The model of circular and longitudinal smooth muscle motions in the motility of intestinal segment

Coordination in circular and longitudinal muscle motions are of crucial importance in the motor function of gastrointestinal (GI) tract. Intestinal wall motions depend on myogenic-active properties of smooth muscles layers of intestinal wall, which is the ability to create active contractile forces in response to distension. Considering the stress in the circular and longitudinal smooth muscles as a sum of passive, depending on muscle deformations, and active, depending on muscle tone, components, and also assuming that the change in the muscle tone depends on the current stress-strain condition, the system of four ordinary differential equations (ODE) is obtained which describes fillingemptying cycle of intestinal segment as a process of coordinated activities of circular and longitudinal muscles of intestinal wall. A general approach in formulating the modelling conditions is based on the previously described model restricted to the circularly distensible reservoir of constant length. Obtained results illustrate the character of coordinated activities of two orthogonal muscle layers, which are alternating phases of reciprocally and uniformly changing modalities such as stretching of the wall and muscle tone. The results also contribute to the existing understanding of the roles of Auerbach and Meisner’s intermuscular and submucous neural plexuses in regulations of autonomous intestinal motility, as well as clarify functional mechanisms of the interstitial cells of Cajal (ICC) in triggering of smooth muscle contractions. Introduction. Under the normal physiological conditions gastrointestinal (GI) motility provides accumulation and emptying of the intestinal content. The character of the movements of chime depends on the activity of smooth muscles of the intestinal wall, which include coordinated contractions and relaxations of intestinal longitudinal and circular muscular layers. The mechanism coordinating the activities of the muscular layers of the intestinal wall is not understood. Previously, using the system of two ordinary differential equations (ODE), we described filling-emptying cycle of an intestinal segment assuming the circular muscles as the only functional component of the model. The equations describing changes in the strain-stress conditions (ε, N) of the reservoir wall have the view [1]: dε /dt = 1/2V0{ P+/z+ + P−/z− − [(Еε + N)/ (2ε +1)] h0 /R0 z0 } ≡Ψ (ε, N) (*) dN/dt = − k3N + k2N − k1N1 + אε + k0(t) ≡ Φ (ε, N), 1/z =1/z+ + 1/z− Variables ε and N are circular stretching and active stress in the wall of the distensible cylinder of constant length L0 and variable radius R with constant pressures P+ and P− and impedances z+ and z− of the inlet and outlet tubes at the ends of the system. The stress-strain condition of the reservoir wall were assumed to be a sum of the passive (Eε), (N), depending on the muscle tone, components. Although the qualitative analysis of the system helps to explain some physiologic phenomena such as the dependence of propulsive (active) contractions from the threshold values of the smooth muscle stretching and tone [5], fixed reservoir length restricts the the model to the radial wall movements and prevents to show relations with longitudinal stretching (deformations) [2, 3, 4]. The similarities of the activation and relaxation mechanisms of smooth muscle elements allow to define additional conditions for longitudinal deformations of intestinal segment, remaining, at the same time, within the frames of the main assumptions of the previous model [1]. Statements of the model. As a physical analogue of intestinal segment consider distensible elastic tube (reservoir) having variable radius R and length L. The reservoir wall has myogenic properties, i.e. the ability to shrink (contract) after its stress-strain condition reaches some threshold. There are two rigid tubes, inlet and outlet, attached to the both ends of the reservoir. Both tubes contain valves with impedance z±. The pressures at the ends of the system are considered to be P±. The ends of the reservoir can move along the reservoir’s axis and also can change their radii remaining rigid. Conditions when input and output valves are open and closed are defined as the threshold odds in the pressures inside the reservoir and the tubes [1]. The system is a non-inertial (Fig 1). Fig 1. Distensible cylinder of variable length L and radius R with inlet and outlet tubes containing valves is considered as a physical model of intestinal segment. P± , z± are pressures and impedances at the ends of the system. ppressure inside the reservoir. (P+ − p) > p*+ , (p − P− ) > p*− are the conditions when inlet and outlet valves are open. p*+ , p*− are inlet and outlet threshold pressure drops considered constants. In some conditions when differences in pressures make the input valve open (P+ − p) > p*+ , and output one closed (p − P− ) ≤ p* , the filling of the reservoir begins. Filling of the reservoir will cause the elevation in the intraluminal pressure due to reactive forces caused by circular and longitudinal distensions of the wall. It is assumed that besides the differences in pressures at the ends of the system change in the sizes of the reservoir also depends on pressure drops within the reservoir due to the differences in the radial and longitudinal forces. It becomes more demonstrative when both valves are closed. Different contractile efforts of circular and longitudinal muscles will cause the pressure drops within the reservoir and redistribution of the contents until radial and longitudinal pressure components are aligned. Independently on whether or not equilibrium is achieved in the closed reservoir, its filling or emptying will depend on the pressure drops exceeding the threshold levels at the ends of the system. Because the reservoir wall has myogenic-active properties, reaching the threshold conditions during reservoir filling will initiate active contractions of longitudinal or/and circular muscles and evacuation of the contents into the outlet or inlet tubes, depending on the conditions of the valves. After the emptying and reaching the equilibrium state a new cycle may begin [5]. If the outlet valve is always closed, the contents may undergo undulating movements until the equilibrium is reached [1]. Conditions determining the character of relationships between longitudinal and circular muscular fibers are defined in the frames of basic assumptions of the first model (*) described in [1]. Considering a reservoir wall being thin enough, stretching forces will be distributed along the longitudinal axis and circularly. For anisotropic thin shells (transversely isotropic materials), if tangential forces are considered zero, conditional equations are: σ1= Е1ε1 + (ν2 Е1/ Е2) σ2 (1.1) P+ P−