A model-based approach for assessing in vivo combination therapy interactions

We present an approach for evaluating the efficacy of combination antitumor agent schedules that accounts for order and timing of drug administration. Our model-based approach compares in vivo tumor volume data over a time course and offers a quantitative definition for additivity of drug effects, relative to which synergism and antagonism are interpreted. We begin by fitting data from individual mice receiving at most one drug to a differential equation tumor growth/drug effect model and combine individual parameter estimates to obtain population statistics. Using two null hypotheses: (i) combination therapy is consistent with additivity or (ii) combination therapy is equivalent to treating with the more effective single agent alone, we compute predicted tumor growth trajectories and their distribution for combination treated animals. We illustrate this approach by comparing entire observed and expected tumor volume trajectories for a data set in which HER-2/neu-overexpressing MCF-7 human breast cancer xenografts are treated with a humanized, anti-HER-2 monoclonal antibody (rhuMAb HER-2), doxorubicin, or one of five proposed combination therapy schedules.

The paradox of alloreactivity and self MHC restriction: Quantitative analysis and statistics

Although 1–24% of T cells are alloreactive, i.e., respond to MHC molecules encoded by a foreign haplotype, it is generally believed that T cells cannot recognize foreign peptides binding foreign MHC molecules. We show using a quantitative model that, if T cell selection and activation are affinity-driven, then an alloreactivity of 1–24% is incompatible with the textbook notion that self MHC restriction is absolute. If an average of 1% of clones are alloreactive, then according to our model, at most 20-fold more clones should, on average, be activated by antigens presented on self MHC than by antigens presented on foreign MHC. This ratio is at best 5 if alloreactivity is 5%. These results describe average properties of the murine immune system, but not the outcome of individual experiments. Using supercomputer technology, we simulated 100,000 MHC restriction experiments. Although the average restriction ratio was 7.1, restriction was absolute in 10% of the simulated experiments, greater than 100, although not absolute, in 29%, and below 6 in 24%. This extreme variability agrees with experimental estimates. Our analysis suggests that alloreactivity and average self MHC restriction both cannot be high, but that a low average restriction level is compatible with high levels in a significant number of experiments.

A modified gambler's ruin model of polyethylene chains in the amorphous region.

Polyethylene chains in the amorphous region between two crystalline lamellae M unit apart are modeled as random walks with one-step memory on a cubic lattice between two absorbing boundaries. These walks avoid the two preceding steps, though they are not true self-avoiding walks. Systems of difference equations are introduced to calculate the statistics of the restricted random walks. They yield that the fraction of loops is (2M - 2)/(2M + 1), the fraction of ties 3/(2M + 1), the average length of loops 2M - 0.5, the average length of ties 2/3M2 + 2/3M - 4/3, the average length of walks equals 3M - 3, the variance of the loop length 16/15M3 + O(M2), the variance of the tie length 28/45M4 + O(M3), and the variance of the walk length 2M3 + O(M2).