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Macro-scale models for fluid flow in tumour tissues: impact of microstructure properties

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Abstract

Understanding the dynamics underlying fluid transport in tumour tissues is of fundamental importance to assess processes of drug delivery. Here, we analyse the impact of the tumour microscopic properties on the macroscopic dynamics of vascular and interstitial fluid flow. More precisely, we investigate the impact of the capillary wall permeability and the hydraulic conductivity of the interstitium on the macroscopic model arising from formal asymptotic 2-scale techniques. The homogenization technique allows us to derive two macroscale tissue models of fluid flow that take into account the microscopic structure of the vessels and the interstitial tissue. Different regimes were derived according to the magnitude of the vessel wall permeability and the interstitial hydraulic conductivity. Importantly, we provide an analysis of the properties of the models and show the link between them. Numerical simulations were eventually performed to test the models and to investigate the impact of the microstructure on the fluid transport. Future applications of our models include their calibration with real imaging data to investigate the impact of the tumour microenvironment on drug delivery.

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Notes

  1. The details of the proof of this behaviour is out of the scope and given in Vaghi et al. (2022).

References

  • Allaire G (1992) Homogenization and two-scale convergence. SIAM J Math Anal 23(6):1482–1518

    Article  MathSciNet  MATH  Google Scholar 

  • Amar M, Andreucci D, Bisegna P, Gianni R (2006) On a hierarchy of models for electrical conduction in biological tissues. Math Methods Appl Sci 29(7):767–787

    Article  MathSciNet  MATH  Google Scholar 

  • Apelblat A, Katzir-Katchalsky A, Silberberg A (1974) A mathematical analysis of capillary-tissue fluid exchange. Biorheology 11(1):1–49

    Article  Google Scholar 

  • Arbogast T, Lehr HL (2006) Homogenization of a Darcy-Stokes system modeling vuggy porous media. Comput Geosci 10(3):291–302

    Article  MathSciNet  MATH  Google Scholar 

  • Baish JW, Gazit Y, Berk DA, Nozue M, Baxter LT, Jain RK (1996) Role of tumor vascular architecture in nutrient and drug delivery: an invasion percolation-based network model. Microvasc Res 51(3):327–346

    Article  Google Scholar 

  • Baish JW, Netti PA, Jain RK (1997) Transmural coupling of fluid flow in microcirculatory network and interstitium in tumors. Microvasc Res 53(2):128–141

    Article  Google Scholar 

  • Balanis CA (2012) Advanced engineering electromagnetics, 2nd edn. Wiley, Hoboken

    Google Scholar 

  • Bartha K, Rieger H (2006) Vascular network remodeling via vessel cooption, regression and growth in tumors. J Theor Biol 241(4):903–918

    Article  MathSciNet  MATH  Google Scholar 

  • Baxter LT, Jain RK (1989) Transport of fluid and macromolecules in tumors. I. Role of interstitial pressure and convection. Microvasc Res 37(1):77–104

    Article  Google Scholar 

  • Beavers GS, Joseph DD (1967) Boundary conditions at a naturally permeable wall. J Fluid Mech 30(1):197–207

    Article  Google Scholar 

  • Blake TR, Gross JF (1982) Analysis of coupled intra- and extraluminal flows for single and multiple capillaries. Math Biosci 59(2):173–206

    Article  MathSciNet  MATH  Google Scholar 

  • Boucher Y, Jain RK (1992) Microvascular pressure is the principal driving force for interstitial hypertension in solid tumors: implications for vascular collapse. Cancer Res 52(18):5110–5114

    Google Scholar 

  • Boucher Y, Baxter LT, Jain RK (1990) Interstitial pressure gradients in tissue-isolated and subcutaneous tumors: implications for therapy. Cancer Res 50(15):4478–4484

    Google Scholar 

  • Chauhan VP, Stylianopoulos T, Boucher Y, Jain RK (2011) Delivery of molecular and nanoscale medicine to tumors: transport barriers and strategies. Annu Rev Chem Biomol Eng 2(1):281–298

    Article  Google Scholar 

  • Conca C (1985) On the application of the homogenization theory to a class of problems arising in fluid mechanics. J. Math. Pures Appl. 64:31–75

    MathSciNet  MATH  Google Scholar 

  • Discacciati M, Quarteroni A (2009) Navier-Stokes/darcy coupling: modeling, analysis, and numerical approximation. Rev Mat Complut 22(2):315–426

    Article  MathSciNet  MATH  Google Scholar 

  • Dvorak HF, Brown LF, Detmar M, Dvorak AM (1995) Vascular permeability factor/vascular endothelial growth factor, microvascular hyperpermeability, and angiogenesis. Am J Pathol 146(5):1029–1039

    Google Scholar 

  • Forster JC, Harriss-Phillips WM, Douglass MJ, Bezak E (2017) A review of the development of tumor vasculature and its effects on the tumor microenvironment. Hypoxia (Auckl) 5:21–32

    Article  Google Scholar 

  • Guyton AC, Granger HJ, Taylor AE (1971) Interstitial fluid pressure. Physiol Rev 51(3):527–563

    Article  Google Scholar 

  • Haddar H, Joly P, Nguyen H-M (2005) Generalized impedance impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case. Math Models Methods Appl Sci 15(08):1273–1300

    Article  MathSciNet  MATH  Google Scholar 

  • Helmlinger G, Netti PA, Lichtenbeld HC, Melder RJ, Jain RK (1997) Solid stress inhibits the growth of multicellular tumor spheroids. Nat Biotechnol 15(8):778–783

    Article  Google Scholar 

  • Hilmas DE, Gillette EL (1974) Morphometric analyses of the microvasculature of tumors during growth and after x-irradiation. Cancer 33(1):103–110

    Article  Google Scholar 

  • Jain RK (1987) Transport of molecules in the tumor interstitium: a review. Cancer Res 47(12):3039–3051

    Google Scholar 

  • Jain RK (1994) Barriers to drug delivery in solid tumors. Sci Am 271(1):58–65

    Article  Google Scholar 

  • Jain RK, Baxter LT (1988) Mechanisms of heterogeneous distribution of monoclonal antibodies and other macromolecules in tumors: significance of elevated interstitial pressure. Cancer Res 48(24 Pt 1):7022–7032

    Google Scholar 

  • Jain RK, Tong RT, Munn LL (2007) Effect of vascular normalization by antiangiogenic therapy on interstitial hypertension, peritumor edema, and lymphatic metastasis: insights from a mathematical model. Cancer Res 67(6):2729–2735

    Article  Google Scholar 

  • Kamoun WS, Chae S-S, Lacorre DA, Tyrrell JA, Mitre M, Gillissen MA, Fukumura D, Jain RK, Munn LL (2010) Simultaneous measurement of RBC velocity, flux, hematocrit and shear rate in vascular networks. Nat Methods 7(8):655–660

    Article  Google Scholar 

  • Kanit T, Forest S, Galliet I, Mounoury V, Jeulin D (2003) Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int J Solids Struct 40(13–14):3647–3679

    Article  MATH  Google Scholar 

  • Kato T (1995) Perturbation theory for linear operators. Classics in mathematics, vol 132. Springer, Berlin

    Book  Google Scholar 

  • Krogh A (1922) The anatomy and physiology of capillaries. Yale University Press, New Haven

    Google Scholar 

  • Layton WJ, Schieweck F, Yotov I (2002) Coupling fluid flow with porous media flow. SIAM J Numer Anal 40(6):2195–2218

    Article  MathSciNet  MATH  Google Scholar 

  • Less JR, Skalak TC, Sevick EM, Jain RK (1991) Microvascular architecture in a mammary carcinoma: branching patterns and vessel dimensions. Cancer Res 51(1):265–273

    Google Scholar 

  • Netti PA, Baxter LT, Boucher Y, Skalak R, Jain RK (1995) Time-dependent behavior of interstitial fluid pressure in solid tumors: implications for drug delivery. Cancer Res 55(22):5451–5458

    Google Scholar 

  • Netti PA, Berk DA, Swartz MA, Grodzinsky AJ, Jain RK (2000) Role of extracellular matrix assembly in interstitial transport in solid tumors. Cancer Res 60(9):2497–2503

    Google Scholar 

  • Owen MR, Alarcón T, Maini PK, Byrne HM (2009) Angiogenesis and vascular remodelling in normal and cancerous tissues. J Math Biol 58(4–5):689–721

    Article  MathSciNet  MATH  Google Scholar 

  • Penta R, Merodio J (2017) Homogenized modeling for vascularized poroelastic materials. Meccanica 52(14):3321–3343

    Article  MathSciNet  MATH  Google Scholar 

  • Penta R, Ambrosi D, Shipley RJ (2014) Effective governing equations for poroelastic growing media. Q J Mech Appl Math 67(1):69–91

    Article  MathSciNet  MATH  Google Scholar 

  • Penta R, Ambrosi D, Quarteroni A (2015) Multiscale homogenization for fluid and drug transport in vascularized malignant tissues. Math Models Methods Appl Sci 25(01):79–108

    Article  MathSciNet  MATH  Google Scholar 

  • Perrussel R, Poignard C (2013) Asymptotic expansion of steady-state potential in a high contrast medium with a thin resistive layer. Appl Math Comput 221:48–65

    MathSciNet  MATH  Google Scholar 

  • Pozrikidis C, Farrow DA (2003) A model of fluid flow in solid tumors. Ann Biomed Eng 31(2):181–194

    Article  Google Scholar 

  • Pries AR, Secomb TW, Gessner T, Sperandio MB, Gross JF, Gaehtgens P (1994) Resistance to blood flow in microvessels in vivo. Circ Res 75(5):904–915

    Article  Google Scholar 

  • Rand PW, Lacombe E, Hunt HE, Austin WH (1964) Viscosity of normal human blood under normothermic and hypothermic conditions. J Appl Physiol 19(1):117–122

    Article  Google Scholar 

  • Rippe B, Kamiya A, Folkow B (1978) Simultaneous measurements of capillary diffusion and filtration exchange during shifts in filtration-absorption and at graded alterations in the capillary permeability surface area product (PS). Acta Physiol Scand 104(3):318–336

    Article  Google Scholar 

  • Saffman PG (1971) On the boundary condition at the surface of a porous medium. Stud Appl Math 50(2):93–101

    Article  MATH  Google Scholar 

  • Schuff MM, Gore JP, Nauman EA (2013) A mixture theory model of fluid and solute transport in the microvasculature of normal and malignant tissues. I. Theory. J Math Biol 66(6):1179–1207

    Article  MathSciNet  MATH  Google Scholar 

  • Sevick EM, Jain RK (1989) Viscous resistance to blood flow in solid tumors: effect of hematocrit on intratumor blood viscosity. Cancer Res 49(13):3513–3519

    Google Scholar 

  • Sevick EM, Jain RK (1991) Measurement of capillary filtration coefficient in a solid tumor. Cancer Res 51(4):1352–1355

    Google Scholar 

  • Shipley RJ, Chapman SJ (2010) Multiscale modelling of fluid and drug transport in vascular tumours. Bull Math Biol 72(6):1464–1491

    Article  MathSciNet  MATH  Google Scholar 

  • Shipley RJ, Sweeney PW, Chapman SJ, Roose T (2020) A four-compartment multiscale model of fluid and drug distribution in vascular tumours. Int J Numer Method Biomed Eng 36(3):e3315

    Article  MathSciNet  Google Scholar 

  • Soltani M, Chen P (2011) Numerical modeling of fluid flow in solid tumors. PLoS ONE 6(6):e20344

    Article  Google Scholar 

  • Stamatelos SK, Kim E, Pathak AP, Popel AS (2014) A bioimage informatics based reconstruction of breast tumor microvasculature with computational blood flow predictions. Microvasc Res 91:8–21

    Article  Google Scholar 

  • Starling EH (1896) On the absorption of fluids from the connective tissue spaces. J Physiol 19(4):312–326

    Article  Google Scholar 

  • Swabb EA, Wei J, Gullino PM (1974) Diffusion and convection in normal and neoplastic tissues. Cancer Res 34(10):2814–2822

    Google Scholar 

  • Sweeney PW, d’Esposito A, Walker-Samuel S, Shipley RJ (2019) Modelling the transport of fluid through heterogeneous, whole tumours in silico. PLoS Comput Biol 15(6):e1006751

  • Vaghi C, Benzekry S, Poignard C (2022) Asymptotic analysis of a biphase tumor fluid flow: the weak coupling case. Appl Math Comput 413:126635

    MathSciNet  MATH  Google Scholar 

  • Vogel AW (1965) Intratumoral vascular changes with increased size of a mammary adenocarcinoma: new method and results. J Natl Cancer Inst 34:571–578

    Google Scholar 

  • Voutouri C, Stylianopoulos T (2014) Evolution of osmotic pressure in solid tumors. J Biomech 47(14):3441–3447

    Article  Google Scholar 

  • Walker-Samuel S, Roberts TA, Ramasawmy R, Burrell JS, Johnson SP, Siow BM, Richardson S, Gonçalves MR, Pendse D, Robinson SP, Pedley RB, Lythgoe MF (2018) Investigating low-velocity fluid flow in tumors with convection-MRI. Cancer Res 78(7):1859–1872

    Article  Google Scholar 

  • Welter M, Rieger H (2013) Interstitial fluid flow and drug delivery in vascularized tumors: a computational model. PLoS ONE 8(8):e70395

    Article  Google Scholar 

  • Wijeratne PA, Hipwell JH, Hawkes DJ, Stylianopoulos T, Vavourakis V (2017) Multiscale biphasic modelling of peritumoural collagen microstructure: the effect of tumour growth on permeability and fluid flow. PLoS ONE 12(9):e0184511

    Article  Google Scholar 

  • Young JS, Lumsden CE, Stalker AL (1950) The significance of the tissue pressure of normal testicular and of neoplastic (Brown-Pearce carcinoma) tissue in the rabbit. J Pathol Bacteriol 62(3):313–333

    Article  Google Scholar 

  • Zhao J, Salmon H, Sarntinoranont M (2007) Effect of heterogeneous vasculature on interstitial transport within a solid tumor. Microvasc Res 73(3):224–236

    Article  Google Scholar 

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Acknowledgements

Funding was provided by the Plan Cancer NUMEP (Grant No. PC201615) and Plan Cancer QUANTIC (Grant No. 19CM148-00).

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Authors and Affiliations

  1. Team MONC, Inria, Institut de Mathématiques de Bordeaux, CNRS, Bordeaux INP, Univ. Bordeaux, Bordeaux, France

    Cristina Vaghi, Sébastien Benzekry & Clair Poignard

  2. SMARTc, CRCM Inserm UMR1068, CNRS UMR7258, Aix Marseille University, Marseille, France

    Raphaëlle Fanciullino

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  1. Cristina Vaghi
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Correspondence to Clair Poignard.

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Vaghi, C., Fanciullino, R., Benzekry, S. et al. Macro-scale models for fluid flow in tumour tissues: impact of microstructure properties. J. Math. Biol. 84, 27 (2022). https://doi.org/10.1007/s00285-022-01719-1

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