Green’s function method for simulation of oxygen and drug transport to tissue
The Green’s function method, for simulating oxygen transport from a network of
vessels to a finite volume of tissue, has been modified to include diffusion of a drug,
which is irreversibly metabolized under hypoxic conditions. A numerical method
is used in which vessels are treated as distributions of oxygen and drug sources
and the tissue is represented as a distribution of oxygen and drug sinks. described
in:
- 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. Journal of the National Cancer Institute, 2006, 98(16), 1118-1128. | Full Text |
This version of the FORTRAN implementation of the drug diffusion model uses the
'no flux -at the boundaries' method, and hence is not as up to date as the
latest oxygen transport model.
An '‘infinite-domain solution', in which the network of vessels and the associated
oxygen and drug-consuming tissue domain are effectively embedded in an infinite
domain without other oxygen sources or sinks, is under development.
All necessary files, including FORTRAN source code and the data file for the vascular
network, are included in a zip file:
Read a description of the
Oxygen transport model
Data files with example oxygen and drug parameters are also included in the root
directory. The 'instructions' folder contains a readme file with step-by-step instructions
and a MS excel file for generating simple graphs of the output. The 'output' folder
contains data files generated from the example in the readme document.
Routines for visualizing the results using Mathematica™ are also provided in the
instructions folder. We have tested this package on a personal computer using Windows
XP and Digital Visual Fortran v6.0, Compaq visual FORTRAN v6.6 and Lahey Fortran
95. For error reporting and suggestions please contact:
This program is freely available for non-commercial use, provided appropriate acknowledgement
is given. Commercial users please contact Dr. Timothy W. Secomb, before using
this program. No assurance is given that it is free of errors and any use is at
the user’s risk. Results may differ slightly depending on choices made in running
the programs.
Notes
The tumor microvascular network was mapped as originally described in Secomb et
al., 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. The network of vessels is approximated by a
set of straight uniform segments located inside a reference cuboidal (box) shape.
This cuboid must include all oxygen-consuming regions associated with the given
network. Each segment is then divided into sub-segments with equal length. The midpoint
of each sub-segment represents a source. The product of the source strength and
the sub-segment's length equals the oxygen (or drug) efflux from this sub-segment.
The number of source points on each vessel should be roughly proportional to its
length, with a minimum of 2.
Oxygen levels are expressed in terms of partial pressure of oxygen (PO2). The rate
of oxygen consumption in tissue is assumed to depend on tissue PO2 with Michaelis-Menten
kinetics:
M = Mmax * PO2/( PO2 + Pcr ) where Pcr is the Michaelis constant (typically 1 mmHg).
The tissue domain is divided into small subregions, each centered on a tissue node
point. The tissue node points form a three dimensional matrix. Inside each subregion,
the consumption rate is assumed to be a constant, and equals the value at the tissue
node point, which depends on the PO2 at the nodal point according to the above equation.
Drug concentrations are expressed in terms of micromolar (C, μM). The rate of drug
metabolism in tissue depends on both the drug concentration and the oxygen PO2,
V = f(O2) * Vmax * C/(Km + C), where Km is the Michaelis constant and f(O2) = KO2/(KO2
+ PO2) and KO2 is the oxygen concentration at which the rate of drug metabolism
is halved relative to its maximum value (under anoxia). If another model for drug
metabolism has been used, such as first order (non-saturable) metabolism, then this
must be approximated by the Michaelis-Menten equation, for example by using a vary
large Km as was done in the Hicks JNCI paper.
Input files supplied by user:
- network.dat -- gives network geometry, vessel diameter, flow rate relative to a
reference value, discharge hematocrit. Do not modify this file.
- flow.dat -- is generated during program execution but must then be modified by the
user to include inflow PO2 and drug concentration.
- para.dat -- gives values of blood-oxygen related parameters:
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- dalfa -- the product of oxygen diffusivity and solubility in tissue.
- P50 -- PO2 at which 50% saturation occurs in the Hill's equation.
- n -- exponent in the Hill's equation.
- C0 -- oxygen binding capacity of blood.
- alfab -- effective oxygen solubility in blood.
- Pcr -- Michaelis constant in oxygen consumption rate.
- L0 -- length scale (cm), used to normalize all geometric lengths.
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This file also has a table of intravascular resistance to material transport as
a function of diameter. This file is supplied by the user, but should be formatted
as in the sample, which is based on values provided by Hellums et al. (1996).
- drug.dat - with the following drug parameters (example values given).
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- L0 -- length scale (cm), used to normalize all geometric lengths.
- D -- diffusivity of drug in
cm^2/s Vmax -- Michaelis constant for drug consumption in micro M.
- KO2 -- constant
in oxygen dependence of drug consumption, the oxygen concentration at which the
rate of drug metabolism is reduced to half its value under anoxia; in mm Hg.
- alpha,
beta, lambda, infusion time -- to determine amount of metabolism and cell kill.
Note:
α, β and λ are the coefficients of the survival model
-log SF = α λ M + β (λM)2
where M is the cumulative metabolised drug concentration and λ is the fraction of
the metabolism that is responsible for cell kill.
*At present the program only calculates
cell kill for the M (β = 0) and M2 (α = 0) and linear-quadratic (α ≠ 0, β ≠ 0) models.
The C × M model must be calculated manually (for example in the spreadsheet below).
These files are supplied in the download. Do not modify network.dat. Example parameter
values are included in para.dat and drug.dat. These can be changed as appropriate.
[Do not confuse network.dat and para.dat with the files for the oxygen-only model,
as the format has changed in later versions of the oxygen program].
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