Skip to main content
SpringerLink
Log in

A mathematical method for parameter estimation in a tumor growth model

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we present a methodology for estimating the effectiveness of a drug, an unknown parameter that appears on an avascular, spheric tumor growth model formulated in terms of a coupled system of partial differential equations (PDEs). This model is formulated considering a continuum of live cells that grow by the action of a nutrient. Volume changes occur due to cell birth and death, describing a velocity field. The model assumes that when the drug is applied externally, it diffuses and kills cells. The effectiveness of the drug is obtained by solving an inverse problem which is a PDE-constrained optimization problem. We define suitable objective functions by fitting the modeled and the observed tumor radius and the inverse problem is solved numerically using a Pattern Search method. It is observed that the effectiveness of the drug is retrieved with a reasonable accuracy. Experiments with noised data are also considered and the results are compared and contrasted.

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

Similar content being viewed by others

References

  • Adam JA (1986) A simplified mathematical model of tumor growth. Math Biosci 81(2):229–244

    Article  MATH  Google Scholar 

  • Agnelli J, Barrea A, Turner C (2011) Tumor location and parameter estimation by thermography. Math Comput Model 53(7–8):1527–1534

    Article  MATH  Google Scholar 

  • Audet C, Dennis J (2002) Analysis of generalized pattern searches. SIAM J Optimiz 13(3):889–903

    Article  MathSciNet  MATH  Google Scholar 

  • Bellomo N, Li N, Maini P (2008) On the foundations of cancer modelling: selected topics, speculations, and perspectives. Math Model Methods Appl Sci 18(4):593–646

    Article  MathSciNet  MATH  Google Scholar 

  • Bellouquid A, De Angelis E, Knopoff D (2013) From the modeling of the immune hallmarks of cancer to a black swan in biology. Math Model Methods Appl Sci 23(05):949–978

    Article  MathSciNet  MATH  Google Scholar 

  • Bergstrom M, Monazzam A, Razifar P, Ide S, Josephsson R, Langstrom B (2008) Modeling spheroid growth, PET tracer uptake, and treatment effects of the Hsp90 inhibitor NVP-AUY922. J. Nucl. Med. 49(7):1204–1210

    Article  Google Scholar 

  • Byrne H, Chaplain M (1997) Free boundary value problems associated with the growth and development of multicellular spheroids. Eur J Appl Math 8(06):639–658

    Article  MathSciNet  MATH  Google Scholar 

  • Byrne H, Drasdo D (2009) Individual-based and continuum models of growing cell populations: a comparison. J Math Biol 58:657–687

    Article  MathSciNet  MATH  Google Scholar 

  • Crank J (1984) Free and moving boundary problems. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York

    MATH  Google Scholar 

  • Dolan E, Lewis R, Torczon V (2003) On the local convergence of pattern search. SIAM J Optimiz 14(2):567–583

    Article  MathSciNet  MATH  Google Scholar 

  • Edelman LB, Eddy JA, Price ND (2010) In silico models of cancer. WIREs Syst Biol Med 2(4):438–459

    Article  Google Scholar 

  • Ford DK, Yerganian G (1958) Observations on the chromosomes of Chinese hamster cells in tissue culture. J Natl Cancer Inst 21(2):393–425

    Google Scholar 

  • Freyer JP, Sutherland RM (1986) Regulation of growth saturation and development of necrosis in EMT6/Ro multicellular spheroids by the glucose and oxygen supply. Cancer Res 46(7):3504–3512

    Google Scholar 

  • Greenspan H (1972) Models for the growth of a solid tumor by diffusion. Stud Appl Math 51(4):317–340

    Article  MATH  Google Scholar 

  • Grillo A, Wittum G, Giaquinta G, Mićunović MV (2009) A multiscale analysis of growth and diffusion dynamics in biological materials. Int J Eng Sci 47(2):261–283

    Article  MathSciNet  MATH  Google Scholar 

  • Hamilton G (1998) Multicellular spheroids as an in vitro tumor model. Cancer Lett 131(1):29–34

    Article  Google Scholar 

  • Herrmann R, Fayad W, Schwarz S, Berndtsson M, Linder S (2008) Screening for compounds that induce apoptosis of cancer cells grown as multicellular spheroids. J Biomol Screen 13(1):1–8

    Article  Google Scholar 

  • Hlatky L, Sachs RK, Alpen EL (1988) Joint oxygen-glucose deprivation as the cause of necrosis in a tumor analog. J Cell Physiol 134(2):167–178

    Article  Google Scholar 

  • Hogea C, Davatzikos C, Biros G (2008) An image-driven parameter estimation problem for a reaction-diffusion glioma growth model with mass effects. J Math Biol 56(6):793–825

    Article  MathSciNet  MATH  Google Scholar 

  • Kiran KL, Jayachandran D, Lakshminarayanan S (2009) Mathematical modelling of avascular tumour growth based on diffusion of nutrients and its validation. Can J Chem Eng 87(5):732–740

    Article  Google Scholar 

  • Knopoff D, Fernández D, Torres G, Turner C (2013) Adjoint method for a tumor growth PDE-constrained optimization problem. Comput Math Appl 66(6):1104–1119

    Article  MathSciNet  Google Scholar 

  • Lowengrub J, Frieboes H, Jin F, Chuang Y, Li X, Macklin P, Wise S, Cristini V (2010) Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity 23:R1

    Article  MathSciNet  MATH  Google Scholar 

  • Martin GR, Jain RK (1994) Noninvasive measurement of interstitial pH profiles in normal and neoplastic tissue using fluorescence ratio imaging microscopy. Cancer Res 54(21):5670–5674

    Google Scholar 

  • Monazzam A, Josephsson R, Blomqvist C, Carlsson J, Langstrom B, Bergstrom M (2007) Application of the multicellular tumour spheroid model to screen PET tracers for analysis of early response of chemotherapy in breast cancer. Breast Cancer Res 9(4):R45

    Article  Google Scholar 

  • Oden JT, Prudencio EE, Hawkins-Daarud A (2013) Selection and assessment of phenomenological models of tumor growth. Math Model Methods Appl Sci 23(07):1309–1338

    Article  MathSciNet  MATH  Google Scholar 

  • Perthame B, Zubelli JP (2007) On the inverse problem for a size-structured population model. Inverse Probl 23(3):1037–1052

    Article  MathSciNet  MATH  Google Scholar 

  • Preziosi L, Vitale G (2011) A multiphase model of tumor and tissue growth including cell adhesion and plastic reorganization. Math Model Methods Appl Sci 21(09):1901–1932

    Article  MathSciNet  MATH  Google Scholar 

  • Rejniak K, McCawley L (2010) Current trends in mathematical modeling of tumor-microenvironment interactions: a survey of tools and applications. Exp Biol Med 235(4):411–423

    Article  Google Scholar 

  • Roose T, Chapman S, Maini P (2007) Mathematical models of avascular cancer. SIAM Rev 49(2):179–208

    Article  MathSciNet  MATH  Google Scholar 

  • Sano Y, Hoshino T, Barker M, Deen DF (1984) Response of 9L rat brain tumor multicellular spheroids to single and fractionated doses of 1,3-bis (2-chloroethyl)-1-nitrosourea. Cancer Res 44(2):571–576

    Google Scholar 

  • Sutherland R (1988) Cell and environment interactions in tumor microregions: the multicell spheroid model. Science 240(4849):177–184

    Article  Google Scholar 

  • Taylor DL, Waggoner AS, Lanni F, Murphy RF, Birge RR (1986) Applications of fluorescence in the biomedical sciences. Alan R, Liss Inc, Technical Report

  • Torczon V (1997) On the convergence of pattern search algorithms. SIAM J Optimiz 7(1):1–25

    Article  MathSciNet  MATH  Google Scholar 

  • Tracqui P (2009) Biophysical models of tumour growth. Rep Prog Phys 72:056701

    Article  Google Scholar 

  • van den Doel K, Ascher UM, Pai DK (2011) Source localization in electromyography using the inverse potential problem. Inverse Probl 27(2):025008

    Article  MathSciNet  MATH  Google Scholar 

  • Venkataraman P (2009) Applied optimization with MATLAB programming. Wiley, London

    Google Scholar 

  • Ward J, King J (1997) Mathematical modelling of avascular-tumour growth. Math Med Biol 14(1):39–69

    Article  MATH  Google Scholar 

  • Ward JP, King JR (2003) Mathematical modelling of drug transport in tumour multicell spheroids and monolayer cultures. Math Biosci 181(2):177–207

    Article  MathSciNet  MATH  Google Scholar 

  • Wise S, Lowengrub J, Frieboes H, Cristini V (2008) Three-dimensional multispecies nonlinear tumor growth-I: model and numerical method. J Theor Biol 253(3):524–543

    Article  MathSciNet  Google Scholar 

  • Zubelli JP, Marabini R, Sorzano COS, Herman GT (2003) Three-dimensional reconstruction by chahine’s method from electron microscopic projections corrupted by instrumental aberrations. Inverse Probl 19(4):933–949

    Article  MATH  Google Scholar 

Download references

Acknowledgments

We thank the referees and the Editor for their careful reading of the manuscript and their valuable suggestions. This work was carried out with the aid of grants from ANPCyT, CONICET and SECyT-UNC; and the European Union FP7 Health Research Grant No. FP7-HEALTH-F4-2008-202047-RESOLVE.

Author information

Authors and Affiliations

  1. Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10100, Torino, Italia

    D. Knopoff

  2. Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, CIEM-CONICET, Medina Allende s/n, Ciudad Universitaria, 5000, Córdoba, Argentina

    D. Knopoff, D. Fernández, G. Torres & C. Turner

Authors
  1. D. Knopoff
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Fernández
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. Torres
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. C. Turner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Fernández.

Additional information

Communicated by George S. Dulikravich.

This work has been partially supported by the European Union FP7 Health Research Grant No. FP7-HEALTH-F4-2008-202047-RESOLVE, and ANPCyT, CONICET and SECyT-UNC.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Knopoff, D., Fernández, D., Torres, G. et al. A mathematical method for parameter estimation in a tumor growth model. Comp. Appl. Math. 36, 733–748 (2017). https://doi.org/10.1007/s40314-015-0259-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40314-015-0259-7

Keywords

Mathematics Subject Classification

Search

Navigation