|
| Figure 1: Gompertzian growth shown as log tumor burden (as represented by number of cells) vs. time (days). The equation is N = exp [(A/α ) * (1-exp[- α t])]. At small time the equation is exponential (tA) and at large time, the expression approaches the asymptotic value of exp (A/α). The Gompertzian growth model has long been assumed to describe primary and metastatic tumors. The growth starts as exponential (constant doubling time) which would appear as a straight line on this semi-log scale. There are two parameters in this function that determine the initial growth rate and the ultimate size (N is the number of cells in the tumor). In figure 1, A was chosen as 0.3 per day and α was chosen to be 0.008 per day. Time t is expressed in days. The ultimate size asymptotically attainable in this example is 1.9 × 1016 cells. This value must be larger than 1012 cells, the value that is usually taken as a lethal tumor burden, since untreated cancer is presumed to be uniformly lethal. Thus, the two parameters A and α are not completely independent. Exp (At) is equal to exp [(A/α ) * (1-exp[- α t]) at t = 0 but is larger at all other times greater than zero. |
