In the Green’s function method for simulating solute transport from a network of vessels to a finite volume of tissue, a numerical method is used in which vessels and tissue are represented as distributions of sources and sinks.
Further developments are available on GitHub. Please go to https://github.com/secomb for updated versions, GreensV4 and GreensV4_GPU. June 15, 2019.
We have developed implementations of this method in FORTRAN and C. This version gives the ‘infinite-domain solution’, in which the network of vessels and the associated tissue domain are effectively embedded in an infinite domain without other sources or sinks. The tissue domain is defined by the region of solute consuming or producing tissue, and no explicit boundary condition is applied on the boundaries of this domain. Such an approach has two advantages: it is applicable to tissue domains of arbitrary shape, and it avoids artifacts that can occur when specific boundary conditions, such as the no-flux condition, are imposed.
The method is described in the following publication: Secomb, T.W., Hsu, R., Park, E.Y.H. and Dewhirst, M.W. Green's function methods for analysis of oxygen delivery to tissue by microvascular networks. Annals of Biomedical Engineering, 32: 1519-1529 (2004).
A version of the method has been developed for simulating time-dependent solute transport and reaction.
Secomb, T.W. A Green's function method for simulation of time-dependent solute transport and reaction in realistic microvascular geometries. Mathematical Medicine and Biology 2015; doi: 10.1093/imammb/dqv031;
Our publications showing simulations of oxygen transport by networks of microvessels include:
Hsu, R. and Secomb, T.W. A Green's function method for analysis of oxygen delivery to tissue by microvascular networks. Math. Biosciences. 96: 61-78 (1989).
Hsu, R. and Secomb, T.W. Analysis of oxygen exchange between arterioles and surrounding capillary-perfused tissue. J. Biomech. Eng. 114: 227-231 (1992).
Secomb, T.W., Hsu, R., Dewhirst, M.W., Klitzman, B. and Gross, J.F. Analysis of oxygen transport to tumor tissue by microvascular networks. Int. J. Rad. Onc. Biol. Phys. 25: 481-489 (1993).
Secomb, T.W. and Hsu, R. Simulation of oxygen transport in skeletal muscle: diffusive exchange between arterioles and capillaries. Am. J. Physiol. 267, H1214-1221 (1994).
Secomb, T.W., Hsu, R., Ong, E.T., Gross, J.F. and Dewhirst, M.W. Analysis of the effects of oxygen supply and demand on hypoxic fraction in tumors. Acta Oncologica 34, 313-316 (1995).
Secomb, T.W., Hsu, R., Braun, R.D., Ross, J.R., Gross, J.F. and Dewhirst, M.W. Theoretical simulation of oxygen transport to tumors by three-dimensional networks of microvessels. In "Oxygen Transport to Tissue XX," ed. A.G. Hudetz and D.F. Bruley. Plenum, New York, 1998, pp. 629-634.
Secomb, T.W., Hsu, R., Beamer, N.B. and Coull, B.M. Theoretical simulation of oxygen transport to brain by networks of microvessels: effects of oxygen supply and demand on tissue hypoxia. Microcirculation 7, 237-247 (2000).
Kavanagh, B.D., Secomb, T.W., Hsu, R., Lin, P.-S., Venitz, J. and Dewhirst, M.W. A theoretical model for the effects of reduced hemoglobin-oxygen affinity on tumor oxygenation. Int. J. Rad. Onc. Biol. Phys. 53, 172-179 (2002).
Secomb, T.W., Hsu, R. and Dewhirst, M.W. Synergistic effects of hyperoxic gas breathing and reduced oxygen consumption on tumor oxygenation: A theoretical model. Int. J. Rad. Onc. Biol. Phys., 59: 572-578 (2004).
Secomb, T.W., Hsu, R., Park, E.Y.H. and Dewhirst, M.W. Green's function methods for analysis of oxygen delivery to tissue by microvascular networks. Annals of Biomedical Engineering, 32: 1519-1529 (2004).
Applications to drug transport include:
Hicks, K.O, Pruijn, F.B., Secomb, T.W., Hay, M.P., Hsu, R., Brown, J.M., Denny, W.A., Dewhirst, M.W., Wilson, W.R. Use of three-dimensional tissue cultures to model extravascular transport and predict in vivo activity of hypoxia targeted anticancer drugs. J. Nat. Cancer Inst., 98: 1118-1128 (2006). See editorial: Sausville, E.A. Respecting cancer drug transportability: A basis for successful lead selection. J. Nat. Cancer Inst., 98: 1098-1099 (2006).
Foehrenbacher, A., Patel, K., Abbattista, M., Guise, C.P., Secomb, T.W., Wilson, W.R., Hicks, K.O. The role of bystander effects in the antitumor activity of the hypoxia-activated prodrug PR-104. Front. Oncol. 3:263. doi: 10.3389/fonc.2013.00263 (2013) (18 pages). PMC3791487.
Foehrenbacher, A., Secomb, T.W., Wilson, W.R. and Hicks, K.O. Design of optimized hypoxia-activated prodrugs using pharmacokinetic/pharmacodynamic modeling. Front. Oncol. 3:314. doi: 10.3389/fonc.2013.00314 (2013) (17 pages). PMC3873531.
Applications to angiogenesis modeling include:
Secomb, T.W., Alberding, J.P., Hsu, R., Dewhirst, M.W. and Pries, A.R. Angiogenesis: an adaptive biological patterning problem. PLoS Computational Biology 9:e1002983. doi:10.1371/journal.pcbi.1002983 (2013) (12 pages). PMC3605064.
Pries, A.R. and Secomb, T.W. Making microvascular networks work: angiogenesis, remodeling and pruning. Physiology 29: 446-455 (2014). PMC4280154.
Updated September 20, 2017