Tailoring nanoparticle designs to target cancer based on tumor pathophysiology

Characterization of Tumors.

Pathophysiological changes associated with tumor volume were studied to identify biological parameters that might impact AuNP targeting. The degree of vascularization, cell density, and extracellular matrix (ECM) content of different-sized orthotopic human breast melanoma xenograft tumors derived from MDA-MB-435 cells in CD1 nude athymic mouse models were characterized. These parameters were selected because they have been shown to individually impact nanoparticle uptake rate, accumulation, and retention (20–22). Histological sections stained with CD31 antibodies were used to colorimetrically visualize tumor blood vessels, whereas Movat’s Pentachrome staining was performed to highlight nuclei and ECM components such as proteoglycans, mucopolysaccharides, and collagen. Vascular density was calculated by counting the number of vessels per tumor cross-section. We observed that the concentration of blood vessels increased with tumor volume but plateaued at 44 ± 3 blood vessels/mm² for tumor volumes exceeding 1.0 cm³ (Fig. 1A). Interestingly, the tumor vasculature was only uniformly distributed in small tumors. Tumor blood vessels became increasingly concentrated near necrotic regions and at the tumor perimeter as tumors enlarged (SI Appendix, Fig. S1).

Beyond tumor vascularization, the fraction of the tumor composed of proteoglycans and mucopolysaccharides increased at a rate of 4.2 arbitrary units/cm³ (Fig. 1B), whereas tumor cell density increased at a rate of 1.70 cells/cm³ (Fig. 1C) with tumor volume. Unstained acellular space also proportionally decreased with tumor growth (Fig. 1D). These factors coincided with heightened ECM production at regions surrounding tumor blood vessels and necrotic tissue, whereas ECM content in regions of dense tumor tissue around the core became reduced (SI Appendix, Fig. S2). A closer examination of ECM composition by Picrosirius red staining (Fig. 2A) and second harmonic generation (SHG) imaging (Fig. 2B) identified that these regions contained type I collagen with a density and structure that evolved with tumor growth. Picrosirius red-stained samples spectrally shifted from deep red to pale pink (Fig. 2C), whereas SHG microscopy images decreased in intensity (Fig. 2D) as tumors enlarged. There is a decrease of 9% per cm³ in Picrosirius red intensity and the spectral shift in SHG peak intensity was characteristic of a loss in structural ECM via reduction in collagen fiber thickness and length (23–25).

Together, these results indicate that as tumors mature through growth, their tissue and vasculature become denser and more chaotic. In particular, the ECM appears to remodel during tumor enlargement, thus leading to a more amorphous phenotype. Given that ECM components were observed to encapsulate tumor blood vessels (SI Appendix, Fig. S3) and are known to biologically function as a basal support for blood vessels that interfaces with the stroma, changes in ECM may be a primary mediator of nanoparticle entry into the tumor compartment.

Gold Nanoparticle Model System.

Having characterized the evolution of tumor tissues during growth, we sought to determine whether these physiological changes could be used to tune the tumor targeting efficacy of nanoparticles. Because tumor uptake is dependent on nanoparticle diameter (12, 26, 27), a library of methoxy-PEG–coated AuNPs of varying diameter were designed to examine the effect of tumor growth on particle delivery. Although clinical trials for AuNPs are limited, AuNPs were selected over more clinically appropriate polymeric nanomaterials because AuNPs can be reproducibly and precisely synthesized in a broad range of sub-100-nm sizes. Furthermore, AuNPs provide a nondeformable formulation for testing the effect of core diameter on tumor uptake, are easily surface modified, and can be quantified in tissues with high sensitivity. A schematic illustrating the AuNP design used in this study is depicted in SI Appendix, Fig. S4A.

Spherical AuNPs with core diameters of 15, 30, 45, 60, and 100 nm (SI Appendix, Fig. S4B) were synthesized using standard citrate and hydroquinone reduction techniques (28). These sizes were selected to systematically characterize how the tumor microenvironment impacts a broad range of particle diameters. AuNP surfaces were modified with hetero-bifunctional 5-kDa PEG with methoxy and sulfhydryl termini as well as Alexa Fluor 750-labeled 10-kDa sulfhydryl-PEG to respectively stabilize particles for blood transport and to fluorescently track particles in vivo. Although it is difficult to use fluorescence as an absolute quantification technique, we have shown previously that fluorescence is an accurate modality for monitoring relative changes in nanoparticle biodistribution (26, 29). Surface modifications resulted in AuNPs with a PEG packing density of 0.3–1.5 ligands/nm². At these densities, surface-bound PEG moieties were calculated according to their Flory diameter to be in the brush layer conformation, ensuring that the tested nanoparticles were sufficiently passivated (SI Appendix, Table S1). Surface modifications were also found to increase nanoparticle hydrodynamic diameters by 20–40 nm (SI Appendix, Fig. S4C) and positively shift nanoparticle zeta potentials by 20–30 mV (SI Appendix, Fig. S4D). Particle fluorescence was confirmed by the migration of distinct fluorescent bands during agarose gel electrophoresis (SI Appendix, Fig. S4E). AuNP fluorescence was shown to increase proportionally with particle diameter (SI Appendix, Fig. S5). Fluorescent PEG groups were also confirmed to be stably bound to particle surfaces because the rate of desorption in the presence of serum remained below 0.2 PEG molecules/h (SI Appendix, Fig. S5D). In vivo pharmacokinetics of our functionalized AuNPs was also characterized by analysis of blood plasma at 0, 2, 4, 8, and 24 h posttail vein injection (HPI) in non–tumor-bearing CD1 nude athymic mice. Inductively coupled plasma atomic emission spectroscopy (ICP-AES) analysis of blood samples revealed that the blood half-lives of our AuNPs ranged from 2 to 10 h. A complete characterization of our formulations is presented in SI Appendix, Table S2.

Analysis of Nanoparticle Accumulation in Tumors.

AuNP accumulation was evaluated via tail vein injection of formulations into CD1 nude athymic mice bearing orthotopic MDA-MB-435 human breast melanoma tumors. Tumors volumes evaluated in this study ranged from 0.05–3.00 cm³. AuNP delivery to the different sized tumors was fluorescently profiled in mice to assess tumor accumulation kinetics and to measure total AuNP exposure. Fluorescent tracking was achieved by whole-animal imaging using a Carestream In Vivo Imaging System at time points ranging from 0–24 HPI.

Total area under the curve (AUC) was calculated from the kinetic curves in SI Appendix, Fig. S6 as a metric for AuNP accumulation within the tumor. Overall, AUC values increased with tumor volume (Fig. 3A). Accumulation for 15-, 30-, and 45-nm AuNPs steadily increased with tumor volume from 490 ± 70% to 720 ± 30% ID⋅h, 280 ± 50% to 750 ± 10% ID⋅h, and 480 ± 70% to 960 ± 100% ID⋅h, respectively. Changes in accumulation of larger formulations occurred as step increases at discrete tumor volumes. There was an ∼1.5 times higher accumulation for 60-nm formulations once tumors exceeded 2.2 cm³, whereas 100-nm particles exhibited a ∼4.6 times increase in accumulation for volumes 0.5 cm³ and larger in comparison to smaller tumors. These trends were confirmed by ICP-AES measurements of gold content in tumors at 24 HPI (Fig. 3B). The ICP-AES results indicated that by 24 HPI tumor uptake of 15- and 30-nm particles were consistently higher than all other formulations and steadily increased from 0.39 ± 0.04% to 0.99 ± 0.18% ID and 0.28 ± 0.03% to 0.90 ± 0.18% ID, respectively (two-way ANOVA, P = 0.05), whereas larger particles such as 60 nm trended higher (although statistically not significant) from 0.18 ± 0.02% to 0.26 ± 0.12% ID as tumor volumes increased.

In combination with our histological observations, these results suggest that the higher porosity of the ECM increasingly accommodates the entry of larger nanoparticles at later stages of tumor growth. This implies that a minimum tumor size must be reached to support entry of each AuNP diameter. An AuNP accumulation threshold of 500% was selected to illustrate this point (Fig. 3A). This threshold was defined as the mean AUC of 15-nm AuNPs in sub-0.5-cm³ tumors as particles in this size range would experience the least steric hindrance. AUC values for each AuNP diameter were statistically compared with the threshold (two-way ANOVA, P = 0.05); 15-nm AuNPs have this accumulation threshold at tumor volumes of 0.5 cm³ and larger, whereas 30-nm nanoparticles achieved a similar trend at a threshold of 0.5–1.0 cm³ and above. Similarly, 45-nm formulations attained statistically higher accumulation at tumor volumes above 1.0 cm³, and 60-nm AuNPs exceeded this threshold (although statistically insignificant) when tumor volumes were beyond 2.2 cm³; 100-nm particles never reached the defined threshold accumulation at any of the tumor volumes tested. It has been shown that AuNPs greater than 100 nm in diameter sequester near tumor blood vessels and do not penetrate into MDA-MB-435 tumors (26, 27). Hence, the difference in the accumulation pattern of 100-nm AuNPs over the other tested formulations was attributed to the inability of these AuNPs to diffuse through pores that are smaller than the particle size.

Nanoparticle Kinetics Within the Different-Sized Tumors.

Kinetics of AuNP delivery to tumors were analyzed in an effort to explain the dependence between accumulation and tumor volume. Tumor uptake rates were calculated by taking the instantaneous slope at 3 HPI of the AuNP accumulation profiles presented in SI Appendix, Fig. S6. We observed that the speed of AuNP accumulation (Fig. 3C) was largely insensitive to changes in tumor volume (two-way ANOVA, P = 0.05); 15-, 60-, and 100-nm AuNPs maintained tumor entry rates of 4.2 ± 0.6%, 3.2 ± 0.9%, and 2.9 ± 0.8% ID⋅h⁻¹ as tumors grew to 1.0 cm³. Particles with 30- and 45-nm diameters were the exception as their rate of uptake steadily rose from 2.3 ± 0.3% to 5.7 ± 0.9% ID⋅h⁻¹ and 2.8 ± 0.9 to 7.0 ± 1.0% ID⋅h⁻¹, respectively, as tumors grew beyond 0.5 cm³. Although the rate of delivery did not statistically vary with growth, AuNP entry into the tumor compartment trended higher as tumors increased in size. The 15-, 30-, and 45-nm AuNPs also consistently accumulated in tumors ∼1.2–1.7 times faster than the 60- and 100-nm formulations. However, these differences became less apparent as tumor volumes increased. These results further reinforce the relationship between ECM porosity and particle size whereby smaller pores restrict larger nanoparticles from deep tumor infiltration and conversely become washed out of the tumor at a faster rate than smaller nanomaterials.

Because it is difficult to probe nanoparticle transport through ECM in animal models, we developed an in vitro system to measure diffusion of AuNPs into a hydrogel to mimic the effects of collagen structure on the transport of nanoparticles into the tumor (Fig. 4A). Although this in vitro model only evaluates diffusion through a collagen matrix independent of fluid flow or cellular interactions, the model provides a means to determine how the velocity of transport and quantity of AuNPs within tumors are dictated by the perivascular stroma upon initial AuNP entry. Self-assembled hydrogels composed of either 2.5 or 4.0 mg/mL of type I collagen were used to mimic stromal changes caused by tumor growth. Type I collagen was selected as a stromal phantom because type I collagen is the primary component of the tumor–blood vessel interface (30, 31). Entry of AuNPs from a fluid reservoir into the hydrogel was kinetically monitored by AuNP fluorescence using scanning confocal microscopy at different time points over 900 min. Overall, AuNP transport into the collagen gel occurred in two phases: (i) rapid concentration at the periphery of the hydrogel; and (ii) gradual movement from the concentrated zone to deeper regions of the matrix (Fig. 4B).

The amount of AuNPs infiltrating the hydrogel plateaued within 120–240 min postexposure for all formulations greater than 45 and 15 nm for 2.5 and 4.0 mg/mL collagen hydrogels, respectively (SI Appendix, Fig. S7A). The AuNP diffusion front also plateaued by 480 min postexposure for all particle diameters independent of collagen density (SI Appendix, Fig. S7B). Rather, AuNP penetration into the hydrogel at later time points occurred by diffusing away from the concentrated zone into the surrounding gel (seen as a broadening of the diffusion front in SI Appendix, Fig. S8). This penetration was dictated by particle diameter; 45-nm AuNPs achieved the highest permeation at 17.0 ± 2.0 and 13.2 ± 0.4 µm, whereas 100-nm AuNPs exhibited the poorest penetration at 8.0 ± 1.0 and 5.4 ± 0.6 µm for collagen densities of 2.5 and 4.0 mg/mL, respectively (Fig. 4C). Although AuNP permeation appeared to decrease with collagen concentration, differences were not statistically significant (two-way ANOVA, P > 0.05). These trends were consistent with the tumor-permeation results at 24 HPI, where AuNP infiltration did not vary with tumor volume (Fig. 4D). Particle permeation also did not change statistically between the tumor periphery, regions neighboring necrotic zones, or within the core of the tumor tissue. Despite the lower collagen density, diffusion of 15-nm AuNPs into the 2.5 mg/mL collagen gels was unexpectedly 2.0 and 1.3 times lower than our 45-nm formulation in vitro and in vivo, respectively. These differences in diffusion were similar to previous studies (26, 27) and were attributed to the speed of AuNP uptake by and expulsion from the collagen matrix (Fig. 4 E and F). In comparison to 4.0 mg/mL collagen gels, 15-nm formulations were taken up by the 2.5 mg/mL gels 14% slower and expelled 51% faster. This result leads to a lower overall AuNP concentration within the 2.5 mg/mL collagen gel and, accordingly, slower particle diffusion. Alternatively, because the uptake rate of AuNPs exceeding 45 nm does not vary with collagen density (two-way ANOVA, P = 0.05), their slower depletion from the collagen matrix allows for greater nanoparticle retention and consequently greater infiltration distances.

Computational Modeling of Nanoparticle Diffusion Through Porous Matrices.

To help understand how nanoparticles interact with the collagen matrix, Monte Carlo numerical simulations of AuNP diffusion through collagen matrices were conducted; 2D models were used to examine the frequency of AuNP collisions with collagen fibers within pores of the hydrogel matrix. The frequency of such collisions can influence the retention and path of the AuNPs in the tumor; 3D simulations were also conducted to compare AuNP permeation capacity through stroma with different collagen densities. These computational models were conducted in Matlab using custom algorithms to simulate AuNP interaction and diffusion within collagen matrices. These simulations followed similar strategies used by Stylianopoulos et al. (32). AuNP motility was modeled as step-wise random walk obeying Einstein–Stokes diffusion (Eq. 2), whereas particle–fiber interactions were modeled as elastic collisions. Fig. 5A provides an illustration delineating the path of AuNP motion within a collagen pore. Obeying Brownian motion, AuNPs move randomly and can collide with collagen fibers. For our 3D simulations, collagen fibers were approximated as cylinders with radii between 0.05 and 0.50 µm. Representative images of the collagen matrices of varying collagen density simulated in Matlab are presented in Fig. 5B. The modeled radii were chosen according to measured thicknesses from scanning electron microscopy images of our 2.5 and 4.0 mg/mL collagen hydrogels (SI Appendix, Fig. S9). A detailed summary of our model and its underlying assumptions can be found in Methods.

AuNP movement in our 2D models for 1,000-particle replicates was simulated in 0.005-, 0.020-, 0.108-, and 0.640-µm² square stromal pores for 10,000 steps at 0.1-s intervals. Our simulations determined that AuNP–fiber collision rates increased with reducing pore size and decreasing AuNP diameter (Fig. 5C); 15-nm AuNPs achieved the highest frequency of interaction with collagen fibers at rates between 0.038–0.023 collisions/s (cps), whereas 100-nm formulations ranged from 0.016–0.001 cps for pore sizes between 0.005–0.640 µm². Interestingly, collision rates for 15-, 45-, and 60-nm AuNPs were statistically similar for 0.005-µm² pores (ANOVA, P = 0.05) but became increasingly dissimilar as pores enlarged. These simulations suggest that impact of particle size on Brownian motion is a primary mediator of AuNP motility within the hydrogel over its frequency of collision with the ECM. This result suggests that the greater the nanoparticles interact with collagen, the longer they will be retained within the tumor.

Expanding on these results, AuNP diffusion was also modeled in three dimensions to compare how AuNP diameter and collagen density might impact stromal accumulation and infiltration. Stromal–ECM of increasing collagen density was modeled computationally as 27,000-µm³ cubes containing anisotropically oriented collagen fibers. The number of fibers were chosen to achieve collagen volume fractions (8.72–87.20%), reflective of conditions found within tumors (33, 34). Diffusion distance for 500 AuNP replicates was tracked for 5,000 discrete steps at 1-s intervals. Our simulations indicate that diffusion rates changed with AuNP diameter but did not change with collagen density (Fig. 5D); 15-nm AuNPs exhibited the greatest mobility at 1.95 ± 0.03 nm/s in the simulated hydrogels, whereas 45-, 60-, and 100-nm particles diffused at rates of 0.78 ± 0.01, 0.60 ± 0.01, and 0.36 ± 0.01 nm/s, respectively. These findings support our AuNP-permeation observations from histological tumor sections whereby AuNP diffusion away from blood vessels (SI Appendix, Fig. S10) did not vary with tumor volume (Fig. 5D).

Together, these 2D and 3D models elucidate how the stromal matrix is implicated in particle permeation. Although these simplified 2D and 3D models ignore the effect of fluid flow, oncotic pressure, and inelastic collagen–AuNP interactions, the models provide a mechanism for our in vitro and in vivo permeation observations. They suggest that AuNP permeation is the balance between the effects of particle size on Brownian motion and the frequency of particle collision with the ECM. The increased mobility of smaller AuNPs afforded greater diffusion but was also inhibitory because of the higher frequency of collision with the ECM. Conversely, AuNPs of larger diameters exhibited slower motion but also a lower propensity to interact with the stroma. The volume fractions tested and simulated in this study equate to ECM pore sizes ranging from 0.45 to 1.74 µm. Because these pores exceed the size of our AuNPs, differences in diffusivity associated with collagen density would be negligible for all tested particle diameters. Extended further, these computational findings demonstrate that AuNP transport within the tumor can be distorted through collisions with ECM fibers. These collisions can limit retention within the tumor compartment if AuNP volume approaches the porosity of the stromal ECM.

Nanoparticle Selection According to Tumor Maturity.

Given the complex dependence of tumor AuNP uptake on both particle size and tumor pathophysiology, we asked whether there was a means to rationally select AuNP formulations according to tumor volume. In our proof-of-concept work, we evaluated whether a decision matrix could be used to select nanomaterials for either tumor detection (diagnostic) or drug delivery (treatment). AuNP formulations with rapid delivery and high tumor contrast were defined as effective probes for delineating tumors, whereas AuNPs capable of high tumor retention and homogeneous tissue distribution were anticipated to fare well as drug delivery vehicles.

Relative measurements of AuNP fluorescence in vitro and in vivo were used as an estimate of tumor contrast achievable by each formulation. Surface area-to-volume ratios were also calculated to approximate the drug-loading capacity of each AuNP size (SI Appendix, Fig. S5C). These parameters in conjunction with tumor accumulation, uptake rate, and penetration capacity were ranked from best (4) to worst (1) for each of the four AuNP sizes used in our experiments. Each parameter was also given a multiplier according to the parameter’s importance to a given AuNP function. The weighted sum of these rankings was then calculated for each AuNP design for the different tumor size ranges. Eq. 1 is a summary of the scoring scheme, where µ is the importance multiplier, β represents the ranking factor, and i denotes the ranked AuNP parameters. Fig. 6B highlights these parameters and the associated values used to calculate the scores found in our decision matrices (Fig. 6 A and B).

Overall, smaller (<45-nm) AuNPs were favored for both diagnostic and therapeutic applications across all tumor sizes. Diameters in the 100-nm range were consistently predicted as poor candidates for either application, whereas 15- and 45-nm particles were both expected to be useful for detection and treatment of large (>1.0-cm³) tumors. AuNPs in the 60-nm range were the exception to these trends because these AuNPs were predicted to be better for detection of small, early-stage tumors (<0.5 cm³). This exception for 60-nm nanoparticles was empirically attributed to the statistical similarity in AUC values for particles with diameters between 15 and 60 nm (Fig. 3A) as well as the higher tumor contrast seen for 60-nm particles (SI Appendix, Fig. S5) in the 0.0- to 0.5-cm³ range. Because macrophage uptake of nanoparticles increases with particle diameter (35), the enhanced utility of 60-nm AuNPs may also be related to changes in phagocytic capacity of tumor-associated macrophages (36) because the macrophages’ phenotypes evolve during tumor progression (37).

These results imply that passively targeted AuNPs with smaller diameters would be more applicable for detection and drug delivery when tumor size is unknown. However, 45-nm AuNPs may be the more effective vehicle for later-staged tumors because their larger surface area-to-volume ratio (SI Appendix, Fig. S5C) theoretically allows for 900% greater drug loading than 15-nm particles with merely a drop to tumor accumulation by less than 57.3%. Although these findings are specific to passively targeted AuNPs, the proposed decision matrix schema can be generalized to provide a systematic method for assessing other particle types. A flowchart detailing a potential means of implementing this strategy is outlined in Fig. 6C.

Validation of the Decision Matrix for Personalized Targeting of Prostate Tumors.

Toward validating our results, we evaluated whether our formulated decision matrices could be used to predict the ideal AuNP design for other tumor models. A blinded study was conducted in CD1 nude athymic mice bearing orthotopic human tumors with PC3 prostate cancer cells to verify whether our tumor-size dependent predictions were accurate; 15- and 100-nm AuNPs were tail vein-injected into tumor-bearing mice to evaluate AuNP efficacy for tumor detection and accumulation. Both particle designs were effective at delineating the location of the tumor (Fig. 7A) but at varying efficacies. Tumor detection speed and contrast for 15-nm AuNPs were, respectively, 53.7% and 50.8% higher than 100-nm particles; 15 nm achieved greater tumor accumulation than 100-nm designs and trended higher with increasing tumor size (Fig. 7 C–E). These findings were consistent with our decision matrix, alluding to the potential of our system for use on other particle formulations and tumor types.

Tumor Accumulation Measurements.

Efficiency of AuNP delivery to tumors was measured by ICP-AES. Tumors were harvested at 24 HPI and digested in 1 mL of aqua regia (1:3 vol/vol nitric acid to hydrochloric acid) supplemented with 1 µg/mL yttrium for 2 h at 70 °C. Yttrium was used as an internal reference to account for sample loss during the digestion and purification process. Postdigestion, acidic solutions were diluted with 2 mL of double-distilled water and filtered through 0.22-µm PVDF membranes to remove undigested tissue. Volumes of the digested samples were then adjusted to achieve a final volume of 4 mL via addition of double-distilled water. Gold and yttrium contents in each sample were measured using a Perkin-Elmer Optima 3000. AuNP accumulation in tumors at 24 HPI was determined by normalizing measured gold concentrations to yttrium content and tumor mass.

Analysis of Nanoparticle Infiltration into Collagen Matrices.

Synthesized nanoparticles were tested in vitro for their permeation capacity through type I collagen hydrogels. Self-assembled hydrogels were first prepared by mixing presolubilized rat tail type I collagen on ice with 10× PBS and 1 M sodium bicarbonate at an 8:1:1 volumetric ratio, followed by dilution with double-distilled water to achieve final collagen concentrations of 2.5 and 4.0 mg/mL collagen solutions that were then placed into gel molds and allowed to self-assemble at 37 °C for 3 h. Postpolymerization, hydrogels were equilibrated in double-distilled water for 2 h, followed by immediate water exchange and introduction of AuNPs. AuNP infiltration into hydrogels was monitored every 30 min for 15 h via laser-scanning confocal microscopy using an Olympus Fluoview FV1000. A transillumination lamp was used to determine the collagen edge, whereas differential interference contrast (DIC) and fluorescence were invoked to profile AuNP distribution within the hydrogel. AuNP permeation was profiled along the length of the collagen hydrogel by analyzing confocal images of AuNP fluorescence in ImageJ. Fluorescent-intensity profiles were then placed into GraphPad Prism to calculate total AuNP uptake and track the mean AuNP-infiltration distances. Calculated values were used to determine AuNP-accumulation rates for the different hydrogel densities by taking the slope of the linear regression curves seen in SI Appendix, Fig. S11A.

Analysis of Nanoparticle Expulsion from Collagen Matrices.

Collagen hydrogels were constructed using a similar pH-based self-assembly process as mentioned in Analysis of Nanoparticle Infiltration into Collagen Matrices. Before gelation, AuNPs equivalent to a total surface area of 30 cm² were thoroughly mixed with hydrogel solutions on ice. AuNP–collagen mixtures were then allowed to set overnight at 37 °C, rinsed with PBS, and suspended in 1 mL of PBS. At 0, 1, 2, 3, 5, 6, 8, and 24 h, the PBS suspension solution was sampled (90 µL) to track AuNP expulsion from the hydrogels. AuNP quantity in sample solutions was approximated by measurement of sample fluorescence in 384 fluorescent well plates (Nunc 384-well optical well plates) using a Carestream Multispectral MS Fx Pro in vivo imager (excitation/emission: 750/830 nm) at an exposure time of 10 min. Fluorescent images were analyzed by densitometry in ImageJ. AuNP expulsion rates were obtained by taking the slope of the linear regression curves seen in SI Appendix, Fig. S11B.

Simulation of Nanoparticle Diffusion in Collagen Matrices.

Two-dimensional and 3D stochastic models of AuNP movement in collagen matrices were programmed and simulated in Matlab; 2D models were used to study how AuNP diameter and differences in the available area fraction of collagen matrices would affect the frequency of AuNP–collagen collisions; 3D simulations were conducted to investigate the impact of collagen density on AuNP diffusion distance. For both models, AuNP movement was taken as discrete random walk steps obeying Einstein–Stokes Brownian motion, as dictated by Eq. 2, where K_B, η, T, and r denote the Boltzmann constant, solvent viscosity, temperature in kelvins, and AuNP radii, respectively:

Particle movement was approximated as the mean square displacement according to Fick’s second law (Eq. 3), where δ and dt were taken as the discrete distance and time interval between steps:

AuNPs were approximated as circles and spheres for two and three dimensions, respectively. AuNP–collagen fiber collisions were assumed to be elastic with collagen fibers approximated as immobile cylinders. Simulations were also conducted in the limit of dilute AuNP concentrations where AuNP–AuNP collisions could be neglected and particles could be independently tracked.

For the 2D simulations, collagen matrices of differing available area fractions were approximated as square pores of varying size. Pore sizes were selected based on empirically measured spaces between collagen fibers seen in scanning electron microscopy images of collagen hydrogels of varying concentration (SI Appendix, Fig. S9). Images were imported into ImageJ and thresholded to differentiate collagen fibers from pores. The size of each pore was measured by calculating the rolling ball radius of each pore using ImageJ’s built-in algorithm. Initial AuNP positions were randomized within the collagen pores and were permitted to move stochastically within the square. Upon AuNP movement beyond the dimensions of the pore, collision events were counted and AuNP trajectories were elastically reflected. Particles were tracked for 10,000 steps at 0.1-s intervals. AuNP–collagen collisions were tallied for each condition.

To simulate 3D collagen, matrices composed of 100, 300, 600, and 1,000 cylinders ranging in length from 0 to 30 µm and radii ranging from 0.05 to 0.50 µm were randomly distributed and oriented within 30 × 30 × 30 µm cubes to mimic hydrogels with collagen volume fractions between 8.72% and 87.20%. Collagen volume fractions were determined by calculating the ratio of volume occupied by collagen fibers (approximated as cylinders) versus the total region of interest (30 × 30 × 30 µm cube). Volume fractions were equated to empirical collagen pore sizes by taking a 2D projection of the generated tissue, followed by calculation using the same ImageJ process as mentioned for our 2D simulations. For particle-motility simulations, AuNPs were randomly placed in our 3D matrices, allowed to move freely within the confines of the cube, and reflect off collagen fibers. The direction of AuNP motion was randomized with each step. AuNPs were tracked for 5,000 steps at 0.5-s intervals. Displacement between start and end points were measured to determine AuNP diffusion distances.

Statistical Analysis.

All statistical analysis comparing between groups were performed using one-way ANOVA (one variable per group) and two-way ANOVA (two variables per group) in GraphPad Prism.

Supporting Information.

See the SI Appendix for details regarding materials, gold nanoparticle synthesis and characterization, tumor induction, nanoparticle administration, tumor histology analysis, and whole-animal imaging.