The Self-Adaptation Ability of Zinc Oxide Nanoparticles Enables Reliable Cancer Treatments

Optimal procedures for reliable anti-cancer treatments involve the systematic delivery of zinc oxide nanoparticles, which spread through the circulatory system. The success of these procedures may largely depend on the NPs’ ability of self-adapting their physicochemical properties to overcome the different challenges facing at each stage on its way to the interior of a cancerous cell. In this article, we combine a multiscale approach, a unique nanoparticle model, and available experimental data to characterize the behavior of zinc oxide nanoparticles under different vessels rheology, pH levels, and biological environments. We investigate their ability to prevent aggregation, allow prolonged circulation time in the bloodstream, avoid clearance, conduct themselves through the capillarity system to reach damaged tissues, and selectively approach to target cancerous cells. Our results show that non-functionalized spherical zinc oxide nanoparticles with surface density N = 5.89 × 10−6 mol/m2, protonation and deprotonation rates pKa = 10.9 and pKb = −5.5, and NP size in the range of 20–50 nm are the most effective, smart anti-cancer agents for biomedical treatments.


Nanoparticle Model: pKa and pKb Calculation
We use the Pivovarov formula to calculate a surface site density of locations capable of protonation or deprotonation. The corresponding result is N = 5.89 × 10 −6 mol/m 2 , which is comparable in magnitude to the surface site density of other metal oxides such as silica oxide. [1] We notice that this large value of ZnO surface site density is in agreement with the findings of other work on material pH sensitivity [2].
Subsequently, we use the value for the parameter value N and assign different values for the parameters pKa and pKb, the protonation and deprotonation constants, respectively, into the complexation surface model introduced in reference 27 to predict values for ZnO NP ZPs (see Figure  S1). We find that the fitting values pKa = 10.9 and pKb = −5.5 reproduce all available experimental results on ZnO NP ZP [3][4][5][6][7]. These values fall within the wide range of values found in literature, which suggest values anywhere from −5.7 to 15.7 for pKa alone [2,3,8,9]. with the same experimental input (solid lines). Different colors represent different sources and experimental conditions: Black [10], Green [5], Blue [6], Brown [4] and Purple [11].

Nanoparticle Transport
We use Gentile et. al.'s expression [12] to calculate the effective diffusion of spherical particles where, with kB, the Boltzmann constant, T, temperature,  and N, the blood viscosity, P, the blood plasma viscosity, RNS, the nanoparticle radius, Pe, the Peclet number, Re, the vessel radius, Ue, the average velocity of blood in the vessel (provided in Table 5), z, the position along the axis within the vessel, l, the total length of the vessel, H, the hematocrit level of the blood, , the permeability of the vessel, and In the latter equations, Rv is the vessel radius, , the rheology of the blood, I, the interstitial fluid pressure, and, po and p1, the inlet and outlet pressures of the vessel, respectively. In this way, the greater the effective diffusion, the smaller the circulation time of the NP in the body because it is more readily coming in contact with tissues that will remove it from the blood stream. Thus, for a longer circulation time in the body, we would desire a NP with a low effective diffusion.

Long-Range Electrostatic Interaction
Once the NP approaches the tissues themselves, the long range electrostatic interactions between the NP and the cells becomes the driving force behind NPs approaching cells and discriminating between cancerous and healthy cells. This long-range interaction is modelled by Ohshima et al. [13] using a modified Columbic force to take the effects of the electrolyte into account. The interaction energy is defined as: where, and all other parameters were previously defined in this appendix: q is the charge of an electron, r and 0, the relative permittivity of the substance (water) and free space respectively, r1 and r2, the radii of the two particles involved in the interaction, r, the separation distance of the particles, , the inverse Debye length of the fluid, and s the surface potential of the particle.
When considering ( ) in a normalized form, ( ) , we obtain a simpler (dimensionless) expression which depends only upon the NP and electrolyte characteristics, with no relationship to the separation distance between the NP and cell, or to the cell itself. This expression reads where the subscript N represents the baseline NP upon which the normalization is to occur, namely, a ZnO NP of 4 nm in radius.

DLVO Theory
Trefalt and Borkovec [14] provide a method, based on the DLVO theory, for calculating the association rate k given a function of the potential energy of attraction U(h) between the two particles as a function of center-to-center separate ion distance h.
Ohshima [15] introduces a potential model which is appropriate for materials that react with their environments to exchange surface groups and modulate their surface charge, especially as they approach one another and those surface group's ability to be exchanged becomes inhibited [16]. The approach provides the following expression for the potential energy function with NA being Avogadro's number and The Hamaker constant H = H132 for the NP-NP and NP-Cell interactions may be approximated if some basic information for each medium is known [15,17,18]. e.g.
The cell membrane potential 2 is provided by several sources [19,20]. These sources note that cancerous cells generally have more negative potentials than healthy cells, however, there are many negatively charged healthy cells. For example, healthy skeletal muscles exhibit a potential of as low as −94 mV. On the other hand, healthy cells may additionally reach membrane potential values of nearly 60 mV, as in the case of normal fibroblasts. Thus, we let these extreme negative values represent a minimum potential for cancerous cells, we additionally let a value of +94 mV represent the absolute maximum potential of healthy cells.

Ion Count Calculations
Using normalized density profiles (r) consisting of the local molarity of an ion divided by the bulk molarity of the electrolyte, as a function of distance from the center of the NP, the total excess ion counts eic may be calculated [21]. Integrating over the domain of these figures, we get eic = With this equation, we receive a unit of molÅ 3 /L. To make this into the excess number of ions per molar of bulk electrolyte trapped by the EDL, we multiply it by the Avogadro's number and divide by a factor of 10 27 .

Stokes' Law & Bernoulli's Equation
Stokes' Law relates the velocity of a spherical particle in a fluid v (or the velocity of the fluid around the particle) to the drag force acting on that particle FD. Using this approach, the drag force a NP would experience in a convective current and, thus, the force required to pull the NP against such a current, can be approximated by where  represents the viscosity of the fluid, R, the NS, and v the relative velocity.
The fluid velocity may further be approximated using the Bernoulli expression modified to consider only a change in pressure versus a change in velocity at constant elevation and zero initial relative velocity.