## SGR is a useful tool to identify subtle drugassociated tumor or marker kinetic changes of tumors

Mehrara, as part of his PhD thesis at the Department of Radiation Physics, University of Gothenburg, Goteborg, Sweden presented an analysis of tumor growth kinetics based on the tumor specific growth rate constant (SGR). The analysis assumes that for most practical purposes clinically observable tumor growth follows exponential growth. Additionally, this is true for the surrogate PSA tumor marker. SGR is rapidly calculable by hand-held mobile devices and facilitates the rapid identification of tumor responses easily overlooked in the clinic, many of which are not readily apparent without computer analysis. Occasionally, changes of SGR uncover subtle tumor stimulation.

Construction of the exponential growth curve, similar in shape to the mid portion of the Gompertzian curve Figure 1, requires just two different measurements of tumor volume (or diameter, area, cell number) or a surrogate marker at two different times to satisfy the exponential growth equation: Vt = V0 eat . Here "a" is the exponential growth constant, and Vt and V0 are the tumor volume at times t and t0, respectively. This model implies that tumor volume can increase indefinitely and the growth rate of a tumor is proportional to its volume and dV/dt =aV .

SGR is the relative change in tumor volume per unit time calculable as percent increase or decrease of tumor volume per unit time. Excluding mutations, for exponentially growing tumors, SGR is constant, i.e., SGR or a is independent of tumor volume or age. Faster growing tumors have higher SGR values, SGR=0 represents non-growing tumors; a negative SGR represents tumor regression. In 1956 Collins et al. [9] graphically introduced the concept of tumor doubling time. The DT formulation was proposed in 1961 [10]: DT = (t2-tj) * ln(2) / [ln(V2/Vj)]. Other relationships of importance include the specific growth rate, SGR = ln(V2/V1)/(t2-ti) and DT = ln(2)/SGR. These equations are descendants of the primary exponential growth equation, Vt = V0eat

Mehrara expresses concerns based on his mathematical treatment of SGR and DT suggesting that for clinical studies, SGR is the best indicator of tumor growth. Tumor growth rate, especially but not limited to urology circles, is usually quantified as DT i.e. PSA-DT. Because of the subtle mathematical relationship between SGR and DT, use of DT alone to evaluate therapeutic effects may give erroneous results.

Mehrara's studies revealed that DT has several drawbacks when used to describe tumor or tumor marker growth rates. The shortfalls include 1) for brief measurement time intervals, or high volume and very small measurement uncertainties the mean DT can either overestimate or underestimate the average growth rate; 2) DT approaches infinity for very slow growing tumors and is mathematically limited while SGR is a continuous variable no mater the speed and 3) the non normal frequency distribution of DT values restricts use of parametric statistics thus reducing use of more discriminatory statistics especially when studying small samples [77]. Unlike DT, SGR is definable for all tumor volume changes no matter how small, and it is Gaussian (normally) distributed allowing use of parametric statistics. SGR is more accurate to use when considering growth fraction, cell loss rate, and tumor growth rate heterogeneity. For these reasons, Mehrara opines that SGR be used instead of DT, to quantify tumor growth rate.

Accuracy and clinical outcome analysis comparing SGR and DT would be a valuable area of research in light of the cytostatic changes leading to subtle changes of growth rate characteristic of targeted therapies. Later, an in depth illustration of the differences between DT and SGR will help illuminate this issue.

Collins and Schwartz [9, 10] both analyzed several tumors in patients as they defined the use of tumor volume doubling time. Note that for bronchogenic lung cancers a semi-logarithmic plot of tumor diameter (y-axis) versus a linear time period (x-axis) produces a near straight line Figure 2.

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