Injecting simultaneously a vascular reference and a
diffusive substrate allows one to easily characterize
the permeability of capillary walls of various organs.
The Crone paper characterizes permeabilty of the capillary
walls to sucrose(brain and hind leg) and inulin (brain,
kidney, lung, liver). The sucrose results show no
permeability in the brain and extraction rates of ~ 30%
in the hind limb of a dog. The inulin extraction in the
kidney shows mean extraction rates of ~ 70%.
The outflow curve data for sucrose shows a 'broad' peak
in comparison to the reference curve data that is not
reflected in our model fit. What could cause the extraction
rate to decrease more than our model predicts between
45 and 55 sec? Is there a small amount of consumption in
the ISF or is it mainly due to experimental variance?
Crone mentions that the hind limb extraction data is
relatively constant over the time course of the experiment
and probably due to very low flows in the hind limb
(1.2 l/sec*g*10^6) as compared to the lung (50) or
kidney (66). Why is this?
*******************************
Plot page 'HindLeg_Suc_plot':
Inflow (Cin), and Outflow concentrations. The inflow and outflow concentrations
are displayed as function of time in seconds. Crone 1963 sucrose data from top
portion of Figure 7 is displayed for comparison.
Plot page 'HindLeg_Extract':
Instantaneous extraction and ISF permeability as a function of time.
Crone sucrose data from curve cr63Fig7_Ext_1 (bottom of figure 7),
shown here, is the only one of the three extraction curves which directly
corresponds to outflow curves shown in top of figure 7 and in Plot page 'Outflow'.
The vascular reference outflow curve (Cpout_ref) is shown for timing purposes only.
Plot page 'Kidney_Inulin_plot':
Inflow (Cin), and Outflow concentrations. The inflow and outflow
concentrations are displayed as function of time in seconds.
Crone 1963 inulin data from top portion of Figure 4 is displayed
for comparison.
Plot page 'Kidney_Extract':
Instantaneous extraction and ISF permeability as a function of time.
Extraction data is derived from Crone inulin outflow data from Figure 4.
The vascular reference outflow curve (Cpout_ref) is shown for timing
purposes only.
Permeability is calculated from eq 2 of Crone 1963 paper.
--------------------------
From Table 1 of Crone 1963:
--------------------------
Kidney Perfusion rate: 66 (l/sec*g *10^6)
Kidney Capillary surface area: 350 (cm^2/g)
Crone calculated Inulin Permeability coefficient of Kidney:
14.4*10^-5 cm/sec
Hind Limb Perfusion rate: 1.2 (l/sec*g *10^6)
Hind Limb Capillary surface area: 70 (cm^2/g)
Crone calculated Sucrose Permeability coefficient of hind limb:
0.74*10^-5 cm/sec (based on average of 9 data sets)
---> Our fit: Max Permeability is 1.3*10^-5 cm/sec
// MODEL NUMBER: 0146
// MODEL NAME: Crone63_BTEX
// SHORT DESCRIPTION: Models a tissue cylinder consisting of
// two regions: plasma, and interstitial fluid. Fit to Crone 1963 data.
import nsrunit; unit conversion on;
math Crone1963_BTEX {
// INDEPENDENT VARIABLES
realDomain t sec ; t.min=0; t.max=30; t.delta=0.1;
realDomain x cm; real L=0.1 cm, Ngrid=61; x.min=0; x.max=L; x.ct=Ngrid;
private x.min, x.max, x.ct;
/* PARAMETERS AND KEY TO NAMES
p = PLASMA/BLOOD
isf = INTERSTITIAL FLUID REGION
*/
real Fp = 1 ml/(g*min), // blood flow
// VOLUMES (virtual volume)
Vp = 0.05 ml/g, // capillary blood
Visf = 0.15 ml/g, // isf (Visf')
OrgW = 1g, // Organ weight
DoseS = 1 umol, // Dose inj substrate
DoseR = 1 umol, // Dose inj vascular ref
// Capillary radius in dog, from Birks EK, 1994:
CapRadius = 3.4e-4 cm, // Cap radius in dog
CapArea = Vp*2/CapRadius,
// PS means Permeability-surface area product between two regions
PSg = 1 ml/(g*min), // between p and isf
// G is the consumption rate coefficient in each region (Gulosity)
Gp = 0 ml/(g*min), // p
Gisf = 0 ml/(g*min), // isf
// D is the axial diffusion coefficient
Dp = 1.0e-5 cm^2/sec, // p
Disf = 1.0e-6 cm^2/sec; // isf
// 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);
// INFLOWING CONCENTRATION
extern real Cin(t) 1/sec;
real CinS(t) = DoseS*Cin(t)/(Fp*OrgW),
CinR(t) = DoseR*Cin(t)/(Fp*OrgW);
real CinSsum = sum(t=t.min to t.max, CinS);
real CinRsum = sum(t=t.min to t.max, CinR);
// CONCENTRATION VARIABLES
real Cp(t,x) mM, // p
Cisf(t,x) mM, // isf
Cout(t) mM, // Outflow Concentration from plasma region
Cp_ref(t,x) mM, // Plasma reference
Cpout_ref(t) mM; // Ref outflow concentration from plasma region
// Extraction:
real Extract(t) dimensionless;
Extract = if ((Cpout_ref >0.1) and (Cout*(DoseR/DoseS)<Cpout_ref)) 1- (Cout*(DoseR/DoseS)/Cpout_ref)
else 0.0;
// Calculated instantaneous permeability into ISF (from Crone 1963 eq 2.)
real Perm(t) = (Fp/CapArea)*ln(1/(1-Extract));
// BOUNDARY CONDITIONS (Note total flux BC for inflowing region.)
when (x=x.min) { (-Fp*L/hVp)*(Cp-CinS)+Dp*Cp:x = 0; Cisf:x = 0; }
when (x=x.min) { (-Fp*L/hVp)*(Cp_ref-CinR)+Dp*Cp_ref:x = 0; }
when (x=x.max) { Cp:x = 0; Cisf:x = 0; Cout = Cp;
Cp_ref:x = 0; Cpout_ref = Cp_ref;}
// INITIAL CONDITIONS
when (t=t.min) { Cp = 0; Cisf = 0; Cp_ref =0; }
// PARTIAL DIFFERENTIAL EQUATIONS
Cp:t = -Fp*L/hVp*Cp:x -Gp/hVp*Cp+ Dp*Cp:x:x
+ PSg/hVp*(Cisf-Cp);
Cisf:t = -Gisf/hVisf*Cisf + Disf*Cisf:x:x
+ PSg/hVisf*(Cp-Cisf) ;
Cp_ref:t = -Fp*L/hVp*Cp_ref:x -Gp/hVp*Cp_ref+ Dp*Cp_ref:x:x ;
}
/*
FIGURE:
Fp ________________________________________
Cin(t) ---> |Vp Cp(t)|---> Cout(t)
|Gp ^ |
|Dp | PLASMA|
___________PSg_________________________|
|Visfp | Cisf(t)|
|Gisf V INTERSTITIAL|
|Disf FLUID REGION|
________________________________________
|<----------------L------------------->|
|--> x
DETAILED DESCRIPTION:
These partial differential equations model a "tissue cylinder"
consisting of two regions. The two regions are capillary plasma,
p, and interstitial fluid, isf. There is a diffusional path from plasma
to the isf. A consumption term is included but is set to zero here.
Model is fit to Crone 1963 data for sucrose uptake into
hind leg of a dog and inulin uptake into the kidney. This paper was
one of the first attempts at characterizing capilliary permeability
in various organs and regions of the body. Vascular reference is Evans' Blue Dye.
KEY WORDS:
BTEX20, PDE, convection, diffusion, permeation, reaction, distributed, capillary,
plasma, isf, interstitial fluid, Crone, Indicator dilution, sucrose,
vascular reference, data
REFERENCES:
W.C. Sangren and C.W. Sheppard. A mathematical derivation of the
exchange of a labelled substance between a liquid flowing in a
vessel and an external compartment. Bull Math BioPhys, 15, 387-394,
1953.
C.A. Goresky, W.H. Ziegler, and G.G. Bach. Capillary exchange modeling:
Barrier-limited and flow-limited distribution. Circ Res 27: 739-764, 1970.
J.B. Bassingthwaighte. A concurrent flow model for extraction
during transcapillary passage. Circ Res 35:483-503, 1974.
B. Guller, T. Yipintsoi, A.L. Orvis, and J.B. Bassingthwaighte. Myocardial
sodium extraction at varied coronary flows in the dog: Estimation of
capillary permeability by residue and outflow detection. Circ Res 37: 359-378, 1975.
Christian Crone, The Permeability of Capillaries in Various Organs as
Deterimined by Use of the 'Indicator Diffusion' Method,
Acta physiol.scand. 58:292-305, 1963,
REVISION HISTORY:
Author BEJ 02/15/11
- Based on Model #0080: btex20
- Fit to Crone 1963 data
Revised by: BEJ Date:15may13 : Update optimization params
Copyright (C) 1999-2013 University of Washington. From the National Simulation Resource,
Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.
Academic use is unrestricted. Software may be copied so long as this copyright notice is included.
This software was developed with support from NIH grant HL073598.
Please cite this grant in any publication for which this software is used and send an email
with the citation and, if possible, a PDF file of the paper to: staff@physiome.org.
*/
35.11 36.92 39.02 40.98 43.01 45.05 47.0 49.02
51.03 53.13 55.11 57.04 59.11
0.0166244 3.17421 18.3585 25.2477 25.7395 26.2313 24.7674 22.2377
20.3596 18.6008 16.4855 15.4361 12.8476
Data Sets from Figure 7 - Relative Concentration vs time
Outflow curves for Evans' Blue Dye (vascular ref) and
Sucrose,(Hind limb) of:
Crone, "The Permeability of Capillaries in Various
Organs as Deterimined by Use of the 'Indicator Diffusion' Method, Acta physiol. scand., 1963, 58(292-305)
Also: Outflows are expressed in relative concentrations, which Crone describes as [Concentration] "in each sample is divided by that in the injection fluid. For each sample the relative concentration of the test substance is divided by that of the reference substance giving a value between 0 and 1." (p296)
Note: figure arbitrarily set t=0 at peak of
Reference Out Flow. Data sets added 45 sec.
-------------
From Table 1:
-------------
Perfusion rate: 1.2 l/sec*g *10^6
Capillary surface area: 70 cm^2/g
Crone calculated Permeability coefficient of hind limb:
0.74 (cm/sec *10^5)
35.0 36.924 39.0 40.95 43.01 45.02 46.96 48.98
51.03 53.06 55.06 57.02 59.08
0.017 1.76 10.68 14.14 13.45 14.0 14.67 14.16
13.7 12.17 10.59 9.3 7.89
Data Sets from Figure 7 - Relative Concentration vs time
Outflow curves for Evans' Blue Dye (vascular ref) and
Sucrose,(Hind limb) of:
Crone, "The Permeability of Capillaries in Various
Organs as Deterimined by Use of the 'Indicator Diffusion' Method, Acta physiol. scand., 1963, 58(292-305)
Also: Outflows are expressed in relative concentrations, which Crone describes as [Concentration] "in each sample is divided by that in the injection fluid. For each sample the relative concentration of the test substance is divided by that of the reference substance giving a value between 0 and 1." (p296)
Note: figure arbitrarily set t=0 at peak of
Reference Out Flow. Data sets added 45 sec.
-------------
From Table 1:
-------------
Perfusion rate: 1.2 l/sec*g *10^6
Capillary surface area: 70 cm^2/g
Crone calculated Permeability coefficient of hind limb:
0.74 (cm/sec *10^5)
38.99 40.94 42.98 45.0 46.98 48.99 51.01 53.02
55.03 57.01 59.05
40.62 44.05 47.5 46.67 39.23 36.04 33.33 33.93
35.95 38.44 37.62
Data Sets from Figure 7 - Relative Concentration vs time
Outflow curves for Evans' Blue Dye (vascular ref) and
Sucrose,(Hind limb) of:
Crone, "The Permeability of Capillaries in Various
Organs as Deterimined by Use of the 'Indicator Diffusion' Method, Acta physiol. scand., 1963, 58(292-305)
Note: figure arbitrarily set t=0 at peak of
Reference Out Flow. Data sets added 45 sec.
cr63Fig7_Ext_1: This dataset corresponds to extraction calc for
outflow curve data shown in Fig 7 top.
-------------
From Table 1:
-------------
Perfusion rate: 1.2 l/sec*g *10^6
Capillary surface area: 70 cm^2/g
Crone calculated Permeability coefficient of hind limb:
0.74 (cm/sec *10^5)
42.78 45.0 47.21 49.43 51.61 53.88 56.04 58.23
60.45 62.66
30.64 27.5 32.02 35.59 40.59 46.07 46.76 46.97
47.18 52.18
Data Sets from Figure 7 - Relative Concentration vs time
Outflow curves for Evans' Blue Dye (vascular ref) and
Sucrose,(Hind limb) of:
Crone, "The Permeability of Capillaries in Various
Organs as Deterimined by Use of the 'Indicator Diffusion' Method, Acta physiol. scand., 1963, 58(292-305)
Note: figure arbitrarily set t=0 at peak of
Reference Out Flow. Data sets added 45 sec.
-------------
From Table 1:
-------------
Perfusion rate: 1.2 l/sec*g *10^6
Capillary surface area: 70 cm^2/g
Crone calculated Permeability coefficient of hind limb:
0.74 (cm/sec *10^5)
40.35 42.65 45.0 47.31 49.56 51.95 54.25 56.53
58.84 61.14
20.82 21.52 23.66 21.97 20.75 14.75 17.37 17.11
14.93 11.32
Data Sets from Figure 7 - Relative Concentration vs time
Outflow curves for Evans' Blue Dye (vascular ref) and
Sucrose,(Hind limb) of:
Crone, "The Permeability of Capillaries in Various
Organs as Deterimined by Use of the 'Indicator Diffusion' Method, Acta physiol. scand., 1963, 58(292-305)
Note: figure arbitrarily set t=0 at peak of
Reference Out Flow. Data sets added 45 sec.
-------------
From Table 1:
-------------
Perfusion rate: 1.2 l/sec*g *10^6
Capillary surface area: 70 cm^2/g
Crone calculated Permeability coefficient of hind limb:
0.74 (cm/sec *10^5)
36.92 39.0 40.95 43.01 45.02 46.96 48.98 51.03
53.06 55.06 57.02 59.08
44.55 41.83 43.99 47.75 46.63 40.77 36.32 32.71
34.57 35.76 39.75 38.59
Data Sets from Figure 7 - Relative Concentration vs time
Outflow curves for Evans' Blue Dye (vascular ref) and
Sucrose,(Hind limb) of:
Crone, "The Permeability of Capillaries in Various
Organs as Deterimined by Use of the 'Indicator Diffusion' Method, Acta physiol. scand., 1963, 58(292-305)
Note: figure arbitrarily set t=0 at peak of
Reference Out Flow. Data sets added 45 sec.
Cr63Fig7ExtractCalc:
This extraction data is calculated directly from Evan's dye and sucrose
outflow curves from Fig 7 top.
27.25 28.58 30.04 31.47 32.84 34.25 35.67 37.0
38.41 39.87 41.19 42.62 45.39 46.82 48.16 49.54
50.97
0.206587 2.85357 25.8897 39.4766 30.5038 21.3842 13.486 8.92754
6.51275 4.71285 3.65765 3.68273 1.89385 1.76662 1.32349 1.64758
1.36803
Data Sets from Figure 4 - Relative Concentration vs time
Outflow curves for Evans' Blue Dye (vascular ref) and
Inulin,(Kidney) of:
Crone, "The Permeability of Capillaries in Various
Organs as Deterimined by Use of the 'Indicator Diffusion' Method, Acta physiol. scand., 1963, 58(292-305)
Also: Outflows are expressed in relative concentrations, which Crone describes as [Concentration] "in each sample is divided by that in the injection fluid. For each sample the relative concentration of the test substance is divided by that of the reference substance giving a value between 0 and 1." (p296)
Note: figure arbitrarily set t=0 at peak of
Reference Out Flow. Data sets added 30 sec.
27.18 28.54 30.04 31.43 32.74 34.21 35.7 36.97
38.43 39.83 41.18 42.61 45.38 46.79 48.16 49.61
50.97
0.0 12.1154 88.6144 57.9346 13.659 5.27688 2.84938 1.92494
2.54723 2.55391 1.78979 1.79904 1.19679 0.440382 1.20501 1.06421
1.37099
Also: Outflows are expressed in relative concentrations, which Crone describes as [Concentration] "in each sample is divided by that in the injection fluid. For each sample the relative concentration of the test substance is divided by that of the reference substance giving a value between 0 and 1." (p296)
28.54 30.04 31.43
76.44675 70.78387 31.86006