W., Gessner T., Sperandio M.B., Gross J.F. and Gaehtgens P. (1994). Resistance to blood flow in microvessels in vivo. Circ. Res. 75: 904–915

    Google Scholar 

  33. Pries A.R., Secomb T.W. and Gaehtgens P. (1998). Structural adaptation and stability of microvascular networks: Theory and simulations. Am. J. Physiol. 275: H349–H360

    Google Scholar 

  34. Secomb, T.W., Hsu, R., Dewhirst, M.W.: Models for Oxygen Exchange between Microvascular Networks and Surrounding Tissue. Advances in Biological Heat and Mass Transfer, ASME HTD-231 (1992)

  35. Seppa, N.: A Shot to the Heart Shows Promise. http://www.sciencenews.org/sn_arc98/2_28_98/feb1.html (1998)

  36. Stephanou A., McDougall S.R., Anderson A.R.A. and Chaplain M.A.J. (2005). Mathematical modelling of flow in 2D and 3D vascular networks: applications to anti-angiogenic and chemotherapeutic drug strategies. Math. Comput. Model. 41: 1137–1156

    Article  MATH  MathSciNet  Google Scholar 

  37. Stephanou A., McDougall S., Anderson A. and Chaplain M.A.J. (2006). Mathematical modelling of the influence of blood rheological properties upon adaptive tumour-induced angiogenesis. Math. Comput. Model. 44: 96–123

    Article  MATH  MathSciNet  Google Scholar 

  38. Van Doormaal J.P. and Raithby G.D. (1984). Enhancements of the SIMPLE method for predicting incompressible fluid flows. Numer. Heat Transf. 7: 147–163

    Article  MATH  Google Scholar 

  39. Zheng X., Wise S.M. and Cristini V. (2005). Nonlinear simulation of tumor necrosis, neo-vascularization and tissue invasion via an adaptive finite-element/level-set method. Bull. Math. Bio. 67: 211–259

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. CFD Research Corporation, 215 Wynn Drive, Huntsville, AL, 35805, USA

    Maciej Z. Pindera, Hui Ding & Zhijian Chen

Authors
  1. Maciej Z. Pindera
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  2. Hui Ding
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  3. Zhijian Chen
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Correspondence to Maciej Z. Pindera.

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Pindera, M.Z., Ding, H. & Chen, Z. Convected element method for simulation of angiogenesis. J. Math. Biol. 57, 467–495 (2008). https://doi.org/10.1007/s00285-008-0171-5

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  • DOI: https://doi.org/10.1007/s00285-008-0171-5

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