Tunneling effect with time-dependent effective potential barrier: A semiclassical (WKB) reinterpretation of drug...

Recent models describe drug release from polymeric nanoparticles through an analogy with quantum tunnelling, treating the delivery system as an effective rectangular potential barrier. The present work does not claim that quantum tunnelling has been experimentally established as the microscopic mechanism of drug permeation, nor that classical partition/diffusion models are generally inadequate. Rather, it reassesses whether the barrier formalism introduced in the pharmaceutical literature can be formulated with a physically motivated time dependence. In this work, we argue that this analogy is structurally identical to the standard solution of the Schrödinger equation for a rectangular barrier, and that the introduction of a multifractal formalism to describe time evolution - obtained via a formal Wick rotation (x→t) - lacks direct physical justification. We propose, instead, to treat the barrier height as an effective function of time, U_eff(t)=U₀f(t), with f(t) varying slowly within the barrier region, reflecting the progressive degradation/swelling of the polymeric matrix, under two hypotheses for f(t) - exponential decay and rational decay (Hill-type). Rather than the thick-barrier WKB approximation, the exact transmission formula is used throughout, which is real-analytic in f(t) and continues smoothly into the resonance (over-barrier) regime once the barrier collapses, avoiding the artificial step-like transitions that a thick-barrier WKB treatment would produce near the collapse point. Both hypotheses for f(t) predict a finite barrier collapse time, t^∗, whose dependence on the energy ratio m=U₀/E differs qualitatively between them (t^∗∝lnm vs. t^∗∝(m-1)^1/n), offering a distinguishable criterion from experimental release data. The model was tested against ex-vivo chicken-skin permeation kinetics of 5-FU digitized from Rata et al. (Rata et al., 2021) (three systems: NCA-1-5-FU, G-NCA-1-5-FU, G-5-FU). The exact transmission expression was used for the numerical fits; the WKB expression is retained as an asymptotic reference rather than as the fitted transport law. Because the exact transmission formula is genuinely non-monotonic past the barrier-collapse point, an unconstrained global optimization can return formally excellent but physically invalid fits, in which the predicted cumulative release transiently decreases between digitized time points - an artifact rather than a feature of the model. All fits reported here therefore enforce monotonicity of the predicted release curve during optimization, with parameter uncertainty estimated by bootstrap resampling (since the constrained optimum's local covariance is not a reliable uncertainty estimate). Under this correction, none of the three systems is well-identified in isolation: energy-ratio estimates (m) carry relative uncertainties of 245%-441%, and several parameters are only weakly constrained by the available 13-14 digitized points without replicate error bars. However, the three independently fitted point estimates of m (1.72, 1.46, and 1.46 for NCA-1-5-FU, G-NCA-1-5-FU, and G-5-FU, respectively) are mutually consistent within their broad uncertainty ranges, and a sensitivity check fixing m at a shared value costs essentially nothing in fit quality for any system. This is evidence of compatibility with a shared value, not independent identification of m. The rational (Hill) family achieves comparable or better AIC than the exponential family once monotonicity is enforced, but its own additional parameters (notably the Hill exponent n) are similarly unconstrained, so model selection between the two functional hypotheses for barrier decay is not yet resolved by the available data. Reference values from the original multifractal study (Băcăiţă et al., 2023, Cadinoiu et al., 2019, Raţă et al., 2019) and candidate extensions to further aptamer-functionalized nanocarrier systems (Tanzadehpanah et al., 2022, Zhang et al., 2024, Daraba et al., 2020) are also discussed. We discuss the implications of this treatment and its limits of validity, and point out paths for further empirical validation.