Myron Larrick1, Nasim Nosoudi1, Surinder P Singh2 and Jaime E Ramirez-Vick1*
1Department of Biomedical, Industrial and Human Factors Engineering, Wright State University, Dayton, OH, USA
2CSIR-National Physical Laboratory, Dr. K S Krishnan Marg, New Delhi, India
Received: February 09, 2019
Accepted: March 18, 2019
Version of Record Online: March 29, 2019
Larrick M, Nosoudi N, Singh SP, Ramirez-Vick JE (2019) Physicochemical Properties of Nanoparticles and Their Effect on Transport Across the Microvasculature. J Ann Bioeng 2019(1): 09-25.
Correspondence should be addressed to
Jaime E Ramirez-Vick
Copyright © 2019 Jaime E Ramirez-Vick et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and work is properly cited.
In the past couple of decades nanomedicine has become a well-defined area of interdisciplinary science, in which advances in nanotechnology are allowing the incorporation of multiple therapeutic, sensing and targeting agents into nanostructured materials. This is reflected by the large number of promising nanomedical formulations being currently evaluated by the US Food and Drug Administration. A problem with many, if not all of these, is that mathematical modeling, as a tool to narrow the design parameter space to obtain an optimal formulation, is not part of the design process. Adoption of these tool will bring nanomedicine closer to pharmaceutics, a field in which mathematical modeling has been an essential part of the design process for many years.
Nanoparticle delivery occurs mainly through the microvasculature, comprised of arterioles, capillaries, and venules ranging in diameter from 100 to 300 m, 5 to 10 m, and 7 to 50 m, respectively. The study of how nanoparticle design parameters affect their transport properties as it pertains to targeting and drug delivery has become a growing field in the development of nanomedical formulations. The physicochemical properties of nanoparticles, such as size, shape and surface charge can be optimized to improve their performance in these applications. For instance, it allows modulation of interactions with the immune system, better control over blood clearance and interactions with target tissue, permitting effective delivery of payload within cells or tissues. Their ability to target and enter tissues from the microvasculature is highly dependent on their behavior under blood flow. Here we present a review of the current approaches to mathematical modeling of nanoparticle transport across the vasculature and its effects on targeting efficacy.
Nanomedicine; Nanoparticle Transport; Physicochemical Properties; Vascular Targeting
Cancer has now been noted as the number one cause of death leading those from cardiovascular heart disease for anyone under 85 years old . Nanoparticles (NPs) in the size range of 1 to 100 nm have been effectively used to target tumors like Kapok’s sarcoma, breast cancer, Non-Hodgkin’s lymphoma, ovarian cancer, acute lymphoblastic leukemia, and myeloma cancer [2,3]. Since 1995, there have been 50 nanotherapeutic drugs approved by the US Food and Drug Administration with the global nanomedicine market expected to reach values of $177 billion by 2019, with markets involving in vitro diagnostics, in vivo imaging, drug therapy, and biomaterials. Liposomal drugs, which are spherical in shape, constitute 80% of this market [4,5], due to their biocompatibility, capacity for self-assembly and ability to carry large drug payloads. Some of these include non-PEGylated and PEGylated liposomes for the drugs DaunoXome, Myocet, Onco TCS, Doxil/Caelyx, Oncaspar, etc., among others [3,6]. It has been shown that NPs that are discoidal tend to marginate toward the vessel wall as compared to the quasi-hemispherical and spherical NPs . It has been seen that the biodistribution of spherical silica particles 700 nm to 3 m in size do not accumulate in non-reticulo-endothelial organs, with increasing NP diameter, when compared to discoidal particles . It has been seen in the pulmonary vasculature of mice that anti-ICAM spheres 0.1, 1, 5 and 10 m in diameter were endocytosed quicker than with delivery using anti-ICAM disks 0.1×3 m2 in size, which remained longer in circulation and were more successful targeting the lung versus the liver . Size is a critical parameter in determining NP fate. It has been also shown that delivering the drug methotrexate in 5 nm polyamidoamine-based G5 dendrimers is 10 times more effective targeting tumors than the free drug, with lower toxicity effects . It should also be noted that NPs with diameters smaller than 100 nm have longer circulation time than those larger than 100 nm . NP pharmacokinetics are greatly affected by geometrical factors, for instance high aspect ratios in rods/tubes have been noted to show lower macrophage uptake than spheres due to the large curvature angles, thus allowing longer circulation times .
There are many factors that have been investigated that affect the physicochemical properties of NP transport across the microvasculature, which include the shape, size, surface density, surface charge, bulk concentration, length of hydrodynamic forces, blood circulation half-life, and diffusivity [4,7,13]. Nanoparticles with different shapes, including spherical, rod, and disk are discussed in this review. Special consideration must be taken when designing drugs for particular applications. This review presents findings from mathematical modeling completed over the last 10 years for discussions on NPs traversing the microvasculature (Table 1). It outlines findings focusing on results showing shape and size in relation to the NP’s binding affinity. It also compares size and shape to currently approved FDA drugs made in a NP formulation compared to the results from the models.
|Model Assumptions||Governing Equations||Observations/Results||References|
|Navier-Stokes, laminar shear flow||Small NPs adhesion limited to number of ligand-receptor bonds; larger NPs adhesion related to the force to interaction, Ac. Ellipsoids with aspect ratio of 2 adhere more effectively than spherical NPs|||
|Brownian motion, equation upon total magnetic force on the NPs||βvf,100 is the volume fraction of magnetic NPs required to be in the carrier NP to be captured by the center of the magnet; βvf,100 is approximately 1/d showing that with increased carrier particles fewer magnetic particles need to be contained in the volume fraction; model revealed not NPs can be captured|||
|Adhesion Model developed for Ligand-receptor pair to follow a Hookean elastic spring||Model showed that for a 50% predicted probability of adhesion with increased NP diameter there is a need for an increased amount of ligands in order to bind to the vessel wall; Design maps were created to show where adhesion occurs and can provide how to design NPs based upon size, surface characteristics of ligand-receptor density and affinity and how to prevent NP endocytosed.|||
|Newton’s Law with Stokes Law applied for NP drag force; magnetic force applied to magnetic NPs||NP capture occurs quicker for increased NP size and increased magnetic flux|||
|laminar flow with Brownian, van der Waals, centrifugal, hydrodynamic, electrostatic forces; NPs follows Newton’s law of motion||Reynolds number less than 0.1 due to size of capillaries. Showed as Stokes number increases it is dependent upon size, density, rotation of NPs where Stokes number of 20 was determined to have maximum margination velocity; disc NPs with smallest aspect ratio of 0.20 marginated toward vessel better than ellipsoidal, hemispherical, and spherical NPs|||
|Navier-Stokes equations in COMSOL Multiphysics with laminar flow; fluid as non-Newtonian Carreau-Model||Larger NPs have a greater chance to adhere to the vessel wall with smaller NPs being swept away; model does not account for pressure changes in the vessel|||
|Non-Newtonian fluid model for magnetic NPs following Herschel-Bulkley fluid||Larger radius of NPs has a greater tendency to be captured by the tumor site; trajectory dependent upon NP radius, volume fraction, and magnetic force location.|||
|Navier-Stokes, laminar shear flow||Mid-sized Disks showed the best adhesion by 2× compared to rods and spheres showed lest ability to adhere. Hydrodynamic forces on the spheres and rods was shown to be greater than on the discs.|||
|2D finite element computational model; Brownian motion; Einstein-Stokes equation for diffusion; apply Fick’s law||Convection to diffusion model developed. Probability of adherence is higher for non-spherical NPs (oblate, rod and disc) compared to spheres where discs displayed the highest adhesion by 300×, rods by 20×, and oblate by 10×|||
|Brownian motion; Naiver-Stokes; incompressible fluid||Model captured effect of ligand-receptor binding. Model used aspect ratio of 3 and 5 where using 5 showed highest binding compared to spheres. Also, smaller NPs had stronger binding. Shear rates going past 2,000 only the rods with aspect ratio of 5 could bind to the vessel wall|||
|Agent based model, Navier-Stokes, laminar flow, fluid having non-Newtonian properties||Model noted polydispersity is important in designing NPs for targeting tumors showing smaller NPs had better uptake. Brownian motion negligible on NPs. Center of vessel has high laminar flow but near the wall lower flow velocity allowing greater NPs to adhere to the wall. Cellular trafficking is incorporated into the model with fenestrations as large as 240 nm for those with tumors|||
|Course-grained model incorporating dissipative particle dynamics||NPs 2, 4, or 6 nm in diameter were not affected by shear rates 0-2000 s-1. Model demonstrated larger NPs that were rod shaped compared to spherical in shape have increased binding abilities. Th model showed the bigger the difference between the ligand-receptor and receptor-solvent the stronger the NP adhesion.|||
|Navier-Stokes, nonlinear flow, Brownian motion||Flow is modeled as nonlinear partial differentials including magnetic force effects of magnetic NPs; as magnetic field is increased from 1 to 5 kOe the NP velocity decreases from 0.05 to 0.027 cm/s and the fluid velocity decreases from 0.07 to 0.036 cm/s along the vessel wall|||
Table 1: Comparison of Computational Models.
Decuzzi, et al., modeled the effect of shape, size and adhesion strength within a laminar shear flow . This model developed the relationship between the optimal volume Vopt at which the NP had the maximum capacity to be adhere to the surface of receptors as given as:
where the Vopt is defined as the ratio of surface receptor density, mr, to wall shear stress, S, and the coefficient α is strong function of the aspect ratio γ. For the spherical particles (γ=1), the coefficient α is a function of the separation distance δeq, while the coefficient β is not. As the wall shear stress decreased, or the surface receptor density increases there is a noticeable increase in the Vopt leading to increased probability of adhesion or adhesion strength. The model follows a stochastic approach, with the assumption that NP would follows the probability adhesion, Pa, distribution:
where a is the characteristic NP size, ro is radius of the NP, the aspect ratio is noted as γ, the equilibrium separation distance between NP and substrate is δeq, ml and mr are the ligand and receptor surface density, λ is the bond length between the ligand and receptor, F is the force, T is the torque between the ligand and receptors, kBT is the Boltzmann thermal energy, and the affinity constant is given as .
For small NPs the adhesion is limited by the number of ligand-receptor bonds, while for large NPs adhesion is related to the force to contact area, Ac. The model assumes a wall shear stress in the range of 1 to 10 Pa, with surface receptor density of 1014 m-2, and an aspect ratio of 1 for spherical NPs, giving an optimal volume of 4.4×10-4 m3, which translates to NPs in the range of 100 to 500 nm. For non-spherical particles using an aspect ratio of 2, the optimal volume would be 3.5 m3, leading to greater drug delivery over spherical NPs. It was shown that increasing the aspect ratio increased the probability of adhesion for a given NP volume. This shows that non-spherical (ellipsoids) particles adhere more effective than spherical ones in delivering their payload. For each aspect ratio, there is a particle Vopt for which the adhesive strength has a maximum. Therefore, this model could potentially be used to estimate adhesion throughout the body if the S and mr parameters were known as related to V and γ and lead us to design optimal NPs for intravascular delivery.
Decuzzi et al., derived the probability of adhesion, Pa(f), of the NP following the model of a Hookean elastic spring for the ligand to receptor pair with respect to the hydrodynamic and torque forces, as derived from the equation below as:
where the β is the ratio of the number of ligands, Nl, to receptors, Nr, over their respective densities,
is a dimensionless number regarding the NP surface binding, is the dimensionless NP radius, is the dimensionless radius of circular area of interaction in the vessel with the NP, is the dimensionless area of adhesion, and is the dimensionless separation distance between the NP and substrate surface .
The model showed that for a 50% predicted probability that with increased NP diameter there is a need for an increased number of ligands in order to bind to the surface as hydrodynamic forces increase. The design maps created show three regions where adhesion and endocytosis occur, adhesion and no endocytosis occur, and where no adhesion occurs. These design maps can be used to provide information on design of NPs based upon size, surface characteristics of ligand to receptor density and ligand to receptor binding affinity and how to prevent NPs from being endocytosed. It was revealed that adhesion increases with the surface density of the receptors and ligands.
Gentile et al., developed a mathematical model the flow of NPs here with radius a = 200 nm in capillary vessels with treating blood as a Casson fluid where the blood flows with a plug zone of no radial velocity in the center and a parabolic velocity outer zone [27,28]. Immune and red blood cells flowing within these vessels remain within the central plug region, with the remaining annular volume in contact with the vessel wall being known as the cell-free layer. The governing equations for this model are as follows:
represents the fraction of the vessel flow area occupied by the central plug region. Here the model considers effective longitudinal diffusion where C is the solute concentration, u(r) is the fluid velocity with mean velocity defined as U, Q is the flow rate, and Dm is the solute Brownian diffusion coefficient in a quiescent fluid. A(ξc) is unity when the rheological parameter ξc, the ratio between the plug radius and the capillary radius, goes to zero, bringing the volumetric flow rate and velocity to follow the Poiseuille flow. In the capillaries there is cell free layer near the wall as red blood cells tend to flow in the center of the vessel. Here it is assumed the one-dimensional Casson velocity distribution, since the lateral flow across the capillary walls only affects the flow rate. This model analyzes the transport of the NPs accounting for the permeability across a capillary vessel by the Deff in relation with the blood rheology. The pressure drops, ∂p/∂z , in the capillary is shown below, where Lp is the hydraulic conductivity, Πi is the interstitial pressure, Γ, Π, and are the permeability parameters of the vessel through the equations shown below.
Here the model establishes that Deff is affected by the molecular diffusion coefficient, Dm = 6.1×10-13 m2/s, and a convection term related to Pe0, the Peclet number at the inlet. The temperature T is set at 300K, with Boltzmann constant kB = 1.38065×10-23 J/K. For impermeable vessels, Π = 0 and Γ(ξc) = 0. In the cell-free layer for impermeability of the solute at the walls, the boundary conditions are set as:
The model showed laminar flow in the capillaries, venules, and arterioles reaching Pe numbers of 833-41667, (1.66-4.16)×104, and (1.67-8.33)×105, respectively, where the red blood cells formed into parachute-like configurations showing Newtonian fluid seen in large vessels is not experienced within the microvasculature closely following Casson law. From this model in the microvasculature it showed that as the NPs moved from larger to smaller vessels, there was reduction in Deff with increased rheology of the blood, ξc, and an increase in Π, creating a physical barrier to the delivery of NPs. The barrier to NP delivery becomes stronger with increasing Dm as NPs follow paths of high Deff, and thus with large Pe numbers. Thus, it is postulated here to design NPs with marginating abilities to go towards the cell-free layer. The model specifies margination can be determined from the physicochemical properties of NP size, shape and surface density.
Lee et al., formulated a mathematical model considering NPs as spheres, ellipses, discs, and hemispheres within laminar flow conditions with external forces of Hydrodynamic, centrifugal, electrostatic, van der Waals, and Brownian acting on the NPs . Here hydrodynamic forces showed the strongest effect while centrifugal forces were shown to be almost negligible. The governing model considered the flow of particles moving in orientation along the x-axis with the conservation of mass (19) and momentum (20) equations expressed respectively as:
Here the u is the fluid velocity, ρf is the fluid density, is the fluid dynamic viscosity and p is the pressure in the fluid.
The NP was described to adhere to Newton’s law of motion, (mp mass of particle and mf mass of fluid) (21), rotation (22,23), move with drag force, D (24) and resistance, R (25) as shown below as:
Here the boundary condition of u = 0 is set at the wall, F is the hydrodynamic force and buoyancy force displaced on the particle surface ∂S, σ is hydrodynamic stress tensor, n is unit normal to the particle, L is the moment inertia tensor, is the angular velocity, T is the hydrodynamic moment vector, g is gravitational acceleration, U is particle velocity vector, X is the particle position vector, and a is the radius of the NP.
This model assumes a low Reynolds number of Re ≤ 0.1, consistent with what is found in capillaries, venules and alveoli vessels. An important parameter for describing NP behavior in blood vessels, is the Stokes number (St), which is a dimensionless parameter characterizing the behavior of particles suspended in a fluid flow. As St increases its dependent upon size, density and rotation, which in turn results in the NPs having increased lateral drift velocity towards the wall where a maximum margination velocity at St ~ 20 was determined in the study. Whereas it was seen that the NPs has a minimal margination occur at St equal to 0.1. It was noticed that disc-shaped NPs with the smallest aspect ratio of 0.20 tended to marginate towards the vessel better, when compared to the other shapes, such as ellipsoidal, hemispherical, and spherical. Here the NPs have been shown to increase margination speed, as buoyancy forces are increased through the vessel. Here external hydrodynamic forces need to be applied within the microvasculature for margination to occur as the shear rate is below 100 s-1 and non-spherical NPs tend to oscillate as they approach the desired targeted site.
In nanomedicine, the development of magnetic NPs and techniques for their transport and concentration at specific sites in the human body using magnetic forces has shown great interest for many years [29,30]. Furlani et al., presented a mathematical model to predict magnetic trajectory in the targeting of NPs noninvasively with dominant magnetic, Fm, and fluidic, Ff, forces . Application of the NPs would have the drug injected above the tumor location and then apply a magnetic field at the site concerned with treating. Here the model predicts the volume fraction of magnetic NPs required in meeting the targeted site. This model considers the blood hematocrit (45%) and flow rate, microvasculature vessel parameters, the NP size, and the magnet properties. The model included Brownian motion on the NPs, particle and blood to cell interactions, buoyancy, magnetic and gravitational forces, magnetic dipole interactions, and viscous drag. The total magnetic force on the NPs is expressed as:
where Ha is the magnetic field applied to the center of the carrier NP, Hx and Hz are represented as the magnetic force inside the microvasculature vessel, mp is the susceptibility and permeability of the magnetic NPs, is equal to 4∏×10-7 H/m the permeability of the air, η (0.0012 Ns/m2) and υf (10 mm/s) is viscosity and velocity of blood, Rcp is hydrodynamic radius of the carrier particle (200-1,000 nm), Rmag is 3 cm (NdFeB - field source), and Rmp is the radius of the magnetic particle embedded within the carrier particle. Now with the assumptions that x/d << 1 since the distance from the magnet to the vessel is larger than the vessel diameter and that χmp >> 1 for Fe3O4 (biocompatible magnetic materials commonly used) the magnetic forces in the x- and z-axis respectively as Fmx and Fmz reduces to:
and where the υf within the microvasculature is expressed as:
where is the average blood velocity, and Rbv is vessel radius (i.e., 50 m). While the equation of motion of the carrier NP was given as:
The volume fraction of magnetic particles was derived under the assumption that -Rbv < x0 <Rbv given as:
It should be noted that the βvf,100 is the volume fraction of magnetic NPs required to be in the carrier NP to be captured by the center of the magnet. Also, close examination shows that βvf,100 is approximately 1/R2cp, showing that with increased carrier particles, fewer magnetic particles need to be contained in the volume fraction. In addition, βvf,100 is close in approximation of 1/d showing that as the tumor site is closer to the body surface the volume fraction of magnetic NPs can be reduced. The model revealed that not all carrier NPs can be captured.
Sharma et al., proposed a mathematical model where a magnetic field is applied to understand the effects of magnetic particles, to reduce reaching healthy cells during delivery . Navier-Stokes equations are established as nonlinear partial differential equations, which include effects from magnetic forces, viscous drag, particle and blood-cell, inertia, buoyancy, gravity, Brownian motion, particle-fluid, and magnetic dipole effects. It demonstrates that NPs have the maximum velocity at the center of the vessel. It was also shown that as the magnetic field is increased from 1 to 5 kOe, the NP velocity decreases from 0.05 to 0.027 cm/s and fluid velocity decreases from 0.07 to 0.036 cm/s along the vessel, respectively.
Babincova et al., proposed a finite element method to solve the model for the trajectory of magnetic NPs with a 50 nm radius in a magnetic field consisting of neodymium rare earth magnets with magnetic energy of 37 MG.Oe for a viscous fluid following Newton’s law :
where mp is mass and vp is the velocity of the NP, Fm is the magnetic drag force, and Fs is the Stokes drag force, mp,eff is the effective dipole moment on the particle, Ha is the external magnetic field intensity occurring at the NP center, and Vp and Rp are the magnetic particle radius and volume respectively . This model applied Stokes’ law for drag on spherical NPs as:
where vf (velocity) is considered 0 m/s for an ambient quiescent fluid. The model showed the NPs could be captured faster when the particle size was increased. It was also found that by increasing the magnetic flux will decrease the time it takes for the magnetic NPs to be captured.
Heidsieck et al., developed a model for tracking magnetic NPs through blood flow using hydrodynamic forces where the particle trajectories are modeled with external magnetic forces (Fmag) consisting of NdFeB N50 Neodymium magnets applied to accelerate the magnetic NPs traveling in the direction of the magnetic force until it adheres to the vessel wall . Buoyancy and inertia forces are present but considered negligible. In addition, Brownian motion is present but considered negligible in this model as well. The model assumed a vessel diameter of 1.2 mm with length 30 mm. The magnetic forces acting on the NPs are expressed as:
where the is the magnetic dipole moment and B is the magnetic flux density field. The fluid flow was modeled using Navier-Stokes equations in COMSOL Multiphysics modeling software with laminar flow occurring at 0.2 m/s. Flow around the NPs was defined using the Stokes drag force on spherical NPs given as:
where rnp is radius of the NP, γ is the shear rate, η is the viscosity of the fluid modeled here as a non-Newtonian Carreau-Model . The magnetic NP trajectory is defined as follows:
where v is NP velocity and g is the gravitational force. Here the NP diameter is given by 2rnp = 100 nm with mass m = 2× 10-19 kg with the magnetic moment of the NPs following volume linearly with = 5×10-13 A·m2. The model showed that the NP size was not influential on the final location of the NP, but the size did determine if the NP would adhere to the vessel wall, showing that larger NPs were more likely of getting trapped, while smaller NPs would be carried away. The model does not account for any motion caused by blood pressure changes within the vessel.
Shaw et al., developed a non-Newtonian fluid model for magnetic NPs (Fe3O4) in various sizes with the radius of the carrier NP Rcp = 300 to 1000 nm traveling in blood flow, following the Herschel-Bulkley fluid in micro-vessels with 50 m radius . The applied magnet for this model had a diameter of 6 cm with magnetization Ms = 106 A/m, where it was kept 2.5 cm away from the vessel axis. The size and volume-fraction of the NPs along with the vessel diameter affected the magnetic targeting of the NPs. The NP is affected by the magnetic and fluid forces and the particles motion is given by Newton’s law:
where mcp and vcp are the mass and velocity of the NP, while Fm and Ff are the magnetic and fluid forces, respectively. Here the inertia term is negligible making the force balance as:
Following the Herschel-Bulkley for the fluid the fluid force on a spherical NP and the blood flow laminar the blood velocity can be written as:
where (1-ξc)1+1/n for r < Rc and (1-ξc)1+1/n - ((r/Rv) - ξc)1+1/n for r > Rc, where Rc is the radius of the core region of the vessel, is the non-dimensional radial coordinate, Rv and lv are the vessel radius and length, respectively, Qis the volumetric flow rate, and is the mean velocity of the blood flow. When ξc goes to zero and n = 1, then the A(ξc) terms goes to 1 and follows Poiseuille flow. It was found that the volume fraction decreased for the magnetic NP when the radius of the carrier NP and the non-dimensional parameter ξc is increased. The model showed that the larger the NP, the better at reaching the tumor site. It was also found that when the rheology of the blood is thickened, the carrier particle would have a greater ability to be captured by the magnetic force. Here the trajectory was dependent on the NP radius, volume fraction, positioning of the magnet along the vessel, and blood rheology.
Adriani et al., modeled rods and disks using both an experimental flow chamber and computational modeling . The rods were made using an aspect ratio of 4.5 to keep them with a similar loading volume to the disks. The disks ranged in sizes (diameter × height) from 600 × 200 nm, 1000 × 400 nm, and 1800 × 600 nm. The rods ranged in size from 1500 × 200 nm to 1800 × 400 nm. The experimental method conducted utilized mesoporous silicon particles passed through a flow chamber with fluid dynamic trajectories captured using a fluorescence. The results showed that the disks adhered more than the rods by a factor of 2 with significance of shear to 100 s-1. The computational model assumed an incompressible fluid and unsteady flow and applying laminar flow conditions with the Navier-Stokes equation reduced to:
where the u is the fluid velocity, p is the pressure, f is the mass force, Re is Reynolds number, ρref is the reference density, uref is the reference velocity, dref is the reference length.
The hydrodynamic forces (e.g., shear rate) on NPs was observed to be within the range of 10 to 200 s-1. It was found that as the shear rate at the wall increases, the NP adhesion capacity decreases. It was shown that the hydrodynamic forces on the rods was somewhat greater than on the disks. The disks at 1000 × 400 nm showed the most adhesion, when compared to the experimental results. The computational model demonstrated that the hydrodynamic forces affect spheres more when compared to the rods and disks, as can be seen by spheres showing the lowest adhesion, followed by rods and disks, respectively.
Liu et al. presented a 2D finite element computational model incorporating NPs in Brownian motion with flow velocity, U (0-25 dyne/cm2), towards adhesion with ligand to receptor binding in the microvasculature 5 m long and 2 m in diameter . Here the convection to diffusion, D (10-9 m2/s), governing equation for concentration, c (c0 = 1000 mol/m3), regarding NP with radius, r, within the microvasculature is expressed as:
where the Boltzmann constant kB, T (300K) is the temperature, is the viscosity. The model incorporates the Einstein-Stokes equation for D, as shown above to solve the mathematical expression for the dissolution of drug NPs. With the term ∂cs/∂t , the model incorporates a material balance equation to account for the NP ligand to receptor binding on the vessel with a rate of attachment ka (10-6 m3/mol·s) and detachment kd (10-3 to 10-6 s-1), where cs represents the surface concentration of adsorbed species (mol/m2), Ds (10-11 m2/s) is the surface diffusivity, cw is the bulk concentration (mol/m3) of NPs at the vessel wall, and θ (θ0 = 1000 mol/m2) is surface concentration. The convection to diffusion equation is coupled to the material balance through Fick’s law being expressed as:
The NPs are modeled following Brownian motion given by velocity (i.e., Vs for solid and Vf for fluid) and friction β and expressed as:
where, the correction factors are defined as
The model incorporates the fluid flow as an incompressible fluid with the following Navier-Stokes equation where vf is the fluid velocity:
The model shows a depletion layer when the drug concentration is reduced as the NP adheres to the receptor surface. Here when the flow rate goes from 0.1 mm/s to 1 mm/s the depletion layer becomes smaller. The model incorporated the probability of NP adhesion, Pa, given as:
where mr and ml are the receptor and ligand density respectively, Ac is the contact area of the NP, f is the force per unit ligand to receptor, kBT is the thermal energy of the system, λ (20 nm) is the characteristic length of ligand to receptor bond, is the affinity ligand to receptor constant at zero load. Results showed that for spherical NPs, Pa increases until 200 nm, it then decreases due to the larger volume of the NP. The shape for non-spherical NPs (oblate, rod, or disc) have a Pa substantially higher than for spheres, with discs displaying the highest adhesion at 300× larger, rods at 20×, and oblate NPs 10× higher adhesion efficacy when compared to spheres.
Tan et al., modeled NPs traversing in Brownian motion through bifurcation vessels with parent diameter of 2 m and daughter diameter of 1 m . The model examined the effects of NP spheres between 100 and 200 nm and rods 63 × 189 nm and 126 × 378 nm, with an aspect ratio of 3, and 52 × 261 nm and 104 × 522 nm, with an aspect ratio of 5. The fluid was treated as an incompressible viscous fluid following Navier-Stokes with the NPs having ligand to receptor, Nb, binding along the vessel surface, moving with velocity, vp with the fluid velocity, vf, are governed by the equations below as:
The velocity of a NP moving under a deterministic force is given by
where the friction coefficients derived from Stokes’ law are
With the receptor-ligand binding process can be described by
where kf is forward binding rate, N1 and Nr is the ligand and receptor density respectively, is the fluid viscosity, d is the particle diameter, βt and βr are translational and rotational friction coefficients, ωf is the angular velocity, Vf is the translational velocity, and Fdet and Tdet are the deterministic force and torque experienced by the NP.
The model focused on how the NPs will target a tumor site by having ligands coated on the NPs and they affected the binding on the receptors of the tumor site with a probability of adhesion, Pa, equivalent to equation (2) from  as:
where is the association constant at zero load of the ligand to receptor pair, λ is the bond length of the ligand to receptor, mr and ml are the receptor and ligand surface density respectively, kB is the Boltzmann constant, T is the temperature, Ac is the contact area between the ligand and receptor, Fdis is the dislodging force of the NP based upon hydrodynamic forces.
It was demonstrated that rod-shaped NPs will bind based on orientation to the vessel wall and inversely proportional to shear stress, with a decrease in binding with shear rates due to increased drag force. Rod-shaped NPs with aspect ratio of 5 showed the highest binding capacity compared to spheres or lower aspect ratio rods. It was shown that smaller NPs (i.e., 100 nm versus 200 nm), tend to show stronger binding capacity due to a lower drag force. In bifurcation areas it was seen that NPs tend to accumulate at higher levels, when compared with straight vessels. It was noticed that spheres cannot bind at shear rate above 1,200 s-1 and that shear rates of 2,000 only the NP rods with aspect ratio of 5 were able to bind. Thus, adhesion is affected by flow rate, vessel size, NP geometry, and NP velocity through the fluid.
Hence, the Peclet number (Pe) seen in Equation (16) was noticed to affect the adhesion ability with length as shown as:
where L is the vessel length (0.1 m), D is the NP diffusivity coefficient (4.4×10-12 m2/s and 2.2×10-12 m2/s) for 100 and 200 nm NPs, respectively, and U is the fluid velocity.
Fullstone et al., developed an agent-based mathematical model of NPs being transported in capillaries at 8 m in laminar blood flow to provide method showing how NP size is possible to predict reaching the targeted site .
The governing equations established for momentum, mass, and energy using Navier-Stokes for the fluid flow are shown as:
where ρ is the density, δt is the step time, is the del operator, v is the velocity, p is the pressure, T is the total stress tensor, F is the force, Cp is the specific heat capacity at constant pressure, T is the absolute temperature, q is the heat flux, S is the rate of strain tensor, and Q is the heat sources. Here the Reynolds number (Re) and diffusion coefficient related to the vessel and NPs as shown by equations (57) and (17), respectively. Here in the capillaries the Re is approximately 0.001 due to the high viscosity and length of the vessels. For the laminar flow, the vessel walls experience reduced flow due to the friction and a flow profile that is parabolic in shape as fluid velocity increases. With the expected hematocrit being 10-20% in capillaries, the model used 10.7%, focusing on red blood cells, and assuming a minimum effect of white blood cells and platelets due to the low level expected. Due to the size of red blood cells being about 7.8 m the fluid adopts non-Newtonian properties, resulting in unpredictable shear stress in-turn affecting the Navier-Stokes equations. Here in this model the red blood cells take on the parabolic`` form as it traverses the capillaries following a Poiseuille flow distribution. An agent-based approach provides analysis for the different specific blocks within the system being investigated and then compiles the blocks together for review. The model considered several interactions to include the receptor binding, ligand density effects, and cellular trafficking. The model assumed that the Brownian motion of NPs is considered inert, since it has been shown that proteins tend to play a role in the movement of the NPs in the blood. Here cellular trafficking is incorporated into the model, adopting fenestrations with 60 nm in pore size for healthy vessels and 240 nm for those having tumors and a leaky microvasculature. Here the model showed that NPs in the 50 to 100 nm range, traversed the vascular wall more efficiently, targeting the tumor site, while those in healthy vessels continued to pass through the vessel, agreeing with the Enhanced Permeability and Retention (EPR) effect. The finite element was checked using heat maps for velocity flow, showing velocity was reduced as flow approached the vessel walls and it increased around the red blood cells. The model revealed that the Mean Squared Displacements (MSD) on NPs due to force effects by Brownian motion are almost negligible in contrast to laminar flow on 100 nm NPs, as the center of the vessel revealed laminar flow with v = 550 m/s but the MSD effects near the wall with v = 440 m/s showed the force due to Brownian motion had very comparable force effects on the NPs as from laminar flow permitting adhesion of NPs to the vessel wall. It was shown in the model that red blood cells aid in the distribution of NPs, with binding occurring within 20 nm of the vessel wall. The model showed that NPs pass through the vessel wall with low dispersion if they are close to the vessel wall and with high dispersion when traveling from the vessel center to wall. The model showed that polydispersity is important in designing NPs for targeting tumors when considering the EPR effect, showing that smaller NPs have a greater uptake.
Peng et al., developed a coarse-grained mathematical model incorporating Dissipative Particle Dynamics (DPD) to study the effects of binding spherical NPs to vessel walls, while considering shear flow . Here DPD is used to model the hydrodynamic interaction of the vessel and NPs regarding adhesion under shear flow situations where various aspect ratios are compared. Here NPs had ligands attached to the NPs and receptors attached to the vessel surface. The model observed that shear rates that ranged between 0 - 2000 s-1 were not affecting the ability of NPs 2, 4, or 6 nm in size to adhere to capillaries. The model compared rods with aspect ratios of 5, 10, and 15 and NPs with diameters of 2, 4, and 6 nm. The model demonstrated that larger spherical NPs and rod-shaped NPs with larger aspect ratios, have increased binding capacity in relation to the time it takes for adhesion to take place, confirming the importance of Brownian motion in this process. The model showed that the bigger the difference between the ligand to receptor and receptor to solvent repulsion, the stronger NP adhesion. It was shown that a 2 nm NPs adhere strongly with a difference of 20 at 1,000 s-1. In DPD simulations, the binding factor Δa is defined as the difference between the ligand-receptor repulsion factor and the receptor-solvent repulsion factor. The model showed that for a weak Δa of 5 to 11, the NPs would remain within the middle of the flow while at larger Δa of 13 to 25, binding occurred faster with a linear shift in improvement going from a 10 to 30% linear increase to almost 100% linear when the difference is at 11, representing the point where NP vessel wall adhesion becomes stable.
Physicochemical properties, such as size, shape, and surface charge, are key design parameters, which optimized to improve NP targeting efficacy. This have been shown by in vivo studies in which variation in these parameters influence underlying processes which affect the performance of these nanotherapeutics [32-34]. In this section, we show the effect of size, shape, and surface charge on NP targeting efficacy, based on in vivo results.
The properties of materials change as their size approaches the nanoscale, this includes melting point, fluorescence, electrical conductivity, magnetic permeability, and chemical reactivity. The variation in these properties as a function of size can be put into practical use in nanomedicine. For instance, at this size scale something called quantum confinement occurs, in which the bandgap varies inversely with size, resulting in electrons and holes being confined into a dimension that approaches a critical quantum measurement, called the exciton Bohr radius . These quantum effects allow the tuning of the NP properties, such a changing the color it fluoresces when excited by changing its size, thus allowing the development of multiple unique labels. In addition, as the size of these NPs decreases, the surface area per volume increases, affecting the amount of material it can interact with, in the physiological environment it is immersed, thus making them more chemically reactive, also affecting their strength and electrical properties . This increase in surface area per unit volume increases the NP contact with the biological environment, affecting its capacity to interact with cells and tissues, circulation time, accumulation and penetration in the tumor, the cell entry mechanism used, and the adsorption of proteins forming the corona .
Size significantly influences the blood circulation time and the biodistribution of the NPs, with smaller size having longer circulation times and reduced accumulation in the liver and spleen . In the case of NP accumulation and penetration in the tumor mass, these leak out leak into tumor tissue through permeable tumor vessels and are then retained in the tumor bed due to reduced lymphatic drainage as a result of the enhanced permeability and retention effect . Their accumulation is significantly increased with size, while their penetration decreases with size . When NPs are exposed to a biological environment, proteins start to adsorb on the surface in a competitive manner forming the protein corona. This corona changes over time due to the Vroman effect, where proteins in the environment adsorb on to the NP surface and are replaced over time with other proteins with a higher affinity for the NP surface . It alters the size and interfacial composition of a NP, affecting how these interact with their environment, and specifically cells. This change in size and interfacial composition affects the physiological response including agglomeration, cellular uptake, circulation lifetime, signaling, kinetics, transport, accumulation, and toxicity. The composition of the protein corona is unique to each nanomaterial and depends on many NP parameters (i.e., material, surface, size, charge, and shape) or environmental parameters (i.e., composition, exposure time, pH, temperature, and shear stress) [41,42].
Along with size, NP shape is another key parameter influencing NP behavior in circulation (i.e., fluid dynamics, margination towards the vessel wall) and cell-NP interactions (i.e., binding, internalization, clearance) [42,43]. Although various NP shapes have been synthesized (e.g., spherical, triangular, cubic, rods, platelets, etc.), the NPs under preclinical or clinical studies, are spherical in shape . The aspect ratio is a parameter used to describe the shape of non-spherical NPs, and which has been shown to influence their margination dynamics via lateral drifting toward the blood vessel wall . This hydrodynamic behavior deviating from spherical NPs could be the reason why non-spherical NPs of many shapes have shown extended circulation times, when compared to spherical NPs .
Studies involving macrophage uptake or phagocytosis as a function of NP shape suggested that it plays an important role. Specifically, the point of contact between the NP and macrophage, determining if these are capable of forming the necessary actin structures to initiate phagocytosis, or if the NPs will simply spread over the cell membrane . In addition, mammalian epithelial and immune cells preferentially internalize disc-shaped NPs with high aspect ratios compared with nanorods and lower aspect-ratio disc-shaped NPs. While endothelial cells prefer disc-shaped NPs of an intermediate aspect ratio . With epithelial cells internalizing the NPs using the caveolae-mediated pathway, the human umbilical vein endothelial cells, used a clathrin-mediated mechanism.
The effective electric charge on the NP surface is measured by the zeta potential, which describes the electrostatic interactions of cells and NPs in a fluid environment, with values of ±30 mV representating stable NPs [48,49]. NP internalization into the cancer cells is promoted with a positive surface, since the surface charge of tumor cells is highly negative compared to healthy cells . The NP surface charge also affects the surface binding of serum proteins during corona formation, their selective adsorption on cellular membranes, and transmembrane permeability. As it was mentioned before, the protein composition of the NP corona is dynamic, changing with time, making the NP surface charge also vary with time [41,42]. Current attempts to control the effect of the protein corona on NP on targeting includes surface modifications to control protein adsorption on to the NP surface (i.e., surface charge, hydrophobicity, and smoothness) [40-42].
FDA approved Nano therapeutics benefits have been shown to reduce drug toxicity, provide longer half-life in circulation by reducing immunogenicity and releasing at projected delivery rates in turn reducing the number of drugs taken. Determining the correct physicochemical properties needed is crucial to the targeting and enhancement of drug efficacy. It is therefore important is design the NPs used for medical drug treatment with optimization to further enhance the efficacy already being witnessed by NPs. A major limitation of current modeling approaches is the lack of consideration of the protein corona. Future mathematical models will greatly improve their capacity to predicts NP targeting and drug delivery efficacy, once the dynamic nature of the protein corona is considered. This review focused on the progress development of mathematical models used to predict the ability of a NPs trajectory, adherence in relationship to its physicochemical properties, such as size, shape, and surface charge. Providing valuable design insight regarding the importance of tailoring and optimizing NP physicochemical properties to enhance their margination and wall-adhesion during vascular targeted drug delivery applications.
All authors declare no conflicts of interest in this article.
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