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# Thick_Wall_Cylinder_Variable_E

This model simulates the pressure and geometry resulting from changes in the internal radius of a thick-walled,

Model number: 0221

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## Description

 This model simulates the pressure and geometry resulting from changes in
the internal radius of a thick-walled, distensible, isotropic, cylindrical
vessel.  Stresses in the radial and axial directions are neglected and vessel
wall material is assumed to be incompresible.  An external function "ri" sets
the inner radius values within the time domain.  In the available parameter
sets, time in seconds equals the internal radius in centimeters.

The model is used to fit six pressure-radius curves from Drzewiecki et al. 1997 - three
from blood vessels and three from surgical tubes



## Equations

$\huge Pi = \frac{(r2^2-r_i^2) \cdot E \cdot \frac{r_x-r_{xo}}{r_{xo}}+(P_{OUT} \cdot r2^2 \cdot (1+(\frac{r_i}{r_x})^2)}{r_i^2 \cdot (1+(\frac{r2}{r_x})^2)}$

$\huge E = (\frac{a1}{r_i})^{a2} + a3 \cdot e^{a4 \cdot r_i}$

$\huge r2_o = r_{io} + h_o$

$\huge P_t = P_i + P_{OUT}$

$\huge A_o = \pi \cdot (r_{io}+h_o)^2 - (\pi \cdot r_{io}^2)$

$\huge A = \pi \cdot r2^2 - \pi \cdot r_i^2$

$\huge r_{xo} = r_{io} + (h_o \cdot x)$

$\huge r_x = r_i + (h \cdot x)$

$\huge r2 = \sqrt{(r_i^2-r_{io}^2+r2_o^2)}$

$\huge h = r2 - r_i$

$\huge V_o = \pi \cdot r2_o^2 \cdot L$

$\huge V = \pi \cdot r2^2 \cdot L$

$\huge \sigma_t = E \cdot \frac{r_x-r_{xo}}{r_{xo}}$

$\huge C = \frac{V-V_o}{P_t}$

The equations for this model may 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.

## References

Drzewiecki G, Field S, Moubarak I, and John K.-J. Li: Vessel growth and collapsible
pressure-area relationship. Am J Physiol Heart Circ Physiol 273:H2030-H2043, 1997.

Popov EP.  Mechincs of Materials, SI version.  Prentice Hall.  Englewood Cliffs, NJ. 1978.



## Key Terms

Thick, Variable, Young's modulus, vessel wall, material properties, thick wall, pressure-radius curve, pressure-volume curve, pressure-area curve, Data, Publication

## Model History

Get Model history in CVS.

Posted by: bej

## Acknowledgements

Please cite www.physiome.org 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.
Or send a copy to:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.