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# thick_wall_vessel_tangential_stress

This model simulates the pressure and geometry resulting from changes in the internal radius of a thick-walled, distensible, isotropic, cylindrical vessel.

Model number: 0204

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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. Reference pressures
and volumes are tunable. 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.

This model does not replicate a typical blood vessel Pressure-Radius relationship because
the wall material is characterized by a constant Young's modulus. As blood vessels dilate
a constant Young's modulus cannot capture the relative stiffening of the vessel wall due
to the collagen fibers in the adventitia.



## Equations

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

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

$\huge P_{t} = P_{i} - P_{OUT}$

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

$\huge A = \pi \cdot r_{o}^2 - \pi \cdot r_{i}^2$

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

$\huge r_{x} = r_{i} + h_o \cdot x$

$\huge r_{o} = \sqrt{r_{i}^2 - r_{io}^2 + r2_o^2}$

$\huge h = r_{o} - r_i$

$\huge V_{o} = \pi \cdot r2_o^2 \cdot L$

$\huge V = \pi \cdot r_{o}^2 \cdot L$

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

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

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

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



## Key Terms

cardiovascular system, thick wall, vessel, cylinder, tangential stress, hoop stress, circumferential stress, isotropic, strain, stress profile

## 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.