Computational models for fluid exchange between microcirculation and tissue interstitium

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March 2014, 9(1): 135-159. doi: 10.3934/nhm.2014.9.135

Computational models for fluid exchange between microcirculation and tissue interstitium

Laura Cattaneo ^1, and Paolo Zunino ^2,

1.	MOX, Department of Mathematics "Francesco Brioschi", Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
2.	Department of Mechanical Engineering and Materials Science, University of Pittsburgh, 3700 O'Hara Street, Pittsburgh, PA 15261, United States

Received May 2013 Revised August 2013 Published April 2014

Abstract
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The aim of this work is to develop a computational model able to capture the interplay between microcirculation and interstitial flow. Such phenomena are at the basis of the exchange of nutrients, wastes and pharmacological agents between the cardiovascular system and the organs. They are particularly interesting for the study of effective therapies to treat vascularized tumors with drugs. We develop a model applicable at the microscopic scale, where the capillaries and the interstitial volume can be described as independent structures capable to propagate flow. We facilitate the analysis of complex capillary bed configurations, by representing the capillaries as a one-dimensional network, ending up with a heterogeneous system characterized by channels embedded into a porous medium. We use the immersed boundary method to couple the one-dimensional with the three-dimensional flow through the network and the interstitial volume, respectively. The main idea consists in replacing the immersed network with an equivalent concentrated source term. After discussing the details for the implementation of a computational solver, we apply it to compare flow within healthy and tumor tissue samples.

Keywords: Perfusion, intestitial flow, tumors enhanced permeability and retention, immersed boundary method..

Mathematics Subject Classification: Primary: 76Z05; Secondary: 65M6.

Citation: Laura Cattaneo, Paolo Zunino. Computational models for fluid exchange between microcirculation and tissue interstitium. Networks and Heterogeneous Media, 2014, 9 (1) : 135-159. doi: 10.3934/nhm.2014.9.135

References:

[1]	L. T. Baxter and R. K. Jain, Transport of fluid and macromolecules in tumors. i. role of interstitial pressure and convection, Microvascular Research, 37 (1989), 77-104. doi: 10.1016/0026-2862(89)90074-5.
[2]	L. T. Baxter and R. K. Jain, Transport of fluid and macromolecules in tumors ii. role of heterogeneous perfusion and lymphatics, Microvascular Research, 40 (1990), 246-263. doi: 10.1016/0026-2862(90)90023-K.
[3]	L. T. Baxter and R. K. Jain, Transport of fluid and macromolecules in tumors. iii. role of binding and metabolism, Microvascular Research, 41 (1991), 5-23.
[4]	L. T. Baxter and R. K. Jain, Transport of fluid and macromolecules in tumors: Iv. a microscopic model of the perivascular distribution, Microvascular Research, 41 (1991), 252-272. doi: 10.1016/0026-2862(91)90026-8.
[5]	T. R. Blake and J. F. Gross, Analysis of coupled intra- and extraluminal flows for single and multiple capillaries, Mathematical Biosciences, 59 (1982), 173-206. doi: 10.1016/0025-5564(82)90022-0.
[6]	S. Canic, D. Lamponi, A. Mikelić and J. Tambaca, Self-consistent effective equations modeling blood flow in medium-to-large compliant arteries, Multiscale Modeling and Simulation, 3 (2005), 559-596. doi: 10.1137/030602605.
[7]	P. Carmeliet and R. K. Jain, Angiogenesis in cancer and other diseases, Nature, 407 (2000), 249-257.
[8]	S. J. Chapman, R. J. Shipley and R. Jawad, Multiscale modeling of fluid transport in tumors, Bulletin of Mathematical Biology, 70 (2008), 2334-2357. doi: 10.1007/s11538-008-9349-7.
[9]	C. D'Angelo, Multiscale Modeling of Metabolism and Transport Phenomena in Living Tissues, Ph.D thesis, 2007.
[10]	C. D'Angelo, Finite element approximation of elliptic problems with dirac measure terms in weighted spaces: Applications to one- and three-dimensional coupled problems, SIAM Journal on Numerical Analysis, 50 (2012), 194-215. doi: 10.1137/100813853.
[11]	C. D'Angelo and A. Quarteroni, On the coupling of 1D and 3D diffusion-reaction equations. Application to tissue perfusion problems, Math. Models Methods Appl. Sci., 18 (2008), 1481-1504. doi: 10.1142/S0218202508003108.
[12]	A. Farina, A. Fasano and J. Mizerski, A new model for blood flow in fenestrated capillaries with application to ultrafiltration in kidney glomeruli, Submitted.
[13]	D. A. Fedosov, G. E. Karniadakis and B. Caswell, Steady shear rheometry of dissipative particle dynamics models of polymer fluids in reverse poiseuille flow, Journal of Chemical Physics, 132 (2010). doi: 10.1063/1.3366658.
[14]	M. Ferrari, Frontiers in cancer nanomedicine: Directing mass transport through biological barriers, Trends in Biotechnology, 28 (2010), 181-188. doi: 10.1016/j.tibtech.2009.12.007.
[15]	G. J. Fleischman, T. W. Secomb and J. F. Gross, The interaction of extravascular pressure fields and fluid exchange in capillary networks, Mathematical Biosciences, 82 (1986), 141-151. doi: 10.1016/0025-5564(86)90134-3.
[16]	G. J. Flieschman, T. W. Secomb and J. F. Gross, Effect of extravascular pressure gradients on capillary fluid exchange, Mathematical Biosciences, 81 (1986), 145-164. doi: 10.1016/0025-5564(86)90114-8.
[17]	L. Formaggia, D. Lamponi and A. Quarteroni, One-dimensional models for blood flow in arteries, Journal of Engineering Mathematics, 47 (2003), 251-276. doi: 10.1023/B:ENGI.0000007980.01347.29.
[18]	L. Formaggia, A. Quarteroni and A. Veneziani, Multiscale models of the vascular system, in Cardiovascular Mathematics, MS&A. Model. Simul. Appl., 1, Springer Italia, Milan, 2009, 395-446. doi: 10.1007/978-88-470-1152-6_11.
[19]	A. Harris, G. Guidoboni, J. C. Arciero, A. Amireskandari, L. A. Tobe and B. A. Siesky, Ocular hemodynamics and glaucoma: The role of mathematical modeling, European Journal of Ophthalmology, 23 (2013), 139-146. doi: 10.5301/ejo.5000255.
[20]	K. O. Hicks, F. B. Pruijn, T. W. Secomb, M. P. Hay, R. Hsu, J. M. Brown, W. A. Denny, M. W. Dewhirst and W. R. Wilson, Use of three-dimensional tissue cultures to model extravascular transport and predict in vivo activity of hypoxia-targeted anticancer drugs, Journal of the National Cancer Institute, 98 (2006), 1118-1128. doi: 10.1093/jnci/djj306.
[21]	S. S. Hossain, Y. Zhang, X. Liang, F. Hussain, M. Ferrari, T. J. Hughes and P. Decuzzi, In silico vascular modeling for personalized nanoparticle delivery, Nanomedicine, 8 (2013), 343-357.
[22]	M. Intaglietta, N. R. Silverman and W. R. Tompkins, Capillary flow velocity measurements in vivo and in situ by television methods, Microvascular Research, 10 (1975), 165-179. doi: 10.1016/0026-2862(75)90004-7.
[23]	R. K. Jain, Transport of molecules, particles, and cells in solid tumors, Annual Review of Biomedical Engineering, (1999), 241-263. doi: 10.1146/annurev.bioeng.1.1.241.
[24]	R. K. Jain, R. T. Tong and L. L. Munn, Effect of vascular normalization by antiangiogenic therapy on interstitial hypertension, peritumor edema, and lymphatic metastasis: Insights from a mathematical model, Cancer Research, 67 (2007), 2729-2735. doi: 10.1158/0008-5472.CAN-06-4102.
[25]	J. Lee and T. C. Skalak, Microvascular Mechanics: Hemodynamics of Systemic and Pulmonary Microcirculation, Springer-Verlag, 1989.
[26]	H. Lei, D. A. Fedosov, B. Caswell and G. E. Karniadakis, Blood flow in small tubes: Quantifying the transition to the non-continuum regime, Journal of Fluid Mechanics, 722 (2013), 214-239. doi: 10.1017/jfm.2013.91.
[27]	J. R. Less, T. C. Skalak, E. M. Sevick and R. K. Jain, Microvascular architecture in a mammary carcinoma: Branching patterns and vessel dimensions, Cancer Research, 51 (1991), 265-273.
[28]	W. K. Liu, Y. Liu, D. Farrell, L. Zhang, X. S. Wang, Y. Fukui, N. Patankar, Y. Zhang, C. Bajaj, J. Lee, J. Hong, X. Chen and H. Hsu, Immersed finite element method and its applications to biological systems, Comput. Methods Appl. Mech. Engrg., 195 (2006), 1722-1749. doi: 10.1016/j.cma.2005.05.049.
[29]	Y. Liu and W. K. Liu, Rheology of red blood cell aggregation by computer simulation, Journal of Computational Physics, 220 (2006), 139-154. doi: 10.1016/j.jcp.2006.05.010.
[30]	Y. Liu, L. Zhang, X. Wang and W. K. Liu, Coupling of navier-stokes equations with protein molecular dynamics and its application to hemodynamics, International Journal for Numerical Methods in Fluids, 46 (2004), 1237-1252. doi: 10.1002/fld.798.
[31]	J. Peiró and A. Veneziani, Reduced models of the cardiovascular system, in Cardiovascular Mathematics, MS&A. Model. Simul. Appl., 1, Springer Italia, Milan, 2009, 347-394. doi: 10.1007/978-88-470-1152-6_10.
[32]	Y. Renard and J. Pommier, Getfem++: A generic finite element library in c++, version 4.2 (2012), http://download.gna.org/getfem/html/homepage/.
[33]	A. M. Robertson and A. Sequeira, A director theory approach for modeling blood flow in the arterial system: An alternative to classical id models, Mathematical Models and Methods in Applied Sciences, 15 (2005), 871-906. doi: 10.1142/S0218202505000601.
[34]	A. M. Robertson, A. Sequeira and R. G. Owens, Rheological models for blood. In Cardiovascular Mathematics, MS&A. Model. Simul. Appl., 1, Springer Italia, Milan, 2009, 211-241. doi: 10.1007/978-88-470-1152-6_6.
[35]	T. W. Secomb, A. R. Pries, P. Gaehtgens and J. F. Gross, Theoretical and experimental analysis of hematocrit distribution in microcirculatory networks, in Microvascular Mechanics (eds. J.-S. Lee and T. C. Skalak), Springer, New York, 1989, 39-49. doi: 10.1007/978-1-4612-3674-0_4.
[36]	T. W. Secomb, Microvascular Network Structures, http://www.physiology.arizona.edu/people/secomb.
[37]	T. W. Secomb, R. Hsu, R. D. Braun, J. R. Ross, J. F. Gross and M. W. Dewhirst, Theoretical simulation of oxygen transport to tumors by three-dimensional networks of microvessels, Advances in Experimental Medicine and Biology, 454 (1998), 629-634. doi: 10.1007/978-1-4615-4863-8_74.
[38]	T. W. Secomb, R. Hsu, E. Y. H. Park and M. W. Dewhirst, Green's function methods for analysis of oxygen delivery to tissue by microvascular networks, Annals of Biomedical Engineering, 32 (2004), 1519-1529. doi: 10.1114/B:ABME.0000049036.08817.44.
[39]	R. J. Shipley and S. J. Chapman, Multiscale modelling of fluid and drug transport in vascular tumours, Bulletin of Mathematical Biology, 72 (2010), 1464-1491. doi: 10.1007/s11538-010-9504-9.
[40]	M. Soltani and P. Chen, Numerical modeling of fluid flow in solid tumors, PLoS ONE, (2011). doi: 10.1371/journal.pone.0020344.
[41]	Q. Sun and G. X. Wu, Coupled finite difference and boundary element methods for fluid flow through a vessel with multibranches in tumours, International Journal for Numerical Methods in Biomedical Engineering, 29 (2013), 309-331. doi: 10.1002/cnm.2502.
[42]	C. J. Van Duijn, A. Mikelić, I. S. Pop and C. Rosier, Effective dispersion equations for reactive flows with dominant pclet and damkohler numbers, Advances in Chemical Engineering, 34 (2008), 1-45.
[43]	G. Vilanova, I. Colominas and H. Gomez, Capillary networks in tumor angiogenesis: From discrete endothelial cells to phase-field averaged descriptions via isogeometric analysis, International Journal for Numerical Methods in Biomedical Engineering, 29 (2013), 1015-1037. doi: 10.1002/cnm.2552.
[44]	L. Zhang, A. Gerstenberger, X. Wang and W. K. Liu, Immersed finite element method, Comput. Methods Appl. Mech. Engrg., 193 (2004), 2051-2067. doi: 10.1016/j.cma.2003.12.044.

show all references

References:

[1]	L. T. Baxter and R. K. Jain, Transport of fluid and macromolecules in tumors. i. role of interstitial pressure and convection, Microvascular Research, 37 (1989), 77-104. doi: 10.1016/0026-2862(89)90074-5.
[2]	L. T. Baxter and R. K. Jain, Transport of fluid and macromolecules in tumors ii. role of heterogeneous perfusion and lymphatics, Microvascular Research, 40 (1990), 246-263. doi: 10.1016/0026-2862(90)90023-K.
[3]	L. T. Baxter and R. K. Jain, Transport of fluid and macromolecules in tumors. iii. role of binding and metabolism, Microvascular Research, 41 (1991), 5-23.
[4]	L. T. Baxter and R. K. Jain, Transport of fluid and macromolecules in tumors: Iv. a microscopic model of the perivascular distribution, Microvascular Research, 41 (1991), 252-272. doi: 10.1016/0026-2862(91)90026-8.
[5]	T. R. Blake and J. F. Gross, Analysis of coupled intra- and extraluminal flows for single and multiple capillaries, Mathematical Biosciences, 59 (1982), 173-206. doi: 10.1016/0025-5564(82)90022-0.
[6]	S. Canic, D. Lamponi, A. Mikelić and J. Tambaca, Self-consistent effective equations modeling blood flow in medium-to-large compliant arteries, Multiscale Modeling and Simulation, 3 (2005), 559-596. doi: 10.1137/030602605.
[7]	P. Carmeliet and R. K. Jain, Angiogenesis in cancer and other diseases, Nature, 407 (2000), 249-257.
[8]	S. J. Chapman, R. J. Shipley and R. Jawad, Multiscale modeling of fluid transport in tumors, Bulletin of Mathematical Biology, 70 (2008), 2334-2357. doi: 10.1007/s11538-008-9349-7.
[9]	C. D'Angelo, Multiscale Modeling of Metabolism and Transport Phenomena in Living Tissues, Ph.D thesis, 2007.
[10]	C. D'Angelo, Finite element approximation of elliptic problems with dirac measure terms in weighted spaces: Applications to one- and three-dimensional coupled problems, SIAM Journal on Numerical Analysis, 50 (2012), 194-215. doi: 10.1137/100813853.
[11]	C. D'Angelo and A. Quarteroni, On the coupling of 1D and 3D diffusion-reaction equations. Application to tissue perfusion problems, Math. Models Methods Appl. Sci., 18 (2008), 1481-1504. doi: 10.1142/S0218202508003108.
[12]	A. Farina, A. Fasano and J. Mizerski, A new model for blood flow in fenestrated capillaries with application to ultrafiltration in kidney glomeruli, Submitted.
[13]	D. A. Fedosov, G. E. Karniadakis and B. Caswell, Steady shear rheometry of dissipative particle dynamics models of polymer fluids in reverse poiseuille flow, Journal of Chemical Physics, 132 (2010). doi: 10.1063/1.3366658.
[14]	M. Ferrari, Frontiers in cancer nanomedicine: Directing mass transport through biological barriers, Trends in Biotechnology, 28 (2010), 181-188. doi: 10.1016/j.tibtech.2009.12.007.
[15]	G. J. Fleischman, T. W. Secomb and J. F. Gross, The interaction of extravascular pressure fields and fluid exchange in capillary networks, Mathematical Biosciences, 82 (1986), 141-151. doi: 10.1016/0025-5564(86)90134-3.
[16]	G. J. Flieschman, T. W. Secomb and J. F. Gross, Effect of extravascular pressure gradients on capillary fluid exchange, Mathematical Biosciences, 81 (1986), 145-164. doi: 10.1016/0025-5564(86)90114-8.
[17]	L. Formaggia, D. Lamponi and A. Quarteroni, One-dimensional models for blood flow in arteries, Journal of Engineering Mathematics, 47 (2003), 251-276. doi: 10.1023/B:ENGI.0000007980.01347.29.
[18]	L. Formaggia, A. Quarteroni and A. Veneziani, Multiscale models of the vascular system, in Cardiovascular Mathematics, MS&A. Model. Simul. Appl., 1, Springer Italia, Milan, 2009, 395-446. doi: 10.1007/978-88-470-1152-6_11.
[19]	A. Harris, G. Guidoboni, J. C. Arciero, A. Amireskandari, L. A. Tobe and B. A. Siesky, Ocular hemodynamics and glaucoma: The role of mathematical modeling, European Journal of Ophthalmology, 23 (2013), 139-146. doi: 10.5301/ejo.5000255.
[20]	K. O. Hicks, F. B. Pruijn, T. W. Secomb, M. P. Hay, R. Hsu, J. M. Brown, W. A. Denny, M. W. Dewhirst and W. R. Wilson, Use of three-dimensional tissue cultures to model extravascular transport and predict in vivo activity of hypoxia-targeted anticancer drugs, Journal of the National Cancer Institute, 98 (2006), 1118-1128. doi: 10.1093/jnci/djj306.
[21]	S. S. Hossain, Y. Zhang, X. Liang, F. Hussain, M. Ferrari, T. J. Hughes and P. Decuzzi, In silico vascular modeling for personalized nanoparticle delivery, Nanomedicine, 8 (2013), 343-357.
[22]	M. Intaglietta, N. R. Silverman and W. R. Tompkins, Capillary flow velocity measurements in vivo and in situ by television methods, Microvascular Research, 10 (1975), 165-179. doi: 10.1016/0026-2862(75)90004-7.
[23]	R. K. Jain, Transport of molecules, particles, and cells in solid tumors, Annual Review of Biomedical Engineering, (1999), 241-263. doi: 10.1146/annurev.bioeng.1.1.241.
[24]	R. K. Jain, R. T. Tong and L. L. Munn, Effect of vascular normalization by antiangiogenic therapy on interstitial hypertension, peritumor edema, and lymphatic metastasis: Insights from a mathematical model, Cancer Research, 67 (2007), 2729-2735. doi: 10.1158/0008-5472.CAN-06-4102.
[25]	J. Lee and T. C. Skalak, Microvascular Mechanics: Hemodynamics of Systemic and Pulmonary Microcirculation, Springer-Verlag, 1989.
[26]	H. Lei, D. A. Fedosov, B. Caswell and G. E. Karniadakis, Blood flow in small tubes: Quantifying the transition to the non-continuum regime, Journal of Fluid Mechanics, 722 (2013), 214-239. doi: 10.1017/jfm.2013.91.
[27]	J. R. Less, T. C. Skalak, E. M. Sevick and R. K. Jain, Microvascular architecture in a mammary carcinoma: Branching patterns and vessel dimensions, Cancer Research, 51 (1991), 265-273.
[28]	W. K. Liu, Y. Liu, D. Farrell, L. Zhang, X. S. Wang, Y. Fukui, N. Patankar, Y. Zhang, C. Bajaj, J. Lee, J. Hong, X. Chen and H. Hsu, Immersed finite element method and its applications to biological systems, Comput. Methods Appl. Mech. Engrg., 195 (2006), 1722-1749. doi: 10.1016/j.cma.2005.05.049.
[29]	Y. Liu and W. K. Liu, Rheology of red blood cell aggregation by computer simulation, Journal of Computational Physics, 220 (2006), 139-154. doi: 10.1016/j.jcp.2006.05.010.
[30]	Y. Liu, L. Zhang, X. Wang and W. K. Liu, Coupling of navier-stokes equations with protein molecular dynamics and its application to hemodynamics, International Journal for Numerical Methods in Fluids, 46 (2004), 1237-1252. doi: 10.1002/fld.798.
[31]	J. Peiró and A. Veneziani, Reduced models of the cardiovascular system, in Cardiovascular Mathematics, MS&A. Model. Simul. Appl., 1, Springer Italia, Milan, 2009, 347-394. doi: 10.1007/978-88-470-1152-6_10.
[32]	Y. Renard and J. Pommier, Getfem++: A generic finite element library in c++, version 4.2 (2012), http://download.gna.org/getfem/html/homepage/.
[33]	A. M. Robertson and A. Sequeira, A director theory approach for modeling blood flow in the arterial system: An alternative to classical id models, Mathematical Models and Methods in Applied Sciences, 15 (2005), 871-906. doi: 10.1142/S0218202505000601.
[34]	A. M. Robertson, A. Sequeira and R. G. Owens, Rheological models for blood. In Cardiovascular Mathematics, MS&A. Model. Simul. Appl., 1, Springer Italia, Milan, 2009, 211-241. doi: 10.1007/978-88-470-1152-6_6.
[35]	T. W. Secomb, A. R. Pries, P. Gaehtgens and J. F. Gross, Theoretical and experimental analysis of hematocrit distribution in microcirculatory networks, in Microvascular Mechanics (eds. J.-S. Lee and T. C. Skalak), Springer, New York, 1989, 39-49. doi: 10.1007/978-1-4612-3674-0_4.
[36]	T. W. Secomb, Microvascular Network Structures, http://www.physiology.arizona.edu/people/secomb.
[37]	T. W. Secomb, R. Hsu, R. D. Braun, J. R. Ross, J. F. Gross and M. W. Dewhirst, Theoretical simulation of oxygen transport to tumors by three-dimensional networks of microvessels, Advances in Experimental Medicine and Biology, 454 (1998), 629-634. doi: 10.1007/978-1-4615-4863-8_74.
[38]	T. W. Secomb, R. Hsu, E. Y. H. Park and M. W. Dewhirst, Green's function methods for analysis of oxygen delivery to tissue by microvascular networks, Annals of Biomedical Engineering, 32 (2004), 1519-1529. doi: 10.1114/B:ABME.0000049036.08817.44.
[39]	R. J. Shipley and S. J. Chapman, Multiscale modelling of fluid and drug transport in vascular tumours, Bulletin of Mathematical Biology, 72 (2010), 1464-1491. doi: 10.1007/s11538-010-9504-9.
[40]	M. Soltani and P. Chen, Numerical modeling of fluid flow in solid tumors, PLoS ONE, (2011). doi: 10.1371/journal.pone.0020344.
[41]	Q. Sun and G. X. Wu, Coupled finite difference and boundary element methods for fluid flow through a vessel with multibranches in tumours, International Journal for Numerical Methods in Biomedical Engineering, 29 (2013), 309-331. doi: 10.1002/cnm.2502.
[42]	C. J. Van Duijn, A. Mikelić, I. S. Pop and C. Rosier, Effective dispersion equations for reactive flows with dominant pclet and damkohler numbers, Advances in Chemical Engineering, 34 (2008), 1-45.
[43]	G. Vilanova, I. Colominas and H. Gomez, Capillary networks in tumor angiogenesis: From discrete endothelial cells to phase-field averaged descriptions via isogeometric analysis, International Journal for Numerical Methods in Biomedical Engineering, 29 (2013), 1015-1037. doi: 10.1002/cnm.2552.
[44]	L. Zhang, A. Gerstenberger, X. Wang and W. K. Liu, Immersed finite element method, Comput. Methods Appl. Mech. Engrg., 193 (2004), 2051-2067. doi: 10.1016/j.cma.2003.12.044.

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