Daehun Kang^{1}, Yul-Wan Sung^{1}, and Satoshi Shioiri^{2}

The BOLD signals related to brain activation is often nonlinear with change in TE. In contrast to extravascular component, the nonlinearity is attributable to intravascular component due to chemical exchange between plasma and deoxy-Hb. Recently, activity-evoked pH change on the brain has been demonstrated. Since the chemical exchange is often pH-dependent, the time for the chemical exchange would change. Thus, the two-compartment model that incorporates the change would be more accurate for estimation of parameters than the model with fixed exchange time. In this study, we measured the nonlinearity by multi-echo GRE-EPI and estimated parameters of the proposed model.

**(Two-compartment model) **The IV component
is supposed to originate from venous blood, and the EV
component predominantly originates from tissues.

$$S(t)=(V+\Delta{V})\cdot\exp(-R_{2,IV}^{*}\cdot{t})+(1-V-\Delta{V})\cdot\lambda\cdot\exp(-(R_{2,EV,0}^{*}+\Delta{R_{2,EV}^{*}})\cdot{t})$$

, where the parameters were defined in Table 1.
$$$R_{2,IV}^{*}$$$ is modulated by the chemical
exchange model^{1,2,7,8}.

$$R_{2,IV}^{*}(t)=R_{2,IV,0}^{*}+R_{2,IV,Hb}^{*}\cdot\left[1-\left(Y+\Delta{Y}\right)\right]^2\cdot\left(1+\frac{\Delta\tau_{ex}}{\tau_{ex}}\right)\cdot\left\{1-\frac{2(\tau_{ex}+\Delta\tau_{ex})}{t}\cdot\tanh\left[\frac{t}{2(\tau_{ex}+\Delta\tau_{ex})}\right]\right\}$$

The signal percent change for activation is calculated at some TE.

$$\%S(t)=\frac{S(t)-S_0(t)}{S_0(t)}\times100(\%)$$

$$${S_0}$$$ was a signal when stimulus-dependent parameters were zero. The two-compartment model was used with time-varying $$$\Delta\tau_{ex}$$$ (TV-model) or $$$\Delta\tau_{ex}=0$$$ (zero-model).

**(Parameter estimation) **Combinations of parameters in
Table 2 were used for the estimation with the following resultant limitations to
reject unrealistic results.

$$\begin{cases}{-0.1<\%S_{IV}\space{and}\space\%S_{EV}<\%S}&{when}\space\%S\geq0\\{\%S<\%S_{IV}\space{and}\space\%S_{EV}<0.1}&{when}\space\%S<0\end{cases}$$

A set of parameters was chosen by minimizing the error.

$$Error\space(s^{-1})=\frac{1}{N\cdot100(\%)}\cdot\sum_{t\in{TE}}\mid\%S_{m,t}-\%S_{e,t}\mid/t$$

, where and denote measured and simulated signal changes at echo
time *t*, respectively. *N* is
the number of TEs. Also, the
corrected Akaike information criterion (AICc) was evaluated to validate a goodness of fit with the complexity of the model.^{9}

**(Experiment) **In 3T Skyra-fit system (Siemens, Germany) with a
standard 20-channel head coil, the six-echo-time EPI
was performed with the TEs of 11,26,41,56,71 and 86 ms, TR of 2s, FA of 90°, in-plane resolution of 3.44 mm, slice thickness of 3.4 mm, an acceleration factor of 2 along PE, and fifteen slices
parallel to the calcarine sulcus. Visual
stimulation of 8-Hz flickering checkerboard was given with 8 blocks of 16/36s on/off. Nine
volunteers (three females) participated with the
consent. The multi-echo images were
reconstructed by home-made MATLAB codes (MathWorks, USA) and motion-corrected by SPM12 (London, England) with the first
echo images. Through
Brain Voyager (Brain Innovation B.V., the Netherlands), the event-related
responses were evaluated with high-pass-filtering and ROI-selecting. The ROI was defined as a cluster of voxels that satisfied the statistical condition of t-value>3.0 (p<0.001) for all TEs.

In Fig. 1, the nonlinearity of TE-dependent BOLD signals for TE intervals of 11 to 41 ms was compared with the results from a linear model.

In Fig. 2a, the minimum fitting error for TV-model was significantly lower than for zero-model or for linear model (0.025±0.003s^{-1}, 0.035±0.004s^{-1} and 0.059±0.010s^{-1}, respectively). The values of $$$\tau_{ex}$$$ were chosen to 4 ms in TV-model and 2 ms in zero-model. The average blood volumes (V) were 5.7%±0.5% in TV-model and 6.4%±0.8% in zero-model, which were within a typical range of 2%–8%. AICc values were -66.6±8.3 in TV-model, -59.5±9.0 in zero-model and -56.0±19.9 in linear model. There was a significant difference of -6.6±3.8 (p<0.001, paired Student's t-test) between TV- and zero-models. TV-model was considered to improve the goodness of fit (Fig. 2b).

As the echo time increased from 26 to 86 ms, the ratios of IV signal to the actual BOLD signal decreased from 0.44 to 0.24 in TV-model and from 0.61 to 0.44 in zero-model (calculated by Fig. 2c and 2d). The result from TV-model was consistent with the result of the previous study showing that the IV fraction decreased from 0.40 to 0.22 in TE of 32.7 to 70.7 ms.^{10}

The stimulus-sensitive parameters were estimated (Fig. 3a–d). In Fig. 3b and d, the average values in the post-stimulus undershoot were almost zero for $$$\Delta\tau_{ex}$$$ (0.04±0.19 ms, p=0.54, one-sample t-test) and non-zero for $$$\Delta{Y}$$$ (-1.62%±0.66%, p<0.001, one-sample t-test). These implied that the source of $$$\Delta\tau_{ex}$$$ might be different from that of $$$\Delta{Y}$$$.

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3. Kang D, Choi U-S, Sung Y-W. Microscopic functional specificity can be predicted from fMRI signals in ventral visual areas. Magn Reson Imaging 2014;32:1031–6.

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7. Luz Z, Meiboom S. Nuclear Magnetic Resonance Study of the Protolysis of Trimethylammonium Ion in Aqueous Solution-Order of the Reaction with Respect to Solvent. J Chem Phys 1963;39:366–70.

8. Kang D, Sung YW, Shioiri S. Estimation of physiological sources of nonlinearity in blood oxygenation level-dependent contrast signals. Magnetic Resonance Imaging 2017; Accepted.

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Figure 1. (a) Average BOLD responses on six echo times of 11–86 ms. A transparent
gray-filled box denotes the visual stimulation period. (b) To show the
nonlinearity in BOLD signals, the actual TE-dependent signals on 16 s (solid
line) were compared with the signals (dotted line) estimated by a linear model. (c) The differences
between the actual BOLD signals and the signals generated by a linear model
were plotted. The nonlinear deviation was dominant at TEs of 11 ms to 41 ms
(*;*p*<0.05, ***;*p*<0.001, paired Student’s t-test). The error bar denotes the
standard error (N=18).

Figure 2. (a) The comparison of fitting errors
between the actual and the simulated signals in three models. To determine τ_{ex}, various values were
tested. (b) The comparison of the nonlinearity of TE-normalized signal changes.
The addition of Δτ_{ex} on the two-compartment model influenced the
signal changes in short TEs of 11 to 41 ms. (c) The TE-dependent EV BOLD
signals from the time-varying Δτ_{ex} model (solid line) and from the zero Δτ_{ex} model (dotted line) were plotted. (d) The
TE-dependent IV BOLD signals were plotted as the same way. Error bar denoted
standard error (N=18).

Figure 3. The responses with a visual stimulus paradigm of 16s/36s
control-state/post-stimulus blocks. The responses of (a) ΔR^{*}_{2,EV}, (b) ΔY, (c) ΔV, and (d) Δτ_{ex} were plotted as stimulus-sensitive parameters.
They were estimated based on the time-varying Δτ_{ex} model (solid line) and the zero Δτ_{ex} model (dotted line) and averaged across ROIs.
The visual stimulation period was indicated by a transparent gray-filled box.
Error bar denoted standard error (N=18).

Table 1. Parameters used in two-compartment model

Table 2. Preparation of parameters for estimation