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Biomechanical modelling in nanomedicine: multiscale approaches and future challenges

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Abstract

Nanomedicine is the branch of nanotechnology devoted to the miniaturization of devices and to the functionalization of processes for the diagnosis and the design of tools of clinical use. In the perspective to develop patient-specific treatments and effective therapies against currently incurable diseases, biomechanical modelling plays a key role in enabling their translation to clinical practice. Establishing a dynamic interaction with experiments, a modelling approach is expected to allow investigating problems with lower economic burden, evaluating a larger range of conditions. Since biological systems have a wide range of typical characteristic length and timescales, a multiscale modelling approach is necessary both for providing a proper description of the biological complexity at the single scales and for keeping the largest amount of functional interdependence among them. This work starts with a survey both of the common frameworks for modelling a biological system, at scales from atoms to a continuous distribution of matter, and of the available multiscale methods that link the different levels of investigation. In the following, we define an original approach for dealing with the specific case of transport and diffusion of nanoparticles and/or drug-delivery carriers from the systemic circulation to a target tissue microstructure. Using a macro–micro viewpoint, we discuss the existing multiscale approaches and we propose few original strategies for overcoming their limitations in bridging scales. In conclusion, we highlight and critically discuss the future challenges of multiscale modelling for achieving the long-term objective to assist the nanomedical research in proposing more accurate clinical approaches for improved medical benefit.

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References

  1. Albanese A., Tang P.S., Chan W.C.: The effect of nanoparticle size, shape, and surface chemistry on biological systems. Ann. Rev. Biomed. Eng. 14(1), 1–16 (2012). doi:10.1146/annurev-bioeng-071811-150124

    Google Scholar 

  2. Alber, M., Kiskowski, M., Glazier, J., Jiang, Y.: On cellular automaton approaches to modeling biological cells. In: Rosenthal, J., Gilliam, D. (eds) Mathematical Systems Theory in Biology, Communications, Computation, and Finance, The IMA Volumes in Mathematics and its Applications, vol. 134, pp 1–39. Springer, New York (2003)

  3. Alder B.J., Wainwright T.E.: Studies in molecular dynamics. I. General method. J. Chem. Phys. 31(2), 459–466 (1959). doi:10.1063/1.1730376

    MathSciNet  Google Scholar 

  4. Ambrosi D., Guana F.: Stress-modulated growth. Math. Mech. Solids 12(3), 319–342 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Ambrosi D., Preziosi L.: On the closure of mass balance models for tumor growth. Math. Models Methods Appl. Sci. 12(5), 737–754 (2002). doi:10.1142/S0218202502001878

    MathSciNet  MATH  Google Scholar 

  6. Ambrosi D., Preziosi L., Vitale G.: The insight of mixtures theory for growth and remodeling. Z. Angew. Math. Phys. 61(1), 177–191 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Ambrosi D., Ateshian G.A., Arruda E.M., Cowin S.C., Dumais J., Goriely A., Holzapfel G.A., Humphrey J.D., Kemkemer R., Kuhl E., Olberding J.E., Taber L.A., Garikipati K.: Perspectives on biological growth and remodeling. J. Mech. Phys. Solids 59(4), 863–883 (2011). doi:10.1016/j.jmps.2010.12.011

    MathSciNet  MATH  Google Scholar 

  8. Anderson, B.: Esf scientific forward look on nanomedicine. Technical report. European Science Foundation (2005). URL:http://www.esf.org/fileadmin/Public_documents/Publications/Nanomedicine_01.pdf

  9. Angot P., Bruneau C.H., Fabrie P.: A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math. 81(4), 497–520 (1999). doi:10.1007/s002110050401

    MathSciNet  MATH  Google Scholar 

  10. Ayton G.S., Noid W.G., Voth G.A.: Multiscale modeling of biomolecular systems: in serial and in parallel. Curr. Opin. Struct. Biol. 17(2), 192–198 (2007). doi:10.1016/j.sbi.2007.03.004

    Google Scholar 

  11. Bahar I., Rader A.: Coarse-grained normal mode analysis in structural biology. Curr. Opin. Struct. Biol. 15(5), 586–592 (2005). doi:10.1016/j.sbi.2005.08.007

    Google Scholar 

  12. Baish J.W., Jain R.K.: Fractals and cancer. Cancer Res. 60(14), 3683–3688 (2000)

    Google Scholar 

  13. Bao G., Suresh S.: Cell and molecular mechanics of biological materials. Nat. Mater. 2(11), 715–725 (2003)

    Google Scholar 

  14. Bao G., Mitragotri S., Tong S.: Multifunctional nanoparticles for drug delivery and molecular imaging. Ann. Rev. Biomed. Eng. 15(1), 253–282 (2013). doi:10.1146/annurev-bioeng-071812-152409

    Google Scholar 

  15. Bechtle S., Ang S.F., Schneider G.A.: On the mechanical properties of hierarchically structured biological materials. Biomaterials 31(25), 6378–6385 (2010). doi:10.1016/j.biomaterials.2010.05.044

    Google Scholar 

  16. Bellomo N., De Angelis E., Preziosi L.: Review article: multiscale modeling and mathematical problems related to tumor evolution and medical therapy. J. Theor. Med. 5(2), 111–136 (2003)

    MATH  Google Scholar 

  17. Bellomo N., Li N.K., Maini P.K.: On the foundation of cancer modelling: selected topics, speculations and perspectives. Math. Model. Methods Appl. Sci. 18(4), 593–646 (2008)

    MathSciNet  MATH  Google Scholar 

  18. Bewick S., Yang R., Zhang M.: Complex mathematical models of biology at the nanoscale. WIREs Nanomed. Nanobiotechnol. 1(6), 650–659 (2009)

    Google Scholar 

  19. Bhaskar, S., Tian, F., Stoeger, T., Kreyling, W., de la Fuente, J., Grazu, V., Borm, P., Estrada, G., Ntziachristos, V., Razansky, D.: Multifunctional nanocarriers for diagnostics, drug delivery and targeted treatment across blood-brain barrier: perspectives on tracking and neuroimaging. Part Fibre Toxicol. 7(1) (2010). doi:10.1186/1743-8977-7-3

  20. Biot M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12(2), 155–164 (1941)

    MATH  Google Scholar 

  21. Bonfiglio, A., Leungchavaphongse, K., Repetto, R., Siggers, J.H.: Mathematical modeling of the circulation in the liver lobule. J. Biomech. Eng. 132(11) (2010). doi:10.1115/1.4002563

  22. Brinkman H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Numer. Math. 81(4), 27–34 (1949). doi:10.1007/s002110050401

    Google Scholar 

  23. Buehler M.J., Ackbarow T.: Nanomechanical strength mechanisms of hierarchical biological materials and tissues. Comput. Methods Biomech. Biomed. Eng. 11(6), 595–607 (2008). doi:10.1080/10255840802078030

    Google Scholar 

  24. Carman P.: Fluid flow through granular beds. Trans. Inst. Chem. Eng. 15, 150–166 (1937)

    Google Scholar 

  25. Chaikin P., Rhodes G.R., Bruno R., Rohatagi S., Natarajan C.: Pharmacokinetics/pharmacodynamics in drug development: An industrial perspective. J. Clin. Pharmacol. 40(12), 1428–1438 (2000). doi:10.1177/009127000004001213

    Google Scholar 

  26. Chen Q., Pugno N.M.: Bio-mimetic mechanisms of natural hierarchical materials: a review. J. Mech. Behav. Biomed. 19(0), 3–33 (2013). doi:10.1016/j.jmbbm.2012.10.012

    Google Scholar 

  27. Ciarletta P., Amar M.B., Labouesse M.: Continuum model of epithelial morphogenesis during caenorhabditis elegans embryonic elongation. Philos. Trans. R. Soc. A 367(1902), 3379–3400 (2009)

    MATH  Google Scholar 

  28. Ciarletta P., Foret L., Amar M.B.: The radial growth phase of malignant melanoma: multi-phase modelling, numerical simulations and linear stability analysis. J. R. Soc. Interface 8(56), 345–368 (2011)

    Google Scholar 

  29. Ciarletta P., Ambrosi D., Maugin G.: Configurational forces for growth and shape regulations in morphogenesis. Bull. Pol. Acad. Sci. 60(2), 253–257 (2012)

    MathSciNet  Google Scholar 

  30. Ciarletta P., Dario P., Tendick F., Micera S.: Hyperelastic model of anisotropic fiber reinforcements within intestinal walls for applications in medical robotics. Int. J. Robot. Res. 28(10), 1279–1288 (2009)

    Google Scholar 

  31. Ciarletta P., Izzo I., Micera S., Tendick F.: Stiffening by fiber reinforcement in soft materials: a hyperelastic theory at large strains and its application. J. Mech. Behav. Biomed. Mater. 4(7), 1359–1368 (2011)

    Google Scholar 

  32. Ciarletta P., Ambrosi D., Maugin G.A.: Mass transport in morphogenetic processes: a second gradient theory for volumetric growth and material remodeling. J. Mech. Phys. Solids 60(3), 432–450 (2012)

    MathSciNet  MATH  Google Scholar 

  33. Clark W.H.: Tumor progression and the nature of cancer. Br. J. Cancer 64(4), 631–644 (1991)

    Google Scholar 

  34. Costa, A.: Permeability-porosity relationship: a reexamination of the kozeny-carman equation based on a fractal pore-space geometry assumption. Geophys. Res. Lett. 33(2) (2006). doi:10.1029/2005GL025134

  35. Cowin, S.: On the modeling of growth and adaptation. In: Holzapfel G., Ogden R. (eds.) Mechanics of Biological Tissue, pp 29–46. Springer Berlin (2006)

  36. Cowin S., Doty S.: Tissue Mechanics. Springer, Berlin (2007)

    MATH  Google Scholar 

  37. Cowin S.C., Cardoso L.: Mixture theory-based poroelasticity as a model of interstitial tissue growth. Mech. Mater. 44, 47–57 (2012). doi:10.1016/j.mechmat.2011.07.005

    Google Scholar 

  38. Dada J.O., Mendes P.: Multi-scale modelling and simulation in systems biology. Integr. Biol. 3, 86–96 (2011). doi:10.1039/C0IB00075B

    Google Scholar 

  39. Danquah M.K., Zhang X.A., Mahato R.I.: Extravasation of polymeric nanomedicines across tumor vasculature. Adv. Drug Deliv. Rev. 63(8), 623–639 (2011). doi:10.1016/j.addr.2010.11.005

    Google Scholar 

  40. Decuzzi P., Ferrari M.: The role of specific and non-specific interactions in receptor-mediated endocytosis of nanoparticles. Biomaterials 28(18), 2915–2922 (2007). doi:10.1016/j.biomaterials.2007.02.013

    Google Scholar 

  41. Decuzzi P., Ferrari M.: The receptor-mediated endocytosis of nonspherical particles. Biophys. J. 94(10), 3790–3797 (2008). doi:10.1529/biophysj.107.120238

    Google Scholar 

  42. Decuzzi P., Godin B., Tanaka T., Lee S.Y., Chiappini C., Liu X., Ferrari M.: Size and shape effects in the biodistribution of intravascularly injected particles. J. Control Release 141(3), 320–327 (2010). doi:10.1016/j.jconrel.2009.10.014

    Google Scholar 

  43. Desai T.A., Chu W.H., Tu J.K., Beattie G.M., Hayek A., Ferrari M.: Microfabricated immunoisolating biocapsules. Biotechnol. Bioeng. 57(1), 118–120 (1998). doi:10.1002/(SICI)1097-0290(19980105)57:1<118::AID-BIT14>3.0.CO;2-G

    Google Scholar 

  44. DiCarlo A., Quiligotti S.: Growth and balance. Mech. Res. Commun. 29(6), 449–456 (2002). doi:10.1016/S0093-6413(02)00297-5

    MathSciNet  MATH  Google Scholar 

  45. Discacciati M., Miglio E., Quarteroni A.: Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43(1), 57–74 (2002)

    MathSciNet  MATH  Google Scholar 

  46. Dobson C.M., Ali A., Karplus M.: Protein folding: a perspective from theory and experiment. Angew Chem. Int. Ed. 37(7), 868–893 (1998)

    Google Scholar 

  47. Drugan W., Willis J.: A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites. J. Mech. Phys. Solids 44(4), 497–524 (1996). doi:10.1016/0022-5096(96)00007-5

    MathSciNet  MATH  Google Scholar 

  48. DuFort C.C., Paszek M.J., Weaver V.M.: Balancing forces: architectural control of mechanotransduction. Nat. Rev. Mol. Cell Biol. 12(5), 308–319 (2011). doi:10.1038/nrm3112

    Google Scholar 

  49. Editorial: Regulating nanomedicine. Nat. Mater. 6 (2007). doi:10.1038/nmat1875

  50. Eladdadi A., Isaacson D.: A mathematical model for the effects of HER2 overexpression on cell proliferation in breast cancer. Bull. Math. Biol. 70(6), 1707–1729 (2008). doi:10.1007/s11538-008-9315-4

    MathSciNet  MATH  Google Scholar 

  51. Eshelby J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. Lond. A Math. 241(1226), 376–396 (1957)

    MathSciNet  MATH  Google Scholar 

  52. Evans J.A., Hughes T.J.: Isogeometric divergence-conforming b-splines for the darcy–stokes–brinkman equations. Math. Models Methods Appl. Sci. 23(04), 671–741 (2013)

    MathSciNet  MATH  Google Scholar 

  53. Fang H., Wang Z., Lin Z., Liu M.: Lattice boltzmann method for simulating the viscous flow in large distensible blood vessels. Phys. Rev. E 65, 051,925 (2002). doi:10.1103/PhysRevE.65.051925

    Google Scholar 

  54. Ferrari M.: The mathematical engines of nanomedicine. Small 4(1), 20–25 (2008). doi:10.1002/smll.200701144

    Google Scholar 

  55. Feynman R.P.: There’s plenty of room at the bottom. Eng. Sci. 23, 22–36 (1960)

    Google Scholar 

  56. Flynn T., Wei C.: The pathway to commercialization for nanomedicine. Nanomedicine 1(1), 47–51 (2005). doi:10.1016/j.nano.2004.11.010

    Google Scholar 

  57. Formaggia, L., Quarteroni, A., Veneziani, A.: Multiscale models of the vascular system. In: Cardiovascular Mathematics, vol. 1, pp 395–446. Springer Milan (2009)

  58. Fournier R.L.: Basic Transport Phenomena in Biomedical Engineering. CRC PressI Llc, London (2011)

    Google Scholar 

  59. Freitas, R.A.: Nanomedicine, vol. I: Basic Capabilities, 1st edn. Landes Bioscience (1999)

  60. Freitas R.A.: Current status of nanomedicine and medical nanorobotics. J. Comput. Theor. Nanosci. 2(1), 1–25 (2005). doi:10.1166/jctn.2005.001

    Google Scholar 

  61. Freitas R.A. Jr: What is nanomedicine?. Nanomed. Nanotechnol. 1(1), 2–9 (2005). doi:10.1016/j.nano.2004.11.003

    Google Scholar 

  62. Freund L., Lin Y.: The role of binder mobility in spontaneous adhesive contact and implications for cell adhesion. J. Mech. Phys. Solids 52(11), 2455–2472 (2004). doi:10.1016/j.jmps.2004.05.004

    MATH  Google Scholar 

  63. Fung L.K., Saltzman W.: Polymeric implants for cancer chemotherapy. Adv. Drug Deliv. Rev. 26(23), 209–230 (1997). doi:10.1016/S0169-409X(97)00036-7

    Google Scholar 

  64. Gao Z., Zhang L., Sun Y.: Nanotechnology applied to overcome tumor drug resistance. J. Control Release 162(1), 45–55 (2012). doi:10.1016/j.jconrel.2012.05.051

    MathSciNet  Google Scholar 

  65. Garnett M.C., Kallinteri P.: Nanomedicines and nanotoxicology: some physiological principles. Occup. Med. 56(5), 307–311 (2006). doi:10.1093/occmed/kql052

    Google Scholar 

  66. Gatenby R.A., Maini P.K.: Mathematical oncology: cancer summed up. Nature 421(6921), 321 (2003)

    Google Scholar 

  67. Gazit Y., Berk D.A., Leunig M., Baxter L.T., Jain R.K.: Scale-invariant behavior and vascular network formation in normal and tumor tissue. Phys. Rev. Lett. 75, 2428–2431 (1995). doi:10.1103/PhysRevLett.75.2428

    Google Scholar 

  68. Goodman T.T., Chen J., Matveev K., Pun S.H.: Spatio-temporal modeling of nanoparticle delivery to multicellular tumor spheroids. Biotechnol. Bioeng. 101(2), 388–399 (2008). doi:10.1002/bit.21910

    Google Scholar 

  69. de Groot B.L., Grubmuller H.: Water permeation across biological membranes: Mechanism and dynamics of aquaporin-1 and glpf. Science 294(5550), 2353–2357 (2001)

    Google Scholar 

  70. Grotendorst J., Blugel S., Marxi D.: Computational Nanoscience: Do it yourself!, vol. 31. John von Neumann Institute for Computing, New York (2006)

    Google Scholar 

  71. Guo P.: Dependency of tortuosity and permeability of porous media on directional distribution of pore voids. Transp. Porous Media 95(2), 285–303 (2012). doi:10.1007/s11242-012-0043-8

    Google Scholar 

  72. Halton J.: A retrospective and prospective survey of the monte carlo method. SIAM Rev. 12(1), 1–63 (1970). doi:10.1137/1012001

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  74. Happel J.: Viscous flow in multiparticle systems: slow motion of fluids relative to beds of spherical particles. AIChE J. 4(2), 197–201 (1958). doi:10.1002/aic.690040214

    Google Scholar 

  75. Hawkins-Daarud A., Prudhomme S., van der Zee K.G., Oden J.T.: Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumor growth. J. Math. Biol. 67(6–7), 1457–1485 (2013). doi:10.1007/s00285-012-0595-9

    MathSciNet  MATH  Google Scholar 

  76. Hondow N., Brydson R., Wang P., Holton M., Brown M., Rees P., Summers H., Brown A.: Quantitative characterization of nanoparticle agglomeration within biological media. J. Nanopart. Res. 14(7), 1–15 (2012). doi:10.1007/s11051-012-0977-3

    Google Scholar 

  77. Hori M., Nemat-Nasser S.: On two micromechanics theories for determining micro-macro relations in heterogeneous solids. Mech. Mater. 31(10), 667–682 (1999)

    Google Scholar 

  78. Horstemeyer, M.: Multiscale modeling: a review. In: Leszczynski, J., Shukla, M.K. (eds.) Practical Aspects of Computational Chemistry, vol. 9, pp 87–135. Springer, Netherlands (2010)

  79. Humphrey J.D., Rajagopal K.R.: A constrained mixture model for growth and remodeling of soft tissues. Math. Models Methods Appl. Sci. 12(3), 407–430 (2002). doi:10.1142/S0218202502001714

    MathSciNet  MATH  Google Scholar 

  80. Izvekov S., Voth G.A.: A multiscale coarse-graining method for biomolecular systems. J. Phys. Chem. B 109(7), 2469–2473 (2005). doi:10.1021/jp044629q

    Google Scholar 

  81. Jain R.K.: Transport of molecules across tumor vasculature. Cancer Metastasis Rev. 6(4), 559–593 (1987). doi:10.1007/BF00047468

    Google Scholar 

  82. Jain R.K., Stylianopoulos T.: Delivering nanomedicine to solid tumor. Nat. Rev. Clin. Oncol. 7(11), 653–664 (2010)

    Google Scholar 

  83. Jaramillo-Botero, A., Abrol, R., van Duin, A., Goddard III, W.A.: Multiscale-multiparadigm modeling and simulation of nanometer scale systems and processes for nanomedical applications. In: Nanomedicine: A Systems Engineering Approach, pp 245–300. Pan Stanford Pub. (2009)

  84. Jiang Y., Pjesivac-Grbovic J., Cantrell C., Freyer J.P.: A multiscale model for avascular tumor growth. Biophys. J. 89(6), 3884–3894 (2005). doi:10.1529/biophysj.105.060640

    Google Scholar 

  85. Jones I.: Low reynolds number flow past a porous spherical shell. Proc. Camb. Philos. Soc. 73, 231–238 (1973)

    MATH  Google Scholar 

  86. Kanit T., Forest S., Galliet I., Mounoury V., Jeulin D.: Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Solids Struct. 40(13), 3647–3679 (2003). doi:10.1016/S0020-7683(03)00143-4

    MATH  Google Scholar 

  87. Karplus M., McCammon J.A.: Molecular dynamics simulations of biomolecules. Nat. Struct. Biol. 9(9), 646–652 (2002)

    Google Scholar 

  88. Khadra K., Angotb P., Parneix S., Caltagirone J.P.: Fictitious domain approach for numerical modelling of navierstokes equations. Int. J. Numer. Methods Fluids 34(8), 651–684 (2000)

    MATH  Google Scholar 

  89. Kouznetsova V., Brekelmans W.A.M., Baaijens F.P.T.: An approach to micro-macro modeling of heterogeneous materials. Comput. Mech. 27(1), 37–48 (2001)

    MATH  Google Scholar 

  90. Kozeny J.: Ueber kapillare leitung des wassers im boden. Sitzungsber Akad Wiss Wien 136, 271–306 (1927)

    Google Scholar 

  91. Lai W., Mow V.: Drag-induced compression of articular cartilage during a permeation experiment. Biorheology 17(1-2), 111 (1980)

    Google Scholar 

  92. Lai W.M., Mow V.C., Roth V.: Effects of nonlinear strain-dependent permeability and rate of compression on the stress behavior of articular cartilage. J. Biomech. Eng. 103(2), 61–66 (1981)

    Google Scholar 

  93. Lekszycki T., dell’Isola F.: A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials. ZAMM J. Appl. Math. Mech./Z Angew Math. Mech. 92(6), 426–444 (2012). doi:10.1002/zamm.201100082

    MathSciNet  MATH  Google Scholar 

  94. Lesne A.: Multiscale analysis of biological systems. Acta Biotheor. 61(1), 3–19 (2013). doi:10.1007/s10441-013-9170-z

    Google Scholar 

  95. Li, J.: Basic molecular dynamics. In: Handbook of Materials Modeling, pp 565–588. Springer, Netherlands (2005)

  96. Liu Y., Miyoshi H., Nakamura M.: Nanomedicine for drug delivery and imaging: a promising avenue for cancer therapy and diagnosis using targeted functional nanoparticles. Int. J. Cancer 120(12), 2527–2537 (2007). doi:10.1002/ijc.22709

    Google Scholar 

  97. Macklin, P., Edgerton, M.E.: Discrete cell modeling. In: Multiscale Modeling of Cancer, pp. 88–122. Cambridge University Press, Cambridge (2010)

  98. Madeo A., Lekszycki T., dellIsola F.: A continuum model for the bio-mechanical interactions between living tissue and bio-resorbable graft after bone reconstructive surgery. Comptes Rendus Mécanique 339(10), 625–640 (2011). doi:10.1016/j.crme.2011.07.004

    Google Scholar 

  99. Malo M., Cartier-Michaud A., Fabre-Guillevin E., Hutzler G., Delaplace F., Barlovatz-Meimon G., Lesne A.: When a collective outcome triggers a rare individual event: a mode of metastatic process in a cell population. Math. Pop. Stud. 17, 136–165 (2010)

    MathSciNet  MATH  Google Scholar 

  100. Malossi A., Blanco P., Crosetto P., Deparis S., Quarteroni A.: Implicit coupling of one-dimensional and three-dimensional blood flow models with compliant vessels. Multiscale Model Simul. 11(2), 474–506 (2013). doi:10.1137/120867408

    MathSciNet  MATH  Google Scholar 

  101. Mardal K.A., Tai X.C., Winther R.: A robust finite element method for Darcy–Stokes flow. SIAM J. Numer. Anal. 40(5), 1605–1631 (2002)

    MathSciNet  MATH  Google Scholar 

  102. Maugin G.A.: On inhomogeneity, growth, ageing and the dynamics of materials. J. Mech. Mater. Struct. 4(4), 731–741 (2009)

    MathSciNet  Google Scholar 

  103. McGuire S., Zaharoff D., Yuan F.: Nonlinear dependence of hydraulic conductivity on tissue deformation during intratumoral infusion. Ann. Biomed. Eng. 34(7), 1173–1181 (2006). doi:10.1007/s10439-006-9136-2

    Google Scholar 

  104. Merkle R.C., Freitas R.A.: Theoretical analysis of a carbon-carbon dimer placement tool for diamond mechanosynthesis. J. Nanosci. Nanotechnol. 3(4), 24–319 (2003)

    Google Scholar 

  105. Moeendarbary E., Valon L., Fritzsche M., Harris A.R., Moulding D.A., Thrasher A.J., Stride E., Mahadevan L., Charras G.T.: The cytoplasm of living cells behaves as a poroelastic material. Nat. Mater. 12(3), 253–261 (2013). doi:10.1038/nmat3517

    Google Scholar 

  106. Moghimi S.M., Farhangrazi Z.S.: Nanomedicine and the complement paradigm. Nanomed. Nanotechnol. 9(4), 458–460 (2013). doi:10.1016/j.nano.2013.02.011

    Google Scholar 

  107. Moghimi S.M., Peer D., Langer R.: Reshaping the future of nanopharmaceuticals: Ad iudicium. ACS Nano 5(11), 8454–8458 (2011). doi:10.1021/nn2038252

    Google Scholar 

  108. Mognetti B.M., Virnau P., Yelash L., Paul W., Binder K., Muller M., MacDowell L.G.: Coarse-graining dipolar interactions in simple fluids and polymer solutions: Monte carlo studies of the phase behavior. Phys. Chem. Chem. Phys. 11(12), 1923–1933 (2009). doi:10.1039/B818020M

    Google Scholar 

  109. Mori N., Kumagae M., Nakamura K.: Brownian dynamics simulation for suspensions of oblong-particles under shear flow. Rheol. Acta 37, 151–157 (1998). doi:10.1007/s003970050101

    Google Scholar 

  110. Muller M., Albe K.: Kinetic lattice monte carlo simulations on the ordering kinetic of free and supported fept L10-nanoparticles. Beilstein J. Nanotechnol. 2, 40–46 (2011). doi:10.3762/bjnano.2.5

    Google Scholar 

  111. Murtola T., Bunker A., Vattulainen I., Deserno M., Karttunen M.: Multiscale modeling of emergent materials: biological and soft matter. Phys. Chem. Chem. Phys. 11, 1869–1892 (2009). doi:10.1039/B818051B

    Google Scholar 

  112. Nelson K.E., Ginn T.R.: Colloid filtration theory and the happel sphere-in-cell model revisited with direct numerical simulation of colloids. Langmuir 21, 2173–2184 (2005). doi:10.1021/la048404i

    Google Scholar 

  113. Netti P.A., Baxter L.T., Boucher Y., Skalak R.K., Jain R.K.: Time-dependent behavior of interstitial fluid pressure in solid tumors: implications for drug delivery. Cancer Res. 55(22), 5451–5458 (1995)

    Google Scholar 

  114. Netti P.A., Baxter L.T., Coucher Y., Skalak R.K., Jain R.K.: A poroelastic model for interstitial pressure in tumors. Biorheology 32(2), 346–346 (1995). doi:10.1016/0006-355X(95)92330-D

    Google Scholar 

  115. Nitta S.K., Numata K.: Biopolymer-based nanoparticles for drug/gene delivery and tissue engineering. Int. J. Mol. Sci. 14, 1629–1654 (2013). doi:10.3390/ijms14011629

    Google Scholar 

  116. Norris A.N., Grinfeld M.A.: Nonlinear poroelasticity for a layered medium. J. Acoust. Soc. Am. 98, 1138 (1995)

    Google Scholar 

  117. Oberdrster G.: Safety assessment for nanotechnology and nanomedicine: concepts of nanotoxicology. J. Intern. Med. 267(1), 89–105 (2010). doi:10.1111/j.1365-2796.2009.02187.x

    Google Scholar 

  118. Oden T.J., Hawkins A., Prudhomme S.: General diffuse-interface theories and an approach to predictive tumor growth modeling. Math. Modols Methods Appl. Sci. 20(3), 477–517 (2010). doi:10.1142/S0218202510004313

    MathSciNet  MATH  Google Scholar 

  119. Ogden, R.: Anisotropy and nonlinear elasticity in arterial wall mechanics. In: Holzapfel G., Ogden, R. (eds.) Biomechanical Modelling at the Molecular, Cellular and Tissue Levels, CISM International Centre for Mechanical Sciences, vol. 508, pp. 179–258. Springer, Vienna (2009)

  120. Ozcelikkale A., Ghosh S., Han B.: Multifaceted transport characteristics of nanomedicine: Needs for characterization in dynamic environment. Mol. Pharm. 10(6), 2111–2126 (2013). doi:10.1021/mp3005947

    Google Scholar 

  121. Payne M.C., Teter M.P., Allan D.C., Arias T.A., Joannopoulos J.D.: Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients. Rev. Mod. Phys. 64, 1045–1097 (1992). doi:10.1103/RevModPhys.64.1045

    Google Scholar 

  122. Penta R., Ambrosi D., Shipley R.J.: Effective governing equations for poroelastic growing media. Q. J. Mech. Appl. Math. 67(1), 69–91 (2014). doi:10.1093/qjmam/hbt024

    MathSciNet  MATH  Google Scholar 

  123. Penta, R., Ambrosi, D., Quarteroni, A.: Multiscale homogenization for fluid and drug transport in vascularized malignant tissues. Math. Models Methods Appl. Sci. 1-28 (2014, accepted for publication)

  124. Rakesh L., Howell B.A., Chai M., Mueller A., Kujawski M., Fan D., Ravi S., Slominski C.: Computer-aided applications of nanoscale smart materials for biomedical applications. Nanomedicine 3(5), 719–739 (2008). doi:10.2217/17435889.3.5.719

    Google Scholar 

  125. Ramachandran A., Guo Q., Iqbal S.M., Liu Y.: Coarse-grained molecular dynamics simulation of dna translocation in chemically modified nanopores. J. Phys. Chem. B 115(19), 6138–6148 (2011). doi:10.1021/jp101052x

    Google Scholar 

  126. Rees M., Moghimi S.M.: Nanotechnology: from fundamental concepts to clinical applications for healthy aging. Nanomed. Nanotechnol. 8(Supplement 1(0)), S1–S4 (2012). doi:10.1016/j.nano.2012.07.006

    Google Scholar 

  127. Reuss A.: Berechnung der fliegrenze von mischkristallen auf grund der plastizittsbedingung fr einkristalle. ZAMM J. Appl. Math. Mech./Z Angew Math. Mech. 9(1), 49–58 (1929)

    MATH  Google Scholar 

  128. Risser, P.F. Laurent and, Steyer, A., Cloetens, P., Le Duc, G., Fonta, C.: From homogeneous to fractal normal and tumorous microvascular networks in the brain. J. Cerebr. Blood Flow Metab. 27:1–20 (2007). doi:10.1038/sj.jcbfm.9600332

  129. Rodriguez E.K., Hoger A., McCulloch A.D.: Stress-dependent finite growth in soft elastic tissues. J. Biomech. 27(4), 455–467 (1994). doi:10.1016/0021-9290(94)90021-3

    Google Scholar 

  130. Schnell S., Grima R., Maini P.K.: Multiscale modeling in biology. Am. Sci. 95(1), 1629–1654 (2007)

    Google Scholar 

  131. Scianna M., Preziosi L.: Multiscale developments of the cellular potts model. Multiscale Model. Simul. 10(2), 342–382 (2012). doi:10.1137/100812951

    MathSciNet  MATH  Google Scholar 

  132. Shah S., Liu Y., Hu W., Gao J.: Modeling particle shape-dependent dynamics in nanomedicine. J. Nanosci. Nanotechnol. 11(2), 919–928 (2011). doi:10.1166/jnn.2011.3536

    Google Scholar 

  133. Shiekh F.: Personalized nanomedicine: future medicine for cancer treatment. Int. J. Nanomed. 8, 201–202 (2013)

    Google Scholar 

  134. Shipley R., Chapman S.: Multiscale modelling of fluid and drug transport in vascular tumours. Bull. Math. Biol. 72(6), 1464–1491 (2010). doi:10.1007/s11538-010-9504-9

    MathSciNet  MATH  Google Scholar 

  135. Siegel R.W.: Exploring mesoscopia: The bold new world of nanostructures. Phys. Today 46(10), 64–69 (1993)

    Google Scholar 

  136. Sivasankar M., Kumar B.P.: Role of nanoparticles in drug delivery system. Int. J. Res. Pharm. Biomed. Sci. 1(2), 41–66 (2010)

    Google Scholar 

  137. Smit R., Brekelmans W., Meijer H.: Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling. Comput. Methods Appl. Mech. Eng. 155(12), 181–192 (1998). doi:10.1016/S0045-7825(97)00139-4

    MATH  Google Scholar 

  138. Song Y.S., Youn J.R.: Modeling of effective elastic properties for polymer based carbon nanotube composites. Polymer 47(5), 1741–1748 (2006)

    Google Scholar 

  139. Southern J., Pitt-Francis J., Whiteley J., Stokeley D., Kobashi H., Nobes R., Kadooka Y., Gavaghan D.: Multi-scale computational modelling in biology and physiology. Prog. Biophys. Mol. Biol. 96(13), 60–89 (2008). doi:10.1016/j.pbiomolbio.2007.07.019

    Google Scholar 

  140. Su D., Ma R., Salloum M., Zhu L.: Multi-scale study of nanoparticle transport and deposition in tissues during an injection process. Med. Biol. Eng. Comput. 48(9), 853–863 (2010). doi:10.1007/s11517-010-0615-0

    Google Scholar 

  141. Sumer B., Gao J.: Theranostic nanomedicine for cancer. Nanomedicine 3(2), 137–140 (2008). doi:10.2217/17435889.3.2.137

    Google Scholar 

  142. Taber, L.A.: Biomechanics of Growth, Remodeling, and Morphogenesis. (1995). doi:10.1115/1.3005109

  143. Tang, L., Su, J., Huang, D.S., Lee, D.Y., Li, K.C., Zhou, X.Z.: An integrated multiscale mechanistic model for cancer drug therapy. ISRN Biomath 2012 (2012). doi:10.5402/2012/818492

  144. Thomas, G.C., Asanbaeva, A., Vena, P., Sah, R.L., Klisch, S.M.: A nonlinear constituent based viscoelastic model for articular cartilage and analysis of tissue remodeling due to altered glycosaminoglycan-collagen interactions. J. Biomech. Eng. 131(10), 101,002 (2009)

  145. Tirion M.M.: Large amplitude elastic motions in proteins from a single-parameter, atomic analysis. Phys. Rev. Lett. 77, 1905–1908 (1996). doi:10.1103/PhysRevLett.77.1905

    Google Scholar 

  146. Tosco T., Marchisio D.L., Lince F., Sethi R.: Extension of the darcy-forchheimer law for shear-thinning fluids and validation via pore-scale flow simulations. Transp. Porous Med. 96(1), 1–20 (2013). doi:10.1007/s11242-012-0070-5

    Google Scholar 

  147. Tozzini V.: Coarse-grained models for proteins. Curr. Opin. Struct. Biol. 15(2), 144–150 (2005). doi:10.1016/j.sbi.2005.02.005

    Google Scholar 

  148. Turner S., Sherratt J.A.: Intercellular adhesion and cancer invasion: A discrete simulation using the extended potts model. J. Theor. Biol. 216(1), 85–100 (2002). doi:10.1006/jtbi.2001.2522

    MathSciNet  Google Scholar 

  149. Uhrmacher, A., Degenring, D., Zeigler, B.: Discrete event multi-level models for systems biology. In: Priami, C. (ed.) Transactions on Computational Systems Biology I, Lecture Notes in Computer Science, vol. 3380, pp. 66–89. Springer, Berlin (2005)

  150. United, Nations, Department, of, Economic, Social, Affairs, Population, Division: World population prospects: The 2010 revision, highlights and advance tables. Technical report, Working paper No. ESA/P/WP.220. URL:esa.un.org/wpp/documentation/pdf/WPP2010_Highlights.pdf (2011)

  151. Vicini P.: Multiscale modeling in drug discovery and development: future opportunities and present challenges. Clin. Pharmacol. Ther. 88(1), 126–129 (2010)

    Google Scholar 

  152. Voigt A.: ber die bedeutung des schwefels beim zinkhttenprocess. Angew Chem. 20(2), 571–573 (1889)

    Google Scholar 

  153. Walburn, F., Schneck, D.: A Constitutive Equation for Whole Human Blood. 75-WA/Bio-12. American Society of Mechanical Engineers (1975)

  154. Wang J., Lu Z., Gao Y., Wientjes M.G., Au J.L.S.: Improving delivery and efficacy of nanomedicines in solid tumors: Role of tumor priming. Nanomedicine 6(9), 1605–1620 (2011)

    Google Scholar 

  155. Wang S., Dormidontova E.E.: Nanoparticle design optimization for enhanced targeting: Monte carlo simulations. Biomacromolecules 11(7), 1785–1795 (2010). doi:10.1021/bm100248e

    Google Scholar 

  156. Wang S.E., Narasanna A., Perez-Torres M., Xiang B., Wu F.Y., Yang S., Carpenter G., Gazdar A.F., Muthuswamy S.K., Arteaga C.L.: HER2 kinase domain mutation results in constitutive phosphorylation and activation of HER2 and EGFR and resistance to EGFR tyrosine kinase inhibitors. Cancer Cell 10(1), 25–38 (2006). doi:10.1016/j.ccr.2006.05.023

    Google Scholar 

  157. Warner S.: Diagnostics plus therapy = theranostics. Scientist 18(62), 38–39 (2004)

    Google Scholar 

  158. Weinan E., Engquist B., Li X., Ren W., Vanden-Eijnden E.: Heterogeneous multiscale methods: a review. Commun. Comput. Phys. 2(3), 367–450 (2007)

    MathSciNet  MATH  Google Scholar 

  159. Wilson W., van Donkelaar C., van Rietbergen B., Huiskes R.: A fibril-reinforced poroviscoelastic swelling model for articular cartilage. J. Biomech. 38(6), 1195–1204 (2005). doi:10.1016/j.jbiomech.2004.07.003

    Google Scholar 

  160. Winter P.M., Cai K., Caruthers S.D., Wickline S.A., Lanza G.M.: Emerging nanomedicine opportunities with perfluorocarbon nanoparticles. Expert Rev. Med. Devices 4(2), 137–145 (2007). doi:10.1586/17434440.4.2.137

    Google Scholar 

  161. Zhang M., Xi N.: Nanomedicine: A Systems Engineering Approach, Chapter 2. Pan Stanford Pub, Stanford (2009)

    Google Scholar 

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Taffetani, M., de Falco, C., Penta, R. et al. Biomechanical modelling in nanomedicine: multiscale approaches and future challenges. Arch Appl Mech 84, 1627–1645 (2014). https://doi.org/10.1007/s00419-014-0864-8

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