The influence of tumour vasculature on fluid flow in solid tumours: a mathematical modelling study

Abstract

Tumour vasculature is known to be aberrant, tortuous and erratic which can have significant implications for fluid flow. Fluid dynamics in tumour tissue plays an important part in tumour growth, metastasis and the delivery of therapeutics. Mathematical models are increasingly employed to elucidate the complex interplay between various aspects of the tumour vasculature and fluid flow. Previous models usually assume a uniformly distributed vasculature without explicitly describing its architecture or incorporate the distribution of vasculature without accounting for real geometric features of the network. In this study, an integrated computational model is developed by resolving fluid flow at the single capillary level across the whole tumour vascular network. It consists of an angiogenesis model and a fluid flow model which resolves flow as a function of the explicit vasculature by coupling intravascular flow and interstitial flow in tumour tissue. The integrated model has been used to examine the influence of microvascular distribution, necrosis and vessel pruning on fluid flow, as well as the effect of heterogeneous vessel permeability. Our results reveal the level of nonuniformity in tumour interstitial fluid pressure (IFP), with large variations in IFP profile between necrotic and non-necrotic tumours. Changes in microscopic features of the vascular network can significantly influence fluid flow in the tumour where removal of vessel blind ends has been found to reduce IFP and promote interstitial fluid flow. Our results demonstrate the importance of incorporating microscopic properties of the tumour vasculature and intravascular flow when predicting fluid flow in tumour tissue.

Graphical abstract

Keywords

Fluid flow / Vasculature / Tumour / Mathematical modelling

References

[1] Anderson AR , Chaplain M . Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull Math Biol, 1998, 60 : 857– 899

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

[3] Baluk P , Hashizume H , McDonald DM . Cellular abnormalities of blood vessels as targets in cancer. Curr Opin Genet Dev, 2005, 15 : 102– 111

[4] Baxter LT , Jain RK . Transport of fluid and macromolecules in tumors. I. Role of interstitial pressure and convection. Microvasc Res, 1989, 37 : 77– 104

[5] Bhandari A , Bansal A , Singh A , Sinha N . Transport of liposome encapsulated drugs in voxelized computational model of human brain tumors. IEEE Trans NanoBiosci, 2017a, 16 : 634– 644

[6] Bhandari A , Bansal A , Singh A , Sinha N . Perfusion kinetics in human brain tumor with DCE-MRI derived model and CFD analysis. J Biomech, 2017b, 59 : 80– 89

[7] Bhandari A , Bansal A , Singh A , Sinha N . Numerical study of transport of anti- cancer drugs in heterogeneous vasculature of human brain tumors using DCE-MRI. J Biomech Eng, 2018, 140 : 051010

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

[9] Brizel DM , Klitzman B , Cook JM , Edwards J , Rosner G , Dewhirst MW . A comparison of tumor and normal tissue microvascular hematocrits and red cell fluxes in a rat window chamber model. Int J Radiat Oncol Biol Phys, 1993, 25 : 269– 276

[10] Cantelmo AR , Pircher A , Kalucka J , Carmeliet P . Vessel pruning or healing: endothelial metabolism as a novel target?. Exp OpinTher Targ, 2017, 21 : 239– 247

[11] Carmeliet P , Jain RK . Angiogenesis in cancer and other diseases. Nature, 2000, 407 : 249

[12] Chary SR , Jain RK . Direct measurement of interstitial convection and diffusion of albumin in normal and neoplastic tissues by fluorescence photobleaching. Proc Natl Acad Sci USA, 1989, 86 : 5385– 5389

[13] Chauhan VP , Stylianopoulos T , Martin JD , Popović Z , Chen O , Kamoun WS , Bawendi MG , Fukumura D , Jain RK . Normalization of tumour blood vessels improves the delivery of nanomedicines in a size-dependent manner. Nat Nanotechnol, 2012, 7 : 383– 388

[14] Chugh BP , Lerch JP , Lisa XY , Pienkowski M , Harrison RV , Henkelman RM , Sled JG . Measurement of cerebral blood volume in mouse brain regions using micro-computed tomography. Neuroimage, 2009, 47 : 1312– 1318

[15] Dewhirst MW , Secomb TW . Transport of drugs from blood vessels to tumour tissue. Nature Rev Cancer, 2017, 17( 12): 738– 750

[16] Gas P . Temperature inside tumor as time function in RF hyperthermia. Przeglad Elektrotechniczny, 2017, 2010, 86( 12): 42– 45

[17] Gulliksrud K , Brurberg KG , Rofstad EK . Dynamic contrast-enhanced magnetic resonance imaging of tumor interstitial fluid pressure. Radiother Oncol, 2009, 91 : 107– 113

[18] Hashizume H , Baluk P , Morikawa S , McLean JW , Thurston G , Roberge S , Jain RK , McDonald DM . Openings between defective endothelial cells explain tumor vessel leakiness. Am J Pathol, 2000, 156( 4): 1363– 1380

[19] Heldin C-H , Rubin K , Pietras K , Östman A . High interstitial fluid pressure — an obstacle in cancer therapy. Nat Rev Cancer, 2004, 4 : 806– 813

[20] Hobbs SK , Monsky WL , Yuan F , Roberts WG , Griffith L , Torchilin VP , Jain RK . Regulation of transport pathways in tumor vessels: role of tumor type and microenvironment. Proc Natl Acad Sci USA, 1998, 95 : 4607– 4612

[21] Hompland T , Lund KV , Ellingsen C , Kristensen GB , Rofstad EK . Peritumoral interstitial fluid flow velocity predicts survival in cervical carcinoma. Radiother Oncol, 2014, 113 : 132– 138

[22] Jain RK . Determinants of tumor blood flow: a review. Cancer research, 1988, 48 : 2641– 2658

[23] Jain RK , Martin JD , Stylianopoulos T . The role of mechanical forces in tumor growth and therapy. Annu Rev Biomed Eng, 2014, 16 : 321– 346

[24] Jensen MM , Jørgensen JT , Binderup T , Kjær A . Tumor volume in subcutaneous mouse xenografts measured by microCT is more accurate and reproducible than determined by 18 F-FDG-microPET or external caliper. BMC Med imaging, 2008, 8 : 16

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

[26] Kim E , Stamatelos S , Cebulla J , Bhujwalla ZM , Popel AS , Pathak AP . Multiscale imaging and computational modeling of blood flow in the tumor vasculature. Ann Biomed Eng, 2012, 40 : 2425– 2441

[27] Konerding M , Fait E , Gaumann A . 3D microvascular architecture of pre-cancerous lesions and invasive carcinomas of the colon. Br J Cancer, 2001, 84( 10): 1354– 1362

[28] Kurbel S , Kurbel B , Belovari T , Maric S , Steiner R , Bozic D . Model of interstitial pressure as a result of cyclical changes in the capillary wall fluid transport. Med Hypoth, 2001, 57 : 161– 166

[29] Kuszyk BS , Corl FM , Franano FN , Bluemke DA , Hofmann LV , Fortman BJ , Fishman EK . Tumor transport physiology: implications for imaging and imaging-guided therapy. Am J Roentgenol, 2001, 177 : 747– 753

[30] Less JR , Posner MC , Boucher Y , Borochovitz D , Wolmark N , Jain RK . Interstitial hypertension in human breast and colorectal tumors. Cancer Res, 1992, 52 : 6371– 6374

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

[32] Liao D , Johnson RS . Hypoxia: a key regulator of angiogenesis in cancer. Cancer Metast Rev, 2007, 26 : 281– 290

[33] Martin JD , Seano G , Jain RK . Normalizing function of tumor vessels: progress, opportunities, and challenges. Annu Rev Physiol, 2019, 81 : 505– 534

[34] McDougall SR , Anderson A , Chaplain M , Sherratt J . Mathematical modelling of flow through vascular networks: implications for tumour-induced angiogenesis and chemotherapy strategies. Bull Math Biol, 2002, 64 : 673– 702

[35] Mohammadi M , Chen P . Effect of microvascular distribution and its density on interstitial fluid pressure in solid tumors: a computational model. Microvasc Res, 2015, 101 : 26– 32

[36] Munson JM , Shieh AC . Interstitial fluid flow in cancer: implications for disease progression and treatment. Cancer Manag Res, 2014, 6 : 317– 328

[37] Nagy JA , Dvorak AM , Dvorak HF . Vascular hyperpermeability, angiogenesis, and stroma generation. Cold Spring Harb Perspec Med, 2012, 2 : a006544

[38] Netti PA , Roberge S , Boucher Y , Baxter LT , Jain RK . Effect of transvascular fluid exchange on pressure–flow relationship in tumors: a proposed mechanism for tumor blood flow heterogeneity. Microvasc Res, 1996, 52 : 27– 46

[39] Nugent LJ , Jain RK . Extravascular diffusion in normal and neoplastic tissues. Cancer Res, 1984, 44 : 238– 244

[40] Pathak AP , Kim E , Zhang J , Jones MV . Three-dimensional imaging of the mouse neurovasculature with magnetic resonance microscopy. PLoS One, 2011, 6 : e22643

[41] Pozrikidis C . Numerical simulation of blood and interstitial flow through a solid tumor. J Math Biol, 2010, 60 : 75– 94

[42] Pozrikidis C , Farrow D . A model of fluid flow in solid tumors. Ann Biomed Eng, 2003, 31 : 181– 194

[43] Pries A , Reglin B , Secomb TW . Structural adaptation of microvascular networks: functional roles of adaptive responses. Am J Physiol-Heart Circ Physiol, 2001, 281 : H1015– H1025

[44] Pries AR , Cornelissen AJ , Sloot AA , Hinkeldey M , Dreher MR , Höpfner M , Dewhirst MW , Secomb TW . Structural adaptation and heterogeneity of normal and tumor microvascular networks. PLoS Comput Biol, 2009, 5 : e1000394

[45] Sevick EM , Jain RK . Measurement of capillary filtration coefficient in a solid tumor. Cancer Res, 1991, 51 : 1352– 1355

[46] Sitohy B , Nagy JA , Dvorak HF . Anti-VEGF/VEGFR therapy for cancer: reassessing the target. Cancer Res, 2012, 72 : 1909– 1914

[47] Skliarenko JV , Lunt SJ , Gordon ML , Vitkin A , Milosevic M , Hill RP . Effects of the vascular disrupting agent ZD6126 on interstitial fluid pressure and cell survival in tumors. Cancer Res, 2006, 66 : 2074– 2080

[48] Soltani M , Chen P . Numerical modeling of fluid flow in solid tumors. PloS One, 2011, 6 : e20344

[49] Soltani M , Chen P . Effect of tumor shape and size on drug delivery to solid tumors. J Biol Eng, 2012, 6 : 4

[50] Soltani M , Chen P . Numerical modeling of interstitial fluid flow coupled with blood flow through a remodeled solid tumor microvascular network. PLoS One, 2013, 8 : e67025

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

[52] Stohrer M , Boucher Y , Stangassinger M , Jain RK . Oncotic pressure in solid tumors is elevated. Cancer Res, 2000, 60 : 4251– 4255

[53] Stokes CL , Lauffenburger DA . Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis. J Theor Biol, 1991, 152 : 377– 403

[54] Swartz MA , Lund AW . Lymphatic and interstitial flow in the tumour microenvironment: linking mechanobiology with immunity. Nat Rev Cancer, 2012, 12 : 210– 219

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

[56] Tong RT , Boucher Y , Kozin SV , Winkler F , Hicklin DJ , Jain RK . Vascular normalization by vascular endothelial growth factor receptor 2 blockade induces a pressure gradient across the vasculature and improves drug penetration in tumors. Cancer Res, 2004, 64 : 3731– 3736

[57] Vavourakis V , Wijeratne PA , Shipley R , Loizidou M , Stylianopoulos T , Hawkes DJ . A validated multiscale in-silico model for mechano-sensitive tumour angiogenesis and growth. PLoS Comput Biol, 2017, 13 : e1005259

[58] Vavra M , Ali MJ , Kang EW-Y , Navalitloha Y , Ebert A , Allen CV , Groothuis DR . Comparative pharmacokinetics of 14C-sucrose in RG-2 rat gliomas after intravenous and convection-enhanced delivery. Neuro-oncol, 2004, 6 : 104– 112

[59] Welter M , Rieger H . Physical determinants of vascular network remodeling during tumor growth. Eur Phys J E, 2010, 33 : 149– 163

[60] Welter M , Rieger H . Interstitial fluid flow and drug delivery in vascularized tumors: a computational model. PloS One, 2013, 8 : e70395

[61] Wu J , Xu S , Long Q , Collins MW , König CS , Zhao G , Jiang Y , Padhani AR . Coupled modeling of blood perfusion in intravascular, interstitial spaces in tumor microvasculature. J Biomech, 2008, 41 : 996– 1004

[62] Zhan W , Gedroyc W , Xu XY . Effect of heterogeneous microvasculature distribution on drug delivery to solid tumour. J Phys D: Appl Phys, 2014, 47 : 475401

[63] Zhan W , Gedroyc W , Xu XY . The effect of tumour size on drug trasnport and uptake in 3-D tumour models reconstructed frlom magnetic resonance images. PLoS One, 2017, 12 : e0172276

[64] Zhan W , Alamer M , Xu XY . Computational modelling of drug delivery to solid tumour: understanding the interplay between chemotherapeutics and biological system for optimal delivery systems. Adv Drug Deliv Rev, 2018, 132 : 81– 103

Compliance with ethics guidelines

Conflict of interest Moath Alamer and Xiao Yun Xu declare that they have no conflict of interest. Human and animal rights and informed consent This article does not contain any studies with human or animal subjects performed by any of the authors. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

RIGHTS & PERMISSIONS

2021 The Author(s) 2021. Published by Higher Education Press. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0)