In this educational poster, we emphasize the critical role that statistical parameter estimation plays in turning MRI relaxometry into an accurate and precise qMRI modality. In this longstanding goal, most of the efforts have been made on deriving more accurate physical models, but, to the authors’ knowledge, the importance of using modern statistical parameter estimation methods has been overlooked. Here, we present "the do's and the don’ts" of statistical parameter estimation for quantitative MRI relaxometry, illustrating all the concepts with real-case scenarios.
MRI relaxometry holds the promise of providing biomarkers for monitoring, staging and follow up of diseases. Imperative to meet minimum standards for objective, reproducible and reliable biomarkers is the need for accurate, precise, quantitative parameters maps1, such as T1 or T2. While unrealistic physical modeling is often argued as the main cause of lack of accuracy, little effort has been made on discussing the impact that inadequate parameter estimation methods have on the accuracy and precision of MRI relaxometry techniques.
This educational poster attempts to introduce young MR students/researchers into the basics of modern statistical parameter estimation theory, and its application for accurate and precise relaxometry. The poster gives recommendations and "the do’s and the don’ts" of statistical parameter estimation for quantitative relaxometry. All the concepts are illustrated with real-case scenarios, using simulated phantoms and real relaxometry data.
Noise and statistical data distribution modelling
Acquired T1 or T2 weighted datasets are always disturbed by noise. This means that the estimated T1 and T2 parameters will vary if the experiment is repeated under exactly the same conditions. Indeed, the intensity of the MR images is not a deterministic value but a realization of a random variable. Random variables are uniquely characterized by their probability density function (PDF)2,3,4. PDFs are fundamental for conceiving accurate parameter estimators. In the educational poster, we will describe the most typical PDFs that appear in relaxometry. While researchers may be familiar with Gaussian and Rician distributions5, others PDFs come into play as well for modern MR scanners6,7. Furthermore, we will address the fact that for certain MR system architectures some of the PDF parameters, such as the standard deviation of the noise, are spatially dependent. This property is often ignored, but should be taken into account when constructing optimal estimators.
How to optimally estimate T1 and T2: the quest for "good" estimators
An estimator is a rule for calculating an estimate of a given parameter based on observations. In relaxometry, the observations are image voxel intensity values. Since these observations are random variables, the estimator, being a function of the observations, is also a random variable. Hence, a performance assessment of an estimator should always be based on a statistical analysis. Fundamental properties of an estimator are its bias and variance2,3,4, which relate to the familiar concepts of accuracy and precision. We will elaborate on the, often underestimated, importance of assessing both concepts in a synergistic manner. Finally, techniques for improving the accuracy and precision of given estimators will be presented.
Does there exist a perfect estimator for my relaxometry data?
The answer is, unfortunately, no. However, we will guide the reader to the construction of one of those "best" estimators. We will focus on the case of unbiased estimators, and derive the unbiased estimator having the minimum variance. i.e., the uniformly minimum variance unbiased estimator (UMVUE)4. In the quest for the UMVUE, the concept of efficiency and Cramér-Rao Lower Bound (CRLB)2,3,4 will naturally come into play. The importance of knowledge of the PDF will be emphasized and focus will be laid on the importance of the CRLB as a benchmark for estimators. Finally, we will cover the Maximum Likelihood estimator (MLE)8, known to be asymptotically efficient and unbiased2,3,4. We will advocate its use especially when large sets of data samples are available.
Very often estimators result in difficult nonconvex optimization problems: Is this a cause for concern?
Many estimators are formulated in terms of optimization problems. Examples include the MLE or other extremum estimators2,3, such as nonlinear least squares estimators (NLLS). Such optimization problems are often nonlinear and, more importantly, nonconvex. As a result, the presence of a unique local optimum is not guaranteed, and hence optimization algorithms may get stuck into unwanted local optima. In this presentation, we will provide advice on initialization issues for optimization algorithms. Besides, we will emphasize the importance of studying the optimization problem at hand, instead of simply applying optimization algorithms as black box procedures. We will show that, for particular cases of relaxometry, despite the nonlinear character of relaxation models, the landscape of the cost-function is well-behaving and reasonable initialization results in finding the proper global optimum.
1. S. C. L. Deoni, “Quantitative relaxometry of the brain,” Top. Magn. Reson. Imaging, vol. 21, no. 2, p. 101, 2010.
2. A. van den Bos, “Parameter Estimation for Scientists and Engineers.’’ Hoboken, New Jersey, USA: John Wiley & Sons, 2007.
3. A. Papoulis and S. U. Pillai, ‘’Probability, Random Variables and Stochastic Processes’’. 4th ed. NY: McGraw-Hill, 2002.
4. S. M. Kay, ‘’Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory’’. 1st ed. Upper Saddle River, NJ: Prentice Hall, Inc, 1993.
5. H. Gudbjartsson and S. Patz, “The Rician distribution of noisy MRI data,” Magn.Reson. Med., vol. 34, no. 6, pp. 910–914, 1995
6. A. J. den Dekker and J. Sijbers, “Data distributions in magnetic resonance images: A review,” Phys. Medica, vol. 30, pp. 725–741, Nov 2014.
7. S. Aja-Fernández and G. Vegas-Sánchez-Ferrero, ‘’Statistical Analysis of Noise in MRI: Modeling, Filtering and Estimation’’. 1st ed. Springer International Publishing AG Springer Science, 2016.
8. J. Sijbers, A. J. den Dekker, P. Scheunders, and D. Van Dyck, “Maximum-likelihood estimation of Rician distribution parameters,” IEEE Trans. Med. Imaging, vol. 17, pp. 357–361, Jun 1998.