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Ann Biomed Eng. Author manuscript; available in PMC 2014 January 9.

Published in final edited form as:

Published online 2012 August 2. doi: 10.1007/s10439-012-0630-4

PMCID: PMC3886728

NIHMSID: NIHMS539552

Triantafyllos Stylianopoulos,^{1} Konstantinos Soteriou,^{1} Dai Fukumura,^{2} and Rakesh K. Jain^{2}

Address correspondence to Triantafyllos Stylianopoulos, Department of Mechanical and Manufacturing Engineering, University of Cyprus, 1678 Nicosia, Cyprus and Rakesh K. Jain, Edwin L. Steele Laboratory for Tumor Biology, Massachusetts General Hospital, Harvard Medical School, Boston, MA 02114, USA. Email: tstylian/at/ucy.ac.cy, Email: jain/at/steele.mgh.harvard.edu

The publisher's final edited version of this article is available at Ann Biomed Eng

See other articles in PMC that cite the published article.

Despite their great promise, only a few nanoparticle formulations have been approved for clinical use in oncology. The failure of nano-scale drugs to enhance cancer therapy is in large part due to inefficient delivery. To overcome this outstanding problem, a better understanding of how the physical properties (i.e., size, surface chemistry, and shape) of nanoparticles affect their transvascular transport in tumors is required. In this study, we developed a mathematical model for nanoparticle delivery to solid tumors taking into account electrostatic interactions between the particles and the negatively-charged pores of the vessel wall. The model predictions suggest that electrostatic repulsion has a minor effect on the transvascular transport of nanoparticles. On the contrary, electrostatic attraction, caused even by small cationic charges (surface charge density less than 3 × 10^{−3} C/m^{2}) can lead to a twofold or more increase in the transvascular flux of nanoparticles into the tumor interstitial space. Importantly, for every nanoparticle size, there is a value of charge density above which a steep increase in transvascular transport is predicted. Our model provides important guidelines for the optimal design of nanoparticle formulation for delivery to solid tumors.

Nanomedicine is a promising modality for cancer detection and treatment.^{22,40} Advances in nanotechnology have led to the development of nanoparticles whose size, surface chemistry and shape can be easily controlled.^{6,29,36} However, only a few nanoparticle formulations have been approved to date for clinical use in oncology.^{22} Although the enhanced permeability and retention (EPR) effect has served as a key rationale for the use of nano-scale drugs to treat solid tumors, physiological barriers posed by the tumor microenvironment hinder homogeneous delivery of nanoparticles in amounts sufficient to eradicate cancer. ^{7,22} Therefore, we need to better understand these barriers and develop criteria for the optimal design of nanoparticles.

Many experimental studies have investigated the effect of particle size on transvascular and interstitial transport in solid tumors.^{8,9,14,25,26,28,43} These studies have shown a drop in the transport properties of the particles once their size approaches the pore size of the vascular wall or the interstitial space. On the contrary, there is limited work on how the surface charge of the particles affects transvascular flux. Experimental findings have shown that cationic nanoparticles preferentially extravasate from tumor vessels, presumably due to electrostatic interactions with the negatively-charged glycocalyx of vascular endothelial cells.^{5,12,32} However, the parameters affecting transvascular transport such as the size and surface charge density of the nanoparticles and the pores of the vessel wall have not been analyzed rigorously.

In previous research, we developed a mathematical framework for the transvascular transport of nanomedicines taking into account steric and hydrodynamic interactions between the particles and the pores of the vessel wall.^{8} The current paper extends our framework by incorporating the surface charge of the nanoparticles and the resulting electrostatic interactions. We used an existing algorithm for modeling tumor-induced angiogenesis to generate the vascular network in this study.^{24,42} The vasculature consists of two inlets and two outlets (Fig. 1) and the pores of the vessel wall are taken by a unimodal distribution based on our previous work.^{15,16} The model accounts for the size and surface charge density of both the particles and the pores of the vessel wall. To account for steric and hydrodynamic interactions, a theory developed by Bungay and Brenner^{4} was employed, while electrostatic interactions were calculated based on a methodology developed by Smith and Deen.^{34}

We employed an existing algorithm that simulates the process of tumor-induced angiogenesis based on the gradients of vascular endothelial growth factors and fibronectin.^{24,42} The vascular domain consists of two parent vessels, from which neoplastic blood vessels are extending towards the tumor^{42} (Fig. 1).

The mathematical model requires coupling of fluid flow and nanoparticle transport in the vascular and interstitial spaces.

Blood volumetric flow rate in a vessel (*Q*_{vascular}) is assumed to be axial and follows Poiseuille’s law,

(1)

where *d* is the vessel diameter, Δ*p*_{v} is the vascular pressure difference that corresponds to a vascular length Δ*x* and *μ* is the blood viscosity.

Volumetric fluid flow rate across the vessel wall (*Q*_{transvascular}) follows Starling’s law,^{2}

(2)

where *L _{p}* is the hydraulic conductivity of the vessel wall,

Interstitial volumetric fluid flow rate (*Q*_{tissue}) follows Darcy’s law,^{2,37}

(3)

where *K _{t}* is the hydraulic conductivity of the interstitial space, Δ

Coupling of nanoparticle transport between the vascular and interstitial spaces is based on the following assumptions.

Inside the blood vessels diffusion is negligible and the mass balance takes the form:

(4)

where *v* is the fluid velocity which is determined by dividing *Q*_{vascular} in Eq. (1) by the cross-sectional area of the vessel, *c*_{v} is the intravascular concentration of the nanoparticle and Δ*c*_{v} is the concentration difference that corresponds to a vascular length Δ*x*.

In the interstitial space transport of nanoparticles is governed by the convection–diffusion equation,^{2}

(5)

where *c*_{i} is the concentration of the nanoparticle in the interstitial space, *D* is the diffusion coefficient, and *v*_{i} is the interstitial fluid velocity which is calculating by dividing *Q*_{tissue} in Eq. (3) by *A*_{C}.

Transport across the tumor vessel wall, *ϕ*, is given by Starling’s approximation as^{2}

(6)

where *Pe* is the Péclet number across the vessel wall, and *P* is the vascular permeability of the nanoparticle through the pores of the wall. Using theory for transport of particles through cylindrical pores^{11} we calculate the hydraulic conductivity, *L _{p}*, vascular permeability,

(7)

where *γ* is the fraction of vessel wall surface area occupied by pores, *r _{o}* is the pore radius,

The parameters *H* and *W* account for hydrodynamic and electrostatic interactions and for dilute solutions are given by the equations^{11}:

(8)

(9)

where *λ* is the ratio of the particle size over the pore size, *E* is the electrostatic energy of interaction between the nanoparticle and the pore, *k* is the Boltzmann’s constant, *T* is the absolute temperature, *K*(*λ*,*β*) and *G*(*λ*,*β*) are hydrodynamic functions, and *β* is the radial distance in the pore divided by the pore radius (i.e., *r*/*r _{o}* in Supplementary Fig. 1).

To calculate the hydrodynamic functions, *K*(*λ*,*β*) and *G*(*λ*,*β*), the centerline approximation is employed, which suggests that use of the centerline values, *K*(*λ*,0) and *G*(*λ*,0) leads to reasonably accurate estimates of *H* and *W*.^{11} Therefore, Eqs. (8) and (9) are written as:

(10)

(11)

Analytical expressions of the hydrodynamic coefficients *K*(*λ*,0) and *G*(*λ*,0) are given by Bungay and Brenner.^{4,11} These expressions are composites of asymptotic centerline results for small and for closely fitting spheres, are valid for 0 ≤ *λ*<1 and are accurate to within 10% for all values of *λ*. Notice that in the absence of electrostatic interactions (i.e., *E* = 0), Eqs. (10) and (11) are written as *H* = Φ*K*^{−1}(*λ*,0) and *W* = Φ(2 − Φ)*G*(*λ*,0), where Φ is the partition coefficient (for *E* = 0, Φ = (1 − *λ*)^{2}).

To calculate the electrostatic energy between the nanoparticle and each pore of the vessel wall a methodology developed by Smith and Deen^{34} is used. This methodology employs the linear form of the Poisson–Boltzmann equation to provide theoretical results for the electrostatic double-layer interaction between a particle and a cylindrical pore. The particle is assumed to be solid and spherical with a given surface charge density. Analytical expressions of the electrostatic energy are calculated based on the ratio of particle size to pore size *λ*, the ionic strength, and the surface charge density of the particle and the cylindrical pore. The theory accounts for both electrostatic attraction and repulsion.

The electrostatic energy, *E*, between the pore and the solid sphere is related to the free energy, Δ*G* by

(12)

where *R* is the gas constant, *T* is the absolute temperature, *ε* the permittivity, and *r _{o}* the radius of the pore. The free energy, Δ

(13)

where *G*_{sp} is the energy of interaction between the sphere and the cylindrical pore, *G*_{s} the energy of the sphere and *G*_{p} the energy of the pore.

Assuming solid spheres of constant surface charge, *q*, and employing the linearized Poisson–Boltzmann equation, the free energy is given by the surface integral of the electric potential, Ψ, as

(14)

where *A* denotes the area of the surface.

Finally, the electric potential is calculated by the solution of the linearized Poisson–Boltzmann equation:

(15)

which is valid for the space inside the pore and *τ* is the inverse of the Debye length. Whereas, inside the spherical particle and inside the solid material surrounding the pore, the electric potential is described by the Laplace’s equation (Supplementary Fig. 1).

The calculation of the free energy, Δ*G*, is given by the solution of Eqs. (13)–(15). This system of equations has an analytical solution given in the reference (Smith and Deen,^{34} Eq. 29) and was employed in the current study.

We first solve the steady-state fluid problem (Eqs. 1–3) and calculate the pressure distribution in the vascular and interstitial spaces. The vascular and interstitial spaces are discretized by nodes as shown in Supplementary Fig. 2. Each node belonging to the vascular space is assigned a pore diameter taken randomly by a unimodal distribution with a given mean and standard deviation. Therefore, each of the “vascular” nodes has its own values of *L _{p}*,

Subsequently, we solve the transient transport problem to calculate the concentration of the nanoparticles (Eqs. 4–6). The transient nanoparticle transport problem is solved with a finite difference scheme. Central differencing for diffusion, upwind differencing for convection and a fourth-order Runge–Kutta for time integration were used. For boundary conditions, the concentration of the particles at the inlets is specified and decays exponentially with a time constant equal to the blood half-life of the particle. In addition, the concentration at the outlets and at the boundary of the interstitial space is set to zero (Fig. 1).

The transvascular flux or effective vascular permeability, *P*_{eff}, is calculated in a region at the center of the tumor by recording the average concentration of nanoparticles in this region through time and fitting to these data the following equation^{8}:

(16)

where *C* is the average concentration in the region of interest, *C _{o}* is the initial concentration of the particles and

Supplementary Fig. 3 shows how Eq. (16) fits with the model predictions of the averaged concentration.

Nanoparticles might vary considerably in both size and surface charge. In this study, particles with diameters in the range of 6–200 nm were considered, which is the range of size for current nanomedicines. ^{8,29} The surface charge density |*q*_{s}| of the particles ranged from 0.0 to 0.10 C/m^{2}, based on values for gold nanoparticles and consistent with pertinent studies. ^{27,36} The interstitial diffusivities of the nanoparticles, *D*, and the time constant of concentration decay, *K*_{d}, depend on the size and the surface chemistry of the nanoparticles and were determined based on our experimental measurements.^{25,28,29}

The size of the pores in the tumor vessel wall depends on the tumor type and the site of growth. Brain tumor xenografts (U87 glioblastoma) have pore cut-off sizes ranging from 7 to 100 nm, while other cancer cell lines (LS174T, Shionogi, HCaI, ST-12, ST-8) when implanted subcutaneously in mice have a maximum pore cut-off size from 380 to 780 nm in diameter.^{16} In previous research, we found that when a unimodal distribution with mean pore size of 400 nm and a standard deviation of 60 nm was used, the model provided very accurate predictions for the transvascular flux of particles 12, 60, and 120 nm in diameter in two leaky tumor models.^{8} Therefore, we used these values here. Tumor vessels are negatively-charged due to vascular glycocalyx.^{31} Lacking information of the charge of the vessel wall, we assumed a surface charge density of −0.05 C/m^{2} based on other studies.^{34} Except where noted otherwise, the physiological ionic strength of 0.15 M was used, which results in a Debye length in the order of 1 nm. The size of the computational domain was 2 cm for a 1 cm diameter tumor, the blood viscosity was 3 × 10^{−5} mmHg s, the vascular pressure at the inlets and outlets was 25 and 5 mmHg, respectively, the vascular density *S*_{v} was set to 100 cm^{−1}, the vessel wall thickness, *L*, was 5 *μ*m, the vessel diameter was taken to be 15 *μ*m and the conductivity of the interstitial space (normal and tumor) was 8 × 10^{−7} cm^{2}/mmHg s (Table 1).

In previous research, we showed that using these parameters our mathematical model predicted successfully the transvascular flux of neutral nanoparticles of diameters 12, 60, and 120 nm in two different transplanted tumor models.^{8} The interstitial fluid pressure as well as the pressure difference between the vascular and interstitial space (transvascular pressure difference) are shown in Fig. 2. Interstitial fluid pressure is elevated at the center of the tumor, resulting in negligible pressure gradients across the vessel wall. The effect of pore size on transvascular pressure difference is shown in Supplementary Fig. 4.

Electrostatic repulsion decreases the hydrodynamic coefficients *H* and *W* (Eqs. 8 and 9), which in turn reduce the vascular permeability, *P*, and reflection coefficient, *σ* (Eq. 7) and thus, the transvascular flux of nanoparticles (Eq. 6). In tumors, the pores of the vessel wall can vary from a few nanometers to hundreds of nanometers depending on the tumor type and location. Therefore, in Fig. 3, we plot *H* and *W* as a function of the particle surface charge density and for pore diameters, *d*_{p}, ranging from 20 to 300 nm. The ratio of the particle size to pore size, *λ*, was kept constant at 0.3, and only negatively-charged particles were considered. The figure shows that electrostatic repulsion affects significantly the coefficients *H* and *W* for small pore diameters, while the effect is less important for large pores. The range of electrostatic interactions is determined by the Debye length. For our simulations the Debye length was chosen as ~1 nm. When the Debye length is comparable to the diameter of the pores, electrostatic forces are strong because the electrostatic double layers of the pore and the particle are close to each other. As the pore size increases the double layers separate and only particles that are close to the wall of the cylindrical pore will interact.

As the ionic strength increases, the Debye length decreases and the electrostatic interactions diminish. Figure 4a shows the transvascular flux of negatively-charged particles of four different sizes (*λ* = 0.1, 0.2, 0.3, and 0.5) as a function of the ionic strength. The surface charge of the particles is −0.05 C/m^{2}, and the mean pore size is 400 nm with a standard deviation of 60 nm. Low ionic strength causes a steep decrease in the flux of the particles independently of their size. Figure 4b presents the transvascular flux as a function of particle surface charge density and for four values of the ionic strength (*C* = 0.005, 0.01, 0.06, and 0.15 M). The pore distribution is the same as in Fig. 4a. The transvascular flux of the negatively-charged particles decreases as their charge increases. For physiological values of the ionic strength (0.15 M), however, the effect of repulsive interactions on the transport of the particles is not significant, which suggests a minimal excluded volume effect. Furthermore, neutral particles at low ionic strength exhibit a lower flux because even if they do not carry a charge, the interaction potential with the charged pores is not zero.

Figure 5a shows the transvascular flux of negatively-charged particles as a function of the particle’s surface charge density and when the ratio of the particle size to pore size, *λ*, is 0.1, 0.3, and 0.5. The mean pore size is 400 nm with a standard deviation of 60 nm. For neutral particles the flux decreases with the particle size as a result of steric and hydrodynamic interactions. For negatively-charged particles, electrostatic repulsion causes a slight and uniform decrease in the extravasation rate of the particles. More significant is the effect of charge on the transvascular transport for positively-charged nanoparticles. Figure 5b shows the effect of electrostatic attraction on the extravasation of nanoparticles. The transvascular flux is plotted as a function of the surface charge density and for three different sizes of particles, *λ* = 0.1, 0.3, and 0.5. Even relatively small positive surface charges can cause an up to threefold increase in the particle flux across the vessel wall. The effect of electrostatic attraction becomes more important for larger particles as the forces become stronger.

Model predictions for the (a) transvascular flux as a function of surface charge density for negatively-charged particles with sizes *λ* = 0.1, 0.3, and 0.5. (b) Transvascular flux as a function of surface charge density for positively-charged particles **...**

Figure 6 depicts the transvascular flux of nanoparticles 10, 60, and 120 nm in diameter as a function of the mean pore size and varying the surface charge density (positively-charged, neutral and negatively-charged). There is an ideal pore size for the particles that transvascular flux reaches a maximum. At smaller pores, steric, hydrodynamic and electrostatic interactions between the particles and the pores hinder extravasation but transport is maintained due to a pressure gradient across the vessel wall (convection). At larger pores, these interactions diminish which enhances transport, but the transvascular pressure gradient disappears (Supplementary Fig. 4) rendering diffusion as the only transport mechanism. Therefore, there is an optimum pore size where both convection and diffusion contribute to the extravasation of the particles. Electrostatic interactions are significant for smaller pores, while they have no effect as the mean pore size passes a value. This value depends on the size of the particle and the surface charge density. A positive surface charge can significantly increase the flux of nanoparticles and switch the optimal flux to smaller pore sizes.

Nanomedicine is an emerging and promising approach for the treatment of cancer. Nanoparticle formulations might be advantageous over conventional chemotherapeutics because they can incorporate multiple diagnostic and/or therapeutic agents, and can preferentially accumulate in tumors due to the EPR effect and the incorporation of specific targeting moiety. The relatively large size of nano-scale drugs, however, might inhibit their homogeneous distribution within solid tumors and thus, compromise the therapeutic outcome. Research for the optimization of nanoparticle delivery to date has been mainly focused on modifying the tumor microenvironment *via* vascular or interstitial normalization^{8,13,20,21} or the use of targeting ligands.^{17,30,38} A better design of nanoparticles (i.e., optimal size, charge, and shape) could also overcome the transport barriers and further improve intratumoral penetration.^{10,41}

Many components of the tumor micro-environment have an electric charge. The vascular glycocalyx renders the blood vessels negatively-charged, while in the interstitial space the hyaluronic acid consists of highly anionic molecules and the collagen fibers have a slightly positive charge.^{31,36} Therefore, electrostatic interactions between nanoparticles and components of the tumor micro-environment could play an important role on drug delivery. In this paper, we developed a mathematical model to study how the surface charge of nanoparticles can affect transport across the tumor vessel wall. The model predicted that transvascular transport of negatively-charged particles is hindered only when the pore size is comparable to the Debye length. Of note, for pores of the tumor vessel wall, whose size is on the order of hundreds of nanometers, the effect of electrostatic repulsion must be negligible (Figs. 3 and and6).6). On the contrary, electrostatic attraction, caused by positively-charged particles, can significantly increase transvascular flux (Figs. 5b and and6).6). Electrostatic attraction, which enhances transvascular transport, competes steric and hydrodynamic interactions, which hinder transport. As a result, it seems that for every nanoparticle size, there is a value of surface charge density above which electrostatic forces become dominant and a steep increase in transvascular flux is predicted (Fig. 5b). In addition, for smaller pore sizes (<100 nm in Fig. 6a) steric and hydrodynamic forces must dominate and for that reason we do not see important effects of charge on the transvascular transport of nanoparticles. As the pore size increases the effect of electrostatic interactions should become dominant and an increase in transport is predicted. Finally, when vascular pores are getting very large (>300 nm in Fig. 6a) compared to the size of the particles all three types of interactions diminish and the effect of charge disappears. Given the fact that some tumor types, such as brain and pancreatic cancers, have relatively small vascular pores, while in other tumors, such as breast cancers, the pore size might exceed 1 *μ*m in diameter,^{16} the advantage of cationic nanoparticles should depend on the tumor type.

The model predictions are in agreement with the experimental findings that the transvascular flux of cationic molecules is higher than that of their neutral or anionic counterparts.^{12} Also experiments have shown that cationic liposomes selectively accumulate in tumor vessels, presumably due to electrostatic attraction, and improve intratumoral delivery of encapsulated chemotherapeutics.^{5,32} On the other hand, neutral particles have been shown to diffuse faster than the charged ones in the interstitial space of tumors and thus, improve intratumoral penetration.^{36} Therefore, particles with initial positive surface charge that switches to neutral once they enter the tumor interstitial space would be the ideal design as far as transport is concerned.^{41} High positive charges, however, might affect plasma clearance rates and reduce the circulation time of the particles.^{23}

The tumor micro-environment is too complex to be precisely represented by a single mathematical model. Therefore a number of assumptions had to be made. Our model is limited in that it does not account for the three-dimensional structure of the vasculature. In addition, it assumes a uniform vessel diameter and blood viscosity. However, the diameter of tumor vessels is not uniform^{45} and the plasma leakage through the pores of the vessels increases the concentration of red blood cells (hemoconcentration) which might alter the viscosity value.^{33} Another limitation of the model is that the expressions for the hindrance coefficients *H* and *W* (Eqs. 8 and 9) are valid for only dilute solutions. Positive surface charges larger than these employed in the current study might render the dilute solution theory invalid and the predictions of the model might become unrealistic.^{34}

To our knowledge this is the first model for the delivery of nanoparticles to solid tumors that takes into account steric, hydrodynamic and electrostatic interactions between the particles and the pores of the vessel wall. Furthermore, while we employed here a computer generated vascular network, our methodology is general and can be directly applied to vascular networks of any geometry.

The authors thank Dr. Vikash Chauhan for his insightful comments on the manuscript. This work was supported by a Marie-Curie International Reintegration Grant (No. PIRG08-GA-2010-276894), the National Cancer Institute (P01-CA080124, R01-CA126642, R01-CA115767, R01-CA096915, R01-CA085140, R01-CA098706, T32-CA073479, Federal Share Proton Beam Program Income Grant), and a DoD Breast Cancer Research Innovator award (W81XWH-10-1-0016).

Associate Editor Konstantinos Konstantopoulos oversaw the review of this article.

ELECTRONIC SUPPLEMENTARY MATERIAL

The online version of this article (doi:10.1007/s10439-012-0630-4) contains supplementary material, which is available to authorized users.

**CONFLICT OF INTEREST**

R.K.J. received research grants from Dyax, Med-Immune and Roche; consultant fees from Dyax, Enlight, Noxxon and SynDevRx; owns equity in Enlight, SynDevRx and XTuit, serves on the Board of Directors of XTuit and Board of Trustees of H&Q Capital Management. No reagents or funding from these companies was used in these studies. Therefore, there is no significant financial or other competing interest in the work. The other authors declare no competing financial interests.

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