## Mathematical relationships of exponentially growing tumors and projected tumor marker or tumor sizevolume

The mathematical expression for exponential expansion of growth is: Vt = V0eat where the tumor volume at time Vt is predictable and is the product of the starting tumor volume [V0] and [e = 2.71828, the base of the natural logarithm raised to the product of the specific growth rate constant a or (SGR) and the duration of growth At or (trt0)].

This is given as Vt = V0eSGR At and mathematical rearrangement yields SGR =

Inhibitory drug effects slow SGR and are precisely quantifiable by calculating changes of SGR and the tumor size before and after therapy. Tumor size after therapy should be compared to the projected tumor size the same time after therapy. The current standard for clinical oncologists is comparison of tumor size before and after therapy while neglecting comparison with the projected tumor size. Differences between post therapy tumor size and the post therapy projected tumor size are the clue to hidden responses that are almost never evaluated by the clinical oncologist. These often-subtle differences between projected and post therapy tumor sizes may reveal hidden growth stimulation (mutation or idiosyncratic drug effect) as well as subtle growth inhibition, which may lead to prolonged clinical stability.

The following relationships, extracted from Mehrara's analysis [69,70,71] define projected ft Vn Vt tumor volume: J △ SGR (t)*dt = ln(y)-ln(y) where Vn = projected tumor volume, Vt =

volume of tumor at the time of response evaluation and V is the volume at the initiation of therapy. The tumor response or TR = - ln(Vt/Vn) where Vt is the volume of treated tumor and Vn is the hypothetical or projected tumor volume, both evaluated at the time of efficacy assessment. These relationships are the model for the growth kinetics of exponentially growing tumors and generally require the use of at least a handheld computer to facilitate evaluation in the clinic. This is further discussed in the appendix.

Figure 6. In vitro and in vivo, a clear distinction between cytostatic and cytolytic drugs does not exist. Low-dose cytolytic chemotherapy may exert cytostasis or so-called cytolentic slowing of cell proliferation leading to cell lysis, while targeted therapy's prolonged cytostatic metabolic effects (or large doses of targeted therapy) may induce cytolysis and autophagy (autophagocytosis). Regardless of mechanism of cell inhibition, the SGR and the TR (treatment response) calculations clearly and objectively define and quantitate drug efficacy (TR value).

Figure 6. In vitro and in vivo, a clear distinction between cytostatic and cytolytic drugs does not exist. Low-dose cytolytic chemotherapy may exert cytostasis or so-called cytolentic slowing of cell proliferation leading to cell lysis, while targeted therapy's prolonged cytostatic metabolic effects (or large doses of targeted therapy) may induce cytolysis and autophagy (autophagocytosis). Regardless of mechanism of cell inhibition, the SGR and the TR (treatment response) calculations clearly and objectively define and quantitate drug efficacy (TR value).

Picture a 4.0 cm diameter (14.1 cc) pulmonary metastasis. At the time of discovery two months before the start of therapy the tumor was 3 cm (33.5 cc). The pre-therapy SGR for this tumor = 1.46%/d (tumor volume was expanding by 1.46%/d). Sixty-one days of therapy was administered and the tumor grew to 4.5 cm (47.7 cc). SGR decreased from 1.46%/d to 1%/d. Clinicians unaware of SGR and the projected tumor volume at this point might declare drug resistance however; the projected tumor volume was actually 80.6 cc and the tu mor reached only 47.7 cc. Even though the tumor grew, therapy was significantly effective in slowing growth (59% of intrinsic tumor growth was inhibited)! The parameter for treatment efficacy, TR was +0.5. A positive value for TR means that therapy had some inhibitory activity against the tumor, the larger the value the better. A negative value means therapy was associated with growth stimulation. The value of TR is useful as an objective standard comparator to help evaluate efficacy between different treatments.

Figure 7. Tumor size (volume) or Marker (PSA, PAP) Time Line

Figure 7 illustrates potential tumor responses to drug treatment. Some of the responses such as positive and negative deviation from the projected PSA value or projected tumor volume are routinely overlooked in the clinic because projected sizes for these parameters must be calculated in advance (projected volume is illustrated by the largest b-b green circle at t2). SGR is calculated based on tumor or PSA growth between t0 and tj. Deviations from projected values reveal subtle drug-tumor interactions. In the appendix we discuss straightforward evaluation of all five-treatment outcome scenarios illustrated above by a hand-held computer.

Until now, most attempts to capture drug effects vs. prostate tumors employed changes of PSA-DT. However, Mehrara [70] presented newer assessments of PSA-DT compared to PSA-SGR that cast doubt on the validity of that historic collection of work.

What follows is a general listing of consequences of drug-tumor interaction. These potential tumor or marker responses Figures 6, 7 are important to understand because subtle changes in tumor proliferation may be the only drug-induced tumor response and may go unnoticed when evaluating targeted therapy by RECIST/RECIST 1.1 response criteria.

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