Bayesian Analysis of Transverse Signal Decay with Application to Human Brain

Abstract

Purpose:

Transverse relaxation analysis with several signal models has been used extensively to determine tissue and material properties. However, the derivation of corresponding parameter values is notoriously unreliable. We evaluate improvements in the quality of parameter estimation using Bayesian analysis and incorporating Rician noise, as appropriate for magnitude MR images.

Theory and Methods:

Monoexponential, stretched exponential, and biexponential signal models were analyzed using nonlinear least squares (NLLS) and Bayesian approaches. Simulations, and phantom and human brain data, were analyzed using three different approaches to account for noise. Parameter estimation bias, reflecting accuracy, and dispersion, reflecting precision, were derived for a range of signal-to-noise ratios (SNR) and relaxation parameters.

Results:

All methods performed well at high SNR. At lower SNR, the Bayesian approach yielded parameter estimates of considerably greater precision, as well as greater accuracy, than did NLLS. Incorporation of Rician noise greatly improved accuracy and, to a somewhat lesser extent, precision, in derived transverse relaxation parameters. Analyses of data obtained from solution phantoms and from brain were consistent with simulations.

Conclusion:

Overall, estimation of parameters characterizing several different transverse relaxation models was markedly improved through use of Bayesian analysis and through incorporation of Rician noise.

INTRODUCTION

Relatively little work has been undertaken to evaluate the Bayesian probability approach to parameter estimation from noisy MR images. In particular, the performance of Bayesian as compared to nonlinear least squares (NLLS) analysis in the quantification of transverse decay, as described by monoexponential and alternative models, has not been reported. However, these models are of increasing interest in both preclinical and clinical studies in order to characterize tissue ultrastructure, identify tissue compartments, and differentiate between normal and pathologic tissues.

Multiexponential transverse relaxation analysis (METRA) (1-7) is one approach to this type of characterization, in which extraction of transverse relaxation times and component amplitudes from relaxation data is achieved through use of the inverse Laplace transform (ILT). This is typically implemented through the NLLS or the nonnegative least square algorithms. However, the ill-posed nature of the ILT makes the inversion very sensitive to noise (8-9), rendering parameter estimation difficult, especially in the clinical setting.

It has been demonstrated that explicitly incorporating the Rician noise model appropriate to magnitude magnetic resonance (MR) images (10-11) greatly improves accuracy and precision in derived parameter estimates (12-14), including for biexponential signals (15). Despite this, bias and dispersion resulting from application of NLLS increase substantially with decreasing SNR or with increasing disparity between component sizes (15-16).

Parameter estimation using Bayesian probability theory has long been recognized as having several advantages over NLLS minimization (17-18), including the fact that it does not require initial estimates. In addition, the estimate of a given parameter is derived through integration, or marginalization, over the probability distribution functions (PDFs) of the remaining parameters. The resulting PDF of the parameter of interest, the so-called posterior PDF, is therefore not influenced by local minima of a global fitting function defined by multiple parameters as is the case for NLLS (19). The posterior PDF completely defines the uncertainty in the derived parameter estimate (20). Among other benefits, this permits assessment of MR parameter uncertainty within a single voxel (21).

While laying a foundation for Bayesian analyses of MR data, these previous results were restricted to NMR spectroscopic data or to monoexponential transverse decay over a restricted parameter range. In the present work, our goal is to provide a much more comprehensive evaluation of the quality of parameter estimation from transverse relaxation imaging data through use of Bayesian analysis as compared to NLLS. Analyses were performed using the appropriate Rician noise model, as that has been clearly demonstrated to provide improved estimates as compared to the Gaussian noise approximation (12-15, 21). Three distinct signal relaxation models were studied, the monoexponential, stretched exponential, and biexponential, over a large range of parameters and SNR values. All three of these models are used in the analysis of complex systems.

We first review the theory of noise distribution in MR images. We then describe in detail the six different voxel-by-voxel fitting methods we compared, three based on NLLS minimization and three based on Bayesian analysis. Next, extensive Monte Carlo (MC) simulations are presented which give the relative bias and dispersion in the estimation of transverse relaxation parameters using the six methods. Finally, we present data obtained from phantoms at 7 Tesla (T) and from human brain at 3T.

THEORY

MRI Signal Distribution

The PDF of magnitude imaging data S = {S_n} measured at echo times TE_n = n * TE defined by echo time TE and echo number n, with Gaussian-distributed noise of equal variance, σ², in the real and imaginary channels, is given by the Rician distribution (10):

$$P_{Rice}(S_n\mid A(\theta ;T{E}_n),\sigma )=\frac{S_n}{{\sigma }^2}exp(-\frac{A^2(\theta ;T{E}_n)+S_n^2}{2{\sigma }^2})I_0\left(\frac{A(\theta ;T{E}_n)S_n}{{\sigma }^2}\right)u\left(S_n\right)$$ [1]

where A(θ;TE_n), is the magnitude of the underlying noise-free signal as a function of TE_n and signal parameters θ, I_k is the modified k-order Bessel function of the first kind and u is the Heaviside step function. The first moment, or expectation value, of the Rician PDF is given by (22-23)

$$\begin{array}{c}\hfill E_{Rice}\left[S_n\right]={\int }_0^{\infty }{S}_n{P}_{Rice}(S_n\mid A(\theta ;T{E}_n),\sigma )d{S}_n\hfill \\ \hfill =\sigma \sqrt{\frac{\pi }{2}}exp(-\frac{{\varphi }_n}{2})+((1+{\varphi }_n)I_0\left(\frac{{\varphi }_n}{2}\right)+{\varphi }_n{I}_1\left(\frac{{\varphi }_n}{2}\right))\hfill \end{array}$$ [2]

where φ_n = A²(θ;TE_n)/2σ².

From Eq. 1, the likelihood function for the transverse decay data within a voxel is (24)

$$\begin{array}{c}\hfill L_{Rice}(\mathbf{S}\mid \theta ,\sigma )=\underset{n=1}{\overset{N}{\Pi }}{P}_{Rice}(S_n\mid A(\theta ;T{E}_n),\sigma )\hfill \\ \hfill =\frac{1}{{\sigma }^{2N}}exp(-\frac{{\Sigma }_{n=1}^N(A^2(\theta ;T{E}_n)+S_n^2)}{2{\sigma }^2})\underset{n=1}{\overset{N}{\Pi }}{S}_n{I}_0\left(\frac{A(\theta ;T{E}_n)S_n}{{\sigma }^2}\right)\hfill \end{array}$$ [3]

At high SNR, S_nA(θ;TE_n) » σ² so that I₀(A(θ;TE_n)S_n/σ²) approaches $exp(S_{n}A(\theta ;T{E}_n)∕{\sigma }^2)∕\sqrt{2\pi S_{n}A(\theta ;T{E}_n)∕{\sigma }^2}$. In this limit, the Rician PDF approximates the Gaussian distribution with mean A(θ;TE_n) and variance σ² (11, 25):

$$P_{Gauss}(S_n\mid A(\theta ;T{E}_n),\sigma )=\frac{1}{\sigma \sqrt{2\Pi }}exp(-\frac{{(A(\theta ;T{E}_n)-S_n)}^2}{2{\sigma }^2})u\left(S_n\right)$$ [4]

with first moment, or expectation value, given by

$$E_{Gauss}\left[S_n\right]={\int }_{-\infty }^{+\infty }{S}_n{P}_{Gauss}(S_n\mid A(\theta ;T{E}_n),\sigma )d{S}_n=A(\theta ;T{E}_n)$$ [5]

From Eq. 4, the Gaussian likelihood function of the imaging data S is

$$\begin{array}{c}\hfill L_{Gauss}(\mathbf{S}\mid \theta ,\sigma )=\underset{n=1}{\overset{N}{\Pi }}{P}_{Guass}(S_n\mid A(\theta ;T{E}_n),\sigma )\hfill \\ \hfill =\frac{1}{{\left(2\pi {\sigma }^2\right)}^{N∕2}}exp(-\frac{{\Sigma }_{n=1}^N{(A(\theta ;T{E}_n)-S_n)}^2}{2{\sigma }^2})\hfill \end{array}$$ [6]

NLLS and Bayesian Parameter Estimation Approaches

NLLS parameter estimation is generally rapid and accurate for low dimensional problems, that is, those for which the number of parameters to be estimated is small. However, in higher dimensions, NLLS estimation is greatly complicated by the presence of local minima; this difficulty increases with model complexity, with an increasing number of free parameters to be determined, and with decreasing SNR (19).

Bayesian analysis is an alternative approach that can be particularly robust for parameter estimation in the presence of noise (17, 20-21, 26-34). In this framework, parameter probability distributions are calculated from the observed data and prior knowledge. Bayesian parameter estimation is based on the posterior probability density of parameters, θ, given the observed data, S, and the standard deviation of the noise, σ. The formal statement of Bayes theorem in this context is (35-36):

$$\mathrm{P}(\theta \mid \mathbf{S},\sigma )=\frac{\mathrm{P}\left(\theta \right)L(\mathbf{S}\mid \theta ,\sigma )}{\mathrm{P}\left(\mathbf{S}\right)}$$ [7]

where P(θ) represents an a priori parameter distribution based on pre-existing knowledge, L(S|θ,σ) is the likelihood function of S given θ and σ, and P(S) is a normalization constant, which corresponds to our use of normalized probability distributions.

The successive marginalization over all parameters results in a considerable computational burden, but permits the calculation of a posterior PDF for each individual parameter. An estimate ${\widehat{\theta }}_i$ can be derived from the PDF of θ_i in several ways. It can be taken as the mean of the PDF P(θ_i|S,σ), as is widely used in statistical inference, or as its median value. In this paper, we define ${\widehat{\theta }}_i$ as the maximum posterior probability (MAP), as is common in Bayesian image analysis (37):

$${\widehat{\theta }}_i=\arg \underset{{\theta }_i\in \theta }{\max }\left\{\mathrm{P}({\theta }_i\mid \mathbf{S},\sigma )\right\}$$ [8]

METHODS

Transverse Relaxation Signal Models

Monoexponential Signal

The monoexponential model implicitly assumes a single pool of water protons, leading to transverse decay of the form:

$$A(\theta ;T{E}_n)=A_{0}exp(-\frac{T{E}_n}{T_2})$$ [10]

Thus, this model is characterized by the parameter pair θ = (A₀ T₂), with A₀ representing the signal amplitude for TE = 0, incorporating T₁-weighting, proton density, coil sensitivity and other machine factors. T₂ represents the spin-spin relaxation time.

Stretched Exponential Signal

Biological tissues are characterized by a complex physicochemical structure (1-7, 38-41) so that the monoexponential function (Eq. 10), while often a useful approximation, would not be expected to account for important features of transverse relaxation (42). Therefore, alternative relaxation models that permit a more detailed description of the relationship between relaxation or diffusion processes and internal material structure have become of interest (43-47).

The stretched exponential decay model is given by

$$A(\theta ;T{E}_n)=A_{0}exp(-{\left(\frac{T{E}_n}{T_2}\right)}^{\alpha })$$ [11]

with parameters θ = (A₀ g=a T₂; α ∈ (0,1] defines the deviation from monoexponential decay. This arises in several settings, including as an approximation to a continuous distribution of decay constants and in systems with memory as represented by fractional order processes (47).

Biexponential Signal

Restricting METRA to biexponential signals has been found to be a good approximation to transverse relaxation in a number of studies and is particularly appropriate in settings of only moderate SNR, such as is achievable clinically (48-51). For the case of two distinct proton pools that are not exchanging, the signal decay function is

$$A(\theta ;T{E}_n)=A_0(f_{s}exp(-\frac{T{E}_n}{T_{2,s}})+(1-f_s)exp(-\frac{T{E}_n}{T_{2,l}}))$$ [12]

with parameters θ = (A₀f_sT_2,sT_2,l), where f_s is the fraction of the short T₂ component, and T_2,s and T_2,l are the spin-spin relaxation times of the short-T₂ and long-T₂ components, respectively. It is well-known that derivation of accurate parameter values can be highly problematic (15-16) even for this simple biexponential signal model.

Parameter Estimation Methods

For each of the above three signal models, six methods of parameter estimation were implemented. They are defined by use of NLLS versus Bayesian analysis, and by signal models using Gaussian noise, Gaussian noise plus an offset, and Rician noise. More precisely, the first method, denoted NLLS_G, uses the Gaussian estimator in Eq. 5, and consists of NLLS minimization of S_n − E_Gauss[S_n], that is, S_n − A(θ;TE_n), with A(θ;TE_n) defined by Eqs. 10-12. The second method evaluated, NLLS_G,O, uses the Gaussian estimator plus an offset ε, and consists of NLLS minimization of S_n − (A(θ;TE_n) + ε). This is the simplest method used to account for noise in magnitude reconstruction (52-54) (Fig. 1). The third method, NLLS_R, makes full use of the expectation value of the Rician distribution given by Eq. 2, and involves NLLS minimization of S_n − E_Rice[S_n]. The fourth method, MAP_G, is a Bayesian analysis using the Gaussian likelihood function, L_Gauss(S|θ,σ), given by Eq. 6. The fifth parameter estimation method, MAP_G,O, is a Bayesian analysis using the Gaussian likelihood function, L_Gauss(S|θ,σ), given by Eq. 6, but incorporating an offset, so that (A(θ;TE_n) is replaced by (A(θ;TE_n). The sixth method evaluated, MAP_R, is a Bayesian analysis using the Rician likelihood function, L_Rice(S|θ,σ), given by Eq. 3. The marginalization process required in the Bayesian-based methods is described in the Appendix. All numerical calculations were performed using Matlab (MathWorks, Natick, MA, USA), with NLLS fits performed with the Levenberg-Marquardt algorithm.

Fig. 1.

Signals as a function of TE_n generated for T₂ = 30 ms and A₀ = 1, assuming monoexponential transverse decay (Eq. 10) with SNR of 20 (Panel a) and 100 (Panel b). The noisy signal, S(TE_n), was obtained through addition of Rician noise to the underlying noise-free signal,A(θ;TE_n), according to $S\left(T{E}_n\right)=\sqrt{{(A(\theta ;T{E}_n)+N_{re})}^2}+N_{im}^2$ where the noise in the real and imaginary channels, N_re and N_im, was generated from a Gaussian distribution with zero mean and standard deviation σ.

Estimations of Standard Deviation, Variance and Offset of the Noise

We take the standard deviation, σ, variance, σ², and offset, ε, of the noise as known parameters. These were input values in our simulation studies, while in phantom and in vivo studies they were estimated from background regions in which A(θ;TE_n) = 0 so that Eq. 1 reduces to the Rayleigh distribution (11, 55-56):

$$P_{Rayleigh}(S_n\mid \alpha )=\frac{S_i}{{\sigma }^2}exp(-\frac{S_n^2}{2{\sigma }^2})$$ [13]

The first and the second moments of P_Rayleigh(S_n|σ) are (57-58)

$$E_{Rayleigh}\left[S_n\right]={\int }_0^{\infty }{S}_n{P}_{Rayleigh}(S_n\mid \sigma )d{S}_n=\sigma \sqrt{\frac{\pi }{2}}$$ [14]

and

$$E_{Rayleigh}\left[S_n^2\right]={\int }_0^{\infty }{S}_n^2{P}_{Rayleigh}(S_n\mid \sigma )d{S}_n=2{\sigma }^2$$ [15]

from which unbiased estimates of σ, σ² and ε may be obtained through $\widehat{\sigma }\simeq {\left(NK\sqrt{\pi ∕2}\right)}^{-1}{\Sigma }_{n=1}^N{\Sigma }_{k=1}^K{S}_n^k,\widehat{{\sigma }^2}\simeq {\left(2NK\right)}^{-1}{\Sigma }_{n=1}^N{\Sigma }_{k=1}^k{\left(S_n^k\right)}^2\mathrm{and}\widehat{∊}\simeq \widehat{\sigma }\sqrt{\pi ∕2}$ (57-58), where K is the total number of voxels taken as background. Note that to improve the quality of these estimates, the signal-free regions were formed from the background intensity in all of the echo images.

Bias and Dispersion Calculations

For each method, the relative bias, a measure of accuracy, was defined from the difference between the true value, θ_i, and the mean of the estimated value, ${\widehat{\theta }}_i$, by $100\ast \mid {\theta }_i-(1∕M){\Sigma }_{m=1}^M{\widehat{\theta }}_i^m\mid ∕{\theta }_i$. In simulations, M is the total number of noise realizations, while in phantom studies it is the total number of voxels for which parameter maps were generated. In simulations, true parameter values were known, while in phantom studies they were obtained from a monoexponential fit (12) of high SNR data acquired from each phantom compartment separately (see below). Note that the phantom studies were limited to the monoexponential and biexponential models. Relative dispersion, a measure of precision, was similarly defined by $100\ast \sqrt{(1∕M-1){\Sigma }_{m=1}^M{({\widehat{\theta }}_i^m-{\theta }_i^m)}^2}∕{\theta }_i$

Relative bias and dispersion were calculated after removal of voxels with extreme outliers (VEO) using Tukey’s method (59) to correspond to actual practice (12, 15).

In contrast to simulations and phantom studies, exact parameter values for the brain were unknown so that relative bias and dispersion cannot be evaluated. Instead, relative error maps for each estimated parameter were calculated using MAP_R analysis of high SNR imaging data as the reference, although we note that maps were essentially identical for all analysis methods using the high SNR data. For each estimated parameter, errors in the values obtained at lower SNR were calculated from the difference between the estimated value, ${\widehat{\theta }}_i$, and its reference value, ${\widehat{\theta }}_{i,ref}$, according to $error=100\ast \mid {\widehat{\theta }}_i-{\widehat{\theta }}_{i,ref}\mid ∕{\widehat{\theta }}_{i,ref}$.

In simulations, SNR was defined as A(θ;TE_n)/σ, where TE₁ is the shortest echo time at which data were generated. In phantom and in vivo studies, it was defined as $\sqrt{\stackrel{‒}{S}{\left(T{E}_1\right)}^2}-2{\widehat{\sigma }}^2∕\widehat{\sigma }$ (60), where S̄(TE₁) represents the mean signal calculated from a large region of the MR image obtained at TE₁.

Numerical Simulation Studies

MC simulations were performed using monoexponential (Eq. 10), stretched exponential (Eq. 11), and biexponential (Eq. 12) signal models for a wide range of underlying parameters and as a function of SNR, achieved through addition of Rician noise (Fig. 1). For all simulations, a total of 32 data points were generated with TE_n increasing uniformly from 8 to 256 ms. The required starting values needed in the NLLS analyses were set to their true values. For each signal model and for each set of parameters, relative bias and dispersion were calculated over 1000 noise realizations for each estimated parameter using the six analysis methods.

MC Simulations 1: Effect of SNR

MC simulations were performed to examine the effect of SNR on the relative bias and dispersion in derived parameters. The underlying true model parameters were chosen to reflect values obtained from the in vivo study of human brain (below).

For the monoexponential (Eq. 10), T₂ = 70 ms was used, with Rician noise added to achieve SNR ranging from 10 to 60 in increments of 10. For the stretched exponential (Eq. 11), T₂ = 60 ms and α = 0.8 were used, with SNR values ranging from 10 to 100 in increments of 10. For the biexponential analysis (Eq. 12), the underlying parameters were f_s = 0.25, T_2,s = 20 ms and T_2,l = 90 ms, with an SNR range of 30 to 120, again in increments of 10. The different SNR ranges were selected to account for the difference in complexity of these three signal models. For example, the SNR requirement for the monoexponential is less stringent than for the other two models, so lower values of SNR were appropriate for the analysis. The parameter estimates begin to break down severely at the lower limit of SNR, and became relatively stable at the higher value, in each case.

MC Simulations 2: Effect of Underlying Parameters

MC simulations were also used to explore the effect of underlying true parameter values on the relative bias and dispersion in derived parameters. The wide range of underlying parameters investigated is summarized in Fig. 3-5. Again, appropriate values of SNR were selected, with SNR of 10 and 50 for the monoexponential, 10 and 70 for the stretched exponential, and 30 and 100 for the biexponential.

Fig. 3.

Relative bias and dispersion in T₂ estimates obtained using Monte Carlo simulations for the monoexponential signal model at SNR of 10 (left column) and 50 (middle column) using the NLLS_G, NLLS_G,O, NLLS_R, MAP_G, MAP_G,O and MAP_R analysis methods for five different input T₂ values (right column). Decreased bias and dispersion, that is, improved accuracy and precision, were achieved through use of Rician noise model and Bayesian analysis as discussed in detail in the text.

Fig. 5.

Relative bias and dispersion of the estimates of f_s, T_2,s and T_2,l obtained using Monte Carlo simulations for the biexponential signal model at SNR of 30 (left column) and 100 (middle column) using NLLS_G, NLLS_G,O, NLLS_R, MAP_G, MAP_G,O and MAP_R analysis methods for five different combinations of f_s, T_2,s and T_2,l (right column). Decreased bias and dispersion, that is, improved accuracy and precision, were achieved through use of Rician noise model and Bayesian analysis as discussed in detail in the text.

Phantom Studies

Phantom studies were designed to validate the monoexponential and biexponential MC simulations. Three rectangular tubes (5 × 3 cm) filled with deionized water were doped with different concentrations of CuSO₄. The tubes were then placed together in the MRI system, and images were acquired using a 3D CPMG sequence (61-62) on a 7T small animal MRI scanner (Bruker Biospin GmbH, Rheinstetten, Germany) equipped with a linearly polarized birdcage coil. 35 T₂-weighted images with TE_n increasing linearly from 5.1 to 180 ms were acquired. Other imaging parameters included TR = 1000 ms, field of view (FOV) = 70 × 40 × 40 mm³, and matrix = 140 × 80 × 20. Images were obtained with different numbers of signal averages, ranging from 1 to 8. Linear combinations of complex imaging data from each pair of phantoms were created and added with consistent phases to obtain Rician-distributed magnitude images with different compartment fractions (15). Relative bias and dispersion were calculated for each estimated parameter using the six analysis methods. Fiducial T₂ values were obtained from high SNR data acquired from each tube independently and fit to a monoexponential; they were 14 ± 2 ms, 58 ± 1 ms and 112 ± 3 ms from tubes 1, 2 and 3, respectively. These were used as initial estimates for the NLLS analysis.

Phantom Study 1: Effect of SNR

Images acquired from tube 2, with T₂ = 58 ms, were fit to a monoexponential using the six investigated methods. Data from the separate measurements on tube 1, with T₂ = 14 ms, and tube 3, with T₂ =112 ms, were constructed as described above in order to synthesize two-component images with f_s = 0.33 and with a range of SNR's. Each voxel from these composite images was fit to a biexponential function.

Phantom Study 2: Effect of Underlying Parameters

T₂ weighted images were fit to a monoexponential independently for each of the three imaged tubes. For the biexponential, different composite images were created with several f_s values and ratios T_2,l/T_2,s (see Fig. 8). Fits were performed at two different SNR's, as indicated in Fig. 8.

Fig. 8.

Relative bias and dispersion obtained from phantom measurements for the estimation of f_s, T_2,s and T_2,l, with a biexponential signal model, for SNR of ~ 30 (left column) and ~ 95 (middle column), using NLLS_G, NLLS_G,O, NLLS_R, MAP_G, MAP_G,O and MAP_R analysis methods, for four different combinations of input f_s, T_2,s and T_2,l values (right column). Improvements in accuracy and precision are seen through use of Rician noise model and Bayesian analysis, as discussed in the text.

In vivo Study

T₂-weighted images of the brain of a 40 year-old male volunteer were acquired with a 3T whole-body MRI system (Achieva, Philips Medical Systems) equipped with a quadrature head coil. 32 axial MR images were obtained using a 3D CPMG sequence with TE_n increasing linearly from 8 to 256 ms, TR = 700 ms, FOV = 240 × 240 × 15 mm³ and matrix = 320 × 240 × 5. Images were obtained for different numbers of signal averages to vary SNR. Data were fit to monoexponential, stretched exponential and biexponential functions using the six methods investigated. Relative error maps were then calculated as described above in the "Bias and Dispersion Calculations" section. Parameter values obtained at the highest SNR were used as starting values for the NLLS-based methods.

RESULTS

Numerical Simulation Studies

Results of the MC simulations were obtained after removal of VEO. The number of removed VEO was almost identical among the six methods, and was less than 1% at the larger, though still moderate, SNR for each. Even at the lower SNR, corresponding to truly modest image quality, VEO represented only a relatively small fraction of the total voxels. Less than 2% of the voxels were excluded for the monoexponential, less than 5% for the stretched exponential, and less than 8% for the biexponential.

Fig. 2 shows the relative bias, as a measure of accuracy, and the relative dispersion, as a measure of precision, in parameter estimates for the monoexponential, with T₂ as the fit parameter of interest (Fig. 2a), the stretched exponential, with T₂ and α as fit parameters of interest (Fig. 2b) and the biexponential, with T_2,s, T_2,l, and f_s as fit parameters of interest (Fig. 2c). For the monoexponential signal model, all six methods showed bias and dispersion less than 10% in the estimation of T₂ for SNR > 30 (Fig. 2a). However, bias and dispersion increased rapidly with decreasing SNR, particularly with NLLS_G, NLLS_G,O, MAP_G and MAP_G,O. At SNR = 10, NLLS_R and MAP_R showed bias and dispersion less than 10%. Although NLLS_G,O and MAP_G,O were the most precise, showing the smallest dispersion, they were the least accurate, that is, showed the greatest bias, over the entire range of SNR. Overall, all methods showed comparable dispersion, but the methods using the proper noise model, NLLS_R and MAP_R, demonstrated substantially lower bias. It is also of note that the Bayesian analysis always performed equivalently to or better than the corresponding NLLS analysis.

Fig. 2.

Monte Carlo simulation results for all three investigated signal models showing relative bias and dispersion in parameter estimates as a function of SNR using the fitting methods NLLS_G, NLLS_G,O, NLLS_R, MAP_G, MAP_G,O and MAP_R. (a) Results for T₂ assuming the monoexponential signal model with T₂ = 70 ms as the underlying parameter value. (b) Results for T₂ and α assuming the stretched exponential signal model with T₂ = 60 ms and α = 0.8. (c) Results for f_s, T_2,s and T_2,l assuming the biexponential signal model with f_s = 0.25, T_2,s = 20 ms and T_2,l = 90 ms. Overall, bias and dispersion were substantially reduced through use of Rician noise model and Bayesian analysis; see text for detailed discussion.

Results for the stretched exponential signal model are shown in Fig. 2b. For SNR > 20 all six methods showed a bias and dispersion of less than 15% in the estimation of T₂. Achieving these bounds for estimation of α required SNR > 30 for NLLS_G and NLLS_G,O. Even at lower SNR's, NLLS_R and MAP_R remained highly accurate, that is, nearly bias-free, while bias became substantial for all other analyses. In addition, Fig. 2b shows that at the lower values of SNR, the dispersion in the estimates of T₂ and α obtained using the NLLS-based analyses was substantially higher than that obtained using the corresponding Bayesian-based methods.

Analysis of the biexponential signal model (Fig. 2c) indicates that bias becomes negligible as SNR increases for NLLS_G, NLLS_R, MAP_G and MAP_R in the estimation of all parameters. In contrast, dispersion remains substantial, particularly for f_s and T_2,s. In general, at lower SNR's, MAP_R was found to provide the least biased parameter estimates. Similar performance was obtained using NLLS_R for the estimation of T_2,s and T_2,l. In addition, the dispersion in parameters estimated using NLLS-based methods was substantially higher than that obtained using Bayesian-based methods. Finally, bias and dispersion for T_2,l were substantially lower than for T_2,s, presumably due to the greater SNR and superior signal sampling for that component of the combined signal. NLLS_G,O was found to perform well for T_2,l estimation even at low SNR.

Figures 3-5 show relative bias and dispersion in the estimation of parameters for the monoexponential (Fig. 3), stretched exponential (Fig. 4) and biexponential (Fig. 5) signal models, for a range of underlying parameters and for two representative SNR's. Figure 3 shows that at high SNR, all methods performed well, with low bias and dispersion in the estimation of T₂. Differences in performance become apparent at lower SNR, for which the inclusion of the Rician noise model is of greater potential importance. We found that NLLS_R and MAP_R provided estimates with the least bias, that is, the greatest accuracy, while MAP_G,O and MAP_R provided estimates with the least dispersion, that is, the greatest precision, although NLLS_R also provided precise estimates in several cases. Both bias and dispersion were found to decrease with increasing T₂, reflecting the higher SNR throughout the duration of the simulated data acquisition. Results for the stretched exponential signal model are shown in Fig. 4. At high SNR, all investigated methods provide low bias and dispersion in parameter estimation for the combinations of T₂ and α shown. However, at low SNR, MAP_R was the least biased for the estimation of both T₂ and α, with NLLS_R also performing well. This highlights the importance of appropriate noise modeling, regardless of the analytic method used for parameter estimation. Overall, the NLLS-based methods showed higher dispersion than the Bayesian-based methods. The bias and dispersion in estimates of T₂ and α decrease with increasing T₂, again reflecting improved SNR and signal sampling throughout the signal decay. Figure 5 summarizes the results for the biexponential signal model. As shown, the bias and dispersion in the estimation of f_s, T_2,s and T_2,l increase with a decreasing ratio T_2,l/T_2,s or with increasing disparity between component sizes. In general, for low SNR, MAP_R was found to be the least biased for the estimation of f_s, T_2,s and T_2,l, with NLLS_R also performing well in estimating T_2,s and T_2,l. Overall, Fig. 5 indicates that the dispersion in the estimation of f_s, T_2,s and T_2,l obtained using the NLLS-based methods was higher than with the Bayesian-based methods.

Fig. 4.

Relative bias and dispersion of T₂ and α estimates obtained using Monte Carlo simulations for the stretched exponential signal model at SNR 10 (left column) and 70 (middle column) using NLLS_G, NLLS_G,O, NLLS_R, MAP_G, MAP_G,O and MAP_R analysis methods for five different combinations of T₂ and α values (right column). Decreased bias and dispersion, that is, improved accuracy and precision, were achieved through use of Rician noise model and Bayesian analysis as discussed in detail in the text.

Phantom Studies

As for the MC simulations, results of the phantom measurements were obtained after removal of VEO. Figure 6 shows the relative bias and dispersion, as a function of SNR, in parameter estimates determined from phantom measurements using single-phantom monoexponential (Fig 6a) and dual-phantom biexponential (Fig. 6b) data. At high SNR, all methods demonstrated low bias and dispersion. For the monoexponential case, at low to moderate SNR, the bias was substantially lower with NLLS_R and MAP_R than it was using the other methods for the estimation of T₂ (Fig. 6a). The dispersion in the estimation of T₂ was comparable among all methods. Similarly, for the biexponential analysis, bias in the estimates of T_2,s and T_2,l was minimized through use of NLLS_R and MAP_R (Fig. 6b). For f_s, MAP_R was found to demonstrate the least bias, although all of the Bayesian-based methods performed well with the exception of MAP_G,O in the estimation of T_2,l. This is likely attributable to the limited echo acquisition window, resulting in incomplete sampling of the decay of the T_2,l component. This effect is also seen in the poor performance of NLLS_G,O. In addition, the dispersion in estimates of all derived parameters obtained using the NLLS-based methods was substantially greater as compared to Bayesian-based estimates.

Fig. 6.

Phantom study results showing relative bias and dispersion in parameter estimates as a function of SNR using the fitting methods NLLS_G, NLLS_G,O, NLLS_R, MAP_G, MAP_G,O and MAP_R. (a) Results for T₂, assuming the monoexponential signal model with T₂ = 58 ms as the underlying parameter value. (b) Results for f_s, T_2,s and T_2,l assuming the biexponential signal model with f_s = 0.33, T_2,s = 14 ms and T_2,l = 112 ms as underlying parameter values. Overall, accuracy and precision were substantially improved through use of Rician noise model and Bayesian analysis; see text for detailed discussion.

The effect of the underlying parameters T₂, f_s, T_2,sand T_2,l on the relative bias and dispersion in parameter estimates is summarized in Figs. 7-8. As shown, bias resulting from NLLS_R and MAP_R was the lowest, while dispersion using Bayesian-based methods was overall lower than using NLLS-based methods. In general, both bias and dispersion decreased with increasing T₂ (Fig. 7). For the biexponential case, bias and dispersion both increase with a decreasing ratio T_2,l/T_2,s and with an increasing disparity between component sizes (Fig. 8).

Fig. 7.

Relative bias and dispersion obtained from phantom measurements for the estimation of T₂, with a monoexponential signal model, for SNR of ~ 12 (left column) and ~ 46 (middle column), using NLLS_G, NLLS_G,O, NLLS_R, MAP_G, MAP_G,O and MAP_R analysis methods for three different input T₂ values (right column). Improvements in accuracy and precision are seen through use of Rician noise model and Bayesian analysis, as discussed in the text.

In vivo Study

Figure 9 shows parameter maps derived from high SNR (~ 200) data (left hand column) using the six investigated methods, and corresponding parameter maps obtained at moderate SNR (~ 50), for monoexponential (Fig. 9a), stretched exponential (Fig. 9b) and biexponential (Fig. 9c) signal models. Figure 9 shows substantial random variation of estimated values in several image regions. To examine this in more detail, relative error maps, calculated as described in the Methods and describing the deviation of estimated parameter values from those obtained from the high SNR fiducial images, were created (Fig. 10). The mean and standard deviation of the relative errors are reported at the bottom of each error map. At the moderate SNR, all methods performed well for the estimation of T₂ in monoexponential decay, although MAP_R and NLLS_R were the best. These two methods also showed the best performance for the estimation of T₂ and α in the stretched exponential signal model. For biexponential analysis, the SNR was insufficient to provide a high-quality estimate of f_s using any of the analysis approaches, although MAP_R performed the best. T_2,s and T_2,l were estimated more accurately, although only T_2,l could be estimated to within ~ 15%, as achieved by MAP_R.

Fig. 9.

Reference parameter maps obtained at high SNR (~ 200) using the MAP_R method (left column) and parameter maps obtained at moderate SNR (~ 50) using the fitting methods NLLS_G, NLLS_G,O, NLLS_R, MAP_G, MAP_G,O and MAP_R (columns from left to right). (a) T₂ maps assuming the monoexponential signal model. (b) T₂ and α maps assuming the stretched exponential signal model. (c) f_s, T_2,s and T_2,l maps assuming the biexponential signal model. Greater consistency is achieved through use of Rician noise model and Bayesian analysis; see text for detailed discussion.

Fig. 10.

Relative error maps indicating the deviation of the parameter maps obtained at modest SNR (~ 50), shown in columns 2-7 of Fig. 9, from the high SNR fiducial map shown in column 1 of Fig. 9. The columns show the results obtained through use of NLLS_G, NLLS_G,O, NLLS_R, MAP_G, MAP_G,O and MAP_R for (a) the monoexponential signal model, (b) the stretched exponential signal model, and (c) the biexponential signal model. The mean and standard deviation of the relative error is given below each error map. Clear improvements are seen through use of Rician noise model and Bayesian analysis, as discussed in detail in the text.

DISCUSSION

Parameter Estimation Methods

Bayesian Probability vs NLLS Approaches

The elegant work of Neil and Bretthorst (17) evaluated estimates of diffusion parameters for a two-compartment model at two values of SNR, using both simulations and spectroscopic NMR data exhibiting Gaussian noise. Substantial advantages were found for Bayesian analysis as compared to NLLS in terms of both accuracy and precision. Our work builds on this and extends it in several ways, using simulations, and phantom and human brain imaging experiments. First, we evaluate three distinct and widely-used signal models for transverse decay, the monoexponential, stretched exponential and biexponential. In addition, for each of these models, we evaluate the performance of three approaches to incorporating noise, namely, the assumption of Gaussian noise, the assumption of Gaussian noise that asymptotes to a non-zero value, and the assumption of Rician noise, as is most appropriate for magnitude MR images. Further, we evaluated the accuracy and precision with which parameters of interest can be estimated for wide ranges of SNR and underlying parameters.

Overall, we found improved accuracy and precision using the Bayesian-based methods as compared to the NLLS-based methods (Figs. 2-8). This may be attributable to the fact that in Bayesian analysis, a given parameter is estimated independently of the others through marginalization, serving to reduce interdependence of parameter estimates by integrating over all values of the marginal parameters. This is in contrast to NLLS where unknown parameter estimates are highly correlated through the use of simultaneous estimation. This correlation has been previously demonstrated through Cramer-Rao lower bound calculations, MC simulations, and phantom measurements (8, 15-16). In addition, the NLLS algorithm may fail to find global minima due to the presence of local minima. Although such local minima are absent for the monoexponential, they are present for the bi- and stretched-exponential cases. This problem becomes more acute with the flattening of the surface of the least squares residuals seen with increasing model complexity (Supporting Figure S1). In fact, the performance of the Bayesian and NLLS methods were very similar for the monoexponential analysis, with the small differences being potentially attributable to the marginalization over A₀.

Finally, we found that in spite of its additional model complexity, potentially leading to substantial analytic difficulties, the performance of MAP_R provides notable improvement in accuracy and precision compared to NLLS_R.

Rician vs Gaussian noise models

Our analyses show substantial improvements in accuracy and precision when Rician noise model is incorporated into the analysis of magnitude imaging data, with estimates performed through either NLLS or Bayesian analyses. Similar results have been previously reported for the estimation of T₁, T₂ and T₂^* using NLLS (12-13, 15), and for the estimation of T₂ using Bayesian analysis (21).

It is common practice to assume Gaussian-distributed noise in conventional magnitude MR images with high SNR (42, 63-66). Our results (Figs. 2-8) serve to validate this and demonstrate that high-quality parameter estimates may be obtained with the Gaussian approximation. However, we found that with more modest SNR, there is a substantial loss of performance. Attempting to compensate for the asymptotic positivity of the noise by using Gaussian noise with offset model was found to be of benefit when fitting data with small T₂, for which the signal is dominated by noise in the later echoes.

Unknown vs estimated noise parameters

All calculations were performed assuming known values for σ, σ² and, ε, such as would be based on separate measurements in order to reduce the dimensionality of parameter space. As expected, MC simulations (data not shown) indicate that this strategy resulted in increased accuracy, presumably through reducing the number of local minima and the interdependence of parameter estimates. A modest increase in precision was also found. In addition, computational times were considerably reduced, particularly for the Bayesian-based methods since marginalization over σ was not needed.

The quality of parameter estimates of course depends upon the quality of the estimates of $\widehat{\sigma },{\widehat{\alpha }}^2\mathrm{and}\widehat{∊}$. Simulation analysis (Supporting Table S1) indicates that, as expected, moderate errors in the estimation of noise parameters, such as would be the case for estimation from large background regions of the image, have a negligible impact on parameter estimates. This was found to be the case for all fitting methods and for all three signal models. Clearly, the presence of non-random artifacts such as streaking or ghosting requires more sophisticated estimates of noise (67-68). In any event, in actual application, the strategy of independent noise estimation is to be preferred over incorporation of yet another parameter into the Bayesian or NLLS analysis.

Results of NLLS and Bayesian Probability Approaches

Performance of the NLLS-based methods

All three NLLS-based methods were found to be sensitive to initial estimates, highlighting the necessity of establishing a reproducible algorithm for selection of these estimates, especially in the low SNR regime. In contrast, at high SNR, convergence to global minima was attained even for relatively poor initial estimates (data not shown). However, in spite using true parameter values as starting points, we found a loss of performance as the total number of unknown parameters increased, such as in comparing the monoexponential to the biexponential. This is likely attributable to the increase in the number of local minima and to the flattening of the surface of least squares residuals as the number of parameters increases (Supporting Figure S1).

In some cases, we found that NLLS_R compared favorably to MAP_R. However, our approach implemented an unrealistically favorable NLLS_R analysis by providing true values as starting parameters, while in actual experiments these values are of course not known. It is well-known that inaccurate initial guesses render the NLLS algorithm highly susceptible to finding local rather than global minima (69), so that this procedure in effect penalizes the Bayesian analysis as compared to NLLS.

Performance of the Bayesian-based methods

One drawback of the Bayesian approaches is the computational time required to compute the integrals needed for the marginalization. In our direct implementation, computational time increased considerably with increasing dimensionality of the parameter space. To address this, the Markov Chain Monte Carlo method can be used so that the posterior distribution can be sampled, rather than directly computed (20, 70).

Numerical Simulations, Phantom and In vivo Studies

Our simulation results show that in the limit of high SNR, all six investigated methods yielded results that were accurate and precise. However, large differences in performance were seen at low to moderate SNR. Overall, MAP_R performed the best of all methods, although NLLS_R provided comparable results in many cases. As an example, the accuracy and precision of the estimate of the fraction of the rapidly decaying component, f_s, in the biexponential model were greatly improved using MAP_R as compared to the other methods, including NLLS_R. This is of particular importance, since f_s has been proposed as a biomarker of pathology in several biological tissues, including cartilage (1-2, 38-39), bone (51) and brain (3-7, 40).

Our in vivo brain study indicates that, as expected, any of the six methods works well at high SNR. However, MAP_R provides the most robust parameter estimates, especially in the low SNR regime. This can be seen visually from the parameter maps (Fig. 9) and from the error maps (Fig. 10). In contrast to MC simulations and phantom experiments, for the in vivo analyses there was no underlying correct model for comparison, and both monoexponential and biexponential models have been used in the analyses of transverse relaxation data from brain. In diffusion studies of brain, mono-, bi- and stretched-exponentials have all been applied (14, 43, 46, 71-73).

We found that in general, methods incorporating Rician noise performed better than those using Gaussian noise, regardless of the signal model. This is in agreement with previous work on cartilage for monoexponential T₂ (12) and biexponential T₂^* (15) measurements. Our results also indicate that Bayesian-based methods exhibit higher precision than NLLS-based methods.

Our MC simulations, and phantom and in vivo experiments, all highlight the importance of considering the correct form of the noise distribution for magnitude MR images. We have used the Rician distribution, as is appropriate for single-coil signal reception. However, multiple receiver coils are widely used to increase speed or SNR. If uncorrelated, equally distributed and stationary noise is assumed, the noise distribution in the resulting magnitude images is described by a non-central χ-distributions

$$P_{\chi }(S_n\mid A(\theta ;T{E}_n),\sigma ,R)=\frac{A(\theta ;T{E}_n)}{{\sigma }^2}{\left(\frac{S_n}{A(\theta ;T{E}_n)}\right)}^{R}exp(-\frac{A^2(\theta ;T{E}_n)+S_n^2}{2{\sigma }^2})I_{R-1}\left(\frac{S_{n}A(\theta ;T{E}_n)}{{\sigma }^2}\right)$$

where R denotes the number of channels (55, 74). The corresponding likelihood function is given by

$$L_{\chi }(\mathbf{S}\mid \theta ,\sigma )=\underset{n=1}{\overset{N}{\Pi }}{P}_{\chi }(S_n\mid A(\theta ;T{E}_n),\sigma ,R)$$

This expression can be incorporated into a Bayesian approach for parameter estimation in a fashion analogous to the above. This can be compared to an NLLS minimization using the first moment of P_χ(S_n | A(θ;TE_n),σ,R) as the expectation value of the acquired data

$$E_{\chi }\left[S_n\right]=\frac{1\cdot 3\cdot 5\dots (2R-1)}{2^{R-1}(R-1)!}\sqrt{\frac{\pi {\sigma }^2}{2}}{}_1{F}_1(-\frac{1}{2},R,-\frac{A^2(\theta ;T{E}_n)}{2{\sigma }^2})$$

(75-78), where ₁F₁ is a confluent hypergeometric function.

Our results indicate that in general, Bayesian analysis provides more reliable estimates of transverse decay parameters than does NLLS when the underlying signal model is known. In realistic experiments, the correct signal model may be unknown. We believe that previous results (17, 20, 79) and the results presented here provide strong motivation for investigations of model selection using Bayesian probability theory.

APPENDIX

We define details of the marginalization used in Bayesian parameter estimation. To obtain the estimated parameter, ${\widehat{\theta }}_i$, (Eqs. A1-A6), the corresponding posterior probability, P(θ_i | S,σ), was computed for a grid of parameter values, θ_i. The MAP estimate, ${\widehat{\theta }}_i$, corresponds to the value for which P(θ_i | S,σ) attains the maximum of the computed values (17, 20-21, 37). Integrals were performed numerically using the integral, integral2 and integral3 commands in Matlab for mono-, stretched-, and biexponential signal models, respectively. The integration limits were chosen such that the contribution of the relevant likelihood function to the integral outside these limits was negligible.

Bayesian inference requires assignment of a prior probability based on pre-existing knowledge (80). This choice becomes of particular importance when the likelihood function is not the dominant term in the posterior probability such as is often the case when only a limited number of measurements are available (37). In the present work, the number of measurements was much greater than the number of derived parameters, so that results are largely independent of reasonable selections of prior probabilities (data not shown). We assumed a formally improper uniform prior in all cases; that is, the P(θ) term in Eq. 7 was taken as unity. With limited data, the selection of the prior is more critical (37).

Likelihood functions for Bayesian-based methods

Each Bayesian-based method is defined by its likelihood function, L(S|θ,σ) and a uniform prior. For MAP_R , L(S|θ,σ) = L_Rice(S|θ,σ) (Eq. 3), for MAP_G , L(S|θ,σ) = L_Gauss(S|θ,σ) (Eq. 6), and for MAP_G,O, L(S|θ,σ) = L_Gauss(S|θ,σ) with A(θ;TE_n) in Eq. 6 replaced by A(θ;TE_n) + ε.

Marginalization Integrals for Signal Models

Monoexponential Signal Model

The estimate of T₂ was obtained according to

$${\widehat{T}}_2=\arg \underset{T_2}{\max }\left\{\mathrm{P}(T_2\mid \mathbf{S},\sigma )\right\}=\arg \underset{T_2}{\max }\{\int L(\mathbf{S}\mid \theta ,\sigma )d{A}_0\}$$ [A.1]

where A(θ;TE_n) is given by Eq. 10.

Stretched Exponential Signal Model

Estimates of T₂ and α were obtained according to

$${\widehat{T}}_2=\arg \underset{T_2}{\max }\left\{\mathrm{P}(T_2\mid \mathbf{S},\sigma )\right\}=\arg \underset{T_2}{\max }\{\int \int L(\mathbf{S}\mid \theta ,\sigma )d{A}_{0}d\alpha \}$$ [A.2]

$$\widehat{\alpha }=\arg \underset{\alpha }{\max }\left\{\mathrm{P}(\alpha \mid \mathbf{S},\sigma )\right\}=\arg \underset{\alpha }{\max }\{\int \int L(\mathbf{S}\mid \theta ,\sigma )d{A}_{0}d{T}_2\}$$ [A.3]

where A(θ;TE_n) is given by Eq. 11.

Biexponential Signal Model

Estimates of f_s, T_2,s and T_2,l were obtained according to

$${\widehat{f}}_s=\arg \underset{fs}{\max }\left\{\mathrm{P}(f_s\mid \mathbf{S},\sigma )\right\}=\arg \underset{fs}{\max }\{\int \int \int L(\mathbf{S}\mid \theta ,\sigma )d{A}_{0}d{T}_{2,s}d{T}_{2,l}\}$$ [A.4]

$${\widehat{T}}_{2,s}=\arg \underset{T_{2,s}}{\max }\left\{\mathrm{P}(T_{2,s}\mid \mathbf{S},\sigma )\right\}=\arg \underset{T_{2,s}}{\max }\{\int \int \int L(\mathbf{S}\mid \theta ,\sigma )d{A}_{0}d{f}_{s}d{T}_{2,l}\}$$ [A.5]

$${\widehat{T}}_{2,l}=\arg \underset{T_{2,l}}{\max }\left\{\mathrm{P}(T_{2,l}\mid \mathbf{S},\sigma )\right\}=\arg \underset{T_{2,l}}{\max }\{\int \int \int L(\mathbf{S}\mid \theta ,\sigma )d{A}_{0}d{f}_{s}d{T}_{2,s}\}$$ [A.6]

where A(θ;TE_n) is given by Eq. 12.