MID4pde

A model for triple-labeled indicator dilution experiments in a four region tissue exchange model.

Model number: 0292

Description

This is a model for triple-labeled indicator dilution experiments in a four region tissue exchange model. A BTEX10 model is used for the vascular tracer (plasma), a BTEX20 model is used for the extracellular tracer (plasma and interstitial fluid region), and a BTEX40 model is used for the permeant tracer (plasma ,interstitial fluid region, and endothelial and parenchymal cell regions. Extraction is calculated and used to calculate estimated permeability-surface area factors between various regions. See MID4ode for a version using compartment models

Equations

Vascular Equations

Differential Equation

$\frac{\partial v\underline{\ } C_p}{\partial t} = \frac{-F_p \cdot L}{V_p} \cdot \frac{\partial v\underline{\ }C_p}{\partial x} - \frac{-v\underline{\ }G_p}{V_p} \cdot v\underline{\ }C_p +v\underline{\ }D_p \cdot \frac{\partial^2 v\underline{\ }C_p}{\partial x^2}$

Left Boundary Condition

$-{\frac {{\it F_p}\cdot L \cdot \left( {\it v\underline{\ }C_p}-{\it C_{in}} \right) }{{\it V_p}}}+{v\underline{\ }D_{p} \cdot \it {\frac {\partial}{\partial x}} v\underline{\ }C_p=0$ .

Right Boundary Conditions

${\it {\frac {\partial }{\partial x}}v\underline{\ }C_p=0$ , ${\it v\underline{\ }C_{out}={\it v\underline{\ }C_{p}$ .

Initial Conditions

$v\underline{\ }C_p=v\underline{\ }C_p0$ or $v\underline{\ }C_p=v\underline{\ }C_p0(x)$ .

Extracellular Equations

Differential Equations

$\frac{\partial e\underline{\ } C_p}{\partial t} = \frac{-F_p \cdot L}{V_p} \cdot \frac{\partial e\underline{\ }C_p}{\partial x} - \frac{-e\underline{\ }G_p}{V_p} \cdot e\underline{\ }C_p -\frac{e\underline{\ }PS_g}{V_{p}}\cdot (e\underline{\ }C_{p}-e\underline{\ }C_{isf}) +e\underline{\ }D_p \cdot \frac{\partial^2 e\underline{\ }C_p}{\partial x^2}$
$\frac{\partial e\underline{\ }C_{isf}}{\partial t} = \frac{-e\underline{\ }G_{isf}}{V'_{isf}} \cdot e\underline{\ }C_{isf} - \frac{e\underline{\ }PS_g}{V'_{ist}}\cdot (pC\underline{\ }_{isf}-e\underline{\ }C_p) +e\underline{\ }D_{isf} \cdot \frac{\partial^2 e\underline{\ }C_{isf}}{\partial x^2}$

Left Boundary Conditions

$-{\frac {{\it F_p}\cdot L \cdot \left( {\it e\underline{\ }C_p}-{\it C_{in}} \right) }{{\it V_p}}}+{e\underline{\ }D_{p} \cdot \it {\frac {\partial}{\partial x}} e\underline{\ }C_p=0$ , ${\it {\frac {\partial}{\partial x}}e\underline{\ }C_{isf}=0$ .

Right Boundary Conditions

${\it {\frac {\partial }{\partial x}}e\underline{\ }C_p=0$ , ${\it {\frac {\partial}{\partial x}}e\underline{\ }C_{isf}=0$ , ${\it e\underline{\ }C_{out}={\it e\underline{\ }C_{p}$ .

Initial Conditions

$e\underline{\ }C_p=e\underline{\ }C_p0$ , $e\underline{\ }C_{isf}=e\underline{\ }C_{isf}0$ or
$e\underline{\ }C_p=e\underline{\ }C_p0(x)$ , .

Permeant Equations

Differential Equations

$\frac{\partial p\underline{\ } C_p}{\partial t} = \frac{-F_p \cdot L}{V_p} \cdot \frac{\partial p\underline{\ }C_p}{\partial x} - \frac{-p\underline{\ }G_p}{V_p} \cdot p\underline{\ }C_p -\frac{p\underline{\ }PS_g}{V_{p}}\cdot (p\underline{\ }C_{p}-p\underline{\ }C_{isf}) -\frac{p\underline{\ }PS_{ecl}}{V_p}\cdot (p\underline{\ }C_{p}-p\underline{\ }C_{ec}) +p\underline{\ }D_p \cdot \frac{\partial^2 p\underline{\ }C_p}{\partial x^2}$ $\frac{\partial p\underline{\ }C_{ec}}{\partial t} = \frac{-p\underline{\ }G_{ec}}{V'_{ec}} \cdot p\underline{\ }C_{ec} -\frac{p\underline{\ }PS_{ecl}}{V'_{ec}} \cdot (p\underline{\ }C_{ec}-p\underline{\ }C_{p}) -\frac{p\underline{\ }PS_{eca}}{V'_{ec}} \cdot (p\underline{\ }C_{ec}-p\underline{\ }C_{isf}) + p\underline{\ }D_{ec} \cdot \frac{\partial^2 p\underline{\ } C_{ec}}{\partial x^2}$ $\frac{\partial p\underline{\ }C_{isf}}{\partial t} = \frac{-p\underline{\ }G_{isf}}{V'_{isf}} \cdot p\underline{\ }C_{isf} - \frac{p\underline{\ }PS_g}{V'_{ist}}\cdot (pC\underline{\ }_{isf}-p\underline{\ }C_p) - \frac{p\underline{\ }PS_{pc}}{V'_{isf}} \cdot (p\underline{\ }C_{isf}-p\underline{\ }C_{pc}) -\frac{p\underline{\ }PS_{eca}}{V'_{isf}} \cdot (p\underline{\ }C_{isf}-p\underline{\ }C_{ec}) +p\underline{\ }D_{isf} \cdot \frac{\partial^2 p\underline{\ }C_{isf}}{\partial x^2}$ $\frac{\partial p\underline{\ }C_{pc}}{\partial t} = \frac{-p\underline{\ }G_{pc}}{V'_{pc}} \cdot p\underline{\ }C_{pc} -\frac{p\underline{\ }PS_pc}{V'_{pc}} \cdot (p\underline{\ }C_{pc}-p\underline{\ }C_{isf}) +p\underline{\ }D_{pc} \cdot \frac{\partial^2 p\underline{\ }C_{pc}}{\partial x^2}$

Left Boundary Conditions

$-{\frac {{\it F_p}\cdot L \cdot \left( {\it p\underline{\ }C_p}-{\it C_{in}} \right) }{{\it V_p}}}+{p\underline{\ }D_{p} \cdot \it {\frac {\partial}{\partial x}} p\underline{\ }C_p=0$ , ${\it {\frac {\partial}{\partial x}}p\underline{\ }C_{ec}=0$ , ${\it {\frac {\partial}{\partial x}}p\underline{\ }C_{isf}=0$ , ${\it {\frac {\partial}{\partial x}}p\underline{\ }C_{pc}=0$ .

Right Boundary Conditions

${\it {\frac {\partial }{\partial x}}p\underline{\ }C_p=0$ , ${\it {\frac {\partial }{\partial x}}p\underline{\ }C_{ec}=0$ , ${\it {\frac {\partial}{\partial x}}p\underline{\ }C_{isf}=0$ , ${\it {\frac {\partial}{\partial x}}p\underline{\ }C_{pc}=0$ , ${\it p\underline{\ }C_{out}={\it p\underline{\ }C_{p}$ .

Initial Conditions

$p\underline{\ }C_p=p\underline{\ }C_p0$ , $p\underline{\ }C_{ec}=p\underline{\ }C_{ec}0$ $p\underline{\ }C_{isf}=p\underline{\ }C_{isf}0$ $p\underline{\ }C_{pc}=p\underline{\ }C_{pc}0$ or
$p\underline{\ }C_p=p\underline{\ }C_p0(x)$ , , , .

The equations for this model may also be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.

Download JSim project file

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.

C.P. Rose, C.A. Goresky, and G.G. Bach.  The capillary and
sarcolemmal barriers in the heart--an exploration of labelled water
permeability.  Circ Res 41: 515, 1977.

J.B. Bassingthwaighte, C.Y. Wang, and I.S. Chan.  Blood-tissue
exchange via transport and transformation by endothelial cells.
Circ. Res. 65:997-1020, 1989.

Poulain CA, Finlayson BA, Bassingthwaighte JB.,Efficient numerical methods 
for nonlinear-facilitated transport and exchange in a blood-tissue exchange 
unit, Ann Biomed Eng. 1997 May-Jun;25(3):547-64.

Related Models

Blood Tissue Exchange (BTEX) models

Key Terms

reaction, distributed, capillary, plasma, isf, interstitial fluid, endothelial, parenchymal, cell blood, tissue, exchange, extraction, crone-renkin, PS, BTEX10, BTEX20, BTEX40, optimization, confidence limits, vascular, extracellular, permeant, MID4, multi-tracer, albumin, sucrose, tritiated water

Model History

Get Model history in CVS.

Acknowledgements

Please cite www.physiome.org in any publication for which this software is used and send one reprint to the address given below:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

[This page was last modified 10Sep10, 9:31 am.]

Model development and archiving support at physiome.org provided by the following grants: NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.

MID4pde

Further reading: Distributed Blood Tissue Exchange Models Explained Download PDF file.

Description

Equations

Vascular Equations

Differential Equation

Left Boundary Condition

Right Boundary Conditions

Initial Conditions

Extracellular Equations

Differential Equations

Left Boundary Conditions

Right Boundary Conditions

Initial Conditions

Permeant Equations

Differential Equations

Left Boundary Conditions

Right Boundary Conditions

Initial Conditions

Download JSim project file

References

Related Models

Key Terms

Model History

Acknowledgements