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Effect of Dose, Molecular Size, Affinity, and Protein Binding on Tumor Uptake of Antibody or Ligand: A Biomathematical Model¹

Immunodiagnostic Research Laboratory, Department of Immunology, University of Birmingham Medical School, Birmingham B15 2TJ [G. D. T., P. W. D., J. R. M. E., A. R. B.]; University Department of Medicine, Queen Elizabeth Hospital, Birmingham B15 2TH [D. B. R.]; and Department of Engineering, University of Warwick, Coventry CV4 7AL [M. J. C., K. R. G.], England

A mathematical model has been developed to determine the best approach to improving tumor targeting with antibody. The amount of antibody in the tumor (tumor content) and the tumor:normal tissue antibody concentration ratio (uptake ratio) were calculated over 12 days from injection, using the computer program FACSIMILE to solve the stiff nonlinear differential equations describing the system. Results indicate that success requires an optimal combination of dose, size, and binding affinity of antibody. Increasing the dose to 100 times that presently used for scanning increased both the percentage of injected antibody in the tumor and the uptake ratio by up to 2 orders of magnitude to maximal values determined by affinity. This result could be achieved by coinjecting unlabeled antibody. Increasing affinity from K_eq = 10⁹ to 10¹³ M^-1 increased the uptake ratio from 5 to 100 for whole antibody and to 550 for a small ligand, at the calculated optimal dose, but had no effect at the current scanning dose. With decreasing molecular size at average affinity, the same maximum tumor content and uptake ratio were achieved but progressively earlier. At high affinity there was a substantial advantage for a small ligand compared with whole antibody in terms of uptake ratio (550 versus 100) and tumor:normal tissue integral dose ratio (330 versus 60). The uptake of a small ligand was not increased by binding to plasma protein but with increasing time the tumor content was higher than without protein binding.

¹ G. D. T. and J. R. M. E. were supported by the Cancer Research Campaign (Grant CEF 412) and M. J. C. by the Science and Engineering Research Council (Grant GR/D90642).

² To whom requests for reprints should be addressed.

Received 10/ 6/88. Revised 3/21/89. Accepted 3/22/89.

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