Estimation of Solid Tumor Doubling Times from Progression-Free Survival Plots Using a Novel Statistical Approach


Tumor doubling time can significantly affect the outcome of anticancer therapy, but it is very challenging to determine. Here, we present a statistical approach that extracts doubling times from progression-free survival (PFS) plots, which inherently contains information regarding the growth of solid tumors. Twelve cancers were investigated and multiple PFS plots were evaluated for each type. The PFS plot showing fastest tumor growth was deemed to best represent the inherent growth kinetics of the solid tumor, and selected for further analysis. The exponential tumor growth rates were extracted from each PFS plot, along with associated variabilities, which ultimately allowed for the estimation of solid tumor doubling times. The mean simulated doubling times for pancreatic cancer, melanoma, hepatocellular carcinoma (HCC), renal cell carcinoma, triple negative breast cancer, non-small cell lung cancer, hormone receptor positive (HR+) breast cancer, human epidermal growth factor receptor-2 positive (HER-2+) breast cancer, gastric cancer, glioblastoma multiforme, colorectal cancer, and prostate cancer were 5.06, 3.78, 3.06, 2.67, 2.38, 2.40, 4.31, 4.12, and 3.84 months, respectively. For all cancers, clinically reported doubling times were within the estimated ranges. For all cancers, except HCC, the growth rates were best characterized by a log-normal distribution. For HCC, the gamma distribution best described the data. The statistical approach presented here provides a qualified method for extracting tumor growth rates and doubling times from PFS plots. It also allows estimation of the distributional characteristics for tumor growth rates and doubling times in a given patient population.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. 1.

    Okazaki N, Yoshino M, Yoshida T, Suzuki M, Moriyama N, Takayasu K, et al. Evaluation of the prognosis for small hepatocellular carcinoma based on tumor volume doubling time. A preliminary report. Cancer. 1989;63:2207–10.

    CAS  Article  Google Scholar 

  2. 2.

    Arai T, Kuroishi T, Saito Y, Kurita Y, Naruke T, Kaneko M. Tumor doubling time and prognosis in lung cancer patients: evaluation from chest films and clinical follow-up study. Japanese Lung Cancer Screening Research Group. Jpn J Clin Oncol. 1994;24:199–204.

    CAS  PubMed  Google Scholar 

  3. 3.

    Furukawa H, Iwata R, Moriyama N. Growth rate of pancreatic adenocarcinoma: initial clinical experience. Pancreas. 2001;22:366–9.

    CAS  Article  Google Scholar 

  4. 4.

    Kuroishi T, Tominaga S, Morimoto T, Tashiro H, Itoh S, Watanabe H, et al. Tumor growth rate and prognosis of breast cancer mainly detected by mass screening. Jpn J Cancer Res. 1990;81:454–62.

    CAS  Article  Google Scholar 

  5. 5.

    Spratt JS Jr, Spratt TL. Rates of growth of pulmonary metastases and host survival. Ann Surg. 1964;159:161–71.

    Article  Google Scholar 

  6. 6.

    Stensjoen AL, Solheim O, Kvistad KA, Haberg AK, Salvesen O, Berntsen EM. Growth dynamics of untreated glioblastomas in vivo. Neuro-Oncology. 2015;17:1402–11.

    Article  Google Scholar 

  7. 7.

    Shackney SE, McCormack GW, Cuchural GJ Jr. Growth rate patterns of solid tumors and their relation to responsiveness to therapy: an analytical review. Ann Intern Med. 1978;89:107–21.

    CAS  Article  Google Scholar 

  8. 8.

    Ollila DW, Stern SL, Morton DL. Tumor doubling time: a selection factor for pulmonary resection of metastatic melanoma. J Surg Oncol. 1998;69:206–11.

    CAS  Article  Google Scholar 

  9. 9.

    Singh AP, Shah DK. Application of a PK-PD modeling and simulation-based strategy for clinical translation of antibody-drug conjugates: a case study with Trastuzumab Emtansine (T-DM1). AAPS J. 2017;19:1054–70.

    CAS  Article  Google Scholar 

  10. 10.

    Schwartz M. A biomathematical approach to clinical tumor growth. Cancer. 1961;14:1272–94.

    CAS  Article  Google Scholar 

  11. 11.

    Mehrara E, Forssell-Aronsson E, Ahlman H, Bernhardt P. Specific growth rate versus doubling time for quantitative characterization of tumor growth rate. Cancer Res. 2007;67:3970–5.

    CAS  Article  Google Scholar 

  12. 12.

    Gutman SI, Piper M, Grant MD, Basch E, Oliansky DM, Aronson N. Progression-free survival: what does it mean for psychological well-being or quality of life? Rockville, MD: Agency for Healthcare Research and Quality; 2013.

  13. 13.

    Magazines T, Savers S. Guidance for industry: clinical trial endpoints for the approval of cancer drugs and biologics. Biotechnol Law Rep. 2007;26:375–86.

    Article  Google Scholar 

  14. 14.

    World Health Organization. WHO handbook for reporting results of cancer treatment. Geneva: World Health Organization; 1979.

  15. 15.

    Therasse P, Arbuck SG, Eisenhauer EA, Wanders J, Kaplan RS, Rubinstein L, et al. New guidelines to evaluate the response to treatment in solid tumors. J Natl Cancer Inst. 2000;92:205–16.

    CAS  Article  Google Scholar 

  16. 16.

    R Core Team. R: a language and environment for statistical computing. Vienna: R Foundation for Statistical Computing; 2016.

  17. 17.

    Yung WK, Albright RE, Olson J, Fredericks R, Fink K, Prados MD, et al. A phase II study of temozolomide vs. procarbazine in patients with glioblastoma multiforme at first relapse. Br J Cancer. 2000;83:588–93.

    CAS  Article  Google Scholar 

  18. 18.

    Ohtsu A, Shimada Y, Shirao K, Boku N, Hyodo I, Saito H, et al. Randomized phase III trial of fluorouracil alone versus fluorouracil plus cisplatin versus uracil and tegafur plus mitomycin in patients with unresectable, advanced gastric cancer: the Japan Clinical Oncology Group Study (JCOG9205). J Clin Oncol. 2003;21:54–9.

    CAS  Article  Google Scholar 

  19. 19.

    Dutang MLD-MC. Fitdistrplus: an R package for fitting distributions. J Stat Softw. 2015;64:1–34.

    Google Scholar 

  20. 20.

    Therneau TM. A package for survival analysis in S. version 2.38. 2015.

  21. 21.

    Kim KB, Kefford R, Pavlick AC, Infante JR, Ribas A, Sosman JA, et al. Phase II study of the MEK1/MEK2 inhibitor Trametinib in patients with metastatic BRAF-mutant cutaneous melanoma previously treated with or without a BRAF inhibitor. J Clin Oncol. 2013;31:482–9.

    CAS  Article  Google Scholar 

  22. 22.

    Oda T, Miyao N, Takahashi A, Yanase M, Masumori N, Itoh N, et al. Growth rates of primary and metastatic lesions of renal cell carcinoma. Int J Urol. 2001;8:473–7.

    CAS  Article  Google Scholar 

  23. 23.

    Friberg S, Mattson S. On the growth rates of human malignant tumors: implications for medical decision making. J Surg Oncol. 1997;65:284–97.

    CAS  Article  Google Scholar 

  24. 24.

    Eskelin S, Pyrhonen S, Summanen P, Hahka-Kemppinen M, Kivela T. Tumor doubling times in metastatic malignant melanoma of the uvea: tumor progression before and after treatment. Ophthalmology. 2000;107:1443–9.

    CAS  Article  Google Scholar 

  25. 25.

    Malaise EP, Chavaudra N, Charbit A, Tubiana M. Relationship between the growth rate of human metastases, survival and pathological type. Eur J Cancer. 1974;10:451–9.

    CAS  Article  Google Scholar 

  26. 26.

    Garland LH, Coulson W, Wollin E. The rate of growth and apparent duration of untreated primary bronchial carcinoma. Cancer. 1963;16:694–707.

    CAS  Article  Google Scholar 

  27. 27.

    Breur K. Growth rate and radiosensitivity of human tumours. I Growth rate of human tumours. Eur J Cancer. 1966;2:157–71.

    CAS  Article  Google Scholar 

  28. 28.

    Gomez-de la Fuente FJ, Martinez-Rodriguez I, de Arcocha-Torres M, Quirce R, Jimenez-Bonilla J, Martinez-Amador N, et al. Contribution of (11)C-Choline PET/CT in prostate carcinoma biochemical relapse with serum PSA level below 1 ng/ml. Rev Esp Med Nucl Imagen Mol. 2018;37:156–62.

    CAS  PubMed  Google Scholar 

  29. 29.

    Watanabe H. Relapse of prostate cancer from the viewpoint of total gland volume kinetics theory. Asian J Androl. 2015;17:904–7 discussion 7.

    Article  Google Scholar 

  30. 30.

    de Bono JS, Oudard S, Ozguroglu M, Hansen S, Machiels J-P, Kocak I, et al. Prednisone plus cabazitaxel or mitoxantrone for metastatic castration-resistant prostate cancer progressing after docetaxel treatment: a randomised open-label trial. Lancet. 2010;376:1147–54.

    Article  Google Scholar 

  31. 31.

    Hutchinson L, Kirk R. High drug attrition rates--where are we going wrong? Nat Rev Clin Oncol. 2011;8:189–90.

    Article  Google Scholar 

  32. 32.

    Begley CG, Ellis LM. Drug development: raise standards for preclinical cancer research. Nature. 2012;483:531–3.

    CAS  Article  Google Scholar 

  33. 33.

    Shah DK, Haddish-Berhane N, Betts A. Bench to bedside translation of antibody drug conjugates using a multiscale mechanistic PK/PD model: a case study with brentuximab-vedotin. J Pharmacokinet Pharmacodyn. 2012;39:643–59.

    Article  Google Scholar 

  34. 34.

    Havelaar IJ, Sugarbaker PH, Vermess M, Miller DL. Rate of growth of intraabdominal metastases from colorectal cancer. Cancer. 1984;54:163–71.

    CAS  Article  Google Scholar 

  35. 35.

    Plesnicar S, Klanjscek G, Modic S. Actual doubling time values of pulmonary metastases from malignant melanoma. Aust N Z J Surg. 1978;48:23–5.

    CAS  Article  Google Scholar 

  36. 36.

    Carlson JA. Tumor doubling time of cutaneous melanoma and its metastasis. Am J Dermatopathol. 2003;25:291–9.

    Article  Google Scholar 

  37. 37.

    Fujimoto N, Sugita A, Terasawa Y, Kato M. Observations on the growth rate of renal cell carcinoma. Int J Urol. 1995;2:71–6.

    CAS  Article  Google Scholar 

  38. 38.

    Ozono S, Miyao N, Igarashi T, Marumo K, Nakazawa H, Fukuda M, et al. Tumor doubling time of renal cell carcinoma measured by CT: collaboration of Japanese Society of Renal Cancer. Jpn J Clin Oncol. 2004;34:82–5.

    Article  Google Scholar 

  39. 39.

    Motzer RJ, Escudier B, Oudard S, Hutson TE, Porta C, Bracarda S, et al. Efficacy of everolimus in advanced renal cell carcinoma: a double-blind, randomised, placebo-controlled phase III trial. Lancet. 2008;372:449–56.

    CAS  Article  Google Scholar 

Download references


This work was supported by NIH grant GM114179 and AI138195 to D.K.S., and the Centre for Protein Therapeutics at University at Buffalo.

Author information



Corresponding author

Correspondence to Dhaval K. Shah.

Ethics declarations

Conflict of Interest

RB has served as an expert witness through Belmore Neidrauer LLP funded by Janssen Pharmaceutical. All other authors declare they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Keith Dolcy is the co-first author.



(DOCX 1196 kb)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kay, K., Dolcy, K., Bies, R. et al. Estimation of Solid Tumor Doubling Times from Progression-Free Survival Plots Using a Novel Statistical Approach. AAPS J 21, 27 (2019).

Download citation


  • PK/PD modeling and simulation
  • preclinical-to-clinical translation
  • progression-free survival
  • solid tumor doubling time
  • tumor growth rate