# Joint k-ω Space Image Reconstruction and Data Fitting for Chemical Exchange Saturation Transfer Magnetic Resonance Imaging

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model for Joint Reconstruction and Data Fitting

#### 2.2. Numerical Algorithm and Implementation

#### 2.3. Evaluations

## 3. Results

#### 3.1. Choice of Fitting Functions

#### 3.2. Numerical Simulations

#### 3.3. Results of Choline Phantom Case

#### 3.4. Results of Iopamidol Phantom Case

#### 3.5. Reduction in Number of Offset Frequencies

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Illustration for the (

**a**) two-compartment numerical phantom, (

**b**) the experimental two-compartment phantom which contained choline water solutions with concentrations: 10, 25, 50, and 100 mM in the Fomblin background, and (

**c**) the experimental multi-compartment phantom which contained iopamidol (IsOvue-300, Bracco Diagnostics, Milan, Italy) solutions at pH 6.0, 6.5, 7.0, and 7.5. (

**d**–

**f**) Z-spectra (solid lines) and corresponding MTR asymmetry (dotted lines) for the phantom in (

**a**–

**c**), respectively.

**Figure 3.**LGfitting (solid red line) and LL fitting (blue dashed line) for (

**a**,

**b**) the choline phantom and (

**c**,

**d**) the iopamidol phantom, respectively. The figures in the upper row (

**a**,

**c**) are water pools and those in the lower row (

**b**,

**d**) are CEST pools, respectively.

**Figure 4.**GT MR images, MR images reconstructed by the FFT-based algorithm and our algorithm, and MTR asymmetry map at ${\omega}_{2}$ = 450 Hz for the simulated phantom case with a noise level of 4.5%. The image windows at 0 Hz are [0 $5\times {10}^{-6}$], [0 0.05] at 450∼900 Hz, and [0 0.2] for MTR asymmetry map.

**Figure 5.**GT MR images, MR images reconstructed by the FFT-based algorithm and our algorithm, and MTR asymmetry map at ${\omega}_{2}$ = 450 Hz for the simulated phantom case with a noise level of 25%. The image windows at 0 Hz are [0 $5\times {10}^{-6}$], [0 0.05] at 450∼900 Hz, and [0 0.2] for MTR asymmetry map.

**Figure 6.**(

**a**) SNR of reconstructed MR images as a function of noise levels. (

**b**–

**h**) MTR asymmetry maps at ${\omega}_{2}$ = 450 Hz by the conventional algorithm (upper row) and our algorithm (lower row) for the simulated phantom with various noise levels of $0.5\%$, $4.5\%$, $8.5\%$, $10.5\%$, $15\%$, $25\%$, and $40\%$, respectively. The display window of MTR asymmetry map is [0 0.2].

**Figure 7.**GT MR images, MR images reconstructed by the FFT-based algorithm and our algorithm, and MTR asymmetry map at ${\omega}_{2}$ = 130 Hz of the choline phantom with noise level 15% at offset frequencies $\omega $ = 0, 130, and 260 Hz. Image windows at 0 Hz are [0 0.1] and [0 1] at 130∼260 Hz. MTR asymmetry window is [0 0.2].

**Figure 8.**MTR asymmetry maps at ${\omega}_{2}$ = 130 Hz of conventional algorithm (upper row) and our algorithm (lower row) of the experimental choline phantom with various noise levels (

**a**) 0.5%, (

**b**) 2.5%, (

**c**) 4.5%, (

**d**) 10.5%, (

**e**) 15%, (

**f**) 25%, and (

**g**) 40%, respectively. MTR asymmetry map window is [0 0.2].

**Figure 9.**Mean and standard deviation of MTR asymmetry in each ROI at ${\omega}_{2}$ = 130 Hz for water and choline concentrations of 10, 25, 50, and 100 mM of the GT; results reconstructed by the conventional method and our method with noise levels of (

**a**) 0.5%, (

**b**) 4.5%, and (

**c**) 10.5%, respectively.

**Figure 10.**GT MR images, MR images reconstructed by the FFT-based algorithm and our algorithm, and MTR asymmetry map at ${\omega}_{2}$ = 530 Hz and ${\omega}_{3}$ = 700 Hz of the iopamidol phantom with noise level 15% at offset frequencies 0, 530, and 700 Hz. Image windows at 0 Hz are [0 0.35] and [0.5 1.5] at 530∼700 Hz. MTR asymmetry map window is [0 0.5].

**Figure 11.**MTR asymmetry maps at ${\omega}_{3}$ = 700 Hz of the conventional algorithm (upper row) and our algorithm (lower row) of the experimental iopamidol phantom with various noise levels (

**a**) 0.5%, (

**b**) 2.5%, (

**c**) 4.5%, (

**d**) 10.5%, (

**e**) 15%, (

**f**) 25%, and (

**g**) 40%, respectively. MTR asymmetry map window is [0 0.5].

**Figure 12.**Mean and standard deviation of MTR asymmetry in each insert at (

**a**–

**c**) ${\omega}_{2}$ = 530 Hz and (

**d**–

**f**) ${\omega}_{3}$ = 700 Hz for water and iopamidol solutions of various pH values 6.0, 6.5, 7.0, and 7.5 of the GT; results reconstructed by the conventional method and our method with noise levels of 0.5%, 4.5%, and 10.5%, respectively.

**Figure 13.**MTR asymmetry maps at ${\omega}_{2}$ = 450 Hz from the conventional algorithm (

**a**,

**c**) and our algorithm (

**b**,

**d**) for the simulated phantom with (

**a**,

**b**) 15.6% and (

**c**,

**d**) 27.5% reduction in offset frequencies. MTR asymmetry map window is [0 0.2].

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## Share and Cite

**MDPI and ACS Style**

Peng, Y.; Dai, Y.; Zhang, S.; Deng, J.; Jia, X.
Joint *k*-*ω* Space Image Reconstruction and Data Fitting for Chemical Exchange Saturation Transfer Magnetic Resonance Imaging. *Tomography* **2024**, *10*, 1123-1138.
https://doi.org/10.3390/tomography10070085

**AMA Style**

Peng Y, Dai Y, Zhang S, Deng J, Jia X.
Joint *k*-*ω* Space Image Reconstruction and Data Fitting for Chemical Exchange Saturation Transfer Magnetic Resonance Imaging. *Tomography*. 2024; 10(7):1123-1138.
https://doi.org/10.3390/tomography10070085

**Chicago/Turabian Style**

Peng, Yuting, Yan Dai, Shu Zhang, Jie Deng, and Xun Jia.
2024. "Joint *k*-*ω* Space Image Reconstruction and Data Fitting for Chemical Exchange Saturation Transfer Magnetic Resonance Imaging" *Tomography* 10, no. 7: 1123-1138.
https://doi.org/10.3390/tomography10070085