Understanding the Connection between Nanoparticle Uptake and Cancer Treatment Efficacy using Mathematical Modeling

Nanoparticles have shown great promise in improving cancer treatment efficacy while reducing toxicity and treatment side effects. Predicting the treatment outcome for nanoparticle systems by measuring nanoparticle biodistribution has been challenging due to the commonly unmatched, heterogeneous distribution of nanoparticles relative to free drug distribution. We here present a proof-of-concept study that uses mathematical modeling together with experimentation to address this challenge. Individual mice with 4T1 breast cancer were treated with either nanoparticle-delivered or free doxorubicin, with results demonstrating improved cancer kill efficacy of doxorubicin loaded nanoparticles in comparison to free doxorubicin. We then developed a mathematical theory to render model predictions from measured nanoparticle biodistribution, as determined using graphite furnace atomic absorption. Model analysis finds that treatment efficacy increased exponentially with increased nanoparticle accumulation within the tumor, emphasizing the significance of developing new ways to optimize the delivery efficiency of nanoparticles to the tumor microenvironment.

(800-1000 m 2 /g), facile surface modification to enable targeting, low toxicity, therapeutic effectiveness 9,10 , and increased circulation time, therefore increasing total tumor drug uptake 10 . As a result, the effective therapeutic drug dosage is reduced when delivered using MSNPs relative to the free drug delivery case, minimizing treatment side effects.
Using an integrated mathematical modeling and experimental approach 11 , our group has shown that nanocarrier mediated drug delivery of doxorubicin achieves equal cell kill efficacy at a dose only 20% of that of the corresponding free doxorubicin in a hepatocellular carcinoma cell model in vitro. In the present study, we use 50-nm diameter acetylated MSNPs, modified with polyethylene glycol (PEG) and polyethyleneimine (PEI) as first reported in Townson et al. 12 . These acetylated MSNPs were found to be colloidally stable and non-toxic. They were shown to have reduced non-specific binding to cell types A549, A431, Hep3b and hepatocytes in vitro, as well as to endothelium and white blood cells, and were observed to remain in circulation over 6 hours post injection in a chorioallantoic membrane (chicken embryo) model 12 . The increased circulation time of the acetylated MSNPs (>6 hrs) greatly improves their likelihood of entering the tumor and delivering effective drug dosages to cancer cells when compared to the circulation time of free drug (<2 hrs, t 1/2 = 1.68 hrs) 13 .
A number of techniques have been developed to monitor the pharmacokinetics and biodistribution of nanoparticles in living animals, including optical fluorescent microscopy imaging [14][15][16] , ultrasound 17 , in vivo bioluminescence (IVIS ® using luciferase) 18,19 , and PET, SPECT, and CT (as well as combinations of these techniques) using appropriate contrast agents 14,15,[20][21][22] . However, successful prediction of treatment outcome based solely on parameters measured from the biodistribution of nanoparticles is challenging because the distribution of nanoparticles and of drug are different and heterogeneous across tissues, organs, and even whole organisms. We have previously developed a mathematical model 11,23 that predicts the fraction of tumor killed by chemotherapeutic treatment (denoted by f kill ) based on drug uptake and nanoparticle drug flux. f kill was shown to be quadratic with respect to time, especially in the initial phase of a treatment. In the present study, we develop a mathematical theory based on this model 11,23 in order to predict nanoparticle-based treatment efficacy using quantitative experimental data obtained from measured nanoparticle biodistribution.
Measurement of drug distribution in vivo is often difficult and expensive. Typically, quantitative analysis of nanoparticle biodistribution is done through organ dissection after injection of labeled nanoparticles, followed by imaging, using methods such as transmission electron microscopy or optical microscopy, or by elemental analysis using inductively coupled plasma-atomic emission 24 . Nanoparticle distribution can also be measured in vivo using magnetic resonance imaging and magnetic particle imaging 24 . Here, capitalizing on the low natural abundance of elemental silicon in mammals, we performed Si elemental analysis of the major organs in order to measure MSNP concentrations, and then compared these quantified MSNP accumulation values to changes in tumor volume. By using our theory to link measured tumor growth with nanoparticle distribution and concentration, and considering the effects of vasculature and diffusion characteristics, we were able to successfully predict the in vivo therapeutic efficacy of MSNP-delivered doxorubicin to 4T1 breast cancer tumors in mice.

Results and Discussion
Cancer stage at the time of diagnosis is demonstrated to be an important predictor of morbidity and survival 25 .
Five-year survival rates for breast cancer are 99% when diagnosed pre-metastasis, but with a significant reduction to only 25% five-year survival rate when the tumor has metastatized 25 . In order to better understand the difficulties of effective theraputic treatment when patients are diagnosed at a later stage, our experiments focused on treating tumors that were relatively large with distant metastases prior to the start of a treament. Accordingly, we implemented a 4T1 cell line experiment in BALB/c mice, in order to study stage IV human breast cancer with metastasis in the presence of an active immune system.
Treatment efficacy between MSNPs loaded with doxorubicin (Dox) and free Dox was compared at the same dosage; phosphate buffered saline (PBS) was used as a control in a third treatment group. Tumor size measurements in mm 3 are shown in Fig. 1 for three groups of seven mice each treated using PBS (control), free Dox, or MSNPs loaded with Dox. Due to this treatment being adminstered at a later stage of breast cancer, it can be observed that free Dox treatment is not effective at a late stage; but the nanoparticle treatment shows some efficacy (see Supplementary Fig. S1 for each individual measurement). However, we note that statistically significant difference in tumor response between different treatment groups was not identified. This warrants additional Each group's average tumor size is shown above. Measurements were taken on days 0, 3, 7, 8, and 9. Treatments were given on days 0, 2, 4, and 7. Individual mice data are shown in Supplementary Fig. S1. experiments on a larger scale with more data points to demonstrate the treatment efficacy for this particular nanoparticle.
In vivo biodistribution of nanoparticles based on size, shape, composition, and surface characteristics is still not well understood, as are the details of nanoparticle removal from circulation by the reticuloendothelial system 26 . To gain a better understanding of these important parameters, we used graphite furnace atomic absorption (GFAA) to measure silicon (Si) concentration and distribution within tissues of interest 27 . The Si concentration in the control (PBS) group was used as a baseline for the background Si concentration which occurs naturally in tissues (Si has important biofunctionality, and thus is found in trace quantities in many tissues). As expected, we observed that the amount of naturally occurring Si in the control tissues was low relative to the signal of MSNP in treated tissues (Fig. 2, control). Figure 2 shows measured values of elemental Si determined by GFAA, presented as the mass percentage of Si in the corresponding tissue being tested in mouse organs from the control and Dox loaded MSNP treatment groups after 9 days of treatment and sacrifice. The tissues tested were tumor, kidney, liver, spleen, as well as a measured sum of all organs. The sum of all organs tested corresponded to ~2-4.4% of the total injected Si dose, as shown in Supplementary Fig. S2.
Mice, even within the same treatment group, demonstrated a wide range of f kill responses, indicated by the large standard deviation seen in tumor volumes in Fig. 1 (error bars). Moreover, the concentrations of Si deposited in the liver, spleen, kidney, and tumor were shown to vary without correlation between uptake in the tissues measured in our study (Fig. 2). For example, mice that had greater MSNP uptake in the liver did not have greater or lesser uptake in the tumor, see Fig. 3 for tumor delivery efficiency (%ID), and mice with less MSNP uptake in the tumor did not have less or more uptake in other organs. From our experiments here, we find that the MSNP delivery efficiency to the tumor is about 0.22%ID on average. Based on an extensive review of papers over the past ten years, Wilhelm et al. reports 0.7%ID as being the average tumor delivery efficacy using a multitude of particles, cancer types, and measurement methods, and mentions that, "it is possible that the amount of nanoparticles  reaching cancer cells and their subcellular compartments in vivo is much less than 0.7%ID because nanoparticles need to cross the tumour extracellular matrix to reach the cancer cells" 28 . Hence, our data seems well consistent with Welhelm et al.'s paper. One paper used elemental analysis, ICP-AES, to measure silica nanoparticle concentration, of which the %ID was determined to be 0.29, 1.6, and 10.8%ID 29 , most of the quantifications used to calculate the average of 0.7%ID in a tumor used PET scans as a method of quantification 16,21,30,31 . Thus, the biodistribution of nanoparticles was observed to show significant variability, even amongst similar mice under the same treatment protocol. Blood volume fraction (i.e., the volume of the tumor occupied by blood vessels) was previously shown to be an important factor in predicting f kill in human colorectal cancer metastasized to the liver 32 .
Many current nanoparticles have achieved the capability to release drugs at a nearly constant, sustained rate for a period of days, weeks, or even months [33][34][35] , although this rate may be dependent on other physiochemical properties of the particles, such as the surface chemistry and pore size 36 . In vivo, this nearly constant drug release rate results in an approximately unchanged rate of change of drug flux (denoted by F) across blood vessels. By further assuming a linear drug uptake by cancer cells, we have devleoped a special form of f kill for predicting treatment efficacy for nanoparticle-based drug delivery systems 23 . We found that tumor response in vivo to Dox loaded nanoparticles occurs quadratically over time, at least for the first several days ( θ = ⋅ f t f kill 2 , i.e., Eq. 4, where θ f is the tumor f kill coefficient; also see Methods), and have further validated this model using experiments on a breast cancer mouse model. Accordingly, we used this quadratic tumor response model to link the MSNP deposition with measured changes in tumor volume (relative to control). The quadratic tumor response coefficient (i.e., f kill coefficient: θ f ) was determined to have an exponential relationship with MSNP deposition in the tumor tissue (Fig. 4); note that we simply applied a phenomenological approach to identification of the relationship between θ f and Si deposition. θ f was found to be predictable based on tumor silicon content with 95% confidence (R 2 = 0.817, p .05 = 0.0007) based on the quadratic curve for f kill described in Eq. 4. This indicates that increasing chemotherapy drug delivery, using a MSNP transport vector, results in an exponentially greater rate of tumor kill. Values for comparison are shown in Table 1, along with statistics that indicate that most model values are statistically significant at 95% confidence except for MSNP 5. MSNP 4 and 7 did not show measurable response to treatment until day 7, resulting in only two data points; therefore, p-values for θ f could not be determined due to an insufficient number of points. As such, these two mice were removed from further analysis. Together, this indicates that MSNP uptake is an important factor in determining tumor treatment efficacy. However, more tumor measurements and/or a longer experiment would be beneficial to validate statistical significance with the model predictions. Si concentration and f kill coefficient (θ f ) determination. θ f , explained in Eq. 4, is predicted based on Si absolute mass percentage in tumor, as measured using graphite furnace atomic absorption (GFAA). R 2 = 0.817, p .05 = 0.0007 for quadratic f kill coefficients calculated after 9 days of treatment and sacrifice. This was determined by the best fit between f kill and Si concentration using an exponential fit (which represents the best fitting function among many we have tested, including linear and nonlinear functions, e.g., quadratic, sigmoidal, the Michaelis-Menton, and the Hill function). Experimental mouse tumor f kill was determined using Eq. 2 for mice treated with MSNPs. Model mouse tumor f kill was determined by optimizing values for t 0 (days in integer values), described in Eq. 4. Standard deviation for measured Si % in each mouse's tumor treated with MSNPs is shown for experimental values on the x-axis.

Mouse
Si % in tumor t 0 (time tumor begins to respond to treatment in days) θ f (quadratic response coefficient) p-value R 2 (quadratic) , where Si is tumor Si mass %) and the corresponding time course MSNP experimental data as shown in Fig. 1; also see Fig. S3. We obtained correlation coefficient r = 0.89 and p < 0.001, and thus consider the model to be acceptable in predicting MSNP-based treatment outcome with absolute Si mass percent as input.
Cancer treatment may be improved by using MSNPs as chemotherapy delivery vessels, which allow for an increased effective drug dose to be delivered to the tumor site due to drug sequestration within the particles during transit to the site, resulting in reduced uptake by the reticuloendothelial system and longer circulation time 37 . We have provided further insight into this drug delivery system using MSNPs in a murine 4T1 in vivo tumor model combined with a mathematical modeling description of drug efficacy. In particular, our mathematical model demonstrates that an increase in MSNPs delivered to the tumor exponentially increases the cell kill at early times in the treatment, leading to improvements in overall treatment outcome. The major hurdle is thus increasing tumor MSNP delivery over current methods, as any increase in delivery to the tumor is expected to significantly improve treatment efficiency. In this perspective, active tumor targeted delivery of chemotherapy via targeting ligand modified nanoparticles 5,38 is expected to enhance treatment efficacy. We also clarify that, as described in our prior work 23 , the drug concentration used to derive the f kill model is the concentration at the blood vessel wall, which is assumed to be a constant and is averaged over the duration of treatment, i.e., the actual drug concentration in the tissue which reaches the tumor was not measured previously. In the present work, we have obtained an actual measurement for the estimated drug concentration in the tissue through measuring the carrier (MSNPs) loaded with Dox in the tumor and MSNP uptake in organs.
Morover, if the quadratic f kill coefficient could be determined early on, the treatment efficacy could be predicted by our mathematical model. It has been demonstrated that tumor exponential growth rate constants were correlated to patient survival 39 . Here, we demonstrate that, due to external stresses and regression in tumor size due to treatment, the quadratic coefficient following treatment is predictive of treatme006Et uptake and therefore treatment efficacy. Note that θ f is a combination of parameters, inlcuding drug flux across blood vessels, cell death due to accumulated drug, and initial tumor volume (see 23 for details). This implies that θ f may have a nonlinear relationshiop with Si content, which has been confirmed in this study where an exponential relation was found. In future efforts, a better understanding of the biodistribution of nanoparticle and Si across organs examined on a larger dataset will allow us to have a more adaptive use of the model presented here. Additionally, further model validation against information obtained from non-invasive imaging modalities such as MRI, PET/CT 40 will help to quantify nanoparticle-based treatment outcome without the need to sacrifice animals. This important next step will also progress the model towards a more clinical functionality, where it may be implemented as a predictive tool without the need for invasive diagnostic procedures.

Materials and Methods
Mathematical modeling. We recently developed a series of mathematical models in closed form for predicting tumor response to treatment based on time-and space-dependent drug diffusion and perfusion properties 11,23,32,[41][42][43][44][45][46][47] . A generalized model presented in 23 can provide predictions of outcome for both conventional chemotherapy with a specific dosing and timing regimen and nanoparticle-based treatment. f kill (i.e., the fraction of tumor killed by treatment) is defined as: where V is tumor volume at time t, i indicates drug treatment group (Dox or Dox loaded MSNPs), and C indicates the control group. Normalizing tumor volume at a given time t to initial volume, we have where t 0 is determined to be the initial day when treatment was started; in our analysis here, i simply indicates either free DOX or MSNP treatment group. Assuming that drug-loaded nanoparticles can accumulate within tumors and continuously release drugs at a nearly constant rate over a certain time interval (especially at the initial phase of a treatment), we derived a special form of f kill 23 : where F is the flux of drug across blood vessel walls, λ k is the death rate of tumor cells, V T,0 is the tumor volume when tumor begins to respond to treatment (positive f kill ), and t is time. Note that there is another key assumption we made to the original f kill model, which is composed of a system of differential equations, in order to develop this simplified form: a drug administered as bolus at a certain dose level has the same effect as the same total amount of drug administered over several months at a constant, smaller dose level. That is, this model functions for a number of situations, including (1) single drug injection at the beginning, (2) multiple drug injections over the course of treatment, and (3) continuous drug administration. Regardless of how we administer the drug, the treatment system can be modeled as a continuous drug delivery system. This assumption has been validated previously in vivo and in patients across different types of cancer 23,32 . Parameters for best model fits to experimental data (determined by Eq. 2) with the model's predictions from Eq. 3 are derived from: . From our experimental analysis, we obtain:  Fig. 1. Mice were randomly divided into three treatment groups (7 mice/group, 21 mice total): control (PBS), free doxorubicin 1 mg/kg per treatment, 1 mg acetylated MSNPs (50 nm, 2.5 nm pores) loaded with doxorubicin (equivalent 1 mg/kg doxorubicin per treatment). Acetylated MSNPs were made using the protocol described in Townson et al. 12 , and loaded with doxorubicin using the drug loading protocol for water soluble doxorubicin in Lin et al. 48 . Treatment was given starting 2 weeks after tumor cell injections. Treatment days were as follows (t = 0 is 2 weeks following tumor injection): 0, 2, 4, and 7. All mice were sacrificed on day 9. Tumor measurements were taken on days: 0, 3, 7, 8, and 9. Tissues were excised and fixed in 4% formaldehyde diluted in PBS, and then Si contents were determined using graphite furnace atomic absorption spectrometry as described below. Statistical analysis was conducted using Matlab, Excel, and Graphpad Prism.

Graphite Furnace Atomic Absorption Spectrophotometry (GFAA). Analysis of Si concentration in
MSNP (mesoporous silica nanoparticle) and control (PBS) treated mice tissue was tested using a THGA graphite furnace on a PinAAcle 900 T Atomic Absorption Spectrophotometer (Perkin Elmer, USA). Tissues tested were tumor, kidney, spleen, and liver digested using aqueous tetramethylammonium hydroxide. Absolute mass percentages of silicon in the specific tissue were measured using the standard additions method, commonly used for samples prepared within a matrix that has a strong influence on the analyte signal (the matrix is defined as anything present in the solution that is not the analyte). In the standard additions method, tissues are spiked with differing, known concentrations of Si in order to determine the Si mass % in the tissue measured by extrapolation of the obtained curve of Si signal versus addition to zero addition; a unique signal vs. addition curve is generated for each tissue sample. Tumor (or organ) delivery efficiency, %ID, was determined by using BALB/c standard organ values from Tsai et al. 49 to estimate total organ Si mass deposited. Total Si delivered was calculated based on Si percentage (39.36% of nanoparticle mass) added during synthesis using the protocol in Townson el al. 12 .
Tumor delivery efficiency, %ID, was determined using Si absolute mass percentage in the average control mice each respective organ subtracted from that of the MSNP treated mouse and dividing it by the total administered dose over the treatment duration. The delivery efficiency was measured based on the mass of Si measured in each respective organs divided by the total injected mass of Si (in the form of MSNPs). See Supplementary Information for more details on GFAA methods.
Data availability. All data generated or analyzed during this study are included in this published article and its Supplementary Information files.