A fractional derivative model to describe arterial viscoelasticity

Abstract

Arterial viscoelasticity can be described with a complex modulus (E^*) in the frequency domain. In arteries, E^* presents a power-law response with a plateau for higher frequencies. Constitutive models based on a combination of purely elastic and viscous elements can be represented with integer order differential equations but show several limitations. Recently, fractional derivative models with fewer parameters have proven to be efficient in describing rheological tissues. A new element, called “spring-pot”, that interpolates between springs and dashpots is incorporated. Starting with a Voigt model, we proposed two fractional alternative models with one and two spring-pots. The three models were tested in an anesthetized sheep in a control state and during smooth muscle activation. A least squares method was used to fit E^*. Local activation induced a vascular constriction with no pressure changes. The E^* results confirmed the steep increase from static to dynamic values and a plateau in the range 2–30 Hz, coherent with fractional model predictions. Activation increased E^*, affecting its real and imaginary parts separately. Only the model with two spring-pots correctly followed this behavior with the best performance in terms of least squares errors. In a context where activation separately modifies E^*, this alternative model should be considered in describing arterial viscoelasticity in vivo.