Modeling Magnitude and Phase Neuronal Current MRI Signal Dependence on Echo Time

Abstract

To enhance sensitivity in measuring neuronal current MRI (ncMRI) signal using T₂*-weighted sequences, appropriate selection of echo time (TE) is vital for optimizing data acquisition strategy. The purpose of this study is to establish the contrast-to-noise-ratio of ncMRI signal dependence on TE and determine the optimum TE (TE_opt) in achieving its highest detection power. The TE_opt in human brain and tissue preparation at 1.5, 3, and 7T are estimated with different voxel sizes. Our results show that TE_opt values are different between magnitude and phase images, and TE_opt is larger in magnitude than phase imaging. This suggests that a dual-echo data acquisition strategy would provide the best efficiency in detecting magnitude and phase ncMRI signals simultaneously. Our results also indicated that the detection sensitivity will be stronger at lower magnetic fields for human brain, while the sensitivity will be enhanced/reduced as field strength increases for phase/magnitude imaging on tissue preparation.

Numerous efforts have been made to use MRI for directly detecting neuronal currents (termed ncMRI) in human subjects (for review see (1)) and tissue preparations (2–4). To date, the reliability of normal sensory-evoked ncMRI has not been demonstrated, partially due to the limited sensitivity of MRI towards neuronal current alternations. To improve the sensitivity of ncMRI, optimization of the imaging parameters for ncMRI sequences is one of the critical steps need to be carefully taken (1).

The alternations in neuronal activity-evoked ionic currents produce transient time and localized space-varying magnetic fields. The field inhomogeneity sensitive T₂*-weighted gradient echo EPI pulse sequences were used in most previous ncMRI experiments for detecting this field alternations. For such sequences, the amplitudes of ncMRI magnitude and phase signals rely heavily on the phase changes of individual spins accumulated over the echo time (TE). To best of our knowledge, the relationship between the contrast-to-noise ratio (CNR) of ncMRI signal and TE has not been fully exploited yet.

In this study, the TE dependences of the CNR of the normal sensory-evoked ncMRI signal will be formulated and the optimum TE values that maximize the CNR will be determined. This work will provide theoretical foundation and guidance in the pulse sequence optimization and magnetic field dependence in future ncMRI experiments.

Methods

The sensitivity of ncMRI is specified by its CNR, which is the product of ncMRI signal change and temporal signal-to-noise ratio (tSNR) (5). The TE dependence of CNR of ncMRI signal and the optimal TE are modeled through the following procedures: (I) Formulate the TE dependences of ncMRI signal changes induced by evoked neuronal responses; (II) Utilize the relationship between tSNR and TE from the MRI noise model (6); (III) Establish the dependence of CNR of ncMRI signal on TE; (IV) Calculate the optimal TE and its corresponding maximum CNR.

(I) Dependence of ncMRI signal on TE

The relationship between ncMRI signal and TE is generalized as (7):

$$|\delta |=A·{(\mathit{\text{TE}})}^{\alpha }$$ [1]

$$|\mathrm{\Delta }\varphi |=B·{(\mathit{\text{TE}})}^{\beta }$$ [2]

where δ and Δϕ are the relative magnitude change and the phase shift in MRI signal produced by evoked neuronal currents, respectively; TE is the echo time in ms; A, B, α and β are constants and are independent of imaging parameters. Blagoev et al. (8) used real neuronal morpholoical and physiological properties to simulate the ncMRI signal, and their results would provide more accurate information regarding the relationship between ncMRI signal change and TE than those obtained from previous simplified current dipole models (7,9). By fitting the results from Blagoev et al. study (8) to Eqs [1–2], it was found that values for A, B, α and β are 0.0017, 0.0067, 1.8 and 1.2, respectively (Fig. 1).

Figure 1.

Fitting results for the relationships of relative magnitude change (δ) (a) and phase shift (Δϕ) (b) of ncMRI signal with TE. The data points are selected from the reference 8.

(II) Dependence of temporal SNR on TE

The tSNR of magnitude MRI signal (tSNR_M) is defined as (6):

$${\mathit{\text{tSNR}}}_M=\{\begin{array}{cc}\frac{S}{\sqrt{{{\sigma }_T}^2+{{\sigma }_P}^2}}\hfill & (\text{human brain})\hfill \\ \frac{S}{{\sigma }_T}\hfill & (\text{tissue preparation})\hfill \end{array}$$ [3]

where S is the MRI signal intensity, σ_T is the noise originated from thermal and scanner instability, and σ_P is the physiological noise associated with hemodynamic fluctuation in human subject and it can be expressed as (6):

$${{\sigma }_P}^2={\lambda }^2·S^2=[{(c_1·\mathrm{\Delta }{R}_2^{*})}^2·{\mathit{\text{TE}}}^2+{c_2}^2]·S^2$$ [4]

where c₁ and c₂ are constants, and ΔR₂^* is the fluctuation in transverse relaxation rate resulting from physiological noise. In the gray matter, c₂ = 0.001 (6), and the values of $c_1·\mathrm{\Delta }{R}_2^{*}$ at 1.5, 3 and 7 T were measured in the previous study (10) and are listed in Table 1.

Table 1.

Parameter values used in the calculation of CNR.

B0 (T)	$c_1·\mathrm{\Delta }{R}_2^{*}$ (s−1)	T1 (ms)	T2* (ms)	θ (°)	k for different voxel sizes
					1×1×3	2×2×3	3×3×3	4×4×3	5×5×3
1.5	0.274±0.020	1000	65	82	0.02	0.06	0.08	0.13	0.19
3	0.365±0.027	1331	42	77	0.05	0.10	0.15	0.27	0.39
7	0.548±0.060	1939	25	69	0.13	0.24	0.39	0.70	1.07

The values of $c_1·\mathrm{\Delta }{R}_2^{*}$ and k are measured from Triantafyllou et al. study (10). The T₁ at 1.5 and 3 T, and all the T₂* values are taken from (11). The T₁ data at 7 T are obtained from (12). $c_1·\mathrm{\Delta }{R}_2^{*}$ is represented by mean ± standard error.

Insert Eq. [4] into Eq. [3], the tSNR_M for gradient echo EPI sequence can be expressed as:

$${\mathit{\text{tSNR}}}_M=\{\begin{array}{cc}\frac{{\mathit{\text{SNR}}}_T}{\sqrt{1+{\lambda }^2·{{\mathit{\text{SNR}}}_T}^2}}\hfill & (\text{human brain})\hfill \\ {\mathit{\text{SNR}}}_T\hfill & (\text{tissue preparation})\hfill \end{array}$$ [5]

where SNR_T is the thermal signal-to-noise ratio. In ncMRI experiments, spoiled gradient echo EPI sequence is mostly used and its SNR_T is expressed as:

$${\mathit{\text{SNR}}}_T=\frac{S}{{\sigma }_T}=\frac{S_0·f·e^{-\mathit{\text{TE}}/T_2*}}{{\sigma }_T}$$ [6]

f is the scale factor related to TR, flip angle (θ), and T₁:

$$f=\frac{\text{sin}(\theta )·(1-e^{-\mathit{\text{TR}}/T_1})}{1-\text{cos}(\theta )·e^{-\mathit{\text{TR}}/T_1}}$$ [7]

S₀ in Eq. [6] is the signal obtained for full equilibrium magnetization, i.e., the signal intensity in the image acquired with TE=0ms, TR>>T₁, and θ=90°; The SNR_T in such an full equilibrium image (= S₀/σ_T) is approximately proportional to voxel volume (V_vox) and field strength (10), then Eq. [6] can be rewritten as:

$${\mathit{\text{SNR}}}_T=p·V_{\mathit{\text{vox}}}·B_0·f·e^{-\mathit{\text{TE}}/{T_2}^{*}}$$ [8]

where p is a constant.

In the fully equilibrium image of human brain, the relationship between S₀ and physiological noise (σ_P,0) can be obtained from Eq. [4]:

$$S_0=\frac{1}{c_2}·{\sigma }_{P,0}=\frac{1}{c_2}·{\sigma }_P{|}_{\mathit{\text{TE}}=0,\mathit{\text{TR}}>>T_1,\theta =90°}$$ [9]

Let k (=σ_P,0/σ_T) denote the ratio of physiological noise to thermal noise, and then with Eqs. [6] and [9], SNR_T can be further expressed as:

$$\begin{array}{cc}{\mathit{\text{SNR}}}_T=\frac{1}{c_2}·k·f·e^{-\mathit{\text{TE}}/T_2*}& (\text{human brain})\end{array}$$ [10]

Triantafyllou et al (10) measured k and its value varies with voxel volume and field strength.

(III) Dependence of CNR on TE

The CNRs of magnitude (CNR_M) and phase (CNR_ϕ) signals are expressed as (5):

$${\mathit{\text{CNR}}}_M=|\delta |·{\mathit{\text{tSNR}}}_M$$ [11]

$${\mathit{\text{CNR}}}_{\varphi }=\frac{|\mathrm{\Delta }\varphi |}{{\sigma }_{\varphi }}\approx |\mathrm{\Delta }\varphi |·{\mathit{\text{tSNR}}}_M$$ [12]

where σ_ϕ is the phase noise. By plugging Eqs. [1–2], [4–5], [8], and [10] into Eqs. [11–12], the relationship between CNR and TE will be:

$${\mathit{\text{CNR}}}_M=\{\begin{array}{cc}\frac{A·{(\mathit{\text{TE}})}^{\alpha }}{\sqrt{{\left(\frac{c_2·e^{\mathit{\text{TE}}/T_2*}}{k·f}\right)}^2+{(c_1·\mathrm{\Delta }{R}_2^{*})}^2·{\mathit{\text{TE}}}^2+{c_2}^2}}\hfill & (\text{human brain})\hfill \\ A·p·f·V_{\mathit{\text{vox}}}·B_0·{(\mathit{\text{TE}})}^{\alpha }·e^{-\mathit{\text{TE}}/T_2*}\hfill & (\text{tissue preparation})\hfill \end{array}$$ [13]

$${\mathit{\text{CNR}}}_{\varphi }=\{\begin{array}{cc}\frac{B·{(\mathit{\text{TE}})}^{\beta }}{\sqrt{{\left(\frac{c_2·e^{\mathit{\text{TE}}/T_2*}}{k·f}\right)}^2+{(c_1·\mathrm{\Delta }{R}_2^{*})}^2·{\mathit{\text{TE}}}^2+{c_2}^2}}\hfill & (\text{human brain})\hfill \\ B·p·f·V_{\mathit{\text{vox}}}·B_0·{(\mathit{\text{TE}})}^{\beta }·e^{-\mathit{\text{TE}}/T_2*}\hfill & (\text{tissue preparation})\hfill \end{array}$$ [14]

From Eqs. [13–14], the analytical expression for optimal TE (TE_opt, the TE at which CNR reaches maximum) of magnitude and phase signals in tissue preparation can be obtained as follows:

$$\begin{array}{cc}{\mathit{\text{TE}}}_{\mathit{\text{opt}},M}=\alpha ·{T_2}^{*}& (\text{tissue preparation})\end{array}$$ [15]

$$\begin{array}{cc}{\mathit{\text{TE}}}_{\mathit{\text{opt}},\varphi }=\beta ·{T_2}^{*}& (\text{tissue preparation})\end{array}$$ [16]

(IV) Calculation of CNR and optimal TE values

The CNR corresponding to the TE range of 0–150 ms were calculated at 1.5, 3, and 7T with Eqs. [13–14], using the typical values for T₁ and T₂* (Table 1). TR was selected as 2s and the corresponding Ernst angles was used as flip angles (Table 1). The CNR was calculated with five different voxel sizes that cover the range typically used in ncMRI experiments: 1×1×3mm³, 2×2×3mm³, 3×3×3mm³, 4×4×3mm³, and 5×5×3mm³. The typical k values for these voxel sizes at 1.5, 3, and 7 T were adopted from the previous study (10) and were used in the present calculation (Table 1). From the curves of CNR vs. TE, the TE_opt values in human brain were obtained by selecting the TEs corresponding to the peak CNRs. The TE_opt in tissue preparation were directly calculated with Eqs. [15–16].

Results

Fig. 2a–b shows the field dependent CNR vs. TE curves when voxel size=3×3×3mm³. It can be observed that magnitude and phase CNRs increase with TE initially, reach to their peaks at the corresponding optimum TE, and then decreases with further increase of TE. The curves also demonstrate that the CNR of phase signal reaches its peak earlier than that of magnitude signal. The CNR vs. TE curves for other voxel sizes exhibit the similar behavior.

Figure 2.

TE dependences of CNR of magnitude and phase ncMRI signals for voxel size of 3×3×3 mm³ in human brain (a) and tissue preparation (b) at 1.5, 3, and 7 T. In the figure, the CNR data of magnitude/phase signal are normalized to the peak CNR of magnitude/phase signal at 1.5 T. (c) and (d) show the optimum TE values for magnitude and phase signals respectively for different voxel volumes at 1.5, 3, and 7T. The data points indicate the results for voxel size=1×1×3, 2×2×3, 3×3×3, 4×4×3, and 5×5×3 mm³.

Fig. 2c–d shows the dependence of optimal TE on voxel volume at 1.5, 3, and 7T. The results indicate that TE_opt in human brain increases with voxel volume and decreases with field strength. In tissue preparation, TE_opt also reduces with increase in B₀, but they are independent of voxel volume (see Eqs. [15–16]). As B₀ increases from 1.5 to 7T, TE_opt,M (TE_opt,ϕ) in tissue preparation decreases from 117ms (78ms) to 45ms (30ms).

Fig. 3 demonstrates the dependence of CNR at TE_opt (CNR_max, maximum CNR) on voxel volume (V_vox) in human brain at different field strengths, and CNR_max is found to increase with increasing voxel volume. When V_vox is extended from 3 to 75mm³, the maximum magnitude/phase CNR at 3T is improved by factor of 4.5/3.4 (Fig. 3a–b). From Eqs. [13–14], it is known that CNR_max in tissue preparation linearly increases with V_vox.

Figure 3.

(a) and (b) show the dependences of maximum CNR (i.e., CNR at optimal TE) for magnitude (CNR_M) and phase (CNR_ϕ) signals on voxel volume in human brain. The maximum CNR values are normalized to that of voxel size of 3×3×3 mm³ at 1.5T.

Discussion

In this study, we have modeled the TE dependence of CNR of normal sensory-evoked ncMRI signal and estimated the optimum TEs in human brain and tissue preparation. The physiological noise and its dependence on TE are considered in the present CNR model to describe the relationship between CNR of ncMRI and TE more accurately. The physiological noise originates from the physiological processes such as respiration-related motion, cardiac pulsation, and fluctuations in arterial carbon dioxide (13). In magnitude images, the physiological noise in gray matter is mainly contributed by low frequency signal fluctuations (< 0.13 Hz) at local scale (< 1 cm), while the physiological noise in phase images presents at large-scale (> 1 cm) and results from respiration effects (~0.3 Hz) primarily (14). In this study, the characteristics of physiological noise are incorporated into the ncMRI CNR model. Our results indicate that at a given imaging condition, the CNRs of magnitude and phase signals reach the maximum at different TE values, and in general TE_opt,M > TE_opt,ϕ.

Our results imply that magnitude and phase ncMRI signals should be acquired at two different TEs to attain their corresponding highest CNRs. Therefore, to achieve the best detection sensitivity and efficiency simultaneously, a dual-echo data acquisition strategy with two separate TEs (TE_opt,M and TE_opt,ϕ) should be adopted. It is noted that the interval between TE_opt,M and TE_opt,ϕ reduces with the increase of B₀ (Fig. 2). Due to the limitation of the minimum achievable echo spacing, at B₀ ≥ 7 T, TE_opt,M and TE_opt,ϕ may be too close to be set as the two echo times in a dual-echo EPI sequence with full k-space acquisition. The keyhole and partial k-space fMRI data acquisition strategies (15,16) could be utilized as a solution to compensate the short echo spacing at high fields (7T and up).

According to Eqs. [11–12], the maximum CNR of ncMRI is the product of ncMRI signal change and temporal SNR (tSNR_M) at the optimal TE and these two factors depend on B₀. (1) ncMRI Signal vs. B₀: ncMRI signal originates from the perturbations to B₀ from the neuronal magnetic field (NMF) generated by neuronal currents. Since NMF is unrelated to B₀, the NMF-induced field inhomogeneity does not depend upon the field strength. Therefore, B₀ will have no impact on ncMRI signal if other imaging parameters (e.g. TE) remain the same. However, since T₂* is shortened at higher B₀ and the optimum TE decreases with T₂* decrease (Fig. 4a–b), the ncMRI signal changes at TE_opt,M and TE_opt,ϕ are reduced with increasing B₀ (see Eqs. [1–2] and Fig. 4c). (2) tSNR_M vs. B₀: due to the superior thermal SNR at higher B₀, the tSNR_M at TE_opt will be improved with B₀ increase (Fig. 4d). Consequently, the use of higher field will produce two competing effects simultaneously: decrease of ncMRI signal and increase of tSNR_M.

Figure 4.

(a) and (b) illustrate the T₂*-dependences of optimal TEs for magnitude and phase signals in human brain. Crosses indicate the data points corresponding to the typical T₂* values. (c) shows the field dependences of ncMRI magnitude and phase signal changes at the optimal TEs, and (d) shows the tSNR_M at the TE_opt,M and TE_opt,ϕ for different B₀. All the data in (c) and (d) are normalized to the corresponding results at 1.5 T.

It is found that the maximum magnitude and phase CNRs in human brain decrease with B₀ increase (Fig. 2a). In tissue preparation, the magnitude signal shows the same behavior, but the CNR of phase signal increases with B₀ (Fig. 2b). This suggests that lower field strength would provide the better sensitivity for detecting ncMRI signal in human subjects. In in vitro experiments, the detection sensitivity of magnitude/phase signal will be improved by decreasing/increasing B₀. As shown in Fig. 4c–d, when B₀ increases, the ratio of magnitude signal reduction exceeds that of tSNR increase, resulting in the smaller CNR at higher B₀. Compared to the magnitude signal, the phase signal decreases with B₀ more gradually (Fig. 4c), and the ratio of signal decrease is less than that of tSNR_M elevation in tissue preparation. But in human brain, due to the presence of physiological noise, the rate of tSNR increase is significantly reduced and is smaller than the decrease rate of phase signal. As a result, the phase CNR increases in tissue preparation while decreases in human brain with the increase in B₀.

In the calculation of TE_opt, it is assumed that the evoked neuronal response continues throughout the TE. In human brain, the neuronal activity evoked by a brief sensory stimulation can last from tens to hundreds of ms (17). In an ncMRI experiment using brief stimuli (i.e., event-related paradigm), a situation may happen in which the evoked neuronal response does not last long enough to enable CNR to reach its global maximum as shown in Fig. 2. In this case, the CNR of ncMRI signal will increase with TE initially and then reach its peak when TE is equal to the time point at which the evoked neuronal response stops. After that point, the CNR of ncMRI signal will decrease with further increase in TE.

Figure 3 indicates that the maximum CNR of ncMRI signal increases with increase in voxel volume. This suggests that the sensitivity of ncMRI is enhanced by using a larger imaging voxel. However, present model assumes partial volume effects to be negligible in the imaging voxels. Hence, these results imply that the relative magnitude and the phase changes of ncMRI signal are independent of voxel size (Eqs. [1–2]) (18). Nevertheless, if the size of imaging voxel exceeds the region of neuronal activation, then partial volume effects will occur. In the presence of partial volume effects, the CNR of ncMRI signal may depend on the ratio of tissue to voxel volume, the density and the locations of firing neurons in the voxel, etc. (1). It is our intention that we will investigate how these factors impact the dependence of maximum CNR of ncMRI on voxel volume in the near future.

In Eqs. [1–2] and [13–14], A and B are the scale factors that govern the amplitudes of magnitude and phase ncMRI signals. They are independent of imaging parameters, and the variation of A and B would not change the optimum TE values and the relative proportions between the CNRs for different TEs, voxel volumes, and field strengths (Figs. 2–4). Nevertheless, the magnitude of A and B will affect the ratios of magnitude to phase CNRs (see Eqs. [13–14]). From the previous ncMRI models (8, 18), it is known that the values of A and B would heavily rely on the biophysical properties of activated brain tissue such as density of synchronously firing neurons, and intensity and distribution of neuronal currents. To compare the CNRs between magnitude and phase signals, it is necessary to know the exact relationship between these biophysical factors and the values of A and B. Unfortunately, this has not been completely investigated to date. Since the purpose of the present study is to formulate the dependence of CNR on TE for ncMRI magnitude and phase signals separately, it is out of our scope to determine whether magnitude or phase imaging is more sensitive to ncMRI effects. Further theoretical modeling and experimental work need to be performed to address this question.

The MRI noise model (6) used in this study assumes that the noise resulting from scanner instability is independent of MRI signal as well as of physiological noise. A more recent study indicated that the correlation may exist between the scanner noise and the hemodynamic fluctuation (19). A further study to investigate the effects of this noise correlation on the TE dependence of CNR of ncMRI is underway in our laboratory.

Conclusions

The dependence of CNR of ncMRI magnitude and phase signals on TE has been established in this study. Based on the CNR model, the optimal TE values for detecting ncMRI signals are estimated in human brain and tissue preparation. The modeling results show that the optimal TE for magnitude signal is larger than that for phase signal. In addition, this model also indicates that optimal TE decreases with B₀ and increase with voxel volume. This work will be beneficial in the design of future ncMRI experiments by providing the theoretical foundation and guidance in selecting appropriate data acquisition strategy and imaging parameters.