Cellular and Molecular Bioengineering

, Volume 8, Issue 1, pp 119–136 | Cite as

Strategies for Efficient Numerical Implementation of Hybrid Multi-scale Agent-Based Models to Describe Biological Systems

  • Nicholas A. Cilfone
  • Denise E. Kirschner
  • Jennifer J. Linderman
Article

Abstract

Biologically related processes operate across multiple spatiotemporal scales. For computational modeling methodologies to mimic this biological complexity, individual scale models must be linked in ways that allow for dynamic exchange of information across scales. A powerful methodology is to combine a discrete modeling approach, agent-based models (ABMs), with continuum models to form hybrid models. Hybrid multi-scale ABMs have been used to simulate emergent responses of biological systems. Here, we review two aspects of hybrid multi-scale ABMs: linking individual scale models and efficiently solving the resulting model. We discuss the computational choices associated with aspects of linking individual scale models while simultaneously maintaining model tractability. We demonstrate implementations of existing numerical methods in the context of hybrid multi-scale ABMs. Using an example model describing Mycobacterium tuberculosis infection, we show relative computational speeds of various combinations of numerical methods. Efficient linking and solution of hybrid multi-scale ABMs is key to model portability, modularity, and their use in understanding biological phenomena at a systems level.

Keywords

Multi-scale modeling Hybrid modeling Agent-based modeling Numerical implementation Linking models Tuneable resolution 

Notes

Acknowledgments

We thank Paul Wolberg and Joe Waliga for computational assistance. This research was supported in part through computational resources and services provided by Advanced Research Computing at the University of Michigan, Ann Arbor. This research was funded by the following NIH Grants: R01 EB012579 (DEK and JJL) and R01 HL 110811 (DEK and JJL).

Conflict of Interest

Nicholas Cilfone, Denise Kirschner, and Jennifer Linderman declare no conflicts of interests.

Ethical Standards

No human or animal studies were carried out by the authors for this article.

References

  1. 1.
    Adra, S., T. Sun, S. MacNeil, M. Holcombe, and R. Smallwood. Development of a three dimensional multiscale computational model of the human epidermis. PLoS ONE 5:e8511, 2010.CrossRefGoogle Scholar
  2. 2.
    Alarcón, T., H. M. Byrne, and P. K. Maini. Towards whole-organ modelling of tumour growth. Prog. Biophys. Mol. Biol. 85:451–472, 2004.CrossRefGoogle Scholar
  3. 3.
    An, G., Q. Mi, J. Dutta-Moscato, and Y. Vodovotz. Agent-based models in translational systems biology. Wiley Interdiscip. Rev. Syst. Biol. Med. 1:159–171, 2009.CrossRefGoogle Scholar
  4. 4.
    Anderson, A. R. A., M. A. J. Chaplain, and K. A. Rejniak (eds.). Single-Cell-Based Models in Biology and Medicine. Birkhäuser Basel: Basel, 2007.MATHGoogle Scholar
  5. 5.
    Angermann, B. R., et al. Computational modeling of cellular signaling processes embedded into dynamic spatial contexts. Nat. Methods 9:283–289, 2012.CrossRefGoogle Scholar
  6. 6.
    Athale, C. A., and T. S. Deisboeck. The effects of EGF-receptor density on multiscale tumor growth patterns. J. Theor. Biol. 238:771–779, 2006.CrossRefMathSciNetGoogle Scholar
  7. 7.
    Athale, C., Y. Mansury, and T. S. Deisboeck. Simulating the impact of a molecular “decision-process” on cellular phenotype and multicellular patterns in brain tumors. J. Theor. Biol. 233:469–481, 2005.CrossRefGoogle Scholar
  8. 8.
    Bailey, A. M., M. B. Lawrence, H. Shang, A. J. Katz, and S. M. Peirce. Agent-based model of therapeutic adipose-derived stromal cell trafficking during ischemia predicts ability to roll on P-selectin. PLoS Comput. Biol. 5:e1000294, 2009.CrossRefGoogle Scholar
  9. 9.
    Barakat, H. Z., and J. A. Clark. On the solution of the diffusion equations by numerical methods. J. Heat Transfer 88:421, 1966.CrossRefGoogle Scholar
  10. 10.
    Basak, S., M. Behar, and A. Hoffmann. Lessons from mathematically modeling the NF-κB pathway. Immunol. Rev. 246:221–238, 2012.CrossRefGoogle Scholar
  11. 11.
    Bauer, A. L., C. A. Beauchemin, and A. S. Perelson. Agent-based modeling of host-pathogen systems: the successes and challenges. Inf. Sci. (NY) 179:1379–1389, 2009.CrossRefGoogle Scholar
  12. 12.
    Berg, E. L. Systems biology in drug discovery and development. Drug Discov. Today 19:113–125, 2013.CrossRefGoogle Scholar
  13. 13.
    Bird, R. B., W. E. Stewart, and E. N. Lightfoot. Transport Phenomena. New York: Wiley, 1994.Google Scholar
  14. 14.
    Braun, D. A., M. Fribourg, and S. C. Sealfon. Cytokine response is determined by duration of receptor and signal transducers and activators of transcription 3 (STAT3) activation. J. Biol. Chem. 288:2986–2993, 2013.CrossRefGoogle Scholar
  15. 15.
    Chakrabarti, A., S. Verbridge, A. D. Stroock, C. Fischbach, and J. D. Varner. Multiscale models of breast cancer progression. Ann. Biomed. Eng. 40:2488–2500, 2012.CrossRefGoogle Scholar
  16. 16.
    Choi, T., M. R. Maurya, D. M. Tartakovsky, and S. Subramaniam. Stochastic operator-splitting method for reaction–diffusion systems. J. Chem. Phys. 137:184102, 2012.CrossRefGoogle Scholar
  17. 17.
    Christley, S., and G. An. Agent-Based Modeling in Translational Systems Biology. In: Complex Systems and Computational Biology Approaches to Acute Inflammation SE—3, edited by Y. Vodovotz, and G. An. New York, NY: Springer, 2013, pp. 29–49.CrossRefGoogle Scholar
  18. 18.
    Cilfone, N. A., C. R. Perry, D. E. Kirschner, and J. J. Linderman. Multi-scale modeling predicts a balance of tumor necrosis factor-α and interleukin-10 controls the granuloma environment during Mycobacterium tuberculosis infection. PLoS ONE 8:e68680, 2013.CrossRefGoogle Scholar
  19. 19.
    Costa, B. Spectral methods for partial differential equations. Cubo - Revista de Matemática 6:1–32, 2004.Google Scholar
  20. 20.
    Coveney, P. V., and P. W. Fowler. Modelling biological complexity: a physical scientist’s perspective. J. R. Soc. Interface 2:267–280, 2005.CrossRefGoogle Scholar
  21. 21.
    Csomós, P., I. Faragó, and Á. Havasi. Weighted sequential splittings and their analysis. Comput. Math. Appl. 50:1017–1031, 2005.CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Dada, J. O., and P. Mendes. Multi-scale modelling and simulation in systems biology. Integr. Biol. (Camb) 3:86–96, 2011.CrossRefGoogle Scholar
  23. 23.
    Daubechies, I. The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inf. Theory 36:961–1005, 1990.CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Deisboeck, T. S., Z. Wang, P. Macklin, and V. Cristini. Multiscale cancer modeling. Annu. Rev. Biomed. Eng. 13:127–155, 2011.CrossRefGoogle Scholar
  25. 25.
    Duhamel, P., and M. Vetterli. Fast fourier transforms: a tutorial review and a state of the art. Signal Process. 19:259–299, 1990.CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Fallahi-Sichani, M., M. El-Kebir, S. Marino, D. E. Kirschner, and J. J. Linderman. Multiscale computational modeling reveals a critical role for TNF-α receptor 1 dynamics in tuberculosis granuloma formation. J. Immunol. 186:3472–3483, 2011.CrossRefGoogle Scholar
  27. 27.
    Fallahi-Sichani, M., J. L. Flynn, J. J. Linderman, and D. E. Kirschner. Differential risk of tuberculosis reactivation among anti-TNF therapies is due to drug binding kinetics and permeability. J. Immunol. 188:3169–3178, 2012.CrossRefGoogle Scholar
  28. 28.
    Fallahi-Sichani, M., D. E. Kirschner, and J. J. Linderman. NF-κB signaling dynamics play a key role in infection control in tuberculosis. Front. Physiol. 3:170, 2012.CrossRefGoogle Scholar
  29. 29.
    Figueredo, G. P., T. V. Joshi, J. M. Osborne, H. M. Byrne, and M. R. Owen. On-lattice agent-based simulation of populations of cells within the open-source Chaste framework. Interface Focus 3:20120081, 2013.CrossRefGoogle Scholar
  30. 30.
    Flynn, J. L., and J. Chan. Immunology of tuberculosis. Annu. Rev. Immunol. 19:93–129, 2001.CrossRefGoogle Scholar
  31. 31.
    Fornberg, B. A practical guide to pseudospectral methods. Cambridge: Cambridge University Press, 1996.MATHGoogle Scholar
  32. 32.
    Frieboes, H. B., et al. Computer simulation of glioma growth and morphology. Neuroimage 37(Suppl 1):S59–S70, 2007.CrossRefGoogle Scholar
  33. 33.
    Frigo, M., and S. G. Johnson. The Design and Implementation of FFTW3. Proc. IEEE 93:216–231, 2005.CrossRefGoogle Scholar
  34. 34.
    Geiser, J., G. Tanoğlu, and N. Gücüyenen. Higher order operator splitting methods via Zassenhaus product formula: theory and applications. Comput. Math. Appl. 62:1994–2015, 2011.CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Gong, C., J. J. Linderman, and D. Kirschner. Harnessing the heterogeneity of T cell differentiation fate to fine-tune generation of effector and memory T cells. Front. Immunol. 5:1–15, 2014.CrossRefGoogle Scholar
  36. 36.
    Gong, C., J. T. Mattila, M. Miller, J. L. Flynn, J. J. Linderman, and D. Kirschner. Predicting lymph node output efficiency using systems biology. J. Theor. Biol. 335C:169–184, 2013.CrossRefGoogle Scholar
  37. 37.
    Gottlieb, D., and C.-W. Shu. On the Gibbs phenomenon and its resolution. SIAM Rev. 39:644–668, 1997.CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Guo, Z., P. M. A. Sloot, and J. C. Tay. A hybrid agent-based approach for modeling microbiological systems. J. Theor. Biol. 255:163–175, 2008.CrossRefMathSciNetGoogle Scholar
  39. 39.
    Hedengren, J. D., and T. F. Edgar. Order reduction of large scale DAE models. Comput. Chem. Eng. 29:2069–2077, 2005.CrossRefGoogle Scholar
  40. 40.
    Hedengren, J. D., and T. F. Edgar. In situ adaptive tabulation for real-time control. Ind. Eng. Chem. Res. 44:2716–2724, 2005.CrossRefGoogle Scholar
  41. 41.
    Heidlauf, T., and O. Röhrle. Modeling the chemoelectromechanical behavior of skeletal muscle using the parallel open-source software library OpenCMISS. Comput. Math. Methods Med. 2013:517287, 2013.CrossRefGoogle Scholar
  42. 42.
    Holcombe, M., et al. Modelling complex biological systems using an agent-based approach. Integr. Biol. (Camb) 4:53–64, 2012.CrossRefGoogle Scholar
  43. 43.
    Hou, T. Y., and R. Li. Computing nearly singular solutions using pseudo-spectral methods. J. Comput. Phys. 226:379–397, 2007.CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Hunt, C. A., R. C. Kennedy, S. H. J. Kim, and G. E. P. Ropella. Agent-based modeling: a systematic assessment of use cases and requirements for enhancing pharmaceutical research and development productivity. Wiley Interdiscip. Rev. Syst. Biol. Med. 5:461–480, 2013.CrossRefGoogle Scholar
  45. 45.
    Karlsen, K. H., K.-A. Lie, J. Natvig, H. Nordhaug, and H. Dahle. Operator splitting methods for systems of convection–diffusion equations: nonlinear error mechanisms and correction strategies. J. Comput. Phys. 173:636–663, 2001.CrossRefMATHMathSciNetGoogle Scholar
  46. 46.
    Kaul, H., Z. Cui, and Y. Ventikos. A multi-paradigm modeling framework to simulate dynamic reciprocity in a bioreactor. PLoS ONE 8:e59671, 2013.CrossRefGoogle Scholar
  47. 47.
    Kim, M., R. J. Gillies, and K. A. Rejniak. Current advances in mathematical modeling of anti-cancer drug penetration into tumor tissues. Front. Oncol. 3:278, 2013.Google Scholar
  48. 48.
    Kirschner, D. E., S. T. Chang, T. W. Riggs, N. Perry, and J. J. Linderman. Toward a multiscale model of antigen presentation in immunity. Immunol. Rev. 216:93–118, 2007.Google Scholar
  49. 49.
    Kirschner, D. E., C. A. Hunt, S. Marino, M. Fallahi-Sichani, and J. J. Linderman. Tuneable resolution as a systems biology approach for multi-scale, multi-compartment computational models. Wiley Interdiscip. Rev. Syst. Biol. Med. 6:289–309, 2014.CrossRefGoogle Scholar
  50. 50.
    Krinner, A., I. Roeder, M. Loeffler, and M. Scholz. Merging concepts—coupling an agent-based model of hematopoietic stem cells with an ODE model of granulopoiesis. BMC Syst. Biol. 7:117, 2013.CrossRefGoogle Scholar
  51. 51.
    Lauffenburger, D. A., and J. J. Linderman. Receptors: Models For Binding, Trafficking, and Signaling. New York: Oxford University Press, 1993.Google Scholar
  52. 52.
    LeVeque, R. J. Finite Difference Methods for Ordinary and Partial Differential Equations. Society for Industrial and Applied Mathematics, 2007.Google Scholar
  53. 53.
    Linderman, J. J., and D. E. Kirschner. In silico models of M. tuberculosis infection provide a route to new therapies. Drug Discov. Today Dis. Model. 1–5, 2014.Google Scholar
  54. 54.
    Lucas, T. A. Operator splitting for an immunology model using reaction–diffusion equations with stochastic source terms. SIAM J. Numer. Anal. 46:3113–3135, 2008.CrossRefMATHMathSciNetGoogle Scholar
  55. 55.
    Marino, S., M. El-Kebir, and D. Kirschner. A hybrid multi-compartment model of granuloma formation and T cell priming in tuberculosis. J. Theor. Biol. Elsevier 280:50–62, 2011.CrossRefGoogle Scholar
  56. 56.
    Marino, S., I. B. Hogue, C. J. Ray, and D. E. Kirschner. A methodology for performing global uncertainty and sensitivity analysis in systems biology. J. Theor. Biol. 254:178–196, 2008.CrossRefMathSciNetGoogle Scholar
  57. 57.
    Marino, S., J. J. Linderman, and D. E. Kirschner. A multifaceted approach to modeling the immune response in tuberculosis. Wiley Interdiscip. Rev. Syst. Biol. Med. 3:479–489, 2011.CrossRefGoogle Scholar
  58. 58.
    Materi, W., and D. S. Wishart. Computational systems biology in drug discovery and development: methods and applications. Drug Discov. Today 12:295–303, 2007.CrossRefGoogle Scholar
  59. 59.
    Mitha, F., T. A. Lucas, F. Feng, T. B. Kepler, and C. Chan. The multiscale systems immunology project: software for cell-based immunological simulation. Source Code Biol. Med. 3:6, 2008.CrossRefGoogle Scholar
  60. 60.
    Mugler, D. H., and R. A. Scott. Fast fourier transform method for partial differential equations, case study: the 2-D diffusion equation. Comput. Math. Appl. 16:221–228, 1988.CrossRefMATHMathSciNetGoogle Scholar
  61. 61.
    Palsson, S., et al. The development of a fully-integrated immune response model (FIRM) simulator of the immune response through integration of multiple subset models. BMC Syst. Biol. BMC Syst. Biol. 7:95, 2013.CrossRefGoogle Scholar
  62. 62.
    Peaceman, D. W., and H. H. Rachford, Jr. The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3:28–41, 1955.CrossRefMATHMathSciNetGoogle Scholar
  63. 63.
    Petersen, B. K., G. E. Ropella, and C. A. Hunt. Toward modular biological models: defining analog modules based on referent physiological mechanisms. BMC Syst. Biol. 8:95, 2014.CrossRefGoogle Scholar
  64. 64.
    Pienaar, E., et al. A computational tool integrating host immunity with antibiotic dynamics to study tuberculosis treatment. J. Theor. Biol. 2014 (in Press).Google Scholar
  65. 65.
    Pope, S. B. Computationally efficient implementation of combustion chemistry using in situ adaptive tabulation. Combust. Theory Model. 1:41–63, 1997.CrossRefMATHMathSciNetGoogle Scholar
  66. 66.
    Press, W. H. Numerical recipes in C++: the art of scientific computing (2nd ed.). Cambridge, UK: Cambridge University Press, 2002.Google Scholar
  67. 67.
    Qutub, A. A., F. Mac Gabhann, E. D. Karagiannis, P. Vempati, and A. S. Popel. Multiscale models of angiogenesis. IEEE Eng. Med. Biol. Mag. 28:14–31, 2009.CrossRefGoogle Scholar
  68. 68.
    Qutub, A. A., and A. S. Popel. Elongation, proliferation & migration differentiate endothelial cell phenotypes and determine capillary sprouting. BMC Syst. Biol. 3:13, 2009.CrossRefGoogle Scholar
  69. 69.
    Rao, S., A. van der Schaft, K. van Eunen, B. M. Bakker, and B. Jayawardhana. A model reduction method for biochemical reaction networks. BMC Syst. Biol. 8:52, 2014.CrossRefGoogle Scholar
  70. 70.
    Rapin, N., O. Lund, M. Bernaschi, and F. Castiglione. Computational immunology meets bioinformatics: the use of prediction tools for molecular binding in the simulation of the immune system. PLoS ONE 5:e9862, 2010.CrossRefGoogle Scholar
  71. 71.
    Ray, J. C. J., J. L. Flynn, and D. E. Kirschner. Synergy between individual TNF-dependent functions determines granuloma performance for controlling Mycobacterium tuberculosis infection. J. Immunol. 182:3706–3717, 2009.CrossRefGoogle Scholar
  72. 72.
    Riley, K. F., M. P. Hobson, and S. J. Bence. Mathematical Methods for Physics and Engineering: A Comprehensive Guide. Cambridge: Cambridge University Press, 2002.CrossRefGoogle Scholar
  73. 73.
    Santoni, D., M. Pedicini, and F. Castiglione. Implementation of a regulatory gene network to simulate the TH1/2 differentiation in an agent-based model of hypersensitivity reactions. Bioinformatics 24:1374–1380, 2008.CrossRefGoogle Scholar
  74. 74.
    Segovia-Juarez, J. L., S. Ganguli, and D. Kirschner. Identifying control mechanisms of granuloma formation during M. tuberculosis infection using an agent-based model. J. Theor. Biol. 231:357–376, 2004.CrossRefMathSciNetGoogle Scholar
  75. 75.
    Singer, M., S. Pope, and H. Najm. Modeling unsteady reacting flow with operator splitting and ISAT. Combust. Flame 147:150–162, 2006.CrossRefGoogle Scholar
  76. 76.
    Sloot, P. M. A., and A. G. Hoekstra. Multi-scale modelling in computational biomedicine. Brief. Bioinform. 11:142–152, 2010.CrossRefGoogle Scholar
  77. 77.
    Southern, J., et al. Multi-scale computational modelling in biology and physiology. Prog. Biophys. Mol. Biol. 96:60–89, 2008.CrossRefGoogle Scholar
  78. 78.
    Stefanini, M. O., F. T. H. Wu, F. Mac Gabhann, and A. S. Popel. The presence of VEGF receptors on the luminal surface of endothelial cells affects VEGF distribution and VEGF signaling. PLoS Comput. Biol. 5:e1000622, 2009.CrossRefMathSciNetGoogle Scholar
  79. 79.
    Stern, J. R., S. Christley, O. Zaborina, J. C. Alverdy, and G. An. Integration of TGF-β- and EGFR-based signaling pathways using an agent-based model of epithelial restitution. Wound Repair Regen. 20:862–871, 2012.CrossRefGoogle Scholar
  80. 80.
    Strang, G. On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5:506–517, 1968.CrossRefMATHMathSciNetGoogle Scholar
  81. 81.
    Sun, T., S. Adra, R. Smallwood, M. Holcombe, and S. MacNeil. Exploring hypotheses of the actions of TGF-beta1 in epidermal wound healing using a 3D computational multiscale model of the human epidermis. PLoS ONE 4:e8515, 2009.CrossRefGoogle Scholar
  82. 82.
    Sundnes, J., G. T. Lines, and A. Tveito. An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso. Math. Biosci. 194:233–248, 2005.CrossRefMATHMathSciNetGoogle Scholar
  83. 83.
    Swat, M. H., G. L. Thomas, J. M. Belmonte, A. Shirinifard, D. Hmeljak, and J. A. Glazier. Multi-scale modeling of tissues using CompuCell 3D. Methods Cell Biol. 110:325–366, 2012.CrossRefGoogle Scholar
  84. 84.
    Tay, S., J. J. Hughey, T. K. Lee, T. Lipniacki, S. R. Quake, and M. W. Covert. Single-cell NF-kappaB dynamics reveal digital activation and analogue information processing. Nature 466:267–271, 2010.CrossRefGoogle Scholar
  85. 85.
    Trefethen, L. N. Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations. Unpublished Text, 1996.Google Scholar
  86. 86.
    Walker, D. C., and J. Southgate. The virtual cell–a candidate co-ordinator for “middle-out” modelling of biological systems. Brief. Bioinform. 10:450–461, 2009.CrossRefGoogle Scholar
  87. 87.
    Walpole, J., J. A. Papin, and S. M. Peirce. Multiscale computational models of complex biological systems. Annu. Rev. Biomed. Eng. 15:137–154, 2013.CrossRefGoogle Scholar
  88. 88.
    Wang, Z., V. Bordas, J. Sagotsky, and T. S. Deisboeck. Identifying therapeutic targets in a combined EGFR-TGFβR signalling cascade using a multiscale agent-based cancer model. Math. Med. Biol. 29:95–108, 2012.CrossRefMATHGoogle Scholar
  89. 89.
    Wang, Z., J. D. Butner, R. Kerketta, V. Cristini, and T. S. Deisboeck. Simulating cancer growth with multiscale agent-based modeling. Semin. Cancer Biol. 1–9, 2014. doi: 10.1016/j.semcancer.2014.04.001
  90. 90.
    Wang, J., et al. Multi-scale agent-based modeling on melanoma and its related angiogenesis analysis. Theor. Biol. Med. Model. 10:41, 2013.CrossRefGoogle Scholar
  91. 91.
    Wise, S., J. Kim, and J. Lowengrub. Solving the regularized, strongly anisotropic Cahn–Hilliard equation by an adaptive nonlinear multigrid method. J. Comput. Phys. 226:414–446, 2007.CrossRefMATHMathSciNetGoogle Scholar
  92. 92.
    Wise, S. M., J. S. Lowengrub, and V. Cristini. An adaptive multigrid algorithm for simulating solid tumor growth using mixture models. Math. Comput. Model. 53:1–20, 2011.CrossRefMATHMathSciNetGoogle Scholar
  93. 93.
    Wolff, K., C. Barrett-Freeman, M. R. Evans, A. B. Goryachev, and D. Marenduzzo. Modelling the effect of myosin X motors on filopodia growth. Phys. Biol. 11:016005, 2014.CrossRefGoogle Scholar
  94. 94.
    Yoshida, H. Construction of higher order symplectic integrators. Phys. Lett. A 150:262–268, 1990.CrossRefMathSciNetGoogle Scholar
  95. 95.
    Zhang, L., et al. Developing a multiscale, multi-resolution agent-based brain tumor model by graphics processing units. Theor. Biol. Med. Model. 8:46, 2011.CrossRefGoogle Scholar
  96. 96.
    Zingg, D. W., and T. T. Chisholm. Runge-Kutta methods for linear ordinary differential equations. Appl. Numer. Math. 31:227–238, 1999.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Biomedical Engineering Society 2014

Authors and Affiliations

  • Nicholas A. Cilfone
    • 1
  • Denise E. Kirschner
    • 2
  • Jennifer J. Linderman
    • 1
  1. 1.Department of Chemical EngineeringUniversity of MichiganAnn ArborUSA
  2. 2.Department of Microbiology and ImmunologyUniversity of Michigan Medical SchoolAnn ArborUSA

Personalised recommendations