Mathematical Modeling Unveils Optimization Strategies for Targeted Radionuclide Therapy of Blood Cancers

Abstract

Mathematical modeling yields general principles for optimization of TRT in mouse models of multiple myeloma that can be extrapolated to other cancer models and clinical settings.

Figure 1

A, Schematic view of the main processes considered in the mathematical model of TRT of blood cancer. Note that antibody–receptor binding is considered irreversible. B, Illustration of three types of radiation damage, considered in the model. (B, Created with BioRender.com.)

Figure 2

Model simulations of treatment by pure radioconjugates (coefficient of drug impurity η = 0) for the basic set of parameters with the dose A_cur leading to minimal number of viable cancer cells N_m = 0.01, which is considered as the curative threshold. A, Dynamics of active antibodies and their fragments in plasma. B, Dynamics of cancer cells. C, Dependence of minimal single curative dose, A_cur, on relative significance of self-damage, k_s. Star marks the case with basic value of k_s = 0.3. See Supplementary Section S.2.1.6 for analytical estimation of A_cur. D, Dynamics of occupied receptors on viable cancer cells. The density of radionuclides on them decreases due to the decay of nuclides, rate of which is λ, and due the to the redistribution of nuclides among the newborn cells, characterized by cell proliferation rate ρ. E, Dynamics of occupied receptors on damaged cancer cells. After a transient period, the density of radionuclides on them decreases only due to the decay of nuclides.

Figure 3

Dependence of minimal single curative dose, A_cur, on the coefficient of drug impurity, η, i.e., the ratio of unlabeled antibodies to radioconjugates. A, Moderate variation in η. Under significant self-damage, saturation of specific receptors leads to redirection of radioconjugates and deposited doses toward still viable cells, which continue generating specific receptors. This effect yields the decrease in A_cur with an increase in η. B, Extended variation of η. The curative treatment becomes lethally toxic shortly after the amount of injected antibodies, (η + 1)A_cur, exceeds the total amount of specific receptors expressed on cancer cells during treatment.

Figure 4

A, Scatter plot of cancer-binding capacity, i.e., total amount of specific receptors on cancer cells at the moment of drug injection, and minimal single curative dose produced by the global parameter sweep within the training set of 1,000 virtual mice. Cancer-binding capacity represents the product of the number of specific receptors on each cancer cell, γ, and the initial number of cancer cells, N₀. Maximal safe dose is based on the exact value of only one parameter, cancer-binding capacity, and on physiologic ranges of other parameters. Coefficient of drug impurity, η, corresponds to the labeling ratio of 1.85 kBq/μg. B, Survival curves and (C) toxicity curves for a single test set of 1,000 virtual mice, treated by different approaches.

Figure 5

Survival curves for the test set of virtual mice, grouped by relative significance of self-damage, k_s: (A) low k_s; (B) intermediate k_s; and (C) high k_s. Personalized schedules based on cancer-binding capacity, γN₀, yield robust treatment optimization via initial saturation of cancer-binding capacity with the first dose, enabling redistribution of further injected small doses toward still viable cells that continue expressing specific receptors. Coefficient of drug impurity, η, corresponds to the labeling ratio of 1.85 kBq/μg.