# Is Drug Delivery System a Deterministic or Probabilistic Approach? A Theoretical Model Based on the Sequence: Electrodynamics–Diffusion–Bayes

## Abstract

**:**

## 1. Introduction

## 2. The Electrodynamic Model

## 3. The Diffusion Model

## 4. Efficiency of Drug Delivery Based on Probabilities

#### 4.1. Gaussian Distributions

#### 4.2. Weibull and Lorentzian Distributions

#### 4.3. The Diffusion–Coulomb Efficiency

## 5. Identification of Stochastic Events through Bayes’s Theorem

- ${P}_{A}$, the probability that N nanoparticles arrive at the tumor;
- ${P}_{B}$, the probability that all of them achieve to internalize the tumor;
- $1-{P}_{A}$, the probability that M nanoparticles fail to reach the tumor;
- ${P}_{W}$, the probability that $1-{P}_{A}$ is wrong and nanoparticles were scattered off the tumor.

#### Bayesian Behavior Inside Linear Scenarios

## 6. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Singh, R.; Lillard, J.W., Jr. Nanoparticle-based targeted drug delivery. Exp. Mol. Pathol.
**2009**, 86, 215–223. [Google Scholar] [CrossRef] [PubMed] - Veiseh, O.; Gunn, J.W.; Zhang, M. Design and fabrication of magnetic nanoparticles for targeted drug delivery and imaging. Adv. Drug Deliv. Rev.
**2010**, 62, 284–304. [Google Scholar] [CrossRef] [PubMed] - Kaushik, A.; Jayant, R.D.; Sagar, V.; Nair, M. The potential of magneto-electric nanocarriers for drug delivery. Expert Opin. Drug Deliv.
**2014**, 11, 1635–1646. [Google Scholar] [CrossRef] [PubMed] - Zhang, N.; Xiong, G.; Liu, Z. Toxicity of metal-based nanoparticles: Challenges in the nano era. Front. Bioeng. Biotechnol.
**2022**, 10, 1001572. [Google Scholar] [CrossRef] [PubMed] - Dobrovolskaia, M.A.; Patri, A.K.; Zheng, J.; Clogston, J.D.; Ayub, N.; Aggarwal, P.; Neun, B.W.; Hall, J.B.; McNeil, S.E. Interaction of colloidal gold nanoparticles with human blood: Effects on particle size and analysis of plasma protein binding profiles. Nanomedicine
**2009**, 5, 106–117. [Google Scholar] [CrossRef] [PubMed] - Fam, S.Y.; Chee, C.F.; Yong, C.Y.; Ho, K.L.; Mariatulqabtiah, A.R.; Tan, W.S. Stealth Coating of Nanoparticles in Drug-Delivery Systems. Nanomaterials
**2020**, 10, 787. [Google Scholar] [CrossRef] - Mosesson, M.W. Fibrinogen and fbrin structure and functions. J. Trombos. Haemost.
**2005**, 3, 1894–1904. [Google Scholar] [CrossRef] - Kharazian, B.; Lohse, S.E.; Ghasemi, F.; Raoufi, M.; Saei, A.A.; Hashemi, F.; Farvadi, F.; Alimohamadi, R.; Jalali, S.A.; Shokrgozar, M.A. Bare surface of gold nanoparticle induces inflammation through unfolding of plasma fibrinogen. Sci. Rep.
**2018**, 8, 12557. [Google Scholar] [CrossRef] - Dobrovolskaia, M.A.; Neun, B.W.; Man, S.; Ye, X.; Hansen, M.; Patri, A.K.; Crist, R.M.; McNeil, S.E. Protein corona composition does not accurately predict haemocompatibility of colloidal gold nanoparticles. Nanomedicine
**2014**, 10, 1453–1463. [Google Scholar] [CrossRef] - Deng, Z.J.; Liang, M.; Monteiro, M.; Toth, I.; Minchin, R.F. Nanoparticle-induced unfolding of fibrinogen promotes Mac-1 receptor activation and infammation. Nat. Nanotechnol.
**2011**, 6, 39–44. [Google Scholar] [CrossRef] - Fernandes, H.P.; Cesar, C.L.; Barjas-Castro Mde, L. Electrical properties of the red blood cell membrane and immunohematological investigation. Rev. Bras. Hematol. Hemoter.
**2011**, 33, 297–301. [Google Scholar] [CrossRef] [PubMed] - Le, W.; Chen, B.; Cui, Z.; Liu, Z.; Shi, D. Detection of cancer cells based on glycolytic-regulated surface electrical charges. Biophys. Rep.
**2019**, 5, 10–18. [Google Scholar] [CrossRef] - Forest, V.; Pourchez, J. Preferential binding of positive nanoparticles on cell membranes is due to electrostatic interactions: A too simplistic explanation that does not take into account the nanoparticle protein corona. Mater. Sci. Eng. C
**2017**, 70, 889–896. [Google Scholar] [CrossRef] [PubMed] - Ajdari, N.; Vyas, C.; Bogan, S.L.; Lwaleed, B.A.; Cousins, B.G. Gold nanoparticle interactions in human blood: A model evaluation. Nanomed. Nanotechnol. Biol. Med.
**2017**, 13, 1531–1542. [Google Scholar] [CrossRef] [PubMed] - Clogston, J.D.; Patri, A.K. Zeta potential measurement. Methods Mol. Biol.
**2011**, 697, 63–70. [Google Scholar] [CrossRef] [PubMed] - Liu, Z.; Clausen, J.R.; Rao, R.R.; Aidun, C.K. Nanoparticle diffusion in sheared cellular blood flow. J. Fluid Mech.
**2019**, 871, 636–667. [Google Scholar] [CrossRef] - Nieto-Chaupis, H. Success and Fail at the Internalization and Expelling of Nanoparticles off Tumor Cells through Electrodynamics and Diffusion Equation. In Proceedings of the (2022) IEEE International Conference on Bioinformatics and Biomedicine (BIBM), Las Vegas, NV, USA, 6–8 December 2022; pp. 3556–3561. [Google Scholar] [CrossRef]
- Bian, T.; Gardin, A.; Gemen, J.; Houben, L.; Perego, C.; Lee, B.; Elad, N.; Chu, Z.; Pavan, G.M.; Klajn, R. Electrostatic co-assembly of nanoparticles with oppositely charged small molecules into static and dynamic superstructures. Nat. Chem.
**2021**, 13, 940–949. [Google Scholar] [CrossRef] - Hu, P.; Qian, W.; Liu, B.; Pichan, C.; Chen, Z. Molecular Interactions Between Gold Nanoparticles and Model Cell Membranes: A Study of Nanoparticle Surface Charge Effect. J. Phys. Chem. C
**2016**, 120, 39. [Google Scholar] [CrossRef] - Panariti, A.; Miserocchi, G.; Rivolta, I. The effect of nanoparticle uptake on cellular behavior: Disrupting or enabling functions? Nanotechnol. Sci. Appl.
**2012**, 5, 87–100. [Google Scholar] - Barisik, M.; Atalay, S.; Beskok, A.; Qian, S. Size Dependent Surface Charge Properties of Silica Nanoparticles. J. Phys. Chem. C
**2014**, 118, 1836–1842. [Google Scholar] [CrossRef] - Jose, L.; Baalrud, S.D. A generalized Boltzmann kinetic theory for strongly magnetized plasmas with application to friction. Phys. Plasmas
**2020**, 27, 112101. [Google Scholar] [CrossRef] - Nagy, A. Shannon entropy density as a descriptor of Coulomb systems. Chem. Phys. Lett.
**2013**, 556, 355–358. [Google Scholar] [CrossRef] - Finbloom, J.A.; Huynh, C.; Huang, X.; Desai, T.A. Bioinspired nanotopographical design of drug delivery systems. Nat. Rev. Bioeng.
**2023**, 1, 139–152. [Google Scholar] [CrossRef] - Webber, M.J.; Pashuck, E.T. (Macro)molecular self-assembly for hydrogel drug delivery. Adv. Drug Deliv. Rev.
**2021**, 172, 275–295. [Google Scholar] [CrossRef] [PubMed] - Lee, M.H. Fick’s Law, Green-Kubo Formula, and Heisenberg’s Equation of Motion. Phys. Rev. Lett.
**2000**, 85, 2422–2425. [Google Scholar] [CrossRef] [PubMed] - Martens, S.; Schmid, G.; Schimansky-Geier, L.; Hänggi, P. Entropic particle transport: Higher-order corrections to the Fick-Jacobs diffusion equation. Phys. Rev. E
**2011**, 83, 051135. [Google Scholar] [CrossRef] - Roxhed, N.; Samel, B.; Nordquist, L.; Griss, P.; Stemme, G. Painless Drug Delivery Through Microneedle-Based Transdermal Patches Featuring Active Infusion. IEEE Trans. Biomed. Eng.
**2008**, 55, 1063–1071. [Google Scholar] [CrossRef] - McGinty, S.; McKee, S.; Wadsworth, R.M.; McCormick, C. Modelling drug-eluting stents. Math. Med. Biol. J. IMA
**2011**, 28, 1–29. [Google Scholar] [CrossRef] - Chahibi, Y.; Pierobon, M.; Song, S.O.; Akyildiz, I.F. A Molecular Communication System Model for Particulate Drug Delivery Systems. IEEE Trans. Biomed. Eng.
**2013**, 60, 3468–3483. [Google Scholar] [CrossRef] - Chahibi, Y.; Pierobon, M.; Akyildiz, I.F. Pharmacokinetic Modeling and Biodistribution Estimation Through the Molecular Communication Paradigm. IEEE Trans. Biomed. Eng.
**2015**, 62, 2410–2420. [Google Scholar] [CrossRef] - Liu, X.; Huang, N.; Li, H.; Jin, Q.; Ji, J. Surface and size effects on cell interaction of gold nanoparticles with both phagocytic and nonphagocytic cells. Langmuir
**2013**, 29, 9138–9148. [Google Scholar] [CrossRef] - Zhao, F.; Zhao, Y.; Liu, Y.; Chang, X.; Chen, C.; Zhao, Y. Cellular uptake, intracellular trafficking, and cytotoxicity of nanomaterials. Small Weinh. Bergstr. Ger.
**2011**, 7, 1322–1337. [Google Scholar] [CrossRef] - Cobanoglu, M.C.; Liu, C.; Hu, F.; Oltvai, Z.N.; Bahar, I. Predicting drug-target interactions using probabilistic matrix factorization. J. Chem. Inf. Model.
**2013**, 53, 3399–3409. [Google Scholar] [CrossRef] [PubMed] - Chahibi, Y.; Akyildiz, I.F. Molecular Communication Noise and Capacity Analysis for Particulate Drug Delivery Systems. IEEE Trans. Commun.
**2014**, 62, 3891–3903. [Google Scholar] [CrossRef] - Chahibi, Y.; Akyildiz, I.F.; Balasubramaniam, S.; Koucheryavy, Y. Molecular Communication Modeling of Antibody-Mediated Drug Delivery Systems. IEEE Trans. Biomed. Eng.
**2015**, 62, 1683–1695. [Google Scholar] [CrossRef] [PubMed] - Available online: https://www.wolframalpha.com/ (accessed on 14 August 2023).
- Wolpert, D.H. Uncertainty Relations and Fluctuation Theorems for Bayes Nets. Phys. Rev. Lett.
**2020**, 125, 200602. [Google Scholar] [CrossRef] - McDonald, P.C.; Chafe, S.C.; Brown, W.S.; Saberi, S.; Swayampakula, M.; Venkateswaran, G.; Nemirovsky, O.; Gillespie, J.A.; Karasinska, J.M.; Kalloger, S.E.; et al. Regulation of pH by carbonic anhydrase 9 mediates survival of pancreatic cancer cells with activated KRAS in response to hypoxia. Gastroenterology
**2019**, 157, 823–837. [Google Scholar] [CrossRef] - Gao, Q.; Zhang, J.; Gao, J.; Zhang, Z.; Zhu, H.; Wang, D. Gold Nanoparticles in Cancer Theranostics. Front. Bioeng. Biotechnol.
**2021**, 9, 647905. [Google Scholar] [CrossRef] - Femminella, M.; Reali, G.; Vasilakos, A.V. A Molecular Communications Model for Drug Delivery. IEEE Trans. Nanobiosci.
**2015**, 14, 935–945. [Google Scholar] [CrossRef] - Muz, B.; de la Puente, P.; Azab, F.; Azab, A.K. The role of hypoxia in cancer progression, angiogenesis, metastasis, and resistance to therapy. Hypoxia
**2015**, 3, 83–92. [Google Scholar] [CrossRef] - Yang, Y.; Zheng, X.; Chen, L.; Gong, X.; Yang, H.; Duan, X.; Zhu, Y. Multifunctional Gold Nanoparticles in Cancer Diagnosis and Treatment. Int. J. Nanomed.
**2022**, 17, 2041–2067. [Google Scholar] [CrossRef] [PubMed] - Bharti, S.; Kaur, G.; Jain, S.; Gupta, S.; Tripathi, S.K. Characteristics and mechanism associated with drug conjugated inorganic nanoparticles. J. Drug Target.
**2019**, 27, 813–829. [Google Scholar] [CrossRef] [PubMed] - Kim, H.; Nguyen, V.P.; Manivasagan, P.; Jung, M.J.; Kim, S.W.; Oh, J.; Kang, H.W. Doxorubicin-fucoidan-gold nanoparticles composite for dual-chemo-photothermal treatment on eye tumors. Oncotarget
**2017**, 8, 113719–113733. [Google Scholar] [CrossRef] [PubMed] - Darweesh, R.S.; Ayoub, N.M.; Nazzal, S. Gold nanoparticles and angiogenesis: Molecular mechanisms and biomedical applications. Int. J. Nanomed.
**2019**, 14, 7643–7663. [Google Scholar] [CrossRef] - Carmeliet, P.; Jain, R.K. Molecular mechanisms and clinical applications of angiogenesis. Nature
**2011**, 473, 298–307. [Google Scholar] [CrossRef] - Gacche, R.N.; Meshram, R.J. Angiogenic factors as potential drug target: Efficacy and limitations of anti-angiogenic therapy. Biochim. Biophys. Acta Rev. Cancer
**2014**, 1846, 161–179. [Google Scholar] [CrossRef] - Yan, L.; Shen, J.; Wang, J.; Yang, X.; Dong, S.; Lu, S. Nanoparticle-Based Drug Delivery System: A Patient-Friendly Chemotherapy for Oncology. Dose Response
**2020**, 18, 1559325820936161. [Google Scholar] [CrossRef] - Yang, Z.; Xie, J.; Zhu, J.; Kang, C.; Chiang, C.; Wang, X.; Wang, X.; Kuang, T.; Chen, F.; Chen, Z.; et al. Functional exosome-mimic for delivery of siRNA to cancer: In vitro and in vivo evaluation. J. Control. Release
**2016**, 243, 160–171. [Google Scholar] [CrossRef]

**Figure 1.**Sketch of a three-phases process: (a) injection, (b) diffusion, transit, (c) arrival and internalization, of nanoparticles into blood stream. It is hypothesized the hybrid view by which the entire process would consist in electrodynamics, diffusion and global probability. The circle indicates that while angiogenesis has started, nanoparticles can also travel through the created vessels.

**Figure 2.**The theoretical efficiency of arrival up to for 3 different critical distances. While these distances are increasing, the efficiencies turns out to be reduced as a consequence of scattering of nanoparticles at the blood stream. Thus one observes a transition from Weibull (sc = 3.5 and sc = 4.0) to Gaussian distributions (sc = 5.0).

**Figure 3.**Contour plots of Equation (42) as function of argument $\sqrt{v/D\times s}$. (

**Left-side**): the case when $v=s/t\approx 0$ displaying the maximum efficiency of order of 12%. (

**Right-side**): the case with the approximation $\mathrm{sin}$($\kappa s$) is applied (see text below) displaying zones of a null efficiency due Coulomb effects at the injected nanoparticles. Arrows are indicating the transition of a null efficiency to one of order of 20%. Plots were done with package of Ref. [37].

**Figure 4.**Sketch for the probabilistic interpretation of Equation (42) by which it is argued that the events of internalization and rejection might to be dictated by the Bayes’s theorem. It should be noted that the hypoxic region is analogue to the case where nanoparticles are not arriving to tumor.

**Figure 5.**Contour plots of Equation (47) showing the apparition of nonlinearities despite the linear relation between time and space traveled by nanoparticles. Left-side: the case where the exponential was approximated to be sinusoidal sin(x). Here it is noted a linearity between the relation space-time (dashed line). Right-side: the case where it was opted by the sin${}^{2}$(x) expressing nonlinearity between space and time as consequence of electrical forces hypothetically due to either rejection or attraction. Plots were done with package of Ref. [37].

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Nieto-Chaupis, H.
Is Drug Delivery System a Deterministic or Probabilistic Approach? A Theoretical Model Based on the Sequence: Electrodynamics–Diffusion–Bayes. *Mathematics* **2023**, *11*, 4528.
https://doi.org/10.3390/math11214528

**AMA Style**

Nieto-Chaupis H.
Is Drug Delivery System a Deterministic or Probabilistic Approach? A Theoretical Model Based on the Sequence: Electrodynamics–Diffusion–Bayes. *Mathematics*. 2023; 11(21):4528.
https://doi.org/10.3390/math11214528

**Chicago/Turabian Style**

Nieto-Chaupis, Huber.
2023. "Is Drug Delivery System a Deterministic or Probabilistic Approach? A Theoretical Model Based on the Sequence: Electrodynamics–Diffusion–Bayes" *Mathematics* 11, no. 21: 4528.
https://doi.org/10.3390/math11214528