A Linear Least Squares Method for Unbiased Estimation of T1 from SPGR Signals

Abstract

The longitudinal relaxation time, T₁, can be estimated from two or more spoiled gradient recalled echo images (SPGR) acquired with different flip angles and/or repetition times. The function relating signal intensity to flip angle and TR is non-linear; however, a linear form proposed 30 years ago is currently widely used. Here, we show that this linear method provides T₁ estimates that have similar precision but lower accuracy than those obtained with a nonlinear method. We also show that T₁ estimated by the linear method is biased due to improper accounting for noise in the fitting. This bias can be significant for clinical SPGR images, for example, T₁ estimated in brain tissue (800ms <T₁< 1600ms) can be over-estimated by 10–20%. We propose a weighting scheme that correctly accounts for the noise contribution in the fitting procedure. Monte Carlo simulations of SPGR experiments are used to evaluate the accuracy of the estimated T₁ from the widely-used linear, the proposed weighted-uncertainty linear, and the nonlinear methods. We show that the linear method with weighted uncertainties reduces the bias of the linear method, providing T₁ estimates comparable in precision and accuracy to those of the nonlinear method while reducing computation time significantly.

INTRODUCTION

Clinical imaging of the longitudinal relaxation time, T₁, in human subjects has several potential applications, including perfusion imaging (1), dynamic contrast imaging (2), assessment of Parkinson’s disease (3), assessment of schizophrenia (4) and multiple sclerosis (5), and quantification of myocardial blood flow (6). In spite of its potential clinical utility quantitative T₁ mapping is not routinely used due to the long scanning time inherent in inversion recovery sequences. Recently, however, the acquisition of high resolution T₁ maps in a clinically feasible timeframe has been demonstrated with Driven Equilibrium Single Pulse Observation of T₁ (DESPOT1) (7,8). DESPOT1 derives T₁ from two or more spoiled gradient recalled echo (SPGR) images acquired with a constant TR and different flip angles. The T₁ maps have been computed on a voxel-by-voxel basis using linear least squares (LLS) fitting of a linear transformation of the function relating signal intensity, flip angle, TR, T_1, and equilibrium longitudinal magnetization, M₀. This transformed linear least squares fitting method—first described by Gupta in 1977 (9), which we denote as GLLS, and used in many previous works (7,8,10–12)—has the advantage of being computationally efficient. However, our preliminary study found that the estimated T₁ using GLLS was generally biased and over estimated (13).

In this paper, we study systematically the bias in T₁ computed from SPGR signals when different fitting methods are used. Monte Carlo simulations provide a detailed evaluation of the accuracy of T₁ using Gupta’s linear least squares (GLLS) and nonlinear least squares (NLS) methods in several experimental conditions. We also evaluate the performance of an intensity-based weighted linear least squares approach (ILLS) proposed by Deoni (14). The ILLS method assigns greater weight to high SNR points in order to increase the precision of estimated T₁. Finally, we propose a new linear least squares approach that uses weighted uncertainties in the fitting (WLLS). The proposed WLLS method weights each image with the uncertainty that takes into account the adjustment of noise contribution due to the rearrangement of a nonlinear model into a linear one. Numerical and human brain data simulations are used to compare the accuracy of T₁ estimates using the GLLS, ILLS, WLLS, and NLS methods.

THEORY

Nonlinear Least Squares Method

The measured SPGR signal intensity, s_i, is a function of the longitudinal relaxation time, T₁, the repetition time, TR, the flip angle, α_i, and the equilibrium longitudinal magnetization, M₀,

$$s_i=\frac{M_0(1-exp(-\frac{TR}{T_1}))sin({\alpha }_i)}{1-exp(-\frac{TR}{T_1})cos({\alpha }_i)}.$$ [1]

The NLS approach estimates T₁ and M₀ from Eq[1] by minimizing the following χ² objective function:

$${\chi }_{\mathit{NLS}}^2(M_0,T_1)=\sum _{i=1}^n\frac{1}{{\sigma }_i^2}{\left(s_i-M_{0}sin({\alpha }_i)\frac{1-exp(-\frac{TR}{T_1})}{1-exp(-\frac{TR}{T_1})cos({\alpha }_i)}\right)}^2,$$ [2]

where σ_i is the expected signal standard deviation due to noise. When the signal to noise ratio, s_i / σ_i, is greater than five, σ_i can be considered as a constant in all images, i.e., σ₁ = σ₂ =…=σ_n = σ. But, when s_i /σ_i is low, this assumption becomes invalid (15).

Linear Least Squares Method

In the LLS method, the linear equation is obtained by assuming that TR is a constant. By rearranging Eq[1] and denoting exp(−TR / T₁) by E₁, we have:

$$\frac{s_i}{sin({\alpha }_i)}=E_1\frac{s_i}{tan({\alpha }_i)}+M_0(1-E_1).$$ [3]

Eq[3] can be rewritten in a more explicit linear notation as y=bx+a with y_i = f(s_i) = s_i / sin(α_i), x_i = g(s_i) = s_i / tan(α_i), b = E₁, and a = M₀ (1−E₁).

The slope, b, and the y-intercept, a, can be estimated by linear regression (16,17), i.e., by minimizing the χ² objective function:

$$\begin{array}{l}{\chi }_{\mathit{LLS}}^2(a,b)=\sum _{i=1}^n{w}_i{\left(y_i-{bx}_i-a\right)}^2\\ =\sum _{i=1}^n{w}_i{\left(\frac{s_i}{sin({\alpha }_i)}-E_1\frac{s_i}{tan({\alpha }_i)}-M_0(1-E_1)\right)}^2.\end{array}$$ [4]

The GLLS method proposed by Gupta (9) uses the linearization approach of Eq[3] with $w_{\mathit{GLLS}}(i)=1/{\sigma }_i^2$, which has been used in many previous works (7,8,10–12). Note that σ_i is the standard deviation of the measurement error on y_i and is generally assumed to be a constant in a linear model when measurements of y_i are independent. The assumption in the widely-used GLLS method that the standard deviation of y_i is the same for all data points (i.e., σ_i =σ). is incorrect because the experimental errors are distorted by the nonlinear to linear transformation. T₁ and M₀ can then be calculated with the representation of a and b (10,16): T₁ = −TR / ln b and M₀ = a /(1−b).

An empirical weighting approach for the LLS fitting was introduced in (14). This method, which we call the intensity-based linear least squares (ILLS) method, assigns greater weight to data points with higher signal intensity in an attempt to increase the precision of estimated T₁, and solves Eq[3] by minimizing Eq[4] with a different weighting function defined in (14): w_ILLS (i) = s_αi / s_αE, where s_αi is the signal intensity acquired with flip angle α_i, and s_αE is the signal intensity acquired with the Ernst angle. T₁ and M₀ can be calculated as above.

Experimental errors are generally distorted when a nonlinear model is transformed to a linear one. A weighting function that accommodates the distorted uncertainties can be derived by using the error propagation theory.(16). (See Appendix A.) Here, we can rewrite the χ² objective function for the NLS approach (Eq[2]) as:

$$\begin{array}{l}{\chi }_{\mathit{NLS}}^2=\sum _{i=1}^n\frac{1}{{\sigma }_i^2}{\left(s_i-M_{0}sin({\alpha }_i)\frac{1-E_1}{1-E_{1}cos({\alpha }_i)}\right)}^2\\ =\sum _{i=1}^n\frac{1}{{\sigma }_i^2}{\left(\frac{sin({\alpha }_i)}{1-E_{1}cos({\alpha }_i)}\right)}^2{\left(\frac{s_i}{sin({\alpha }_i)}-E_1\frac{s_i}{tan({\alpha }_i)}-M_0(1-E_1)\right)}^2\end{array}$$ [5]

Notice that Eq[5] is similar in form to the objective function in LLS (i.e., Eq[4]), and can be considered as a new weighted LLS method, which we denote as WLLS. By comparing Eq[4] with Eq[5], the weighting function of WLLS can therefore be defined as $w_{\mathit{WLLS}}(i)=\frac{1}{{\sigma }_i^2}{\left(\frac{sin({\alpha }_i)}{1-E_{1}cos({\alpha }_i)}\right)}^2$.

Here, we have established the theoretical connection between the nonlinear and proposed weighted linear methods based on the approach used in (18). The proposed WLLS method is equivalent to the NLS method in principle because Eq[5] is derived directly from Eq[2], and no transformation or approximation is applied during the rearrangement.

We now want to minimize Eq[5] with respect to a and b. However, the occurrence of b (= E₁) in the denominator of Eq[5] makes the task of fitting considerably harder. An iterative least squares fitting approach can be used here to find the optimal solution (19) and the GLLS method can be used as an initial solution. There are strategies for fitting a line in this situation (17,20,21), any of which can be applied here. We use the Brent’s method as described in (22,23) which is a reasonable strategy for minimizing a general one-dimensional function that the minimization with respect to b is also minimized with respect to a.

METHODS

Numerical Simulations

We evaluated the accuracy and precision of different fitting algorithms in several experimental conditions by performing Monte Carlo simulations. Noise-free SPGR signals were generated using Eq[1] given a fixed TR value, multiple flip angles, a single expected value of the equilibrium longitudinal magnetization M₀, and various expected values of T₁. Different signal to noise (SNR₀) levels were simulated by adding Gaussian noise (s_noise) in quadrature with zero mean and variable standard deviation, σ, to the noise-free SPGR signals, i.e., $s_{\mathit{spgr}}=\sqrt{{(s_{\mathit{noise}-\mathit{free}-\mathit{spgr}}+s_{\mathit{noise}})}^2+s_{\mathit{noise}}^2}$ (24). The noise level is defined as M₀ divided by the signal standard deviation, i.e., SNR₀=M₀/σ. We calculated T₁ by fitting the synthetic SPGR signals using the GLLS, WLLS, and NLS fitting approaches. Results reported in the next section are computed with TR = 10 ms, M₀ = 3000 and T₁ = 600, 800, 1000, 1200, 1600, or 2000 ms. The two optimal flip angles for the chosen T₁ were computed according to the formula in (10,14). For example, given T₁ = 1000 ms, TR = 10 ms, and M₀ = 3000, the optimal flip angles are 3.35 and 19.38 degrees. Six SPGR images were used in simulations with repeated experiments of the two optimal angles without averaging. Noise levels were tested with SNR₀ ranging from 30 to 300. Each set of parameters was repeated 131,072 times.

Human Brain Simulation

A separate set of simulations was aimed at assessing the accuracy of estimated T₁ for experimental conditions compatible with clinical studies of the human brain at 1.5T. We collected a high quality SPGR dataset of the brain of a healthy male volunteer. Images were acquired with a DESPOT1 sequence (7) in axial view with 0.9375 mm² in-plane resolution, 2mm slice thickness, TR = 8 ms, and 20 different flip angles. The scan time was about 20 minutes (one minute per flip angle). This high quality SPGR dataset was used to compute a T₁- and an M₀-map using the NLS method and the values obtained in each voxel of the brain were assumed to be error-free. From these “gold standard” T₁- and M₀-maps, and a given set of TR and flip angles, we then generated synthetic noise-free SPGR images by computing the signal intensity in each voxel using Eq[1]. Gaussian noise in quadrature with zero mean and a given standard deviation, σ = 100, was added to the noise-free synthetic SPGR images as described in the previous paragraph. In the resultant SPGR brain images SNR₀ = M₀/ σ ranging from 90–150 throughout most areas of the brain tissue. We then computed the T₁ maps by fitting this synthetic human brain data with GLLS, WLLS, and ILLS methods. This procedure was repeated 500 times and an averaged T₁-map was created by taking the mean value of T₁ from the 500 repeats on a voxel-by-voxel basis. Results reported in the next section are computed using TR = 10 ms and 6 SPGR images, with the flip angles in each image equal to 3, 6, 9, 12, 15, and 18 degrees respectively.

RESULTS

Figure 1 demonstrates the behavior of T₁ bias when estimating T₁ from SPGR signals using the GLLS method. Fig. 1(a) shows that the distributions of T₁ obtained with GLLS are biased, with the bias more pronounced at low SNR₀ since the distribution of T₁ is shifted more to the right when the M₀/σ ratio is lower (Group B in Fig. 1(a)). The precision of estimated T₁ from GLLS and NLS is similar since the distributions under the same SNR₀ are similar. Fig. 1(b) shows the bias as a function of T₁ and SNR₀. For a given SNR₀, the relative error of T₁ is found to be proportional to the value of T, although the accuracy of T₁ is relatively unaffected by the value of T₁ in a high SNR₀ regime (> 200). When given the same T₁, the relative error of T₁ was found to be inversely proportional to SNR₀. The precision of estimated T₁ from GLLS and NLS is also similar (data not shown).

Fig. 1.

(a) Distribution of T₁ using 6 SPGR images, three replicates of two flip angles, with two different noise levels (Group A: M₀/σ=200, Group B: M₀/σ=100). The true T₁ value is 1000 ms. (b) Relative errors of T₁ using the GLLS approach at different noise levels (SNR₀ = 60, 100, 200, and 300) with the true T₁ value set to 600, 800, 1000, 1200, 1600 and 2000 ms. Relative error of T₁ = 100×(Estimated T₁ − True T₁) / True T₁.

The results in Figure 2 show that the WLLS virtually eliminates the T₁ bias for a broad range of SNR₀ and T₁. Fig. 2(a) shows that WLLS and NLS produce comparable accuracy of T₁ at all SNR₀ tested, while GLLS overestimates T₁ progressively as SNR₀ decreases. Fig. 2(b) shows that the bias of T₁ is corrected in WLLS with the relative error less than 5% regardless of the T₁ value. T₁ estimated using WLLS and NLS have similar precision; T₁ estimated using GLLS has slightly lower precision than that of T₁ estimated using NLS, in agreement with previous reports in (10,12) (data not shown).

Fig. 2.

(a) Estimated T₁ using GLLS, WLLC, and NLS methods assuming a true T₁ value of 1000ms, and (b) Relative error of T₁ using GLLS, WLLS, and NLS methods with SNR₀ = 100. Six SPGR images, consisting of three replicates of two flip angles without averaging, were used in both (a) and (b).

Figure 3 shows maps of the relative error on the estimated T₁ in the synthetic human brain data using the GLLS, ILLS, and WLLS methods. The results shown in Figure 3 were scaled in the range of negative/positive 20% and the gray background corresponds to zero. The relative error of GLLS is consistently higher than that of WLLS in the brain tissue, and is positive indicating that T₁ is overestimated. The error of ILLS is higher than the error of WLLS in most of the brain tissue. Although the error of ILLS is generally lower than that of GLLS, it shows a strong dependency on T₁, resulting in an undesirable tissue-dependent pattern. It is also notable that the bias in the cerebral spinal fluid (CSF) regions is higher than the bias in the brain parenchyma for all three methods with GLLS and WLLS showing positive bias and ILLS negative bias.

Fig. 3.

Relative error of T₁ on selected slice of synthetic human brain data using (b) GLLS, (c) WLLS, and (d) ILLS methods in the fitting procedure. The true T₁-map of the same slice is shown in (a) for reference.

DISCUSSION AND CONCLUSION

The linear model for T₁ estimation presented by Gupta (9) and used in many previous works (7–12) is an example of linearization. In general, weighted uncertainties must be used with linearly transformed data because the transformation distorts the experimental errors (16). However, Gupta’s linear regression assumes that the scatter of points around the line follows a Gaussian distribution and that the standard deviation is the same at every data point. In this work we show that neglecting such adjustments to the uncertainty produces significant errors in the T₁ estimation. The magnitude of T₁ bias can be related to the true T₁ value, and the experiment design affecting SNR such as the TR, flip angles, image resolution, and receiver coil. For a clinical whole-brain SPGR data acquired at 1.5T with single channel receiver coil (TR = 8 msec, 1 mm³ resolution, and flip angles = 2, 3, 14, and 17 degrees), the SNR₀ ranges from 100 to 200 in brain tissue; T₁ value can be overestimated by 10–20%.

In many applications, achieving an unbiased estimation of the desired parameters from transformed data is achieved by computing weighted uncertainties with error analysis and error propagation techniques. We show such an approach in Appendix A. Moreover, we have derived a new weighted linear least squares (WLLS) model for T₁ estimation directly from the nonlinear model without using any transformation or approximation. This is an interesting result because converting a nonlinear model to a linear one using direct analytical derivation without approximations is not always possible.

It is worth emphasizing that the bias of the linear model not only depends on the measurement errors in the abscissa and the ordinate, but also on their covariance since the measurements in both axes are no longer independent. Our simulations show that using correct weights on the abscissa and the ordinate in Gupta’s formula will not result in unbiased estimates of T₁; a covariance term must be included (see Appendix A).

The previously proposed intensity-based weighting approach, ILLS, improves the accuracy of the estimated T₁ in some brain areas. However, with this method the bias shows an undesirable strong dependence on the values of T₁. In general, all linear methods—GLLS, ILLS, and WLLS—are least accurate in estimating T₁ in regions with high T₁ such as in CSF, but GLLS and WLLS overestimate while ILLS underestimates. This poor performance in CSF has little practical relevance because accuracy in CSF is generally biologically less important than in brain parenchyma. Moreover, the estimation of T₁ from SPGR data in regions of high T₁ such as CSF is already problematic, given that the standard deviation of T₁ measured from SPGR signals is proportional to the square of T₁ (10). The SPGR simulations we performed in the human brain used experimental parameters aimed at optimizing typical values of T₁ in gray (T₁ ~ 950 ms) and white matter (T₁ ~ 600 ms), but not in CSF (T₁ ~ 4500 ms), resulting in a magnification of potential problems in regions with high T₁. Although the results of the WLLS method may be affected by rectified noise, which is not accounted for in all models previously studied, signal correction methods such as those of Henkelman (24) and of Koay (15) may be used to resolve this problem. For scanning protocols used in clinical applications, signal-to-noise ratio in brain parenchyma is generally high enough to avoid effects from rectified noise.

T₁ bias can also originate from sources that are not addressed here. For example, flip angle variations caused by B1 inhomogeneity can cause additional errors in T₁ estimation; approaches have been proposed to correct these inaccuracies (12,25). To get a robust estimation of T₁ from clinical SPGR signals, correcting B1 inhomogeneity or the flip angles should be considered in addition to using the proposed weighted linear method in the fitting.

In terms of computation speed, WLLS is slower than GLLS since WLLS may need several iterations of linear regression while GLLS needs only one; however, WLLS is still computationally faster than NLS. For example, the numerical simulation (Figure 2) we performed took 11~12 second for GLLS, 59 ~ 65 seconds for WLLS, but 17 to 29 minutes for NLS. The difference in time for the same method varied due to different noise levels.

WLLS and NLS are comparable in terms of both precision and accuracy in estimating T₁ and M₀. WLLS, however, was found to be more stable than NLS at low SNR₀ (i.e. lower occurrence of T₁ outliers such as negative T₁ values). We suspect that this instability of the nonlinear least squares approach is due to the known large-residuals problem in nonlinear regression (26). The Newton method for nonlinear fitting is known to be more robust for noisy data than the Levenberg-Marquardt based approaches (18). In future experiments we plan to compare systematically WLLS and NLS using the Newton method in testing the instability in the very low SNR regime. In general, clinical SPGR signals have higher SNR₀ and do not have the problem described above.

In this paper, we have shown that the relaxation time, T₁, estimated from the SPGR signals using the widely accepted linear least squares model is biased. The bias stems from neglecting to adjust uncertainties when transforming a nonlinear model into a linear one. We propose a weighting approach for the linear model that can be derived from the nonlinear model without any approximation. The proposed weighted linear least squares method yields estimated T₁ with precision and accuracy comparable to that obtained from nonlinear fitting while reducing the computation time significantly, enabling the generation of robust T₁ maps at the scanner console.

Appendix A

Nonlinear regression is generally done without weighting or with constant weighting for all experimental data. For example, the σ_i in Eq[2] are the same for all SPGR data points. Giving equal weight to all data points is appropriate when the experimental uncertainty is expected to be the same in all measurements. When transforming a nonlinear function into a linear function, however, we must use weighted (or adjusted) uncertainties ${\sigma }_i^{\prime }$ instead of σ_i to account for the transformation of the dependent variables (16). In general, if we fit the function f(u) rather than u, the uncertainties in the measured quantities must be modified using the following formula (16)

$${\sigma }_i^{\prime }={\sigma }_{f(u_i)}=\frac{\partial f(u_i)}{\partial u_i}{\sigma }_i.$$ [A1]

Note that the uncertainties should be modified in both abscissa and ordinate in the linear form of LLS (Eq[3]) since both y_i and x_i are now subject to measurement errors on signals, s_i. Therefore, we have

$${\sigma }_{y_i}^2={\left(\frac{\partial f(s_i)}{\partial s_i}\right)}^2{\sigma }_i^2={\left(\frac{1}{sin({\alpha }_i)}\right)}^2{\sigma }^2$$ [A2]

$${\sigma }_{x_i}^2={\left(\frac{\partial g(s_i)}{\partial s_i}\right)}^2{\sigma }_i^2={\left(\frac{1}{tan({\alpha }_i)}\right)}^2{\sigma }^2.$$ [A3]

Also note that the measurement errors due to noise in the abscissa and the ordinate are not independent; therefore, the covariance term between the measurement errors of y_i and x_i should also be taken into account

$${\sigma }_{x_i{y}_i}^2=\frac{\partial f(s_i)}{\partial s_i}\times \frac{\partial g(s_i)}{\partial s_i}\times {\sigma }_i^2=\left(\frac{1}{sin({\alpha }_i)}\right)\left(\frac{1}{tan({\alpha }_i)}\right){\sigma }^2.$$ [A4]

When applying the correct weighted uncertainties to Gupta’s linear model—which we denote as WLLS—using the error propagation equation described in (16,25), we use the new variance that takes the uncertainties in both y_i and x_i, and their covariance term into account.

$$\begin{array}{l}\mathit{Var}({\sigma }_i^{\prime })=\mathit{Var}(y_i-{bx}_i-a)\\ ={\left(\frac{1}{sin({\alpha }_i)}\right)}^2{\sigma }_i^2+b^2{\left(\frac{1}{tan({\alpha }_i)}\right)}^2{\sigma }_i^2-2b\left(\frac{1}{sin({\alpha }_i)}\right)\left(\frac{1}{tan({\alpha }_i)}\right){\sigma }_i^2.\\ ={\left(\frac{1-bcos({\alpha }_i)}{sin({\alpha }_i)}\right)}^2{\sigma }_i^2\end{array}$$ [A5]

The transformed linear equation therefore has the following χ² objective function (16,17,20)

$${\chi }_{\mathit{WLLS}}^2(a,b)=\sum _{i=1}^n{w}_i{(y_i-a-{bx}_i)}^2,$$ [A6]

where $w_{\mathit{WLLS}}(i)=\frac{1}{\mathit{Var}({\sigma }_i^{\prime })}=\frac{1}{{\sigma }_i^2}{\left(\frac{sin({\alpha }_i)}{1-bcos({\alpha }_i)}\right)}^2$

Note that σ_i =σ is a constant and therefore can be factored out.