NOTES written December 3, 2005 by J. Bassingthwaighte
Comments:
There is a flow term for W and WS in the capillary region.
This is thus a tank model for infusion of GdDTPA into an
inflowing stream of water and converting (by reaction) the
water to water spins (so to speak), and washing both W and WS
as well as GdDPTA out of the field of the ROI into the effluent.
Function generator 1, fgen_1, defines the form of the Gd
infusion or bolus injection. fgen_2 must set start at t = 0,
and amplitude 1 (or other choice of water concentration for
the whole of the simulation.
Notes:
1. The permeation of GdDPTA into V2, giving concentration
GdDPTA2, creates WS2, so the tail of the residue curve and
the outflow curve GdDPTA1 are governed by the length of
time that GdDPTA2 is above zero.
2. W1 +WS1 = 1, and W2 + WS2 = 1, as required for
conservation.
3. The residue function in this model would formally be
available only for GdDTPA, and would be
(V1*GdDPTA1 + V2*GdDPTA2)/Dose injected as a Dirac delta function at t= 0 into the inflow.
The residue function observed by MRI is for WS:
Observed Residue = [V1*WS1 + V2*WS2] , modified by a
non-linear calibration, which is not included in this
program.
4. Figure 2: WS1 vs GdDPTA1, WS2 vs GdDPTA2: Default
parameter set
Plot of [WS1] versus [GdDTPA1] gives a counterclockwise
loop. Plot of [WS2] vs [GdDTPA2] gives a clockwise loop.
Presumably the area of the loops is governed by the
combination of the rate of WS production and the spin
relaxation rate.
5. Figure 3: TOTAL QUANTITIES: Default parameter set
Figure 3 shows moles of GdDTPA and of WS versus time.
Figure 4: Total WS vs Total GdDPTA: Default parameter set
Figure 4 shows a counter-clockwise loop of moles WS
versus moles GdDPTA.
6. Run the loops, set up for starting with 1 mM GdDTPA
infusion and increase the concentration by 1 mM for each
of 5 runs.
Now it becomes evident from Figure 2 and Figure 4 that
the calibration is non-linear,a straightforward result of
the bimolecular reaction equation;
dWS1/dt = G*W1*GdDPTA1 + ...
and the corresponding terms for WS2, W1, W2, GdDPTA1, and
GdDPTA2. In this reaction, GdDTPA is not consumed, but W
is consumed to form WS and the reverse transformation
occurs on spin relaxation.
7. Here, W1 was arbitrarily set to 1 mM. What is really
being set is the fractional rate of conversion of W to WS.
Increasing the values of W1(t=0) and W2(t=0) and the
amplitude of the Water inflow, Win (fgen_2), raises WS1
and WS2 to 100 times as high relative to GdDPTA1, but
does not change the fraction of W ->WS so long as
G is not changed.
8. Since GdDTPA does not enter RBC, but water spins will
enter RBC, hematocrit needs to be taken into account
separately from ISF(since RBC are flowing). The effect of
raising Hct is to decrease plasma space for GdDTPA, have
an intravascular moving water space for WS permeating
into RBC, and to increase the relative volumes of
distribution of water in ISF and parenchymal cell
compared to the plasma space.
/*MODEL NUMBER: 0265
MODEL NAME: Comp2FlowMRIContrast
SHORT DESCRIPTION: Model for analysis of NMR contrast agents from MRI signal
from an organ region of interest (ROI)
*/
import nsrunit; unit conversion on;
math Comp2FlowMRIContrast {
// INDEPENDENT VARIABLE
realDomain t sec; t.min=0; t.max=1000; t.delta=0.1; // time t
// PARAMETERS
real Flow = 0.02 ml/sec, // Flow rate
scalar = 1, // Multiplier for inflow of GdDPTA
V1 = 0.05 ml, // Volume of compartment 1 (flow compartment)
V2 = 0.10 ml, // Volume of compartment 2
G = 0.80 mM^(-1)/sec, // Conversion of W to WS dependent of [GdDPTA]
PS = 0.002 ml/sec, // Permeability Surface area product for GdDTPA
PSw = 0.10 ml/sec, // Permeability Surface area product for W and WS
Tr = 1 sec; // Water spin relaxation time constant 0.6 to 1 sec
extern real GdDPTAin(t); // Input function for GdDTPA
extern real Win(t); // Input function for water
// VARIABLES
real GdDPTA1(t) mM, GdDPTA2(t) mM, // Concentrations of gdDPTA
W1(t) mM, W2(t) mM, // Concentration of water
WS1(t) mM, WS2(t); // Concentration of water spin
// INITIAL CONDITIONS
when(t=t.min) {GdDPTA1=0; GdDPTA2=0;
W1=1; W2=1;
WS1=0; WS2 =0;}
// ORDINARY DIFFERENTIAL EQUATIONS
GdDPTA1:t = Flow*(scalar*GdDPTAin-GdDPTA1)/V1 + (PS/V1)*(GdDPTA2-GdDPTA1);
GdDPTA2:t = (PS/V2)*(GdDPTA1-GdDPTA2);
WS1:t= Flow*(0-WS1)/V1 + G*GdDPTA1*W1 - WS1/Tr + (PSw/V1)*(WS2-WS1);
WS2:t= G*GdDPTA2*W2 - WS2/Tr - (PSw/V2)*(WS2-WS1);
W1:t = Flow*(Win-W1)/V1 - G*GdDPTA1*W1 + WS1/Tr + (PSw/V1)*(W2-W1);
W2:t = - G*GdDPTA2*W2 + WS2/Tr - (PSw/V2)*(W2-W1);
}
/*
DIAGRAM
+-----------------------------------------+
Flow*GdDPTAin*Scalar----> G -->Flow*GdDPTA1
Flow*Win----------------> GdDPTA1+W1--->WS1 -->Flow*W1
| (conversion) -->Flow*WS1
| |
| TR |
| WS1--->W1 |
| (spin relaxation) |
| |
| GdDPTA1 W1 WS1 |
| ^ ^ ^ V1 |
+-----|PS---------|PSw----|PSw------------+
| v v v |
| GdDPTA2 W2 WS2 |
| |
| G |
| GdDPTA2+W2--->WS2 |
| (conversion) |
| |
| TR |
| WS2--->W2 V2 |
+-----------------------------------------+
DETAILED DESCRIPTION:
Two compartment model for MRI contrast using GdDTPA (gadolinium chelated
with diethylenetriamine penta-acetic acid to convert water (W) to water
spin (WS). The permeability-surface are product for GdDPTA is PS and for
W and WS is PSw. The spin relaxation which converts WS back to W is given
by Tr. V1 is the volume of the flowing region and V2 is the volume of
the exchange region. The conversion rate is governed by G.
Kroll et al. 1996 uses an axially-distributed 1-regional BTEX to analyze data
from polylysine-bound GdDTPA, and intravascular indicator. Kroll did not
account for WS outside of the vascular space. The WS-Gd relationship was
taken to be a saturation curve (Langmuir isotherm). There was no non-exchanging
vascular space distinguished from the capillary plasma. Hematocrit was not accounted for.
KEY WORDS:
Transport physiology, Flow, Compartmental, NMR contrast agent, functional MRI,
Gadolinium, flow estimation from residue curves, permeating GdTPA, GdDpta polylysine,
myocardial blood flow hetergeneity, regional tissue blood flows,
capillary-tissue exchange, CTEX, interstitial fluid space, Tutorial
REFERENCES:
Kroll K, Wilke N, Jerosch-Herold M, Wang Y, Zhang Y, Bache RJ, and
Bassingthwaighte JB. Modeling regional myocardial flows from residue functions of
an intravascular indicator. Am J Physiol: 271:H1643-55, 1996.
REVISION HISTORY:
Dec 2009: GR Revision of variable and parameter names.
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
*/