This model simulates the flow through a passive and actively responding vessel driven by a sinusoidal pressure input.

Description

The model simulates fluid flow, F, through a compliant vessel that actively responds to the pressure in the vessel given a pressure drop across the length of the vessel equivalent to Pin - Pout. The previous compliant vessel models (Compliant vessel, Thin wall compliant vessel and Nonlinear compliant vessel) have behaved passively to the changes in pressure but it is known that arterioles in particular respond actively: dilating when the pressure decreases and constricting when the pressure increases.The active response of a vessel to pressure is known as the myogenic response. A misnomer since responses to all different stimulii are effected through activation or deactivation of the vascular smooth muscle (VSM).

The flows, Fin, Fout and Fcomp, and the vessel volume, V are unknown in this simulation. Therefore four equations are needed to solve for these four variables. The expressions for the flows are as presented in the previous compliant vessel models. To solve for V the myogenic response formulation is used to evaluate the vessel diameter based on relative contributions of the passive and myogenically active responses. The mathematical formulation for the myogenic response is from Carlson and Secomb (Microcirc 12:327-338, 2005). In this study the vessel wall is represented by a nonlinear spring and a contractile VSM element in parallel as shown below.

                                   Tp
                                   
                        --------/\/\/\/\--------
                        |                      |
                 T <----|                      |----> T
                        |      ----------      |             
                        -------|  VSM   |-------
                               ----------

                                 A*Tma

where T is the total tension, Tp is the passive tension and Tma is the maximally active tension in the vessel wall. A is the level of VSM activation with a range of 0 to 1. The passive tension is described by an exponential function of the form:

                   Tp = C1p * exp [ C2p * (D/Dp100 - 1) ]

where C1p and C2p are parameters optimized to fit passive response experimental data, D is the vessel diameter and Dp100 is the reference passive diameter of the vessel at a intraluminal pressure of 100 mmHg. The maximally active tension is described by a gaussian function:

                                 _   _               _ 2 _
                                |   |  D/Dp100 - C2a  |   |
                Tma = C1a *exp <  - | --------------- |    >
                                |_  |_      C3a      _|  _|

where C1a, C2a and C3a are parameters optimized to fit myogenically responding arterioles from experimental data. Finally we have the VSM activation, A, which is represented by a sigmoidally shaped function of the form:

                                     1
                    A = ----------------------------
                         1 + exp(-Cmyo*T + Cglobal)

where Cmyo and Cglobal are also fit to the myogenically active arteriolar data and the total tension, T, is given by the law of Laplace:

                                   Pin*D
                             T = --------
                                     2

where Pin is the pressure at the vessel input and in this case is an sinusoidal input driving the flow in the vessel. All of these expressions can be combined to create an implicit function of diameter as a function of input pressure, Pin.

                      _
              2      |
         D = ---- * <  C1p * exp [ C2p * (D/Dp100 - 1) ]
              Pc     |_
                                 _   _               _ 2 _    _
                                |   |  D/Dp100 - C2a  |   |    |
                       C1a *exp <  - | --------------- |    >  |
                                |_  |_      C3a      _|  _|    |
                    + ----------------------------------------  >
                         1 + exp(-Cmyo*(Pin*D/2) + Cglobal)   _|

This implicit equation is used to solve for the initial diameter and then this diameter and the VSM activation is updated depending on the steady state diamter and activation that the vessel would desire to go to at the current pressure. This update is facilitated by the ordinary diffeerntial equations for D and A:

                 dD/dt = (1/taud)*(Dc/Tc)(T - Ttarget)
     
                     dA/dt = (1/taua)*(Atarget - A)

where taud and taua are the time constants that determine how fast the diameter and activation move towards their respective steady state values, Dtarget and Atarget.

The remaining code is the same as we have developed for the compliant vessel and the compliant vessel thinwall formulation. The flow out of the vessel is related to the resistance by the fluid equivalent of Ohm's Law.

                        Fout = (Pin - Pout) / R

where Pin - Pout is the difference in pressure between the beginning and end of the vessel and R is determined from Poiseuille's Law as:

                         R = 128*mu*L / pi*D^4

where mu is the fluid viscosity.

The flow into the vessel and the flow out of the vessel are different because of the change in volume which adds or subtracts flow from that leaving the vessel depending on whether the pressure is increasing or decreasing in the vessel. So we have:

                          Fin = Fout + Fcomp

where the flow attributed to the vessel compliance, Fcomp, is given by:

                            Fcomp = dV/dt

and where V is the vessel volume and is now purely a function of the diameter

 
                           V = PI * D^2 * L / 4  

The parameters defining the vessels myogenic response are optimized to data for ~200 um arteriolar vessels from Liao and Kuo (Am J Physiol, Heart Circ Physiol 272:H1571-H1581, 1997).

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.