Investigating the effect of tumor vascularization on magnetic targeting in vivo using retrospective design of experiment

Nanocarriers take advantages of the enhanced permeability and retention (EPR) to accumulate passively in solid tumors. Magnetic targeting has shown to further enhance tumor accumulation in response to a magnetic field gradient. It is widely known that passive accumulation of nanocarriers varies hugely in tumor tissues of different tumor vascularization. It is hypothesized that magnetic targeting is likely to be influenced by such factors. In this work, magnetic targeting is assessed in a range of subcutaneously implanted murine tumors, namely, colon (CT26), breast (4T1), lung (Lewis lung carcinoma) cancer and melanoma (B16F10). Passively- and magnetically-driven tumor accumulation of the radiolabeled polymeric magnetic nanocapsules are assessed with gamma counting. The influence of tumor vasculature, namely, the tumor microvessel density, permeability and diameter on passive and magnetic tumor targeting is assessed with the aid of the retrospective design of experiment (DoE) approach. It is clear that the three tumor vascular parameters contribute greatly to both passive and magnetically targeted tumor accumulation but play different roles when nanocarriers are targeted to the tumor with different strategies. It is concluded that tumor permeability is a rate-limiting factor in both targeting modes. Diameter and microvessel density influence passive and magnetic tumor targeting, respectively.


Developing response surface from historical data analysis
Four different solid tumor models i.e., CT26 (colon), 4T1 (breast), LLC (lung), and B16F10 (melanoma) were used to evaluate the tumor accumulation of the m-NCs, with or without the application of a magnetic field. To better understand and visualize which vascular factors (MVD, CO or DM) may have had an impact on magnetic tumor targeting, a retrospective DoE analysis, using historical data, was performed to establish the response surfaces and the predictive model. Raw data that are used to create the Predictive Response Surface for Responses 1 and 2 are summarized in Table S1. Data were analyzed using Design-Expert 9, v9.0.6.2 (Stat-ease, Inc., USA). Suitable predictive models for Responses 1 and 2 were achieved using Sequential Model Sum of Squares (SMSS).

Selecting predictive model using sequential model sum of squares (SMSS)
For Response 1, Box-cox transformation was performed for the raw data to improve model fit (power transformed λ = 0.5).
Step-wise regression (involving forward selection, backwards elimination, and bidirectional elimination) was used to determine the model terms (Alpha in and Alpha out = 0.1). The SMSS was used to select a suitable predictive model for data analysis. The mean square of the model was firstly calculated followed by the addition of a higher level source of term, i.e., a higher degree of the polynomial in the predictive equation. The aim was to include a higher level source of terms only if this could explain a significant amount of variation in the responses when compared with the lower-level model. In other words, when one or more predictor variables (source of term) are included in the model, the error sum of squares (SSE) should be reduced or the regression sum of square (SSR) should be increased. As shown in Table S2, the Model Fvalue of 37.88 implied the model was significant (p-value < 0.0001) and factor B (CO) and C (D) were significant model terms (p-value < 0.0001). The SMSS table for Response 1 is shown in Table S3. Linear predictive model explained a significant amount of variability in the responses when compared to the overall sample mean (p-value <0.001).
Adding the two-factor interaction (2FI) into the model did not explain the rest of the variability i.e., no improvement in the model. Linear model was therefore suggested for Response 1. The provisional models were then evaluated with a lack of fit test. The linear model "lack of fit F value" was 7.97, p-value = 0.2727, which implied that there was a 27.27% chance that the lack of fit F value could have occurred due to noise, i.e., the model had a good fit (Table S4). Finally, all models were assessed by an overall standard deviation of the model, various R 2 and Predicted Residual Sum of Square (PRESS) statistics ( Table S5). The "Predicted R 2 " of 0.7129 was in reasonable agreement with the "Adjusted R 2 " of 0.7623, i.e., the difference was less than 0.2. "Adeq. Precision" of 13.494 indicated an adequate signal, i.e., a good signal-to-noise ratio. The fit test summary for Response 1 is shown in Table S6. This model can be used to navigate the design space ( Table S7). The coefficient of the model terms is shown in Table S8 with a small Variance Inflation Factor (VIF) of 1.04 for both Factor B and C. The normality of the residual from the predictive model was tested and the results are shown in Figure S4, which appeared to be normally distributed.
For Response 2, Box-cox transformation was performed for the raw data to improve model fit (power transformed λ = 0).
Step-wise regression (involving forward selection, backwards elimination, and bidirectional elimination) was used to determine the model terms (Alpha in and Alpha out = 0.1). As shown in Table S9, the Model F-value of 21.97 implied that the model was significant (p-value < 0.0001). Factor A (MVD) and Factor B (CO) were significant model terms with a p-value of 0.0003 and 0.0005, respectively. The Table S10. Linear predictive model explained a significant amount of variability in the responses when compared to the overall sample mean (p-value <0.001). Adding the two-factor interaction (2FI) into the model did not explain the rest of the variability significantly, i.e., no improvement in the model. Linear model was therefore suggested for Response 2. The linear model lack of fit F value for Response 2 was 15.01, p-value = 0.2010, which also indicated a good fit ( Table S11). The "Predicted R 2 " of 0.5951 was in reasonable agreement with the "Adjusted R 2 " of 0.6458, i.e., the difference is less than 0.2. "Adeq. Precision" of 13.631 indicates an adequate signal, i.e., a good signal-to-noise ratio ( Table S12). The fit test summary for Response 2 is shown in Table S13. This model could be used to navigate the design space ( Table   S14). The coefficient of the model terms is shown in Table S15 with a small Variance Inflation Factor (VIF) of 1.04 for both Factor A and B. The normality of the residual from the predicted model was tested and the results are shown in Figure S5, which appeared to be normally distributed.   The percentage injection dose per gram tumors (%ID/g), with (TU+) or without (TU-) exposure of a magnetic field, was assessed with gamma counting, at 1, 4 and 24 h. Results are expressed as mean ± SEM (n = 3). One-way ANOVA was performed using IBM SPSS version 20 followed by Tukey's multiple comparison test (*p > 0.05) and ** p < 0.01).      The "Pred R-Squared" of 0.7129 is in reasonable agreement with the "Adj R-Squared" of 0.7623; i.e. the difference is less than 0.2. "Adeq Precision" measures the signal-to-noise ratio. A ratio greater than 4 is desirable. A ratio of 13.494 indicates an adequate signal. This model can be used to navigate the design space.    The "Pred R-Squared" of 0.5951 is in reasonable agreement with the "Adj R-Squared" of 0.6458; i.e. the difference is less than 0.2.

Supplementary Figures
"Adeq Precision" measures the signal-to-noise ratio. A ratio greater than 4 is desirable. A ratio of 13.631 indicates an adequate signal. This model can be used to navigate the design space.