Collins et al.  reported that the growth of human tumors was exponential, and they proposed that the tumor growth rate is best described as tumor DT. Other research concerning the changes in serum Ct and carcinoembryonic antigen (CEA) values over time in patients with MTC who had persistent disease postoperatively and studies of the serum thyroglobulin values in patients with papillary thyroid carcinoma (PTC) after a total thyroidectomy has demonstrated that these values changed exponentially and that the DTs of these tumor marker values were very strong prognostic factors, supporting the concept described by Collins et al. [1,7]. Thus, the DT is a well-validated parameter for the analyses and expressions of tumor growth.
However, the DT has two major limitations. First, if the volume of some of the tumors decreases over time, the tumors' DTs become negative values, which creates a discontinuity problem among positive and negative DT values. Second, the magnitude of the DT values is the opposite of the magnitude of the tumor growth rate. However, if the inverse of the DT is used (i.e., 1/DT), these limitations are resolved, and these inverse values should indicate the number of doublings that occur per unit of time. We thus proposed calling the 1/DT value the 'doubling rate' .
With the TV-DR we were able to demonstrate tumor volume kinetics of papillary microcarcinomas during active surveillance in individual patients . Very interestingly, the volume of 17% of the tumors in that study decreased over time, showing negative TV-DRs values. Our calculation of the hypothetical TV-DR (an estimate of the lowest tumor growth rate) before presentation using the patient's age and tumor size at presentation showed that this value was much higher than the TV-DR observed during active surveillance in most of the patients, suggesting that the tumor growth before presentation had been much more rapid . We also observed that in patients with papillary thyroid microcarcinoma during active surveillance, the probability of tumor progression greatly decreased with age [8,9]. These data indicated that the spontaneous deceleration of tumor growth with age is a rather common phenomenon in papillary thyroid microcarcinomas.
To determine whether a similar phenomenon occurs in medullary thyroid carcinoma, we calculated the hTV-DR applying our above-described hypothesis, and we also calculated the Ct-DR in the 46 patients with MTC showing biochemically persistent disease postoperatively. The estimated hTV-DRs were significantly higher than the observed Ct-DRs. This difference was more pronounced in the 18 patients with hereditary MTC than in the 28 patients with sporadic MTC. This thought experiment suggests that a rapid growth phase preceded surgery in the vast majority of the present patients with MTC, especially in the 18 patients with hereditary MTC. Even in patients with sporadic MTC whose hTV-DRs were lower than those of hereditary MTC, possibly due to higher age of the former patients, their hTV-DRs were often higher than their Ct-DRs. These data thus suggest that most of the MTCs grew more rapidly before surgery. In other words, the growth rate of most of the MTCs was lower during follow-up than hypothetical maximal value during early stages of tumor development; the reason(s) for this deceleration remain to be clarified. This was especially evident for the hereditary MTCs. Hereditary MTCs arise at younger ages than sporadic MTCs [10–13], yet their prognosis is not worse than that of sporadic MTCs [13–15]. Four of the presented 28 patients with sporadic MTC had Ct-DR values that were higher than their hTV-DR values. This may be due to too small estimates of hTV-DR in these patients since they had large age values (old ages) at surgery. The actual origin of the tumor should have been much later than birth in these patients. Alternatively, the growth rate in these patients became more rapid after surgery.
Kasahara et al. reported a spontaneous decrease in serum thyroglobulin levels in pediatric patients with papillary thyroid carcinoma after total thyroidectomy . We reported a spontaneous postoperative deceleration in the Ct-DT of patients with persistent hypercalcitoninemia in a long-term follow-up . Ito et al. reported that the TV-DR values of the majority of papillary thyroid microcarcinomas decreased during the follow-up, after they showed increases in their tumor size during active surveillance . These data suggest that a spontaneous deceleration of tumor growth following presentation or surgery is a common phenomenon in PTCs and MTCs as well. Of course, some of these tumors show a spontaneous aggressive change or an acceleration in their growth rate . Acceleration in the tumor growth rate is typically seen in cases of poor differentiation and anaplastic change of differentiated thyroid carcinoma.
Our study has several limitations. The study design was retrospective, and the number of patients was small (n=46) because we used only the data of a new Ct measurement method by ECLIA after April 2015. Second, in this thought-experiment study, we compared the Ct-DR with a hypothetical TV-DR. Although both values would reflect the tumor growth rate, the bases of their calculations are different. Since active surveillance for MTCs was not performed (unlike papillary thyroid microcarcinomas), our patients' TV-DR values after surgery based on the actual tumor size measurement were not available. However, our present findings provide some insight into the natural history of MTCs. Third, the number of hereditary patients was small at 18 and it was not possible to analyze according to RET mutations.