Tunneling Effect with Time-Dependent Effective Potential Barrier: A Semiclassical (WKB) Reinterpretation of Drug Release Kinetics in Polymeric Nanocapsules

Recent models describe drug release from polymeric nanoparticles through an analogy with the quantum tunneling effect, treating the delivery system as a static rectangular potential barrier. 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 ) = U ₀ f ( 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 produced by the WKB approximation used in earlier drafts of this work. Both hypotheses for f ( t ) predict a finite barrier collapse time, t *, whose dependence on the energy ratio m = U ₀ / E differs qualitatively between them ( t * ∝ln m 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. [1] (three systems: NCA-1-5-FU, G-NCA-1-5-FU, G-5-FU). The exact formula substantially improved fit quality relative to the WKB approximation for all three systems. Fits were obtained by global optimization (differential evolution, polished with scipy.optimize.curve_fit for covariance estimates) rather than a single local search, which proved necessary: for G-5-FU and NCA-1-5-FU, the exponential family is well-identified (all parameter uncertainties below 11% and 6% of the estimates, respectively; R ² > 0.999), while the rational (Hill) family remained poorly identified for all three systems despite the improved formula – favoring, by parsimony, the simpler exponential model throughout. Only G-NCA-1-5-FU remained non-identified with m free. A sensitivity check fixing m at the G-5-FU-derived value ( m = 1.377) resolves this non-identifiability for G-NCA-1-5-FU at negligible cost in fit quality, consistent with a shared energy ratio for that system; the same constraint applied to NCA-1-5-FU, however, degrades its (already well-identified) fit by a factor of ∼4 in maximum residual; its own energy ratio ( m = 1.188 ± 0.007) differs from the shared value by ∼4 σ , a formally significant difference, so a single universal m is rejected for the complete set of systems studied. Reference values from the original multifractal study [2–4] and candidate extensions to further aptamer-functionalized nanocarrier systems [5–7] are also discussed. We discuss the implications of this treatment and its limits of validity, and point out paths for further empirical validation.