This model describes how a microvessel responds to changes in intraluminal pressure in the steady state. This change in vessel diameter to pressure is known as the myogenic response.

Description

  This model describes how a microvessel responds to changes in intraluminal
  pressure in the steady state.  This change in vessel diameter to pressure 
  is known as the myogenic response.  This response is a local regulatory 
  response and has been characterized in several experimental studies.  Other 
  regulatory responses not included in this model are the shear, metabolic, 
  neuronal and hormonal responses.  Regulatory microvessels are composed of 
  an inner endothelial layer surrounded by circumferentially oriented 
  vascular smooth muscle (VSM) cells and sheathed with a collagen sleeve.  
  The VSM cells actively respond to the pressure difference across the vessel 
  wall. The other components of the vessel wall act as a non-linear spring in 
  parallel with this active component.  Therefore the stress in the vessel 
  wall can be described as the stress in the passive elements of the the 
  vessel wall added to the active VSM stress.  These two components must 
  balance the circumferential stress generated by the pressure difference 
  across the vessel wall.  So:

                         SigTot = SigPass + SigAct

  where SigTot is the pressure generated circumferential stress governed by
  the Law of Laplace:

                                      P * D
                          SigTot = -----------     
                                    2 * tWall

  where P is the intraluminal pressure, D is the vessel diameter and tWall
  is the vessel wall thickness.  The passive stress is nonlinear with 
  respect to D and has been approximated here with an exponential:

           SigPass = (Cp1/tWall) * exp ( Cp2 * (( D/Dp100 ) - 1) )

  where Cp1 is the passive tension at an intraluminal pressure of 100 mmHg,
  Cp2 describes the steepness of the exponential and Dp100 is the diameter
  of the vessel in a passive state at 100 mmHg.  The active stress can be
  further broken down into two components: Act, the degree of activation of 
  the VSM (range from 0 to 1) and SigActMax, the active stress generated by 
  the VSM in a maximally activated state.  The shape of the maximally active 
  VSM stress curve as a function of diameter based on experimental findings 
  has been found to be Gaussian in form and is given here by:

                                     _                        _
                                    |   _                 _ 2  |
                       Ca1          |  |  (D/Dp100) - Ca2  |   |
         SigActMax = ------- * exp < - | ----------------- |    >
                      tWall         |  |_       Ca3       _|   |
                                    |_                        _|

  where Ca1 is the peak active tension, Ca2 is the diameter of the peak
  active stress normalized by the passive vessel diameter at 100 mmHg, 
  and Ca3 is the width of the Gaussian normalized by Dp100.  The VSM 
  activation is approximated by a sigmoidal function and is given by:

          Act = 1 / ( 1 + exp ( -Cmyo*SigTot*tWall + Ctone ) )

  where Cmyo determines the sensitivity of the VSM activation to
  circumferential tension and Ctone is the base level of VSM tone that is 
  in a vessel without these stimuli.  So now the circumferential stress
  in the vessel wall, SigTot, can be given by:

                   SigTot = SigPass + (Act * SigActMax) 

  The vessel that is fit by the parameters used in this code is presented 
  of mesenteric arterioles in Wister-Kyoto rats as presented in Figure 1 of 
   a study by Bund (Clin Sci 101:385-393,2001).  Other data sets can be fit 
  using different values of Cp1, Cp2, Ca1, Ca2, Ca3, Cmyo and Ctone.  This 
  model was modified from the previously developed Regulatory_Vessel model 
  which was formulated only in terms of vessel wall tension.  

Figure: Relationship between pressure within the vessel (intraluminal) and vessel diameter.

Equations

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.