Robust and efficient pharmacokinetic parameter non-linear least squares estimation for dynamic contrast enhanced MRI of the prostate

Soudabeh Kargar, Eric A. Borisch, Adam T. Froemming, Akira Kawashima, Lance A. Mynderse, Eric G. Stinson, Joshua D. Trzasko, Stephen J. Riederer

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


Purpose To describe an efficient numerical optimization technique using non-linear least squares to estimate perfusion parameters for the Tofts and extended Tofts models from dynamic contrast enhanced (DCE) MRI data and apply the technique to prostate cancer. Methods Parameters were estimated by fitting the two Tofts-based perfusion models to the acquired data via non-linear least squares. We apply Variable Projection (VP) to convert the fitting problem from a multi-dimensional to a one-dimensional line search to improve computational efficiency and robustness. Using simulation and DCE-MRI studies in twenty patients with suspected prostate cancer, the VP-based solver was compared against the traditional Levenberg-Marquardt (LM) strategy for accuracy, noise amplification, robustness to converge, and computation time. Results The simulation demonstrated that VP and LM were both accurate in that the medians closely matched assumed values across typical signal to noise ratio (SNR) levels for both Tofts models. VP and LM showed similar noise sensitivity. Studies using the patient data showed that the VP method reliably converged and matched results from LM with approximate 3 × and 2 × reductions in computation time for the standard (two-parameter) and extended (three-parameter) Tofts models. While LM failed to converge in 14% of the patient data, VP converged in the ideal 100%. Conclusion The VP-based method for non-linear least squares estimation of perfusion parameters for prostate MRI is equivalent in accuracy and robustness to noise, while being more reliably (100%) convergent and computationally about 3 × (TM) and 2 × (ETM) faster than the LM-based method.

Original languageEnglish (US)
Pages (from-to)50-61
Number of pages12
JournalMagnetic Resonance Imaging
StatePublished - May 2018


  • Dynamic-contrast-enhanced magnetic resonance imaging
  • Multi-parametric magnetic resonance imaging
  • Perfusion
  • Pharmacokinetic modeling
  • Prostate cancer

ASJC Scopus subject areas

  • Biophysics
  • Biomedical Engineering
  • Radiology Nuclear Medicine and imaging


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