Mathematical analysis of the uptake kinetics of antitumor antibodies in the spheroid model of free floating micrometastases

A mathematical model consisting of two differential equations was developed to estimate the values of 3 physical constants which govern the uptake kinetics of monoclonal antibody 96.5 (isotype IgG2a) in human melanoma spheroids. The constants are the macroscopic binding rate Kf, macroscopic dissociation rate Kr, and macroscopic diffusion constant D. The estimated values of these constants are 8. 8 x 108 (1/Mh) , 0.073 (cm 2/h), and 8.28 x 10-5 (cm2/h ) respectively. The binding valence n of the antibody with its associated antigen was assumed to be 1. When n was assumed to be 2 and with the same value of D, the estimated values are Kf = 6.5 x 108 (1/Mh) and Kr = 0.46 (cm 2 /h). This model gives reasonably good agreement with the experimental antibody uptake by the spheroid in the cases of brief washing and 24 hour washing after different incubation periods with the antibody, even though saturation of binding sites occurred in the peripheral layer of the spheroid. Overall, this model showed better prediction of the uptake of antibody than the simple model that the authors reported previously. The simple model does not allow for saturation of binding sites. A model consisting of three differential equations was also introduced in this work to estimate not only Kf, D, and Kr, but also the rate of shedding of antibody-antigen complex