Theoretical analysis of antibody targeting of tumor spheroids: Importance of dosage for penetration, and affinity for retention

The interplay among antibody/antigen binding kinetics, antibody diffusion, and antigen metabolic turnover together determines the depth of penetration of antitumor antibodies into prevascular tumor spheroid cell clumps. A sharp boundary between an outer shell of bound high-affinity antibody and an inner antibody-free core has been previously observed and mathematically modeled and was termed the "binding site barrier." We show here that this process is well described by a simplified shrinking core model wherein binding equilibration is much more rapid than diffusion. This analysis provides the following experimentally testable predictions: (a) the binding site barrier is a moving boundary whose velocity is proportional to the time integral of antibody concentration at the spheroid surface (i.e. plasma antibody AUC); (b) the velocity of this moving boundary is independent of binding affinity, if the affinity is sufficiently high to strongly favor antibody/antigen complex formation at prevailing antibody concentrations; and (c) maximum tumor retention is achieved when the antibody/antigen dissociation rate approaches the rate of antigen metabolic turnover. The consistency of these predictions with published experimental results is demonstrated. The shrinking core model provides a simple analytic relationship predicting the effects of altered antibody pharmacokinetics, antibody molecular weight, antigen turnover rate, antigen expression level, and micrometastasis size on antibody penetration and retention. For example, a formula is provided for predicting the bolus dose necessary to accomplish tumor saturation as a function of antibody and tumor properties. Furthermore, this analysis indicates certain attributes necessary for an optimal tumor targeting agent.