Example Modeling Tumor Growth

In the development of a new oncolytic, an experiment was conducted wherein mice subcutaneously implanted with A549 human lung tumors were randomized to four treatment groups  

Saline once daily by intraperitoneal (IP) administration (control group) 

Drug X 10 mg/kg once daily by oral (PO) administration for 28 days 

On days 1, 5, 8, 12, 15, 20, 22, 26, 29, and 33 the mouse weight, tumor length, and tumor width were measured. Length and width were converted to tumor volume which was used as a surrogate for tumor growth. Seven (7) mice were randomized into each treatment group. Of interest to the researchers were the following questions  

Subject Period 1 Sequence AUC(0 - oo) Period 2 AUC(0 - oo) Period 1 Cmax Period 2 Cmax 

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While the obvious value of in vivo animal models is clear, there also are instances—especially in cases of inflammatory arthritis, behavior, and tumor growth—where they have failed to be predictive of useful clinical activity in humans [51], For example, leukotriene (LTB4) antagonists showed activity in animal models of inflammatory arthritis yet failed to be useful in rheumatoid arthritis [52]. Similarly, dopamine D4 antagonists showed activity in animal behavior models previously predictive of dopamine D2 antagonists in schizophrenia. However, testing of dopamine D4 antagonists showed no efficacy in humans [53]. 

One further question that has a substantial impact on the application of modeling techniques to biomedical problems is the choice of the design. Suppose that in our Gompertz tumor growth example we wanted to decide, given the results of some pilot experiments, when it is most useful to observe the tumor volume. In other words, we wish to choose the time points at which we obtain tumor volume observations in order to maximize the precision of the resulting parameter estimates. 

Such discrepancy is observed in many other cases [12, 13] as well as in cases of tumors growth considered above. Obviously, mathematical model of growth (8) is a very rough approximation. It may be used for rough estimation, for example for classification of population development [16, 17] and also for description of experimental data on separate sections of growth curves. 

The predictions of this simple model agree surprisingly well with data on tumor growth, as long as N is not too small see Aroesty et al. (1973) and Newton (1980) for examples. 

It can be derived from this model that the reciprocal of k2 (l/fcj) equals a threshold exposure (AUCX) required for complete suppression of tumor growth. This AUCX equals a threshold concentration (Cx) multiplied by the mean tumor growth time, thus it represents a total AUC for the length of mean tumor growth time. For example, 1 lk2 (AUCX) represents a total AUC for four days, when = 0.25 day1. 

Table 6.10 Comparison of ML and REML goodness of fit metrics and p-values for significance of fixed effects for tumor growth example under the final model (unstructured G-matrix, spatial power R-matrix).

Additionally, small molecule inhibitors of the sphingolipid pathway to induce the accumulation of ceramide have been used in some cancers. For example, B13, an inhibitor of acid CDase was used in a metastatic colon cancer mouse model and a prostate cancer xenograph model (Selzner et al., 2001 Samsel et al., 2004). In both cases, BI3 caused the accumulation of ceramide and resulted in prevention of tumor growth. Another effective approach to increase ceramide accumulation in cancer cells has been to inhibit SM synthase, or acid CDase (Meng et al., 2004 Saad et al., 2007). 

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