A modeling analysis of the effects of molecular size and binding affinity on tumor targeting

Abstract

A diverse array of tumor targeting agents ranging in size from peptides to nanoparticles is currently under development for applications in cancer imaging and therapy. However, it remains largely unclear how size differences among these molecules influence their targeting properties. Here, we develop a simple, mechanistic model that can be used to understand and predict the complex interplay between molecular size, affinity, and tumor uptake. Empirical relationships between molecular radius and capillary permeability, interstitial diffusivity, available volume fraction, and plasma clearance were obtained using data in the literature. These relationships were incorporated into a compartmental model of tumor targeting using MATLAB to predict the magnitude, specificity, time dependence, and affinity dependence of tumor uptake for molecules across a broad size spectrum. In the typical size range for proteins, the model uncovers a complex trend in which intermediate-sized targeting agents (MW, approximately 25 kDa) have the lowest tumor uptake, whereas higher tumor uptake levels are achieved by smaller and larger agents. Small peptides accumulate rapidly in the tumor but require high affinity to be retained, whereas larger proteins can achieve similar retention with >100-fold weaker binding. For molecules in the size range of liposomes, the model predicts that antigen targeting will not significantly increase tumor uptake relative to untargeted molecules. All model predictions are shown to be consistent with experimental observations from published targeting studies. The results and techniques have implications for drug development, imaging, and therapeutic dosing.

Figure 1

Size dependent transport parameters. A, B. Relationship between molecular radius and effective diffusivity (D) and available volume fraction (ε) in the tumor. Data points were simultaneously fit to a two pore model of the tumor interstitial space. C. Relationship between molecular radius (R) and effective molecular permeability across the tumor vasculature (P). Data points were fit to a two pore model of the capillary wall. D. Relationship between molecular radius and plasma clearance rate (k_clear). Data points were fit to an empirical model of renal and non-renal clearance. IgG clearance is denoted by an open circle and was not included in the fit. All data fitting was performed using a non-linear least squares method. Data points were collected from experimental results reported in the literature and include measurements of proteins (circles), dextran and PEG polymers (squares), small molecule tracers (diamonds), and liposomes (triangles). Additional descriptions of the experimental data are presented in Supplemental Tables 1–4.

Figure 2

Predicted effect of molecular size on maximum tumor uptake. Simulations were performed using a compartmental model of tumor transport and size-dependent values of P, ε, and k_clear. All size-independent parameters are reported in Table 1. Tumor concentrations are reported as %ID/g. A. Predicted peak tumor concentrations of HER2 binding molecules (K_d = 1 nM) labeled with ¹²⁵I (solid line) or residualizing ^99mTc (dashed line). IgG uptake was simulated independently (open circle - ¹²⁵I, solid circle – ^99mTc) and is predicted to be higher due to FcRn mediated reduction in plasma clearance. The vertical grey lines highlight the size range typical of protein therapeutics that is further analyzed in Figure 2B. B. Comparison to experimental data. Peak uptake simulations were performed as above and plotted as a function of effective molecular weight. The predicted uptake trends for R_Krogh = 50 μm and R_Krogh = 100 μm form the upper and lower bounds respectively of the shaded grey area. Data points were collected from HER2 targeting experiments in the literature including ^99mTc, ¹¹¹In, and ⁶⁴Cu labeled molecules of various sizes. References and additional details for each experimental data point are presented in Supplemental Table 5. The units of radius and effective MW used in panel A and B, respectively, can be related using the relationship MW = 1.32*R_mol³ (for example 7 kDa affibodies, 27 kDa scFvs, 50 kDa Fabs, and 150 kDa IgGs have radii of 1.74 nm, 2.74 nm, 3.47 nm, and 4.86 nm, respectively.)

Figure 3

Predicted effect of molecular size on time course of tumor uptake. Tumor uptake over time was simulated for ¹²⁵I (A) or ^99mTc (B) labeled non-Fc domain containing HER2 binding molecules (K_d = 1 nM) ranging in size from 2–1000 kDa. C. Effect of molecular size on the time of maximum tumor uptake for ¹²⁵I (solid line) or ^99mTc (dashed line) labeled molecules. D. Comparison to experimental data. Tumor uptake simulations were performed for affibodies (MW = 7 kDa), scFvs (27 kDa), tetrabodies (130 kDa), and IgGs (150 kDa) and compared to experimentally measured time courses for ^99mTc labeled HER2 targeting molecules (–32). R_Krogh values were fit to the experimental data for each molecule using the least squares method with results of 57, 80, 101, and 84 μm for the affibody, scFv, tetrabody, and IgG data sets respectively. These values are all in a physiologically reasonable range.

Figure 4

Binding and affinity dependence. A. Predicted tumor uptake at 24 hours for ^99mTc labeled HER2 targeting molecules varying in both size and affinity for the target antigen. B. Affinity necessary to achieve 10% (small dashes), 50% (large dashes), or 90% (solid line) of the maximum tumor uptake at 24 hours as a function of molecular size. C. Comparison to experimental data. The predicted 24 hour tumor concentration for HER2 targeting scFvs (MW = 27 kDa) of various affinities were compared to experimental uptake measurements for affinity variants of the C6.5 scFv (33). Model predictions and experimental data are normalized by their respective uptake values for the highest affinity case. D. EPR mediated non-specific uptake. Predicted tumor concentrations of non-targeted molecules (K_d = 1 M) ranging in radii from 0.5–60 nm were calculated for various times and normalized by the predicted uptake of size matched antigen binding molecules with a K_d of 1 nM (untargeted to targeted uptake ratio). A value of 0 represents fully binding mediated tumor retention, while a value of 1 represents equivalent uptake of targeted and non-targeted molecules. R_Krogh = 100 μm.

Figure 5

Predicted tumor uptake in humans. Simulations were performed as described in Figure 2 except with V_plasma = 3L, and [Ag] and k_e adjusted for targeting CEA. A. Predicted peak tumor concentrations in humans of CEA binding molecules (K_d = 1 nM) labeled with ¹²⁵I. IgG uptake was simulated independently and denoted by the solid circle. B. Comparison to clinical data. Peak uptake simulations were performed as above and plotted as a function of effective molecular weight in the size range typical of proteins (2–500 kDa). The predicted uptake trends for R_Krogh = 50 μm and R_Krogh = 100 μm form the upper and lower bounds respectively of the shaded grey area. The data points represent clinically measured tumor concentrations for scFv, F(ab′)₂, DFM, and IgG molecules targeting CEA expressing tumors in humans (35).