2011

Accounting for bias in estimated metabolite concentrations from cohort studies as caused by limiting the fitting parameter space
Rudy Rizzo1 and Roland Kreis1
1Department of Radiology and Biomedical Research, University of Bern, Bern, Switzerland

Synopsis

Potential problems arising from restricting the fitting algorithm in MR spectroscopy to a limited parameter space of physically meaningful values are investigated via Monte-Carlo approach and theoretical considerations. Three parameter-space configurations are compared to evaluate potential bias in the estimated mean cohort concentration for a simulated cohort study typical for conditions for MRS of hippocampus. The bias found for restrictions to positive concentration values can be ameliorated based on an estimate of the distribution width, e.g., based on Cramer-Rao bounds. Loosening parameter space restrictions can also eliminate bias while maintaining the benefit of parameter restrictions for the search algorithm.

Introduction

Most fit packages for in vivo MR spectroscopy offer ways to restrict fit parameters to remain in a space of physically meaningful values either by enforcing prior knowledge relations and/or setting bounds for the available parameter space. However, random spectral noise and other random factors inherently lead to a Gaussian distribution of values, which is the basis for unbiased averaging to obtain the cohort mean of estimated values. Excluding natural variance-related parts of solution space may hence lead to estimation bias. In particular, limiting concentrations to positive values (i.e., positive peak areas) for low-concentration metabolites or in general for spectra with limited SNR (e.g. from a small VOI inside hippocampus) may lead to overestimation of the cohort average. This can be prevented by excluding metabolites from evaluation based on Cramer Rao Bounds (CRLBs) criteria, which also includes dangers for bias1 and which prevents estimation of cohort averages independent of the cohort size. Here we investigate the size of the effect using Monte-Carlo simulations plus CRLBs calculations and propose a correction term for zero-bounded fits as well as alternative fitting bounds.

Methods

Spectra of a simplified metabolite mixture were simulated by NMR-scopeB2 with SNR and linewidth realistic for hippocampal spectra at 3T3 (Fig. 1). A semiLASER sequence with TE=35 ms, spectral width 4000 Hz, 4096 datapoints was simulated including real pulse shapes. Voigt lines were imposed as lineshapes with T2=130 ms and Gaussian width of 5.5 Hz. Two cohorts of 2000 cases each were designed with small concentration differences including ground truth (GT) values for GABA about equal to 1 or 0.5 times its CRLB value and zero GT value for Lactate (Lac) (Fig. 1). Fit (and CRLBs calculation) was performed using FitAID4 with a Levenberg-Marquardt algorithm without (LM) and with lower area bounds at zero (LMB0) or at minus one CRLB (LMB-1CRLB). The expected distributions of estimates and the calculation of the proposed bias correction term are based on formulas for a truncated Gaussian distribution5 where its natural width is assumed to be equal to the mean estimated CRLB, as illustrated in Fig. 2.

Results & Discussion

Estimates of cohort averages are affected by detectable bias in a 0-bounded LM fit if the relative CRLB for single measurements is >25% and bias becomes substantial for relative CRLB >50%. In our synthetic examples, GABA and Lac show the biggest effect as they were simulated in low or absent concentration. The distributions of estimated concentrations are presented in Fig.3. The proposed corrections work well, but not perfectly, because the fitting algorithm does not yield zero concentrations for those cases that should come out with negative values if not bounded in parameter space. Indeed, the corrected estimated means move close to GT values (Fig. 4) but are only partly included within the confidence intervals around GT ($$$± 2*SEM$$$, standard error of the mean $$$= CRLB / \sqrt{N}$$$, N: cohort size).
  • COHORT 1: GABA $$$ = 0.524 ∈ 0.500 ± 0.024$$$; Lac $$$ = 0.056 ∉ 0.000 ± 0.013$$$.
  • COHORT 2: GABA $$$ = 0.263 ∈ 0.250 ± 0.024$$$; Lac $$$ = 0.084 ∉ 0.000 ± 0.013$$$.
Further optimization can be done when accounting for a non-zero $$$\mu_{TL}$$$ contribution. Using an iterative look up function, $$$\mu_{TL}$$$ can be seen to be close to 0 and can be incorporated in the correction function (Fig. 5). The relationship between $$$\mu_{TL}$$$ and the distribution characteristics would need to be investigated for each fit package.
Extending the allowed parameter space with a lower boundary of -1 CRLB for each estimated metabolite would reduce the bias substantially (Fig. 4) and seems to largely retain the benefits of parameter space restrictions (i.e., prevention of grossly wrong local $$$\chi^2$$$ and easing the path for $$$\chi^2$$$ minimization.)

Conclusions

  • Restricting the parameter space for concentrations to positive values leads to bias in cohort averaging. Large effects are seen for cases where the mean relative CRLB in single spectra is >50% of the GT, but it can also realize with smaller CRLB in small cohorts.
  • Instead of refraining from reporting results for metabolites with large fitting uncertainties, cohort averages can be corrected for the expected bias.
  • The proposed correction term can substantially reduce this bias for large cohort sizes while for small cohorts care has to be used to prevent incidental findings due to small numbers and t-tests would not be appropriate for group comparisons.
  • Here, it was assumed that the CRLB is a valid proxy for the width of the cohort distribution. This assumption can be extended if data for true repeatability of estimates is available and might be deduced from total variance analysis of a set of strongly-represented metabolites.
  • The broadening term $$$\mu_{TL}$$$ for the values expected near zero, which may help to increase the accuracy of the correction, is expected to be fit-package dependent.
  • Lowering the area parameter bound from zero to -1 CRLB essentially eliminates the bias without correction term needed and is proposed as compromise between strict zero-bounded and free fitting.
  • Limiting the fit-parameter space to meaningful values remains appropriate for evaluation of single subjects or small groups but should be reconsidered for larger cohorts.

Acknowledgements

This work is supported by the Marie-Sklodowska-Curie Grant ITN-39 237 (Inspire-Med).

References

  1. R. Kreis, The trouble with quality filtering based on relative Cramer-Rao lower bounds, Magn. Reson. Med. 75, 15-18 (2016).
  2. Z. Starcuk Jr, J. Starcukova, O. Strbak and D. Graveron-Demilly: Simulation of coupled-spin systems in the steady-state free-precession acquisition mode for fast magnetic resonance (MR) spectroscopic imaging. Meas. Sci. Technol. 20, 10, 104033 (2009).
  3. N. Allaili, R. Valabregue, E.J. Auerbach, et al. Single-voxel 1H spectroscopy in the human hippocampus at 3 T using the LASER sequence: characterization of neurochemical profile and reproducibility. NMR Biomed. 28(10), 1209-1217 (2015).
  4. D. G. Q. Chong, R. Kreis, C. S. Bolliger, et al. Two-dimensional linear combination model fitting of magnetic resonance spectra to define the macromolecule baseline using FiTAID, a Fitting Tool for Arrays of Interrelated Datasets. Magn. Reson. Mater. Physics, Biol. Med. 24, 147-164 (2011).
  5. N. Johnson, S. Kotz, N. Balakrishnan, Continuous Univariate Distributions, Second edition, Wiley, 1994.

Figures

Figure 1: Top: ground truth metabolite composition and visualization of synthetic spectrum incl. macromolecular background (MMBG). Asp: Aspartate, NAA: N-Acetylaspartate, mI: myo-Inositol, PCr: Phosphocreatine, PCho: Phosporylcholine, GABA: γ-aminobutyrate, Glu: Glutmate, Gln: Glutamine. Bottom: fitted sample spectrum from cohort 1.

Figure 2: Top: cohort distribution with unbounded fitting algorithm and pertinent formulae. µ: ground truth mean, σ: ground truth std. µTR: mean of right truncated distribution, µTL: mean of left truncated distribution. Bottom: cohort distribution with 0+ fitting boundary. Limiting parameter space skews the Gaussian distribution. The negative tail is mapped to a small interval around 0+. Assuming its contribution to equal 0, the true mean can be reconstructed from its distorted version.

Figure 3: histogram of estimated concentrations for GABA in three different parameter space settings. Cohort 1 is depicted in the left column and cohort 2 in the right. µGT: ground truth concentration. µdistr: distribution estimated concentration.

Figure 4: summary of the main numeric findings. Fit results in different settings (parameter space and cohorts) are compared. Corrected values (assuming 0 contribution from truncated tail of the true distribution) with the proposed algorithm strongly reduce the bias. Conc: concentration, GT: Ground Truth, Diff.: differences as Conc. – GT, Corr.: corrected.

Figure 5: Illustration of iterative search for the contribution in the current example as fitted with FitAID for perfect correction of the cohort averaged concentration. Results displayed for cohort 1, Lactate and GABA. TL: Truncated Left, CORR: corrected, GT: Ground Truth

Proc. Intl. Soc. Mag. Reson. Med. 29 (2021)
2011