Modeling and Analysis of Fractional Maxwell Fluid in Intra-Articular Drug Injection Flow

Intra-articular drug injection is a core therapeutic approach for knee osteoarthritis (KOA), and its efficacy is closely related to the spatial distribution of the injected fluid. Supported by fractional-order derivative theory, this study first constructs an anatomically accurate model of the human knee joint cavity as a foundation for flow simulation. Given that the injected drug exhibits time-dependent viscoelastic relaxation with pronounced memory effects, the fractional Maxwell constitutive equation is adopted to more accurately describe its mechanical behavior. This constitutive model is then embedded in the fluid-dynamic governing equations, which are discretized and solved using the finite volume method combined with the L1 algorithm. Through numerical simulations, we systematically investigate how key parameters—such as the characteristic relaxation time and the fractional derivative order—affect the intra-articular distribution of the injected drug. The results indicate that increasing the fractional derivative order enhances the effective distribution volume of the drug within the joint cavity and increases its retention in peripheral regions, while reducing distribution uniformity in the central region. By introducing fractional-order derivatives, this study enables a more precise representation of drug flow behavior during injection. The results elucidate the regulatory roles of key parameters in shaping drug distribution, thereby offering valuable scientific guidance for the development of personalized KOA injection-based treatment strategies.