Mathematical model of simultaneous diffusion and binding of antitumor antibodies in multicellular human tumor spheroids.

Multicellular tumor spheroids are widely used as in vitro models of poorly vascularized tumor nodules in vivo. The uptake kinetics of tumor-associated antibodies in multicellular tumor spheroids is assumed to be governed by passive diffusion and irreversible binding of the antibodies with binding sites on the cell surface. By further assuming that the spheroids are homogeneous with respect to diffusion and binding, a mathematical model has been developed which permits the extraction of the macroscopic diffusion constant D and the macroscopic binding rate k from empirical studies. The model was applied to uptake kinetics data obtained (a) with a melanoma-associated monoclonal antibody 96.5 (isotype IgG2a)-human multicellular melanoma spheroid system exhibiting strong antibody to cell binding and (b) with the same monoclonal antibody-human multicellular colon adenocarcinoma HT29 spheroid system exhibiting nonspecific binding. The spheroids had approximately 300 microns diameter. The constants D and k were estimated to be 0.45 micron2 s-1 and 2.0 x 10(-3) s-1, respectively, for the system with specific binding. Saturation of binding sites occurred. In the nonspecific binding system, D and k were found to be 0.10 micron2 s-1 and 1.0 x 10(-5) s-1. No saturation of binding sites occurred. D and k were also estimated to be, respectively, 0.52 micron2 s-1 and 6.4 x 10(-5) s-1 for another melanoma-associated monoclonal antibody 140.240 (same isotype as 96.5) in the melanoma spheroid system exhibiting moderate cell binding with the antibody. The mathematical model describes well the system exhibiting nonspecific binding, but requires modifications and further development for the systems exhibiting moderate to strong binding.