Theoretic Criteria for Antibody Penetration into Solid Tumors and Micrometastases

MATERIALS AND METHODS

We present 2 criteria to predict the extent of antibody targeting before and during saturation. The model is validated against published experimental data and is intended to provide a framework in which to analyze and interpret antibody targeting in vivo while also highlighting strategies for improvement in antibody micropharmacokinetics.

Dimensionless ratios of pairs of critical rates, sometimes referred to as moduli, simplify identification of the most dominant rates under particular conditions. We derive here 2 such moduli for tumor targeting.

Thiele Modulus

The Thiele modulus captures the rate of antibody catabolism in the tumor compared with the time required to saturate the tissue by diffusion. If the characteristic time for internalization and catabolism is faster than the time required for antibody to reach the most distant tumor tissue (), antibody will be degraded before it completely saturates the tumor.Eq. 3whereEq. 4Eq. 5where the normal tissue concentration isEq. 6for a constant infusion and is set to the plasma concentration for bolus doses. The internalization rate constant k_e represents the rate of catabolism in the tumor tissue, and [Ag] represents the concentration of internalizing antigen. Bi is the mass transfer Biot number, a dimensionless number characterizing transport resistances in series. For a small Biot number, permeability dominates over diffusion.

These parameters together define necessary conditions for tumor saturation:

Failure of either criterion is sufficient for failure to saturate the tumor volume. In addition to providing criteria for saturation of a given tumor volume, and Γ_tumor enable prediction of the depth of penetration of bound antibody. One of these parameters will approximately equal 1 at the distance of maximum antibody penetration, because by definition that is the point at which the diffusion or extravasation rate is balanced by either systemic clearance or antibody catabolism as illustrated in Figure 2.

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FIGURE 2. 

Antibody penetration distance limited by clearance (A and B) or antigen turnover (C and D). Pharmacokinetic parameters are from Table 1; antigen concentration is 0.1 μM. Formulas for R are located in the supplemental data (available online only at http://jnm.snmjournals.org).

RESULTS

For experimental validation (Fig. 3), 2 datasets that were not compounded by clearance effects were chosen to verify the Thiele modulus. Independent estimates for the various parameters were obtained from the literature (Tables 1 and 2). In the first experiment (10), single-chain variable fragments (scFvs) were dosed in anephric mice, eliminating rapid renal clearance. Before kidney removal, the clearance modulus of 3.8 indicates poor targeting due to the rapid drop in plasma concentration. In the second (11), continuous dosing of a slow-clearing antibody kept the effective concentration high throughout the experiment. Caution must be taken when interpreting other targeting results of tumors with high catabolism rates using radioactive tracers. Additional factors, such as the residualizing or nonresidualizing nature of the radioisotope, must also be taken into consideration (12). Also, for very low doses of radioactive antibodies used in diagnostics, more targeting may occur near the periphery of the tumor where there is some convection due to fluid leakage. By improving the moduli and saturating a larger fraction of the tumor, the imaging signal will be improved. However, this must be carefully weighed against any increased background from larger doses or longer clearance times.

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FIGURE 3. 

Experimental validation. (A) Model prediction (top); image adapted from Adams et al. (9), with yellow fluorescence marking capillary, red fluorescence indicating antibody, and green circle indicating model-predicted distance for penetration without any fitted parameters (k_on = 7 × 10⁵ M⁻¹s⁻¹; other parameters are in Table 1) (center); and fluorescence vs. distance using ImageJ (bottom). (B) Tumor size as fraction of control vs. Thiele modulus (with trastuzumab), defined using 100-μm distance, which corresponds with targeting 95% of tumor tissue (9). (C) Growth attenuation vs. fraction of tumor that is predicted to be saturated by model. The distances given in Baish et al. (9) were used.

The 2 criteria for effective targeting are not independent. In many cases, both clearance and internalization occur on similar time scales. To address the interaction between the two, full numeric simulations were performed with both internalization and clearance (Fig. 4).

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FIGURE 4. 

Full numeric simulations were performed with biexponential clearance and catabolism in tumor tissue. Antibody concentration was varied to change clearance modulus, and endocytosis rate was varied to independently change Thiele modulus. met = metastasis.

DISCUSSION

The moduli provide insight into improving targeting. Increases in antibody dose unambiguously improve both and with capillary permeability necessitating large doses for solid tumors. Practical concerns, such as toxicity, immunogenicity, and economics, set an upper limit to this parameter. Extending the plasma half-life minimizes for agents such as scFvs, which clear too quickly to target all cells. A more rapid diffusivity helps targeting in micrometastases, and this is most easily achieved by smaller molecules, often trading off with faster clearance. To increase penetration into the tumor, should be minimized by using “noninternalizing” antibodies, such as antibodies that are effectively recycled in the endosome, extracellular matrix proteins, shed antigen, trapped necrotic debris, or perhaps tight-junction antigens. However, drugs requiring antibody uptake (e.g., immunotoxins and chemotherapeutic conjugates) clearly cannot apply this strategy; the particular problems with immunotoxin uptake versus diffusion have been quantitatively considered previously (13). Antigen level plays a subtle role in trading off pharmacodynamics vs. micropharmacokinetics. High expression levels provide a large amount of target, but this can limit penetration into the tumor. Although ideal for imaging, therapeutic targeting can be significantly hindered by these levels. Once is high enough to enable delivery of a cell-killing dose of drug to an individual cell, any further increases only serve to worsen the distribution of drug throughout the tumor because of proportional increases in ϕ² and . For highly potent cytotoxic payloads such as α-particle–emitting radionuclides that kill cells with delivery of only 1–10 molecules of drug, antigens with 10⁵–10⁶ molecules per cell present an unnecessarily onerous impediment to diffusion through the tumor tissue. Capillary permeability provides a very large barrier for solid tumor uptake as illustrated by the small Biot number. Attempts to improve P in solid tumors through transcytosis by caveolar targeting have begun to meet with initial success in this regard (14). Even in these scenarios, it may not be possible to effectively target larger tumors with significant necrotic areas lacking functional blood vessels or even vascularized tumors less than the roughly 1-cm detection limit, which are slightly less heterogeneous. Finally, the distance from the circulating source of antibody drug will in general be smaller for micrometastases relative to vascularized bulk tumors, supporting a strategy of applying antibody therapeutics primarily to adjuvant therapy for minimal residual disease after resection (15).

CONCLUSION

Scaling arguments such as those presented in this paper can be useful analytic tools for qualitative consideration of overall systems behavior when multiple different rate phenomena are present (e.g., clearance, binding, endocytosis, diffusion). Arguably, the ability to generalize such simplistic analyses is actually preferable to detailed descriptive models of specific cases, when one is considering highly variable phenomena such as tumor physiology. The scaling analysis indicates that even low levels of antigen turnover can severely limit penetration of antibodies into tumor tissue; clearance of antibodies, especially fragments that are rapidly filtered, restricts penetration; and the large resistance of the tumor capillary wall may make antibodies more suitable for targeting micrometastases than solid tumors.