Input function is a lagged normal density curve of area=1,
tMean = 5 s, RD=0.3, skewn=1.3, frPeak=1e-6, Up
Loops and sensitivity set up re PSg.
BTEX30MM2 is adapted from BTEX30 only by changing the PSpc.
The source code values for PSpcmax and kmpc are set to give
equivalent results to PSpc = 1 in the BTEX30 model.
Par1 is default parameter set, as in source code.
Par2 is available for explorations.
To match the BTEX30.cdata at computed by the BTEX30 model
with intervals at 1.0sec, and PSg= 1 ml/(g*min) and other
parameters the default values, and with Kmpc = Kmisf and
with the same input function giving Cisf in the range of
0.01 to 0.1, and Cpc below 0.01 mmole/ml, the following
PSpcmax / Kmisf pairs gave good matches:
PSpcmax: 8.65e3, 114.6, 5.11, 1.35, 1.03, 0.995, 0.992
Kmisf 1e-4, 1e-3, 1.e-2, 1e-1, 1e-0, 1e+1, 1e+2
Run SENSITIVITY (beside Loops) for Sensitivity plot (Sensit)
SUMMARY of BEHAVIOR:
When the ambient concentrations are well below the Km, and
when the Kmisf = Kmpc, the system behaves as if the conductances
are first order with value PSpcmax, as demonstrated in the above
table where the fit to the passive transport model with PSpc = 1
is obtained with ratios of 1 and high values of Km.
At low values of Km, the effective PS's are reduced, and in
order to fit the reference "data" the values of PSpcmax have to
be increased. This occurs with Km's of 0.1 or lower because the
ambient concentrations in ISF and PC partially saturate the
tranpsorter. Note that using a Km of 0.001 mmol/ml, which is
1/100 of the ambient concentrations, the PSpcmax must be raised
to 1.15e5 times the Km to give the same total flux of solute into
the PC and maintain the magnitude of the tail of the Q, the total
retained solute.
PSpcmax
-------- PSpcmax*Km
PS = C = ----------
1 + ---- Km + C
Km
In comparison with BTEX30MM1, where PSpc is the same in each direction,but the value is governed by Cisf, here the PSpc2isf is governed by Cpc
and PSisf2pc is governed by Cisf, this model requires higher values of
PSpcmax to obtain the same Q at 30 seconds as the parent passive BTEX30
model, e.g. 114.6 ml/(g*min) at Km = 0.001 mmol/g instead of 28 for MM1
model.
/*
MODEL NUMBER: 0020
MODEL NAME: TranspMM2sided_Distrib3f.2Ch
SHORT DESCRIPTION: Capillary-ISF-cell convection diffusion
model, Modified BTEX30 with a Michaelis Menten saturable
transporter on the cell membrane represented by two separate
independent unidirecrtional transporters. Thus 3 region, 2
1-sided transporters of MM type.
*/
import nsrunit; unit conversion on;
math TranspMM_2sided_Distrib3F_2Ch {
//INDEPENDENT DOMAINS, time t and axial position x
realDomain t sec ; t.min=0; t.max=30; t.delta=0.2;
realDomain x cm; real Lcap =0.1 cm, Ngrid=31; x.min=0; x.max=Lcap; x.ct=Ngrid;
private x.min, x.max, x.ct; //set private so it does no appear in the param list
/* PARAMETERS:
Subscripts p = PLASMA, isf= INTERSTITIAL FLUID REGION, pc = parenchymal cell */
real Fp = 1 ml/(g*min), // Plasma flow
Vp = 0.05 ml/g, // Plasma volume
Visf = 0.15 ml/g, // ISF volume of distribution
Vpc = 0.58 ml/g; // PC volume of distribution
// the hVolumes protect against zero divides
private real hVp =if(Vp>0) Vp else (1e-6 ml/g);
private real hVisf =if(Visf>0) Visf else (1e-6 ml/g);
private real hVpc =if(Vpc>0) Vpc else (1e-6 ml/g);
real PSg = 1 ml/(g*min); // Permeability-surface area product between
// plasma and interstitial fluid region (ISF)
real PSisf2pc(t,x) ml/(g*min); // PSisf2pc = PSpcmax/(1 + Cisf/Kmisf) at each x and t
// Note that this depends solely on ISF concn
real PSpc2isf(t,x) ml/(g*min); // PSpc2isf = PSpcmax/(1 + Cpc/Kmpc) at each x and t
// Note that this depends solely on pc concn
real PSpcmax = 1 ml/(g*min);// Max value for PSpc
real Kmpc = 0.05 mmol/ml;// Use of Km is two-sided; both ISF and PC
real Kmisf = 0.05 mmol/ml;// Use of Km is two-sided; both ISF and PC
real Gp = 0 ml/(g*min); // Plasma consumption rate for metabolite
real Gisf = 0 ml/(g*min); // ISF consumption rate for metabolite
real Gpc = 1 ml/(g*min); // PC consumption rate for metabolite
real Dp = 1e-04 cm^2/sec;// Plasma axial diffusion coefficient
real Disf = 0.0 cm^2/sec; // ISF axial diffusion coefficient
real Dpc = 0.0 cm^2/sec; // PC axial diffusion coefficient
real Cp0 = 0 mmol/ml, Cisf0 = 0 mmol/ml, Cpc0 = 0 mmol/ml; //Init.concns in regions
// TIME DEPENDENT VARIABLES:
extern real Cin(t) mmol/ml; //INPUT FUNCTION
real Cp(t,x) mmol/ml, Cisf(t,x) mmol/ml, Cpc(t,x) mmol/ml, Cout(t) mmol/ml;
// Boundary Conditions:
when (x=x.min) { Cisf:x = 0; Cpc:x = 0;}
when (x=x.max) { Cisf:x = 0; Cpc:x = 0;}
when (x=x.max) { Dp*Cp:x = 0; Cout = Cp; } // Right Hand total flux
when (x=x.min) { (-Fp*Lcap/hVp)*(Cp-Cin)+Dp*Cp:x =0;} // Left Hand Total flux BC.
// Initial.Conditions:
when (t=t.min) { Cp = if (x=x.min) Cin else Cp0; Cisf = Cisf0; Cpc = Cpc0;}
// Partial differential equations for the three concentric regions
// Here we use PSg bidirectionally but PSisf2pc and PSpc2isf as independent 1.sided
PSisf2pc = PSpcmax/(1 + Cisf/Kmisf);
PSpc2isf = PSpcmax/(1 + Cpc/Kmpc);
hVp *Cp:t = -Fp*Lcap*Cp:x -Gp*Cp + hVp*Dp*Cp:x:x - PSg*(Cp-Cisf) ;
hVisf*Cisf:t = -Gisf*Cisf+ hVisf*Disf*Cisf:x:x + PSg*(Cp-Cisf) + PSpc2isf*Cpc - PSisf2pc*Cisf;
hVpc *Cpc:t = -Gpc*Cpc + hVpc*Dpc*Cpc:x:x + PSisf2pc*Cisf - PSpc2isf*Cpc;
real Q(t) mmol/g; // quantity within the BTEX unit
when(t=t.min) {Q = Cp0*hVp+Cisf0*hVisf +Cpc0*hVpc;}
Q:t = Fp*(Cin-Cout);
}
/*
Fp ________________________________________
Cin(t) ---> |Vp Cp(t)|---> Cout(t)
|Gp ^ |
|Dp | PLASMA|
___________PSg_________________________|
|Visfp | Cisf(t)|
|Gisf V ^ INTERSTITIAL|
|Disf | | FLUID REGION|
_________PSpc2isf______PSisf2pc_________
|Vpcp | | Cpc(t)|
|Gpc V PARENCHYMAL|
|Dpc CELL|
________________________________________
|<----------------L------------------->|
|--> x
Fp : Plasma Flow Rate, (ml/g)/min
Vp : Plasma Volume, ml/g
Visfp, Vpcp: Volumes of Distribution, ml/g
PSg, Pspc: Permeability-surface area product exchange
coefficients, (ml/g)/min
Gp, Gisf, Gpc: Consumption rates for metabolite, (ml/g)/min
Dp, Disf, Dpc: Axial Diffusion Rate, cm^2/sec
Cin: Plasma metabolite inflow, mmol/ml
Cout: Plasma metabolite outflow, mmol/ml
Cp, Cisf, Cpc: metabolite concentration, mmol/ml
DETAILED DESCRIPTION:
This model is almost the same as TranspMM.2sided.Distrib2F.proj
but differs in the form of the transporter. In this program the
two opposite fluxes are independent. In the .Distrib2F.proj
model the conductances are identical for the two directions.
BTEX stands for blood-tissue exchange, 3 denotes 3 regions,
0 indicates a basic model. The 3 regions are the convecting
plasma, p, and stagnant ISF and parenchymal cell, pc.
Axial gradients exist in all three regions. Each region is
considered radially uniform on the basis that radial diffusion
distances are so short that diffusional relaxation times are at
most a few milliseconds and can be considered instantaneous.
The endothelial cell is considered as a passive barrier without
capacitance and has permeability-surface area product, PSg, where
the subscript g indicates the interendothelial gaps or clefts.
The organ parenchymal cells, pc, have a saturable PSpc.
Consumption in all regions is by simple first order reactions with
rate constants Gp, Gisf, and Gpc. Axial diffusion occurs in all
regions with diffusion coefficients Dp, Disf, and Dpc. Capillary
mean transit time is Vp/Fp. The physical velocity is L/(Vp/Fp).
SHORTCOMINGS/GENERAL COMMENTS:
- Specific inadequacies or next level steps
KEY WORDS: Michaelis-Menten, BTEX30, Blood tissue exchange,
3 region, interstitial fluid, parenchymal cell, axial diffusion,
passive barrier, interendothelial clefts, capillary mean transit
time, tutorial
REFERENCES:
Bassingthwaighte JB. A concurrent flow model for extraction during transcapillary passage. Circ Res 35: 483-503, 1974.
(This gives numerical solutions, which are faster than the analytic solutions, and embeds the model in an organ
with tissue volumes conserved, and with arteries and veins. The original Lagrangian sliding fluid element model with diffusion.)
Klingenberg M. Membrane protein oligomeric structure and transport function. Nature 290: 449-454, 1981.
Stein WD. The Movement of Molecules across Cell Membranes. New York: Academic Press, 1967.
Stein WD. Transport and Diffusion across Cell Membranes. Orlando, Florida: Academic Press Inc., 1986.
Wilbrandt W and Rosenberg T. The concept of carrier transport and its corollaries in pharmacology.
Pharmacol Rev 13: 109-183, 1961.
Schwartz LM, Bukowski TR, Ploger JD, and Bassingthwaighte JB. Endothelial adenosin transporter
characterization in perfused guinea pig hearts. Am J Physiol Heart Circ Physiol 279: H1502-H1511, 2000.
Dawson CA, Linehan JH, Rickaby DA, and Roerig DL. In?uence of plasma protein on the inhibitory effects
of indocyanine green and bromcresol green on pulmonary prostaglandin E1 extraction. Br J Pharmac 81: 449-455, 1984.
Crone C. Facilitated transfer of glucose from blood into brain tissue. J Physiol 181: 103-113, 1965.
REVISION HISTORY:
Original Author : JBB Date: Nov/08
Revised by : BEJ Date: 16/Dec/09
Revision: 1) Update format of comments
Revised by : GMR Date 17/Jun/10
JSim SOFTWARE COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
JSim software was developed with support from NIH grants HL088516,
and HL073598. Please cite these grants in any publication for which
this software is used and send one reprint of published abstracts or
articles to the address given below. Academic use is unrestricted.
Software may be copied so long as this copyright notice is included.
Copyright (C) 1999-2009 University of Washington.
Contact Information:
The National Simulation Resource,
Director J. B. Bassingthwaighte,
Department of Bioengineering,
University of Washington, Seattle, WA
98195-5061
*/
0.0 0.0 5.132560194813628E-10 5.0688680386999035E-6 3.792846822796391E-4 0.003745232985754127 0.029150721288495402 0.08759605687512953
0.11857553669729758 0.09906494380067674 0.06696858359466376 0.04463294074035577 0.029853067023675694 0.022960410546508908 0.018528848095072235 0.015441118072810714
0.013143078094632549 0.01134987207049647 0.009906210171073484 0.008720713616987865 0.007734860698203219 0.006908070527057579 0.006210346322803362 0.005618491437602841
0.005114040614599414 0.004682036842841989 0.004310243081787937 0.003988588552687889 0.003708758352587394 0.0034639114667564413 0.0032483590459406173
0.0 0.0 8.805743649119998E-6 6.295029262258289E-4 0.009110891661684689 0.0349230841218903 0.05902931740152274 0.0694240850566037
0.06810205699261679 0.060989747949424064 0.054016912212918586 0.049030277856649956 0.04558604054229859 0.043225743948872616 0.041371414576154175 0.03985468999774033
0.038580596691537704 0.037491207239125794 0.03654810214837241 0.035723843400256645 0.034997700317765036 0.03435339865891348 0.03377785120779928 0.033260380289737226
0.03279219950837253 0.03236604188854403 0.03197587215164053 0.0316166852281582 0.03128430284876754 0.030975239652718596 0.030686580363822224