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

Computational models for fluid exchange between microcirculation and tissue interstitium

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

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.
Citation: Laura Cattaneo, Paolo Zunino. Computational models for fluid exchange between microcirculation and tissue interstitium. Networks & 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. doi: 10.1016/0026-2862(89)90074-5. Google Scholar

[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. doi: 10.1016/0026-2862(90)90023-K. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/0026-2862(91)90026-8. Google Scholar

[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. doi: 10.1016/0025-5564(82)90022-0. Google Scholar

[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. doi: 10.1137/030602605. Google Scholar

[7]

P. Carmeliet and R. K. Jain, Angiogenesis in cancer and other diseases,, Nature, 407 (2000), 249. Google Scholar

[8]

S. J. Chapman, R. J. Shipley and R. Jawad, Multiscale modeling of fluid transport in tumors,, Bulletin of Mathematical Biology, 70 (2008), 2334. doi: 10.1007/s11538-008-9349-7. Google Scholar

[9]

C. D'Angelo, Multiscale Modeling of Metabolism and Transport Phenomena in Living Tissues,, Ph.D thesis, (2007). Google Scholar

[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. doi: 10.1137/100813853. Google Scholar

[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. doi: 10.1142/S0218202508003108. Google Scholar

[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., (). Google Scholar

[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. Google Scholar

[14]

M. Ferrari, Frontiers in cancer nanomedicine: Directing mass transport through biological barriers,, Trends in Biotechnology, 28 (2010), 181. doi: 10.1016/j.tibtech.2009.12.007. Google Scholar

[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. doi: 10.1016/0025-5564(86)90134-3. Google Scholar

[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. doi: 10.1016/0025-5564(86)90114-8. Google Scholar

[17]

L. Formaggia, D. Lamponi and A. Quarteroni, One-dimensional models for blood flow in arteries,, Journal of Engineering Mathematics, 47 (2003), 251. doi: 10.1023/B:ENGI.0000007980.01347.29. Google Scholar

[18]

L. Formaggia, A. Quarteroni and A. Veneziani, Multiscale models of the vascular system,, in Cardiovascular Mathematics, (2009), 395. doi: 10.1007/978-88-470-1152-6_11. Google Scholar

[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. doi: 10.5301/ejo.5000255. Google Scholar

[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. doi: 10.1093/jnci/djj306. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/0026-2862(75)90004-7. Google Scholar

[23]

R. K. Jain, Transport of molecules, particles, and cells in solid tumors,, Annual Review of Biomedical Engineering, (1999), 241. doi: 10.1146/annurev.bioeng.1.1.241. Google Scholar

[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. doi: 10.1158/0008-5472.CAN-06-4102. Google Scholar

[25]

J. Lee and T. C. Skalak, Microvascular Mechanics: Hemodynamics of Systemic and Pulmonary Microcirculation,, Springer-Verlag, (1989). Google Scholar

[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. doi: 10.1017/jfm.2013.91. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/j.cma.2005.05.049. Google Scholar

[29]

Y. Liu and W. K. Liu, Rheology of red blood cell aggregation by computer simulation,, Journal of Computational Physics, 220 (2006), 139. doi: 10.1016/j.jcp.2006.05.010. Google Scholar

[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. doi: 10.1002/fld.798. Google Scholar

[31]

J. Peiró and A. Veneziani, Reduced models of the cardiovascular system,, in Cardiovascular Mathematics, (2009), 347. doi: 10.1007/978-88-470-1152-6_10. Google Scholar

[32]

Y. Renard and J. Pommier, Getfem++: A generic finite element library in c++, version 4.2 (2012),, , (). Google Scholar

[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. doi: 10.1142/S0218202505000601. Google Scholar

[34]

A. M. Robertson, A. Sequeira and R. G. Owens, Rheological models for blood., In Cardiovascular Mathematics, (2009), 211. doi: 10.1007/978-88-470-1152-6_6. Google Scholar

[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), (1989), 39. doi: 10.1007/978-1-4612-3674-0_4. Google Scholar

[36]

T. W. Secomb, Microvascular Network Structures,, , (). Google Scholar

[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. doi: 10.1007/978-1-4615-4863-8_74. Google Scholar

[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. doi: 10.1114/B:ABME.0000049036.08817.44. Google Scholar

[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. doi: 10.1007/s11538-010-9504-9. Google Scholar

[40]

M. Soltani and P. Chen, Numerical modeling of fluid flow in solid tumors,, PLoS ONE, (2011). doi: 10.1371/journal.pone.0020344. Google Scholar

[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. doi: 10.1002/cnm.2502. Google Scholar

[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. Google Scholar

[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. doi: 10.1002/cnm.2552. Google Scholar

[44]

L. Zhang, A. Gerstenberger, X. Wang and W. K. Liu, Immersed finite element method,, Comput. Methods Appl. Mech. Engrg., 193 (2004), 2051. doi: 10.1016/j.cma.2003.12.044. Google Scholar

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. doi: 10.1016/0026-2862(89)90074-5. Google Scholar

[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. doi: 10.1016/0026-2862(90)90023-K. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/0026-2862(91)90026-8. Google Scholar

[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. doi: 10.1016/0025-5564(82)90022-0. Google Scholar

[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. doi: 10.1137/030602605. Google Scholar

[7]

P. Carmeliet and R. K. Jain, Angiogenesis in cancer and other diseases,, Nature, 407 (2000), 249. Google Scholar

[8]

S. J. Chapman, R. J. Shipley and R. Jawad, Multiscale modeling of fluid transport in tumors,, Bulletin of Mathematical Biology, 70 (2008), 2334. doi: 10.1007/s11538-008-9349-7. Google Scholar

[9]

C. D'Angelo, Multiscale Modeling of Metabolism and Transport Phenomena in Living Tissues,, Ph.D thesis, (2007). Google Scholar

[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. doi: 10.1137/100813853. Google Scholar

[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. doi: 10.1142/S0218202508003108. Google Scholar

[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., (). Google Scholar

[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. Google Scholar

[14]

M. Ferrari, Frontiers in cancer nanomedicine: Directing mass transport through biological barriers,, Trends in Biotechnology, 28 (2010), 181. doi: 10.1016/j.tibtech.2009.12.007. Google Scholar

[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. doi: 10.1016/0025-5564(86)90134-3. Google Scholar

[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. doi: 10.1016/0025-5564(86)90114-8. Google Scholar

[17]

L. Formaggia, D. Lamponi and A. Quarteroni, One-dimensional models for blood flow in arteries,, Journal of Engineering Mathematics, 47 (2003), 251. doi: 10.1023/B:ENGI.0000007980.01347.29. Google Scholar

[18]

L. Formaggia, A. Quarteroni and A. Veneziani, Multiscale models of the vascular system,, in Cardiovascular Mathematics, (2009), 395. doi: 10.1007/978-88-470-1152-6_11. Google Scholar

[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. doi: 10.5301/ejo.5000255. Google Scholar

[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. doi: 10.1093/jnci/djj306. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/0026-2862(75)90004-7. Google Scholar

[23]

R. K. Jain, Transport of molecules, particles, and cells in solid tumors,, Annual Review of Biomedical Engineering, (1999), 241. doi: 10.1146/annurev.bioeng.1.1.241. Google Scholar

[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. doi: 10.1158/0008-5472.CAN-06-4102. Google Scholar

[25]

J. Lee and T. C. Skalak, Microvascular Mechanics: Hemodynamics of Systemic and Pulmonary Microcirculation,, Springer-Verlag, (1989). Google Scholar

[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. doi: 10.1017/jfm.2013.91. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/j.cma.2005.05.049. Google Scholar

[29]

Y. Liu and W. K. Liu, Rheology of red blood cell aggregation by computer simulation,, Journal of Computational Physics, 220 (2006), 139. doi: 10.1016/j.jcp.2006.05.010. Google Scholar

[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. doi: 10.1002/fld.798. Google Scholar

[31]

J. Peiró and A. Veneziani, Reduced models of the cardiovascular system,, in Cardiovascular Mathematics, (2009), 347. doi: 10.1007/978-88-470-1152-6_10. Google Scholar

[32]

Y. Renard and J. Pommier, Getfem++: A generic finite element library in c++, version 4.2 (2012),, , (). Google Scholar

[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. doi: 10.1142/S0218202505000601. Google Scholar

[34]

A. M. Robertson, A. Sequeira and R. G. Owens, Rheological models for blood., In Cardiovascular Mathematics, (2009), 211. doi: 10.1007/978-88-470-1152-6_6. Google Scholar

[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), (1989), 39. doi: 10.1007/978-1-4612-3674-0_4. Google Scholar

[36]

T. W. Secomb, Microvascular Network Structures,, , (). Google Scholar

[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. doi: 10.1007/978-1-4615-4863-8_74. Google Scholar

[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. doi: 10.1114/B:ABME.0000049036.08817.44. Google Scholar

[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. doi: 10.1007/s11538-010-9504-9. Google Scholar

[40]

M. Soltani and P. Chen, Numerical modeling of fluid flow in solid tumors,, PLoS ONE, (2011). doi: 10.1371/journal.pone.0020344. Google Scholar

[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. doi: 10.1002/cnm.2502. Google Scholar

[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. Google Scholar

[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. doi: 10.1002/cnm.2552. Google Scholar

[44]

L. Zhang, A. Gerstenberger, X. Wang and W. K. Liu, Immersed finite element method,, Comput. Methods Appl. Mech. Engrg., 193 (2004), 2051. doi: 10.1016/j.cma.2003.12.044. Google Scholar

[1]

Qiang Du, Manlin Li. On the stochastic immersed boundary method with an implicit interface formulation. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 373-389. doi: 10.3934/dcdsb.2011.15.373

[2]

Harvey A. R. Williams, Lisa J. Fauci, Donald P. Gaver III. Evaluation of interfacial fluid dynamical stresses using the immersed boundary method. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 519-540. doi: 10.3934/dcdsb.2009.11.519

[3]

Sheng Xu. Derivation of principal jump conditions for the immersed interface method in two-fluid flow simulation. Conference Publications, 2009, 2009 (Special) : 838-845. doi: 10.3934/proc.2009.2009.838

[4]

Robert H. Dillon, Jingxuan Zhuo. Using the immersed boundary method to model complex fluids-structure interaction in sperm motility. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 343-355. doi: 10.3934/dcdsb.2011.15.343

[5]

Daniele Boffi, Lucia Gastaldi. Discrete models for fluid-structure interactions: The finite element Immersed Boundary Method. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 89-107. doi: 10.3934/dcdss.2016.9.89

[6]

Champike Attanayake, So-Hsiang Chou. An immersed interface method for Pennes bioheat transfer equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 323-337. doi: 10.3934/dcdsb.2015.20.323

[7]

Jian Hao, Zhilin Li, Sharon R. Lubkin. An augmented immersed interface method for moving structures with mass. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1175-1184. doi: 10.3934/dcdsb.2012.17.1175

[8]

Junde Wu. Bifurcation for a free boundary problem modeling the growth of necrotic multilayered tumors. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3399-3411. doi: 10.3934/dcds.2019140

[9]

Kai Koike. Wall effect on the motion of a rigid body immersed in a free molecular flow. Kinetic & Related Models, 2018, 11 (3) : 441-467. doi: 10.3934/krm.2018020

[10]

So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343

[11]

Hyeuknam Kwon, Yoon Mo Jung, Jaeseok Park, Jin Keun Seo. A new computer-aided method for detecting brain metastases on contrast-enhanced MR images. Inverse Problems & Imaging, 2014, 8 (2) : 491-505. doi: 10.3934/ipi.2014.8.491

[12]

X.H. Wu, Y. Efendiev, Thomas Y. Hou. Analysis of upscaling absolute permeability. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 185-204. doi: 10.3934/dcdsb.2002.2.185

[13]

Hassib Selmi, Lassaad Elasmi, Giovanni Ghigliotti, Chaouqi Misbah. Boundary integral and fast multipole method for two dimensional vesicle sets in Poiseuille flow. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 1065-1076. doi: 10.3934/dcdsb.2011.15.1065

[14]

Fujun Zhou, Junde Wu, Shangbin Cui. Existence and asymptotic behavior of solutions to a moving boundary problem modeling the growth of multi-layer tumors. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1669-1688. doi: 10.3934/cpaa.2009.8.1669

[15]

Joachim Escher, Anca-Voichita Matioc. Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 573-596. doi: 10.3934/dcdsb.2011.15.573

[16]

Junde Wu, Shangbin Cui. Asymptotic behavior of solutions of a free boundary problem modelling the growth of tumors with Stokes equations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 625-651. doi: 10.3934/dcds.2009.24.625

[17]

Luis Alvarez, Jesús Ildefonso Díaz. On the retention of the interfaces in some elliptic and parabolic nonlinear problems. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 1-17. doi: 10.3934/dcds.2009.25.1

[18]

T. Tachim Medjo. On the Newton method in robust control of fluid flow. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1201-1222. doi: 10.3934/dcds.2003.9.1201

[19]

Xiaojiao Tong, Felix F. Wu, Yongping Zhang, Zheng Yan, Yixin Ni. A semismooth Newton method for solving optimal power flow. Journal of Industrial & Management Optimization, 2007, 3 (3) : 553-567. doi: 10.3934/jimo.2007.3.553

[20]

Elena Kosygina. Brownian flow on a finite interval with jump boundary conditions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 867-880. doi: 10.3934/dcdsb.2006.6.867

2018 Impact Factor: 0.871

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]