From Molecular Protection to Targeted Delivery: A Layer 8/9 Digital Twin Framework for pH-Triggered Controlled Release and Multi-Compartment Pharmacokinetics of Allicin/Al₁₂N₁₂ Nanocomplexes against SARS-CoV-2

Jefferson Lorençoni de Morais^1,2*, Heliel Gabriel Borges de Sena³, Lanna Araujo Borges³, Larissa Neres Barbosa³.

1. Polytechnic School of the Alves Faria University Center — UNIALFA, Goiânia, Brazil.

2. American University of Global Technology – AGTU, Orlando, USA.                

3. Institute of Pharmaceutical and Exact Sciences — University Center of Goiás — UNIGOIAS,

   Goiânia, Brazil.

*Correspondence: Jefferson.morais@unialfa.com.br;

DOI: https://doi.org/10.71431/IJRPAS.2026.5708        

INTRODUCTION

Scientific Context and Series Continuity

Allicin (diallyl thiosulfinate, C₆H₁₀OS₂; CAS 539-86-6), the principal bioactive compound of Allium sativum (garlic), exhibits broad anticancer, anti-inflammatory, antiviral, and antibacterial activities ^[1,2]. Its fundamental pharmacological limitation is rapid aqueous degradation (t½ ≈ 3.5–4 days at 37°C, pH 7.4) ^[3,4], which the Digital Twin series has progressively addressed through computational modeling: v2.0 established the Nanocage Protection Factor (NPF = 5,498,905× for Al₁₂N₁₂) from DFT thermodynamics and Gaussian Process surrogate modeling ^[5]; v3.0 integrated real molecular dynamics data (Li & Cheng 2023, CHARMm36) to calibrate Mpro binding dynamics [6]; v3.1 extended the framework to dual-target inhibition of Mpro (6LU7) and S/ACE2 (6M17), introducing the DTIS and SVEI metrics ^[7]. Together, these publications established that Allicin/Al₁₂N₁₂ is thermodynamically stable and biologically potent. What none of them addressed is the pharmacokinetic question that precedes any clinical translation: how does the nanocomplex release its cargo across biologically heterogeneous environments, and what sustained drug concentration does the released allicin achieve at the target site over time?

Figure 1. Three-Compartment Kinetic Resolution and Dual-Target Blockade (Layer 9). Schematic of the mass flow across the three simulated compartments: C₁ (Intact Nanocomplex) → C₂ (Available Free Allicin) → C₃ (Biological Target — Mpro/S/ACE2 occupancy). Rate constants k_eff, k_rel, k_hyd, and k_elim govern the flux at each transition.

Figure 2. pH-Triggered Nanocage Dissociation Mechanism (Layer 8). Molecular illustration of the Al₁₂N₁₂ surface Al–O bond protonation at pH 5.5 (endosomal), which weakens the electrostatic anchor of allicin within the nanocage and triggers desorption. At pH 7.4 (plasma), the Al–O bonds remain intact, maintaining the ultra-slow-release profile characteristic of Scenario A.

The Scientific Gap: From Stability to Pharmacokinetics

   The Korsmeyer–Peppas model ^[9,10] is the standard semi-empirical framework for characterizing drug release from polymeric and colloidal carriers. The dimensionless transport exponent n discriminates between Fickian diffusion (n = 0.45), anomalous diffusion (0.45 < n < 0.89), and Case II transport (n ≥ 0.89). For inorganic nanocage systems such as Al₁₂N₁₂, where desorption is governed by a combination of electrostatic unbinding and diffusion through an aqueous shell, anomalous diffusion (n = 0.65) is physically appropriate ^[11]. Coupling this release model to a pH-dependent Arrhenius barrier — which captures the protonation of Al–O bonds at acidic pH — creates a biologically gated delivery system whose kinetics can be predicted in silico without destructive assays.

Compartmental pharmacokinetic models have been the standard tool in drug development since the 1960s for quantifying absorption, distribution, and elimination ^[12]. Digital Twin frameworks incorporating ODE-based compartmental engines are now recognized as enabling technologies for silico pre-clinical prediction ^[13]. The present work is the first application of this paradigm to a garlic-derived phytopharmaceutical nanocomplex, directly extending the computational infrastructure established in the v2.0–v3.1 series.

Objectives

This work aims to: (i) develop Layer 8 — a pH/T-dependent Korsmeyer–Peppas release engine parametrized by the Al–O protonation desorption barrier; (ii) develop Layer 9 — a three-compartment LSODA-solved ODE pharmacokinetic model coupled to the frozen v3.1 dual-target PD model; (iii) simulate and compare 40-day drug dynamics across Scenario A (pH 7.4, 37°C) and Scenario B (pH 5.5, 39°C) for all four allicin formulations; (iv) quantify the reservoir effect of Al₁₂N₁₂ in terms of sustained free allicin concentration, cumulative DTIS, and SVEI over the simulation window; and (v) identify the critical experimental gaps that the Digital Twin v4.0 predictions motivate for subsequent in vitro and in vivo validation.

MATERIALS AND METHODS

Frozen Parameters Inherited from Prior Publications

A defining design principle of the Digital Twin series is parameter inheritance: once a constant is validated and published, it is frozen and cannot be modified by subsequent layers. This ensures that each new layer adds predictive power without introducing retroactive uncertainty into prior results. Table 1 lists all frozen constants used in v4.0.

Table 1. Frozen parameters inherited from the Digital Twin v2.0–v3.1 series.

Layer 8 — pH-Dependent Controlled Release Engine

Korsmeyer–Peppas Cumulative Release Model

The cumulative release fraction of allicin from the nanocage, , is described by a modified Korsmeyer–Peppas model coupled to a pH-dependent colloidal desorption barrier:

where is the pH- and temperature-dependent release constant, is the release exponent characterizing the transport mechanism, and the operator ensures that the cumulative release fraction remains physically bounded within the interval .

The variable denotes time (days), is the Korsmeyer–Peppas anomalous transport exponent, and is the effective release-rate constant modulated by both pH and temperature:

Here, is the baseline release-rate prefactor under physiological conditions; is the physiological reference pH; represents the effective activation energy associated with colloidal desorption, corresponding to the energetic barrier required to overcome electrostatic Al–O surface anchoring; is the universal gas constant; and is the absolute temperature (K).

Physical Interpretation of the pH Amplification Factor

The exponential pH-dependent term,

mechanistically describes the progressive protonation of surface Al–O anchoring sites under acidic conditions, thereby reducing the electrostatic desorption barrier and enhancing allicin release from the nanocage. Under physiological conditions ( ; Scenario A), this factor is equal to unity, defining the baseline release rate. In the mildly acidic microenvironment ( ; Scenario B), the factor increases to approximately 1.339, corresponding to a 33.9% enhancement in the effective release-rate constant. This behavior reflects the increased protonation of surface Al–O bonds, which weakens ligand–surface interactions and facilitates cargo desorption. Such acid-responsive release mechanisms are consistent with the behavior reported for aluminum-based nanocarriers in the materials science and drug-delivery literature ^[14].

The temperature dependence is described by the Arrhenius term,

which yields values of approximately 0.198 at 37 °C (310.15 K) and 0.200 at 39 °C (312.15 K), corresponding to an increase of approximately 1% in the release-rate constant between the two simulated scenarios. Consequently, pH constitutes the dominant driver of release kinetics, whereas the temperature variation contributes only a minor modulation within the investigated physiological range.

Table 2 summarizes the complete set of parameters adopted for the Layer 8 releases model.

Table 2. Layer 8 — Korsmeyer–Peppas pH-dependent release parameters and scenario outputs.

Layer 9 — Three-Compartment ODE Pharmacokinetic Engine

Compartment 1: Transport Vector (Intact Nanocomplex)

The concentration of the intact nanocomplex, ( ), decreases through two parallel first-order processes: (i) nanocage-mitigated aqueous hydrolysis, characterized by the effective rate constant , and (ii) pH-dependent desorption, governed by the release-rate constant :

Here, denotes the Arrhenius hydrolysis rate constant of free allicin, incorporating the Version 2.0 pH correction,

such that , while NPF represents the Nanocage Protection Factor obtained from the quantum-mechanical stabilization model. Consequently, the effective hydrolysis constant is given by

For the nanocage,

which is several orders of magnitude smaller than the desorption rate constant. Therefore, hydrolytic degradation of the encapsulated compound is effectively suppressed, and the persistence of the intact nanocomplex is governed almost entirely by the pH-dependent desorption process rather than by chemical decomposition.

Compartment 2: Available Drug (Desorbed Free Allicin)

The concentration of free allicin, ( ), is governed by the balance between pH-dependent desorption from the nanocomplex and its subsequent loss through aqueous hydrolysis and systemic elimination:

The first term describes the release of allicin from the intact nanocomplex, whereas the latter two terms account for chemical hydrolysis and first-order metabolic clearance, respectively. The elimination rate constant is defined as

where is the assumed systemic half-life adopted for the in silico pharmacokinetic model. This conservative value is consistent with the rapid systemic clearance reported for thiosulfinate compounds in preclinical pharmacokinetic studies.

Initial conditions differ according to the simulated formulation. For free allicin administration,

whereas for the nanocage formulation,

indicating that the administered dose is initially fully encapsulated and becomes available only through the pH-dependent release process.

Compartment 3: Biological Target — Dual-Target PD (v3.1 frozen)

At each integration time point, the instantaneous free allicin concentration, , obtained from the pharmacokinetic model, dynamically drives the Michaelis-type dual-target occupancy model adopted from Version 3.1 ^[7]:

Here, and are the inhibition constants retained from Version 3.1 ^[7]. The resulting occupancies represent the fractional inhibition of the viral main protease (Mpro) and the ACE2-mediated viral entry pathway, respectively. The Dual-Target Inhibition Score (DTIS) combines both mechanisms through a weighted synergistic formulation, whereas the Spike Viral Entry Inhibition Index (SVEI) is defined as

The coupled pharmacokinetic–pharmacodynamic (PK–PD) model was numerically integrated using the adaptive LSODA algorithm implemented in scipy.integrate.solve_ivp, with relative and absolute tolerances of and , respectively. To ensure complete computational reproducibility, the random number generator was initialized with numpy. random.seed(42), although the deterministic ODE solver itself is independent of stochastic sampling.

Table 3 summarizes the kinetic parameters and the corresponding peak free-allicin concentrations predicted by the model.

Table 3. Layer 9 — Compartmental ODE rate constants and peak free-allicin concentrations.

Note: K_hyd values shown for Scenario A (pH 7.4, 37°C). K_elim = 2.773 day^-1 (T^1/2_sys = 6 h) applied uniformly. Highlighted row (blue) = Allicin/Al₁₂N₁₂.

RESULTS AND DISCUSSION

 Layer 8 — pH-Driven Desorption Profiles

Figure 3 presents the cumulative release fraction F_rel(t) over 40 days for both scenarios. The most immediate result is that K_release is independent of the nanocage species — it depends solely on pH and temperature — which is physically coherent: the Korsmeyer–Peppas model characterizes the release kinetics of the system, not the thermodynamic stability of individual species. This means the pH trigger acts uniformly across all formulations, but its consequences differ profoundly between species because the amount of intact nanocomplex available for desorption depends on NPF.

At pH 7.4 (Scenario A), F_rel(40 days) = 19.5% a genuinely ultra-slow, sustained release compatible with long-term systemic circulation. At pH 5.5 (Scenario B), F_rel (40 days) = 26.4%, representing a 35.3% relative increase in released fraction attributable to the endosomal pH trigger. For Allicin/Al₁₂N₁₂, which maintains 98.3% of its initial 2.0 µM as intact nanocomplex at t = 40 days (Scenario A), this 19.5% cumulative release translates to a continuous, low-level drug supply to Compartment 2 overs the entire window — the quantitative definition of the reservoir effect.

Figure 3. Korsmeyer–Peppas cumulative release profiles F_rel(t) over 40 days. (a) All species at pH 7.4 (Scenario A) and pH 5.5 (Scenario B) for Allicin/Al₁₂N₁₂, with dotted lines showing B₁₂N₁₂ and C₂₄ at pH 5.5 for reference. (b) Detail for Allicin/C₂₄. The pH amplification of 35.3% between scenarios is uniform across all species (K_release depends on pH/T only). n = 0.65 (anomalous diffusion, non-Fickian transport).

Layer 9 — Compartment 1: Nanocomplex Integrity

Figure 4 shows the intact nanocomplex concentration [C₁, µM] over 40 days. The hierarchy of stability is unambiguous and directly reflects the NPF values frozen from v2.0. Al₁₂N₁₂ and B₁₂N₁₂ retain essentially identical C₁ profiles (0.983 µM at t = 40 d, Scenario A), because both have NPF values high enough to make k_eff negligible — the dominant loss pathway from Compartment 1 is desorption (k_rel), not hydrolysis. C₂₄, by contrast, with NPF = 7.01, shows a k_eff = 2.51×10⁻² day⁻¹ that is large enough to visibly deplete the nanocomplex: only 0.361 µM of the initial 2.0 µM remains intact at t = 40 d in Scenario A (82% depletion), and only 0.135 µM in Scenario B (93% depletion). This confirms that C₂₄ should not be considered a long-term depot carrier, while Al₁₂N₁₂ and B₁₂N₁₂ function as genuine reservoir vehicles.

Figure 4. Compartment 1 (Intact Nanocomplex) concentration [µM] over 40 days. (a) Scenario A (pH 7.4, 37°C): Al₁₂N₁₂ and B₁₂N₁₂ retain 98.3% of initial nanocomplex at t = 40 d; C₂₄ retains only 18.0% (NPF = 7.01 insufficient to suppress hydrolysis). (b) Scenario B (pH 5.5, 39°C): faster depletion for all species due to increased k_rel and k_hyd. Free Allicin omitted (no Compartment 1).

Layer 9 — Compartment 2: Free Allicin Dynamics

Figure 5 is the pharmacokinetically critical output of v4.0. It directly compares the temporal profile of bioavailable free allicin [C₂, µM] for all four formulations under both scenarios. The contrast is stark: free allicin starts at C₂ (0) = 2.0 µM and collapses to approximately zero within 5 days under both scenarios, with a half-life of approximately 0.24 days (dominated by k_elim = 2.773 day⁻¹). No C₂ remains detectable beyond day 10.

The nanocomplexes tell a fundamentally different story. For Allicin/Al₁₂N₁₂ and Allicin/B₁₂N₁₂, the desorption-limited supply from Compartment 1 establishes a pseudo-steady-state free allicin concentration of approximately 0.012 µM (12 nM) within the first 2 days, which is then sustained for the entire 40-day window. This is the quantitative manifestation of the reservoir effect: the nanocomplex acts as a slow-release depot that continuously replenishes the bioavailable pool at a rate governed by k_rel rather than k_hyd or k_elim. At pH 5.5 (Scenario B), the higher k_rel (2.405×10⁻² vs 1.777×10⁻² day⁻¹) drives a slightly higher plateau of 0.015 µM, meaning the endosomal environment provides a modest but reproducible 25% enhancement in bioavailable concentration.

Allicin/C₂₄ reaches a lower plateau of 0.002–0.001 µM, significantly below B₁₂N₁₂/Al₁₂N₁₂, because the depletion of Compartment 1 (18% remaining at t=40d) limits the desorption source term.

Figure 5. Compartment 2 (Available Free Allicin) concentration [µM] over 40 days. (a) Scenario A: free allicin (red) collapses to zero within 5 days; Al₁₂N₁₂ and B₁₂N₁₂ sustain a stable plateau of ~0.012 µM for 40 days. (b) Scenario B: endosomal pH (5.5) elevates the plateau to ~0.015 µM (+25%). The plateau concentration is governed by the balance between desorption supply (k_rel) and combined drain (k_hyd + k_elim).

Layer 9 — Compartment 3: DTIS and SVEI Dynamics

Figures 6 and 7 translate the pharmacokinetic profiles into pharmacodynamic outcomes. The Michaelis-type occupancy model (Ki_Mpro = 61.62 µM, Ki_ACE2 = 38.50 µM) converts C₂(t) into fractional inhibition scores. Since the plateau concentration of 0.012–0.015 µM is approximately 2,500–4,000-fold below Ki_Mpro and 2,500–3,000-fold below Ki_ACE2, the instantaneous inhibition values are small in absolute terms. This is, however, a scientifically important and reproducible result: it quantifies the concentration gap that separates current computational predictions from clinically effective doses, providing a concrete engineering target for the next iteration of the framework (dose optimization, multi-dose regimens, or more potent derivatives).

The key pharmacodynamic finding is the cumulative advantage of the nanocomplexes over free allicin. While free allicin achieves a higher DTIS peak at t = 0 (0.0400 vs 0.0002 for nanocomplexes), its DTAUC over 40 days (0.0138 in Scenario A) reflects only the contribution of the first 5–10 days before complete depletion. Al₁₂N₁₂ and B₁₂N₁₂ accumulate DTAUC = 0.00707 (Scenario A) and 0.00824 (Scenario B) from a sustained, consistent contribution across all 40 days. In Scenario B, the DTAUC of Al₁₂N₁₂ represents 62.3% of the free allicin DTAUC with 1/170 of the peak concentration — a clear pharmacokinetic efficiency advantage that would compound over longer timescales.

Figure 6. Compartment 3: Dual-Target Inhibition Score (DTIS, %) over 40 days. Free allicin (red) dominates at t = 0 but decays rapidly. Al₁₂N₁₂ and B₁₂N₁₂ (purple/green) maintain a low, sustained inhibition plateau for 40 days. DTIS = 0.55·I_Mpro + 0.45·I_ACE2 + 0.30·I_Mpro·I_ACE2 (v3.1 frozen). (a) Scenario A; (b) Scenario B.

Figure 7. SVEI — SARS-CoV-2 Viral Entry Inhibition (%) over 40 days. SVEI = I_ACE2(t) = C₂(t)/(C₂(t) + Ki_ACE2). Al₁₂N₁₂ and B₁₂N₁₂ sustain entry blockade throughout the 40-day window, while free allicin achieves complete SVEI depletion by day 10. (a) Scenario A (pH 7.4); (b) Scenario B (pH 5.5).

Accumulated Pharmacodynamic Exposure: DTAUC and SVEI_AUC

Figure 8 and Table 4 provide integrated pharmacodynamic summary. The DTAUC and SVEI_AUC metrics (area under the respective inhibition–time curves over 40 days) capture the total therapeutic exposure — the pharmacodynamic equivalent of bioavailability. Several observations are critical. First, the DTAUC/SVEI_AUC of free allicin is higher than that of nanocomplexes in Scenario A, because the initial 2 µM dose (identical for all formulations) provides a large but transient inhibition peak for free allicin. Second, in Scenario B, nanocomplexes surpass free allicin in DTAUC (Al₁₂N₁₂ DTAUC_B = 0.00824 vs 0.01322 for free allicin), but with 16.5× more temporal distribution — all the allicin inhibition occurs in the first 5 days for the free form, versus distributed evenly over 40 days for the nanocomplexes. Third, Al₁₂N₁₂ DTAUC increases by 16.5% from Scenario A to B (0.00707 → 0.00824), directly demonstrating the clinical utility of acidic microenvironments for improving drug delivery from Al₁₂N₁₂.

Table 4. Layer 9 / Compartment 3 — DTIS and SVEI pharmacodynamic outputs over the 40-day simulation.

DTAUC = ∫DTIS dt over 40 days. SVEI_AUC = ∫SVEI dt over 40 days. Red = Free Allicin; White = C₂₄; Light blue = B₁₂N₁₂; Dark blue = Al₁₂N₁₂. Scen. A=pH 7.4 / B=pH 5.5.

Figure 8. Cumulative pharmacodynamic exposure: DTAUC (a) and SVEI_AUC (b) integrated over 40 days. Solid bars = Scenario A (pH 7.4); hatched bars = Scenario B (pH 5.5). While free allicin achieves higher total AUC from its initial peak, the nanocomplex advantage is temporal distribution — sustained inhibition across the entire window rather than concentrated in the first few days.

The Reservoir Effect: A Quantitative Definition

The central pharmacokinetic finding of v4.0 can be stated precisely: for Allicin/Al₁₂N₁₂ in Scenario A, the ratio of Compartment 1 remaining at t = 40 days to initial dose is 0.982/2.0 = 49.1%. This means that after 40 days, 49.1% of the original allicin dose remains encapsulated and available for future release, while 100% of the free allicin dose has been degraded or eliminated. The steady-state free allicin concentration maintained by this reservoir (0.012 µM) is 167× lower than the initial free allicin concentration but is present for 40 days versus 5 days — a temporal extension of 8×. Expressed as a sustained inhibition advantage: the nanocomplex provides pharmacodynamic coverage for 40 days where free allicin provides only 5, defining an effective temporal extension factor of at least 8× at the sub-therapeutic plateau concentration.

LIMITATIONS AND FUTURE DIRECTIONS

Current Limitations

•           The plateau free allicin concentration of 0.012–0.015 µM is ~2,500–4,000-fold below Ki values (38.5–61.6 µM). While this is a valid in silico result for C₀ = 2 µM, clinical efficacy would require higher doses or more potent derivatives. The Digital Twin v4.0 defines the concentration gap; closing it requires in vitro cytotoxicity data to establish therapeutic windows.

•           The systemic half-life of 6 hours for k_elim is an in-silico estimate. Actual allicin pharmacokinetics in humans are poorly characterized; the model is conservative but not experimentally calibrated.

•           The Korsmeyer–Peppas model assumes a homogeneous carrier matrix. Real Al₁₂N₁₂ nanocages may exhibit surface heterogeneity, aggregation, or protein corona effects that modify release kinetics.

•           No actual GROMACS MD trajectories were used in v4.0; the GP surrogate from v2.0 remains unretrained against experimental MD.

Critical Next Experimental Steps

The Digital Twin v4.0 defines two experimental priorities in strict order:

•         In vitro stability assays: UV-Vis degradation kinetics of Allicin/Al₁₂N₁₂ at pH 7.4 and 5.5, 37–39°C, to validate F_rel(t) predictions and the K_release ratio between scenarios.

•         Cell-based efficacy assays at sub-therapeutic concentrations (0.01–0.1 µM range) to determine whether the sustained plateau can still achieve biologically meaningful SARS-CoV-2 inhibition through prolonged exposure.

Future Computational Layers (Layer 10 and beyond)

Layer 10 is envisioned as a stochastic global optimization engine that searches the parameter space (K_base, n, Eₐ_rel, dose) to maximize DTAUC under constraints of toxicity and manufacturability. Bayesian optimization or genetic algorithms applied to the v4.0 ODE system would identify the optimal nanocage formulation and dosing regimen without additional animal experiments, consistent with the Digital Twin paradigm of minimizing destructive testing through predictive computation ^[13].

CONCLUSION

Digital Twin v4.0 presents the first pharmacokinetic modeling layer of the Allicin/Al₁₂N₁₂ computational series, bridging the gap between thermodynamic stability prediction (v2.0) and actionable drug delivery design. The pH-dependent Korsmeyer–Peppas engine (Layer 8) quantifies the endosomal pH trigger (pH 5.5 → 33.9% K_release amplification) through Al–O protonation. The three-compartment LSODA-ODE pharmacokinetic engine (Layer 9) reveals that Al₁₂N₁₂ sustains 98.3% of the initial nanocomplex intact for 40 days in healthy plasma, continuously feeding a free allicin plateau of 0.012 µM — a pharmacokinetic profile impossible for free allicin, which is fully depleted within 5 days.

The quantitative definition of the reservoir effect (49.1% of dose remaining at t = 40 d, effective temporal extension factor of 8×), the pH-triggered 35.3% release enhancement in Scenario B, and the 16.5% DTAUC gain from Scenario A to B collectively confirm Allicin/Al₁₂N₁₂ as a pharmacokinetically rational candidate for sustained antiviral delivery. The next critical question — whether the predicted 0.012 µM plateau concentration is biologically active against SARS-CoV-2 — is the primary in vitro experimental objective for the next study in this series.

ACKNOWLEDGEMENT

        The authors thank the Polytechnic School of UNIALFA and the Institute of Pharmaceutical Sciences of UNIGOIAS for institutional support. The Digital Twin v4.0 framework was developed entirely with open-source tools (Python 3.12, NumPy, SciPy, Matplotlib, Pandas). The authors acknowledge the foundational work of Korsmeyer & Peppas (1981) and Peppas et al. (2014) that established the mathematical framework for controlled release modeling.

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