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.

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

$r2_o = r_{io} + h_o$

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

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

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

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

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

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

$h = r_{o} - r_i$

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

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

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

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

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.

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References

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

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Key Terms

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

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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.

[This page was last modified 14Sep11, 2:54 pm.]

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.