This model describes the dynamic response of a vessel after a step increase in intraluminal pressure.

Description

  This model describes the dynamic response of a vessel after a step 
  increase in intraluminal pressure.  It has been well documented that an
  initial passive distension occurs which is then followed by a 
  vasoconstriction to a vessel diameter below that of the initial diameter
  at low pressure.  We have previously developed a model of the myogenic 
  response in the resistance vessels which differentiates and defines the
  passive and active diameter responses to pressure (see Ref 1 below). 
  Briefly, the passive and active tensions must balance the circumferential 
  tension generated by the pressure difference across the vessel wall.  So:

                           Ttot = Tpass + Tact

  where Ttot is the pressure generated circumferential tension governed by
  the Law of Laplace:
                                     P * D
                             Ttot = -------     
                                       2

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

               Tpass = Cp1 * 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 tension can be
  further broken down into two components: A, the degree of activation of 
  the VSM (range from 0 to 1) and Tactmax, the active tension generated by 
  the VSM in a maximally activated state.  The maximally active tension is
  given by:
                                  _                        _
                                 |   _                 _ 2  |
                                 |  |  (D/Dp100) - Ca2  |   |
            Tactmax = Ca1 * exp < - | ----------------- |    >
                                 |  |_       Ca3       _|   |
                                 |_                        _|

  where Ca1 is the peak active tension, Ca2 is the diameter of the peak
  active tension 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:

               A = 1 / ( 1 + exp ( -Cmyo*Ttot + Ctone ) )

  where Cmyo determines the sensitivity of the VSM activation to
  circumferential tension and Ctone is the base level of tone that is in 
  a vessel without any stimuli.  This previous formulation produces the 
  steady state diameter as a function of pressure for a vessel defined by
  the parameters Cp1, Cp2, Dp100, Ca1, Ca2, Ca3, Cmyo and Ctone.  In order
  to model the dynamic response we assume that:

               dD      1       Dc
              ---- = ------ * ---- * ( T - Ttarget )
               dt     taud     Tc

  and 

                  dA      1  
                 ---- = ------ * ( Atarget - A )
                  dt     taua  

  where taud and taua are the time constants of the response, Dc and
  Tc are the control diameter and total vessel wall tension for scaling 
  purposes, and Ttarget and Atarget are the steady state tension and 
  activation based on the current diameter, D, pressure, P, vessel wall 
  tension, T, and VSM activation, A.  The expressions for Ttarget and 
  Atarget are given by:
                          _                   _
                         |        (2*T/P)      |
     Ttarget = C1p * exp<  C2p * --------- - 1  >
         		   |_        Dp100      _|
                                     _   _                    _ 2 _
                                    |   | (2*T/P)/Dp100 - C2a  |   |
                    + A * C1a * exp<  - |--------------------- |    >
             	                      |_  |_         C3a        _|  _|

  and

            Atarget = 1 / ( 1 + exp ( -Cmyo*(D*P/2) + Ctone ) )

  In this model the model parameters C1p, C2p, C1a, C2a, C3a, Cmyo and 
  Ctone have been set to those optimized to fit the experimental data of 
  Davis and Sikes (see Ref 3 and Regulatory Vessel model on the Physiome
  site).  An additional parameter set has been developed where taud and 
  taua in addition to the previous parameters and taud and have been 
  optimized in a two step process to fit the dynamic data of Sun et al. 
  (see Ref 2).  The time constants were first optimized to fit the data
  and then the parameters C1p, C2p, C1a, Cmyo and Ctone were optimized
  at the new taud and taua values.  This process was repeated yeilding a
  very good fit to the eperimental data.  To see this fit load the 
  Opt_2_Hill parameter set and run the model.  The fit and data will be
  displayed on the Fit_2_Hill plotpage.	

Figure: Model fit to Hill et al. 2000 data. Vessel diameter as function of time in response to change in intraluminal pressure.

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.