Skip to main content

Advertisement

SpringerLink
Log in

Estimating Tumor Growth Rates In Vivo

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

In this paper, we develop methods for inferring tumor growth rates from the observation of tumor volumes at two time points. We fit power law, exponential, Gompertz, and Spratt’s generalized logistic model to five data sets. Though the data sets are small and there are biases due to the way the samples were ascertained, there is a clear sign of exponential growth for the breast and liver cancers, and a 2/3’s power law (surface growth) for the two neurological cancers.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Bloom HJG, Richardson WW, Harries EJ (1962) Natural history of untreated breast cancer (1805–1933). Br Med J 2:213–221

    Article  Google Scholar 

  • Castorina P, Diesboeck TS, Gabriele P, Guiot C (2007) Growth laws in cancer: implications for radiotherapy. Radiat Res 168:349–356

    Article  Google Scholar 

  • Chingola R, Foroni RI (2005) Estimating the growth kinetics of experimental tumors from as few as two determinations of tumor size: implications for clinical oncology. IEEE Trans Biomed Eng 52:808–815

    Article  Google Scholar 

  • Collins VP et al (1956) Observations on growth rates of human tumors. Am J Roentgeol Radium Ther Nucl Med 76:988–1000

    Google Scholar 

  • Comen E, Morris PG, Norton L (2012) Translating mathematical modeling of tumor growth patterns into novel therapeutic approaches to breast cancer. J Mammay Gland Biol Noeplasia 17:241–249

    Article  Google Scholar 

  • Friberg S, Mattson S (1997) On the growth rates of human malignant tumors: implications for decision making. J Surg Oncol 65:284–297

    Article  Google Scholar 

  • Gerlee P (2013) The model muddle: in search of tumor growth laws. Cancer Res 73:2407–2411

    Article  Google Scholar 

  • Gompertz B (1825) On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of contingencies. Phil Trans Roy Soc 115:513–583

    Article  Google Scholar 

  • Guiot C, Degioris PG, Delsanto PP, Gabriele P, Deisboeck TS (2003) Does tumor growth follow a “universal law”? J Theor Biol 225:147–151

    Article  Google Scholar 

  • Hart D, Shocat E, Agur Z (1998) The growth law of primary breast cancer as inferred from mammography screening trials data. Br J Cancer 78:382–387

    Article  Google Scholar 

  • Heuser L, Spratt JS, Polk HC Jr (1979) Growth rates of primary breast cancers. Cancer 43:1888–1894

    Article  Google Scholar 

  • Kuroshi T et al (1990) Tumor growth rate and prognosis of breast cancer mainly detected by mass screening. Jpn J Cancer Res 81:454–462

    Article  Google Scholar 

  • Laasonen EM, Troupp H (1986) Volume growth of acoustic neurinomas. Neuroradiology 28:203–207

    Article  Google Scholar 

  • Laird AK (1963) Dynamics of tumor growth. Br J Cancer 18:490–502

    Article  Google Scholar 

  • Mandonnet E et al (2003) Continuous growth of mean tumor diameter in a subset of grade II gliomas. Ann Neurol 53:524–528

    Article  Google Scholar 

  • Mayenord WV (1932) On a law of growth of Jensen’s rat sarcoma. Am J Cancer 16:687–693

    Google Scholar 

  • Mehara E, Forsell-Aronsson E (2014) Analysis of inter-patient variability in tumor growth rates. Theoretical Biology and Medical Modeling. 11, paper 21

  • Michaelson JS, Halpern E, Kopans DB (1999) Breast cancer: computer simulation method for estimating optimal intervals for screening. Radiology 212:551–560

    Article  Google Scholar 

  • Nakajima T et al (2002) Simple tumor profile chart based on cell kinetic parameters and histologic grade is useful for estimating the natural growth rate of hepatocellular carcinoma. Human Pathol 33:92–99

    Article  Google Scholar 

  • Nakamura M, Roseer F, Michel J, Jacobs C, Samii M (2003) The natural history of incidental meningiomas. Neurosurgery 53:62–71

    Article  Google Scholar 

  • Norton L (1988) A Gompertzian model of human breast cancer growth. Cancer Res 48:7067–7071

    Google Scholar 

  • Norton L (2005) Conceptual and practical implications of breast tissue geometry: toward a more effective, less toxic therapy. The Oncol 10:370–381

    Article  Google Scholar 

  • Norton L, Simon R, Brereton HD, Bogden AE (1976) Predicting the course of Gompertzian growth. Nature 264:542–545

    Article  Google Scholar 

  • Rodriguez-Brenes IA, Komarova NJ, Wodarz D (2013) Tumor growth dynamics: insights into evolutionary processes. Trends Ecol Evolut 28:597–604

    Article  Google Scholar 

  • Sachs RK, Hlatky LR, Hanfeldt P (2001) Simple ODE models of tumor growth and anti-angiogenic or radiation treatment. Math Comput Model 33:1297–1305

    Article  MATH  Google Scholar 

  • Saito Y et al (1998) Multiple regression analysis for assessing the growth of small hepatocellular carcinomas. J Gastroenterol 33:229–235

    Article  Google Scholar 

  • Sarapata EA, de Piliis LG (2014) A comparison and catalog of intrinsic tumor growth models. Bull Math Biol, to appear

  • Shackeney SE (1970) A computer model for tumor growth with chemotherapy, and its application to l1210 leukemia treated with cytosine arabinose (nsc-63878). Cancer Chemother Rep 54:399–429

    Google Scholar 

  • Spratt JA, von Fournier D, Spratt JS, Weber EE (1992) Decelerating growth and human breast cancer. Cancer 71:2013–2019

    Article  Google Scholar 

  • Spratt JA, von Fournier D, Spratt JS, Weber EE (1992) Mammographic assessment of human breast cancer growth and duration. Cancer 71:2020–2026

    Article  Google Scholar 

  • von Bertalanffy L (1949) Problems of organic growth. Nature 163:156–158

    Article  Google Scholar 

  • Weedon-Fekjaer Lindquist BH, Vatten LJ, Aalen OO, Tretli S (2008) Breast cancer growth estimated through mammography screening data. Breast Cancer Res 10:R41

    Article  Google Scholar 

  • West GB, Brown JH, Enquist BJ (2001) A general model for ontogenic growth. Nature 413:628–631

    Article  Google Scholar 

  • Wodarz D, Komarova N (2014) Dynamics of cancer: mathematical foundations of oncology. World Scientific, Singapore

    Book  Google Scholar 

  • Wright S (1926) Book review. J Am Stat Assoc 21:493–497

    Article  Google Scholar 

Download references

Acknowledgments

This work was begun during an REU in the summer of 2013 associated with an NSF Research Training Grant at Duke University in mathematical biology. Both authors were partially supported by DMS 1305997 from the probability program at NSF. They would also like to thank Natalia Komarova, Marc Ryser, and referees #2 and #3 who made a number of helpful suggestions.

Author information

Authors and Affiliations

  1. Department of Math, Duke University, Durham, NC, USA

    Anne Talkington & Rick Durrett

Authors
  1. Anne Talkington
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rick Durrett
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rick Durrett.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Talkington, A., Durrett, R. Estimating Tumor Growth Rates In Vivo. Bull Math Biol 77, 1934–1954 (2015). https://doi.org/10.1007/s11538-015-0110-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-015-0110-8

Keywords

Search

Navigation