Download PDF
Methods Inf Med 2016; 55(01): 31-41
DOI: 10.3414/ME14-01-0131
Original Articles
Schattauer GmbH

Boosting in Nonlinear Regression Models with an Application to DCE-MRI Data[*]

M. Feilke
1   Institut für Statistik, Ludwig-Maximilians-Universität München, München, Germany
,
B. Bischl
1   Institut für Statistik, Ludwig-Maximilians-Universität München, München, Germany
2   Fakultät Statistik, Technische Universität Dortmund, Dortmund, Germany
,
V. J. Schmid
1   Institut für Statistik, Ludwig-Maximilians-Universität München, München, Germany
,
J. Gertheiss
3   Department für Nutztierwissenschaften, Georg-August-Universität Göttingen, Göttingen, Germany
4   Zentrum für Statistik, Georg-August-Universität Göttingen, Göttingen, Germany
› Author Affiliations
Further Information

Publication History

Received 04 December 2014

Accepted 26 May 2015

Publication Date:
08 January 2018 (online)

    Permissions and Reprints

Summary

Background: For the statistical analysis of dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) data, compartment models are a commonly used tool. By these models, the observed uptake of contrast agent in some tissue over time is linked to physiologic properties like capillary permeability and blood flow. Up to now, models of different complexity have been used, and it is still unclear which model should be used in which situation. In previous studies, it has been found that for DCE-MRI data, the number of compartments differs for different types of tissue, and that in cancerous tissue, it might actually differ over a region of voxels of one DCE-MR image. Objectives: To find the appropriate number of compartments and estimate the parameters of a regression model for each voxel in an DCE-MR image. With that, tumors in an DCE-MR image can be located, and for example therapy success can be assessed. Methods: The observed uptake of contrast agent in a voxel of an image of some tissue is described by a concentration time curve. This curve can be modeled using a nonlinear regression model. We present a boosting approach with nonlinear regression as base procedure, which allows us to estimate the number of compartments and the related parameters for each voxel of an DCE-MR image. In addition, a spatially regularized version of this approach is proposed. Results: With the proposed approach, the number of compartments – and with that the complexity of the model – per voxel is not fixed but data-driven, which allows us to fit models of adequate complexity to the concentration time curves of all voxels. The parameters of the model remain nevertheless interpretable because of the underlying compartment model. Conclusions: The proposed boosting approaches outperform all competing methods considered in this paper regarding the correct localization of tumors in DCE-MR images as well as the spatial homogeneity of the estimated number of compartments across the image, and the definition of the tumor edge.

Keywords

Statistical computing - computer-assisted image processing - statistical models - algorithms - regression analysis - spatial regularization - biological models - magnetic resonance imaging

* Supplementary online material published on our website http://dx.doi.org/10.3414/ME14-01-0131


 

  • References

  • 1 Buckley D, Parker G. Measuring contrast agent concentration in Ti-weighted dynamic contrast-enhanced MRI. In: Jackson A, Buckley D, Parker G. editors. Dynamic Contrast-Enhanced Magnetic Resonance Imaging in Oncology. New York: Springer; 2005. pp 69-79.
    CrossrefGoogle Scholar
  • 2 Padhani AR, Ah-See ML, Makris A. MRI in the detection and management of breast cancer. Expert Rev Anticancer Ther 2005; 5 (Suppl. 02) 239-252.
    CrossrefPubMedGoogle Scholar
  • 3 Liu G, Rugo HS, Wilding G, Mc-Shane TM, Evelhoch JL, Ng C. et al. Dynamic contrast-enhanced magnetic resonance imaging as a pharmacodynamic measure of response after acute dosing of AG-013736, an oral angiogenesis inhibitor, in patients with advanced solid tumors: results from a phase I study. J Clin Oncol 2005; 23 (Suppl. 24) 5464-5473.
    CrossrefPubMedGoogle Scholar
  • 4 Tofts P, Kermode A. Measurement of the blood-brain barrier permeability and leakage space using dynamic MR imaging – 1. Fundamental concepts. Magn Reson Med 1991; 17: 357-367.
    CrossrefPubMedGoogle Scholar
  • 5 Schmid V, Whitcher B, Padhani A, Yang G. Quantitative analysis of dynamic contrast-enhanced MR images based on Bayesian P-Splines. IEEE Trans Med Imaging 2009; 28 (Suppl. 06) 789-798.
    CrossrefPubMedGoogle Scholar
  • 6 Brix G, Zwick S, Kiessling F, Griebel J. Pharmacokinetic analysis of tissue microcirculation using nested models: multimodel inference and parameter identifiability. Med Phys 2009; 36 (Suppl. 07) 2923-2933.
    CrossrefPubMedGoogle Scholar
  • 7 Sourbron S, Buckley D. On the scope and interpretation of the Tofts models for DCE-MRI. Magn Reson Med 2011; 66 (Suppl. 03) 735-745.
    CrossrefPubMedGoogle Scholar
  • 8 Sommer J, Schmid V. Spatial two-tissue compartment model for dynamic contrast-enhanced magnetic resonance imaging. J R Stat Soc C-App 2014; 63: 695-713.
    PubMedGoogle Scholar
  • 9 Port R, Knopp M, Hoffmann U, Milker-Zabel S, Brix G. Multicompartment analysis of gadolinium chelate kinetics: blood-tissue exchange in mam-mary tumors as monitored by dynamic MR imaging. J Magn Reson Imaging 1999; 10: 233-241.
    CrossrefPubMedGoogle Scholar
  • 10 Sommer J, Gertheiss J, Schmid V. Spatially regularized estimation for the analysis of dynamic contrast-enhanced magnetic resonance imaging data. Stat Med 2014; 33: 1029-1041.
    CrossrefPubMedGoogle Scholar
  • 11 Freund Y, Shapire R. Experiments with a new boosting algorithm. In: Proceedings of the Thirteenth International Conference on Machine Learning Theory. 3. San Francisco, CA: Morgan Kaufmann Publishers Inc; 1996. pp 148-156.
    Google Scholar
  • 12 Mayr A, Binder H, Gefeller O, Schmid M. Extending statistical boosting. An overview of recent methodological developments. Methods Inf Med 2014; 53: 428-435.
    Article in Thieme ConnectPubMedGoogle Scholar
  • 13 Friedman J, Hastie T, Tibshi-rani R. Additive logistic regression: a statistical view of boosting (with Discussion). Ann Stat 2000; 28: 337-407.
    PubMedGoogle Scholar
  • 14 Friedman J. Greedy function approximation: A gradient boosting machine. Ann Stat 2001; 29: 1189-1232.
    CrossrefPubMedGoogle Scholar
  • 15 Biihlmann P, Yu B. Boosting with the L2-L0SS: Regression and classification. J Am Stat Assoc 2003; 98: 324-338.
    CrossrefPubMedGoogle Scholar
  • 16 Biihlmann P. Boosting for high-dimensional linear models. Anna Stat 2006; 34: 559-583.
    PubMedGoogle Scholar
  • 17 Biihlmann P, Hothorn T. Boosting algorithms: Regularization, prediction and model fitting. Stat Sci 2007; 22: 477-522.
    CrossrefPubMedGoogle Scholar
  • 18 Mayr A, Binder H, Gefeller O, Schmid M. The evolution of boosting algorithms. From machine learning to statistical modelling. Methods Inf Med 2014; 53: 419-427.
    Article in Thieme ConnectPubMedGoogle Scholar
  • 19 Feilke M. dcemriboost: Boosting for DCE-MRI data. 2015 R package version 0.1. Available from: https://github.com/feilke/dcemriboost.
    PubMedGoogle Scholar
  • 20 Tofts P. Modeling tracer kinetics in dynamic GdDTPA MR imaging. J Magne Reson Imaging 1997; 7 (Suppl. 01) 91-101.
    PubMedGoogle Scholar
  • 21 Tofts P, Brix G, Buckley D, Evelhoch J, Henderson E, Knopp M. et al. Estimating kinetic parameters from dynamic contrast-enhanced T1-weighted MRI of a diffusable tracer: Standardized quantities and symbols. J Magn Reson Imaging 1999; 10: 223-232.
    CrossrefPubMedGoogle Scholar
  • 22 Sommer J. Regularized estimation and model selection in compartment models. Dissertation at the Faculty of Mathematics, Computer Science and Statistics of the LMU Munich. Munich, Germany: Verlag Dr. Hut; 2013
    Google Scholar
  • 23 Yang X, Kopp MV. Quantifying tumor vascular heterogeneity with dynamic contrast-enhanced magnetic resonance imaging: a review. J Biomed Biotech 2011; 2011: 1-12.
    PubMedGoogle Scholar
  • 24 Schmid V, Whitcher B, Padhani A, Taylor N, Yang GZ. A Bayesian hierarchical model for the analysis of a longitudinal dynamic contrast-enhanced MRI oncology study. Magn Reson Med 2009; 61 (Suppl. 01) 163-174.
    CrossrefPubMedGoogle Scholar
  • 25 Seber GAF, Wild CJ. Nonlinear Regression. New York: Wiley; 1989
    Google Scholar
  • 26 Schmid VJ, Whitcher B, Padhani AR, Taylor NJ, Yang GZ. Bayesian methods for pharmacokinetic models in dynamic contrast-enhanced magnetic resonance imaging. IEEE Trans Med Imaging 2006; 25 (Suppl. 12) 1627-1636.
    CrossrefPubMedGoogle Scholar
  • 27 Parker G, Suckling J, Tanner S, Padhani A, Husband J, Leach M. MRIW: Parametric analysis software for contrast-enhanced dynamic MR imaging in cancer. Radiographics 1998; 18 (Suppl. 02) 497-506.
    CrossrefPubMedGoogle Scholar
  • 28 Weinmann H, Laniado M, Miitzel W. Pharmokinetics of Gd-DTPA/dimeglumine after intravenous injection into healthy volunteers. Physiol Chem Phys Med NMR 1984; 16: 167-172.
    PubMedGoogle Scholar
  • 29 Hofner B, Mayr A, Robinzonov N, Schmid M. Model-based boosting in R: a hands-on tutorial using the R package mboost. Comput Stat 2014; 29: 3-35.
    CrossrefPubMedGoogle Scholar
  • 30 R Core Team.. R: A language and environment for statistical computing. Vienna, Austria: 2013 ISBN 3-900051-07-0. Available from: http://www.Rproject.org.
    PubMedGoogle Scholar
  • 31 Elzhov T, Mullen K, Spiess AN, Bolker B. minpack.lm: R interface to the Levenberg-Marquardt nonlinear least-squares algorithm found in MIN-PACK, plus support for bounds. 2013 R package version 1.1-8. Available from: http://CRAN.Rproject.org/package=minpack.lm.
    PubMedGoogle Scholar
  • 32 Ahearn T, Staff R, Redpath T, Semple S. The use of the Levenberg-Marquardt curve-fitting algorithm in pharmacokinetic modelling of DCE-MRI data. Phys Med Biol 2005; 50 (Suppl. 09) N85-N92.
    CrossrefPubMedGoogle Scholar
  • 33 Reich J. On parameter redundancy in curve fitting of kinetic data. Kinetic data analysis: Design and analysis of enzyme and pharmacokinetic experiments. New York: Plenum Press; 1981
    Google Scholar
  • 34 Schmid M, Hothorn T. Boosting additive models using componentwise P-splines. Comput Stat Data An 2008; 53: 298-311.
    CrossrefPubMedGoogle Scholar
  • 35 Mayr A, Hofner B, Schmid M. The importance of knowing when to stop. A sequential stopping rule for component-wise gradient boosting. Methods Inf Med 2012; 51 (Suppl. 02) 178-186.
    Article in Thieme ConnectPubMedGoogle Scholar
  • 36 Hastie T, Tibshirani R, Friedman J. The elements of statistical learning. 2nd ed. New York: Springer; 2009
    Google Scholar
  • 37 Hansen M, Yu B. Model selection and the principle of minimum description length. J Am Stat Assoc 2001; 96 (Suppl. 454) 746-774.
    CrossrefPubMedGoogle Scholar
  • 38 Candes E, Tao T. The Dantzig selector: Statistical estimation when p is much larger than n. Ann Stat 2007; 35 (Suppl. 06) 2313-2351.
    CrossrefPubMedGoogle Scholar
  • 39 Gertheiss J, Tutz G. Sparse modeling of categorial explanatory variables. Ann Appl Stat 2010; 4 (Suppl. 04) 2150-2180.
    CrossrefPubMedGoogle Scholar
  • 40 Tutz G, Groll A. Likelihood-based boosting in binary and ordinal random effects models. J Comput Graph Stat 2013; 22 (Suppl. 02) 356-378.
    CrossrefPubMedGoogle Scholar
  • 41 Groll A. GMMBoost: Likelihood-based Boosting for Generalized mixed models. 2013 R package version 1.1.2. Available from: http://CRAN.Rproject.org/package=GMMBoost.
    PubMedGoogle Scholar
  • 42 Turlach B, Weingessel A. quad-prog: Functions to solve Quadratic Programming Problems. 2013 R package version 1.5–5. Available from: http://CRAN.R-project.org/package=quadprog.
    PubMedGoogle Scholar
  • 43 Zou H, Hastie T. Regularization and variable selection via the elastic net. J Roy Stat Soc B 2005; 67: 301-320.
    CrossrefPubMedGoogle Scholar
  • 44 Ah-See ML, Makris A, Taylor N, Burcombe R, Harrison M, Stirling J. et al. Does vascular imaging with MRI predict response to neoadjuvant chemo-therapy in primary breast cancer?. J Clin Oncol 2004; 22 14S 582.
    CrossrefPubMedGoogle Scholar
  • 45 Schmid V, Whitcher B, Yang G, Taylor N, Padhani A. Statistical analysis of pharmacokinetical models in dynamic contrast-enhanced magnetic resonance imaging. In: Duncan J, Gerig S. editors. Medical Imaging Computing and Computer-Assisted Intervention – MICCAI 2005. New York: Springer; 2005. pp 886-893.
    Google Scholar
  • 46 Bischl B, Lang M, Mersmann O, Rahnenfuehrer J, Weihs C. BatchJobs and BatchExperiments: Abstraction mechanisms for using R in batch environments. J Stat Softw 2015; 64: 1-25.
    PubMedGoogle Scholar