Skip to main content
SpringerLink
Log in

Mathematical Modeling of Inflammatory Processes

  • Chapter
  • First Online:
Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics

  • 562 Accesses

Abstract

The association of inflammation with modern human diseases remains an unsolved mystery of current biology and medicine. Inflammation is a protective response to noxious stimuli that unavoidably destroys normal tissue function. Various types of mathematical models are used, each suited for a different type of disease or to different parts of the immune system, in a multiscale framework. For instance, significant research on the atherosclerosis modeling was done at both, microscale and macroscale levels. In the latter case, the studies address the biomechanical and biochemical blood flow interaction with the vessel walls and tissue, giving rise to complex fluid–structure interaction (FSI) problems. The methods to describe the FSI are of great importance to achieve accurate numerical simulations.

The microscale modeling of atherosclerosis concerns circulating monocytes, which form a small subpopulation of the leukocytes. The rheological properties of monocytes play a significant role in flow dynamics and alterations in cell mechanical properties, such as cell deformability, can significantly influence vascular flow and might lead to vascular complications, such as sequestration of monocytes on the vessel wall.

This preliminary work proposes the development, analysis, and computer implementation of mathematical models of the inflammatory processes in healthy or diseased states. It will be followed by the validation of these approaches making use of the experimental parameters obtained within the frame of collaborative projects with medical institutions.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. D. Okin, R. Medzhitov, Evolution of inflammatory diseases. Curr. Biol. 22(17), 733–740 (2012)

    Article  Google Scholar 

  2. S.S. Dhawan, R.P. Avati Nanjundappa, J.R. Branch, W.R. Taylor, A.A. Quyyumi, H. Jo, M.C. McDaniel, J. Suo, D. Giddens, H. Samady, Shear stress and plaque development. Expert Rev. 8(4), 545–556 (2010)

    Google Scholar 

  3. Y.C. Chang, T.Y. Hou, B. Merriman, S. Osher, A level set formulation of eulerian interface capturing methods for incompressible fluid flows. J. Comput. Phys. 124, 449–464 (1996)

    Article  MathSciNet  Google Scholar 

  4. C.S. Peskin, Numerical analysis of blood flow in the heart. J. Comput. Phys. 25, 220–252 (1977)

    Article  MathSciNet  Google Scholar 

  5. C.S. Peskin, D.M. McQueen, A three-dimensional computational method for blood flow in the heart – I Immersed elastic fibers in a viscous incompressible fluid. J. Comput. Phys. 81, 372–405 (1989)

    Article  MathSciNet  Google Scholar 

  6. R. Glowinski, T.W. Pan, J. Periaux, A Fictitious domain method for Dirichlet problem and applications. Comput. Method. Appl. Mech. 111, 283–303 (1994)

    Article  MathSciNet  Google Scholar 

  7. R. Glowinski, T.W. Pan, J. Periaux, A Fictitious domain method for external incompressible viscous flow modeled by Navier-Stokes equations. Comput. Method. Appl. Mech. 112, 133–148 (1994)

    Article  MathSciNet  Google Scholar 

  8. S. Frei, T. Richter, T. Wick, Eulerian techniques for fluid-structure interactions – part II: applications, in Numerical Mathematics and Advanced Applications, ENUMATH 2013, ed. by A. Abdulle, S. Deparis, D. Kressner, F. Nobile, M. Picasso (Springer, Berlin, 2015), pp. 745–754

    Chapter  Google Scholar 

  9. T. Wick, Flapping and contact FSI computations with the fluid-solid interface-tracking/interface-capturing technique and mesh adaptivity. Comput. Mech. 53, 29–43 (2014)

    Article  MathSciNet  Google Scholar 

  10. T.J.R. Hughes, W.K. Liu, T.K. Zimmermann, Arbitrary Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Comput. Method. Appl. Mech. 29, 329–349 (1981)

    Article  Google Scholar 

  11. J. Donea, S. Giuliani, J.P. Halleux, An Arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions. Comput. Methods. Appl. Mech. Eng. 33, 689–723 (1982)

    Article  Google Scholar 

  12. F. Nobile, Numerical Approximation of Fluid-Structure Interaction Problems with Application to Haemodynamics, Ph.D thesis, École Polytechnique Fédérale de Lausanne (2001)

    Google Scholar 

  13. S. Boujena, O. Kafi, N. El Khatib, A 2D mathematical model of blood flow and its interactions in the atherosclerotic artery. Math. Model. Nat. Phenom. 9, 46–68 (2014)

    Article  MathSciNet  Google Scholar 

  14. S. Boujena, O. Kafi, N. El Khatib, Generalized Navier-Stokes equations with non-standard conditions for blood flow in atherosclerotic artery. Appl. Anal. 95, 1645–1670 (2016)

    Article  MathSciNet  Google Scholar 

  15. A.S. Silva-Herdade, A. Sequeira, A. Calado, C. Saldanha, O. Kafi, Hydrodynamics of a free-flowing leukocyte toward the endothelial wall. Microvasc. Res. 112, 7–13 (2017)

    Article  Google Scholar 

  16. A.M. Gambaruto, J. Janela, A. Moura, A. Sequeira, Sensitivity of hemodynamics in a patient specific cerebral aneurysm to vascular geometry and blood rheology. Math. Biosci. Eng. 8, 409–423 (2011)

    Article  MathSciNet  Google Scholar 

  17. J. Janela, A. Moura, A. Sequeira, Absorbing boundary conditions for a 3D non-Newtonian fluid-structure interaction model for blood flow in arteries. Int. J. Eng. Sci. 48, 1332–1349 (2010)

    Article  MathSciNet  Google Scholar 

  18. L. Formaggia, A. Moura, F. Nobile, On the stability of the coupling of 3D and 1D fluid-structure interaction models for blood flow simulations. ESAIM-Math. Model. Num. 41, 743–769 (2007)

    Article  MathSciNet  Google Scholar 

  19. S. Ramalho, A. Moura, A.M. Gambaruto, A. Sequeira, Sensitivity to out flow boundary conditions and level of geometry description for a cerebral aneurysm. Int. J. Numer. Method. Biomed. Eng. 28, 697–713 (2012)

    Article  Google Scholar 

  20. P.G. Ciarlet, Mathematical Elasticity Three Dimensional Elasticity, vol. 1 (Elsevier, Amsterdam, 2004)

    MATH  Google Scholar 

  21. S. Le Floc’h, J. Ohayon, P. Tracqui, G. Finet, A.M. Gharib, R.L. Maurice, G. Cloutier, R.I. Pettigrew, Vulnerable atherosclerotic plaque elasticity reconstruction based on a segmentation-driven optimization procedure using strain measurements: theoretical framework. IEEE Trans. Med. Imaging 28, 1126–1137 (2009)

    Article  Google Scholar 

  22. S. Glagov, E. Weisenberg, C.K. Zarins, R. Stankunavicius, G.J. Kolettis, Compensatory enlargement of human atherosclerotic coronary arteries. N. Engl. J. Med. 316, 1371–1375 (1987)

    Article  Google Scholar 

  23. Z.Y. Li, S. Howarth, R.A. Trivedi, J.M. U-King-Im, M.J. Graves, A. Brown, L. Wang, J.H. Gillard, Stress analysis of carotid plaque rupture based on in vivo high resolution MRI. J. Biomech. 39, 2611–2622 (2006)

    Article  Google Scholar 

  24. D. Tang, C. Yang, J. Zheng, P.K. Woodard, G.A. Sicard, J.E. Saffitz, C. Yuan, 3D MRI-based multicomponent FSI models for atherosclerotic plaques. Ann. Biomed. Eng. 32, 947–960 (2004)

    Article  Google Scholar 

  25. J. Janela, A. Moura, A. Sequeira, A 3D non-Newtonian fluid-structure interaction model for blood flow in arteries. J. Comput. Appl. Math. 234, 2783–2791 (2010)

    Article  MathSciNet  Google Scholar 

  26. O. Kafi, N. El Khatib, J. Tiago, A. Sequeira, Numerical simulations of a 3D fluid-structure interaction model for blood flow in an athersclerotic artery. Math. Biosci. Eng. 14, 179–193 (2017)

    Article  MathSciNet  Google Scholar 

  27. N. El Khatib, Modélisation mathématique de l’athérosclérose, Ph.D thesis, Université Claude Bernard–Lyon 1 (2009).

    Google Scholar 

  28. D. Oliveira, Numerical Simulation of Dilatation Patterns of the Ascending Aorta in Aortopathies, MSc thesis, Instituto Superior Técnico, Lisbon (2016)

    Google Scholar 

  29. COMSOL Multiphysics, User’s Guide 4.3b, Licence 17073661 (2012)

    Google Scholar 

  30. M. Anand, J. Kwack, A. Masud, A new generalized Oldroyd-B model for blood flow in complex geometries. Int. J. Eng. Sci. 72, 78–88 (2013)

    Article  Google Scholar 

  31. K.E. Jensen, P. Szabo, F. Okkels, Implementation of the Log-conformation Formulation for Two-dimensional Viscoelastic Flow (2016). Preprint. arXiv:1508.01041v2

    Google Scholar 

  32. S. Gross, A. Reusken, Numerical methods for two-phase incompressible flows. Springer Ser. Comput. Math. 40(1) (2011)

    Google Scholar 

  33. S. Boujena, O. Kafi, A. Sequeira, Mathematical study of a single leukocyte in microchannel flow. Math. Model. Nat. Phenom. 13 (2018)

    Google Scholar 

  34. A.S. Silva-Herdade, G. Andolina, C. Faggio, Â. Calado, C. Saldanha, Erythrocyte deformability – a partner of the inflammatory response. Microvasc. Res. 107, 34–38 (2016)

    Article  Google Scholar 

  35. S. Osher, J.A. Sethian, Fronts propagating with curvature-dependent speed: algorithm based on Hamilton-Jacobi formations. J. Comput. Phys. 210, 225–246 (2012)

    Google Scholar 

Download references

Acknowledgements

This work was partially supported by the Portuguese FCT—Fundação para a Ciência e a Tecnologia, through the project UID/Multi/04621/2013 of the CEMAT—Center for Computational and Stochastic Mathematics, IST, University of Lisbon.

Author information

Authors and Affiliations

  1. Department of Mathematics and CEMAT, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal

    O. Kafi & A. Sequeira

Authors
  1. O. Kafi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Sequeira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to O. Kafi .

Editor information

Editors and Affiliations

  1. Federal University of Rio de Janeiro, President, BIOMAT Consortium – International Institute for Interdisciplinary Sciences, Rio de Janeiro, Rio de Janeiro, Brazil

    Rubem P. Mondaini

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Kafi, O., Sequeira, A. (2019). Mathematical Modeling of Inflammatory Processes. In: Mondaini, R. (eds) Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-030-23433-1_17

Download citation

Publish with us

Policies and ethics

Search

Navigation