# A Deep Learning-Based Framework for Highly Accelerated Prostate MR Dispersion Imaging

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

- 1.
- The fMRDI model resembles intravascular dispersion with a simple linear combination of slow and fast AIFs, which is easier to optimize and requires less computation.
- 2.
- The dispersion parameter in our fMRDI model can be used to differentiate csPCa from normal tissue and improve the overall performance of csPCa identification (Section 3.2).
- 3.
- The two-stage estimation framework is fast, accurate, flexible, and more robust against noise and initializations. It does not restrict the form of the AIF or the sampling interval. It operates significantly faster than NLLS and achieves more accurate fitting results.

## 2. Methods and Materials

#### 2.1. From Tofts Model to Fast MRDI Model

#### 2.1.1. The Tofts Model

^{−1}) and the rate constant (${k}_{ep}$, measured in min

^{−1}), that are relevant to tissue perfusion and permeability. ${K}^{trans}$ and ${k}_{ep}$, which measure the CA wash-in and wash-out, are commonly associated with csPCa, as indicated by studies such as those of Fütterer et al. [41], Kuenen et al. [24], and Sung et al. [25]. They can enhance lesion visibility, according to the Prostate Imaging Reporting and Data System (PI-RADS) [7].

#### 2.1.2. MRDI and mMRDI: Dispersion-Applied AIFs

#### 2.1.3. fMRDI: Fast MRDI Model

#### 2.2. Overall Workflows

#### 2.3. Training Data Synthesis

- 1.
- Sample random PK parameters ${K}^{trans},{k}_{ep},{t}_{0}$, and $\lambda $ from designated distributions.
- 2.
- Synthesize smooth time series using the fMRDI formulated in Equation (5).
- 3.
- Add Gaussian noise to the smooth time series to close the gap between synthetical and real data.

#### 2.4. Model Training Workflow

#### 2.4.1. Model Architecture

#### 2.4.2. Preprocessing for Robust Neural Networks

- 1.
- To enhance the model’s robustness against the noise and capture information at various scales.
- 2.
- To increase the data dimension and project the one-dimensional time series into high-dimensional space.
- 3.
- To normalize the time series data into a fixed range with zero mean and constant variance.

#### 2.4.3. Model Training

#### 2.5. Model Inference Workflow

#### 2.5.1. From MRI Signal to CA Concentration

#### 2.5.2. Initial Coarse Estimation

#### 2.5.3. Coarse-to-Fine via Iterative Fitting

#### 2.6. Study Population and DCE-MRI Data

## 3. Experiments and Results

#### 3.1. Running Time and Quality of Fitting

#### 3.1.1. Running Time and Fitting Errors

#### 3.1.2. Compared with MRDI and mMRDI

#### 3.2. csPCa Lesions with ${K}^{trans}$

^{−1}) [7] and the dispersion parameter $\lambda $ [8,24,25] were considered to be indicative of csPCa, we only visualized ${K}^{trans}$ and $\lambda $.

#### 3.2.1. Qualitative Visualization of ${K}^{trans}$ Maps

#### 3.2.2. Quantitative Comparison of Tissue Contrast

#### 3.3. Validation with Digital Reference Objects

## 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 1.**(

**a**) T2-weighted MR slice with annotated prostate (gray), normal tissue (blue), and a tumor (red). (

**b**) Corresponding histopathology image. (

**c**) Concentration curves for the three regions demonstrate different contrast-agent concentrations, ${C}_{t}\left(t\right)$, in three different ROIs.

**Figure 3.**The overall workflow of our proposed method for PK parameter estimation. Circles represent the processed data, and the shape of the data is indicated near each circle. In training, the random PK parameter P is sampled to synthesize the training data, ${C}_{t}$. After preprocessing, the data are fed into the neural network, and the estimated parameter ${\widehat{P}}_{0}$ is compared against P for loss computation. In testing, the model takes clinical DCE-MRI concentration curves, ${C}_{t}$, as input, and the subsequent ‘iterative refinement’ takes ${\widehat{P}}_{0}$ as the starting point of the iteration and refines the initial estimation with NLLS.

**Figure 4.**The pipeline of concentration-curve synthesis. (

**a**) random PK parameters are sampled to synthesize smooth curves (

**b**). (

**c**) noise is added to the smooth curves to simulate the real cases.

**Figure 5.**Histograms of pharmacokinetic parameters (

**top**) and distributions used for data synthesis (

**bottom**).

**Figure 6.**Real (

**top**) and synthetic concentration curves (

**bottom**). Curves of each column share the same PK parameters.

**Figure 7.**The architecture of the transformer neural network used in our experiments. ReLU activation was used in the last of the network to ensure positive estimations.

**Figure 9.**Example fitting results of different methods. Methods with Parker AIF do not fit data with slow uptakes well. fMRDI model achieves the best overall fittings that match the data points the best.

**Figure 10.**The squared error of fitting with different numbers of repeats. We performed NLLS fitting various times with random initializations and then picked the best fitting for each voxel.

**Figure 11.**Visualization of PK parametric maps generated using different methods and the beta ${K}^{trans}$ maps proposed in our study. The T2-weighted images are used as a background, and corresponding histopathological images are provided in the right-most column for reference to identify the location of the lesion.

**Figure 12.**Illustration of how we annotated csPCa and normal tissue ROIs on ${K}^{trans}$ maps. Using the T2W annotations and the histopathological images as references, we annotated hyperintensity areas on ${K}^{trans}$ maps as lesions, and then we annotated normal tissue in the same zone on the ${K}^{trans}$ maps.

**Figure 13.**Scatter plot of ROI-averaged ${K}^{trans}$ and $\lambda $ values in TZ and PZ. When applied individually, both $\lambda $ (

**a**) and ${K}^{trans}$ (

**b**) can differentiate csPCa lesions from normal tissue, while ${K}^{trans}$ performs better. When applied jointly (

**c**), the lesions and normal tissue can be better separated.

**Figure 14.**Specificity–sensitivity curves of csPCa detection in the peripheral zone (PZ), the transitional zone (TZ), and the whole prostate (PZ+TZ). Different colors represent different pharmacokinetic models. ‘Ours’ represents the fMRDI model with NN + refine curve fitting.

**Figure 15.**ROC curves from LDA analysis were generated for the entire prostate (PZ + TZ) using various combinations of PK parameters. (

**a**) When analyzed individually, ${K}^{trans}$ exhibited the highest performance, followed by $\lambda $ and then ${k}_{ep}$. Conversely, ${t}_{0}$ demonstrated the lowest performance and could only differentiate lesions from normal tissue to a limited extent. (

**b**) When analyzed collectively, excluding ${t}_{0}$ barely impacted the performance, and the inclusion of $\lambda $ moderately improved the performance.

**Figure 16.**The fitting error of different methods under different noise levels using DROs. The heat maps show fitting errors at different ${K}^{trans}$ and ${k}_{ep}$ bins. The table summarizes the average errors at different noise levels.

**Table 1.**Squared errors are assessed by incrementally adding model components. The baseline model, labeled ‘NN + Parker’, represents the NN model without the proposed preprocessing or WS AIF.

NN + Parker AIF | $0.6681\pm 2.2015$ |

+WS AIF | $0.5982\pm 1.9383$ |

+Pyramid | $0.5892\pm 1.9012$ |

+Sinusoidal | $0.5801\pm 1.8729$ |

+Refine | $0.4114\pm 1.5181$ |

**Table 2.**Squared errors, iterations required to converge, and per-patient times of different fitting methods and PK models. fMRDI achieves significantly lower errors compared to the Tofts model, and the two-stage fitting method also outperforms NLLS in both fitting errors and running time.

Fitting Method | Ottens | NLLS | NN + NLLS Refine | ||||
---|---|---|---|---|---|---|---|

PK Model | Tofts + Exp | Tofts + Parker | MRDI | fMRDI | Tofts + Parker | MRDI | fMRDI |

Error | 0.6723 ±2.2209 | 0.6184 ±1.9867 | 0.4272 ±1.6618 | 0.4261 ±1.5687 | 0.5917 ±2.1221 | 0.4175 ±1.5212 | 0.4114 ±1.5181 |

Iterations | N/A | 200 | 300 | 200 | 20 | 50 | 30 |

Time (per-patient) | 109 s | 480 s | 644 s | 480 s | 71 s | 115 s | 176 s |

**Table 3.**Sensitivity, specificity, and AUC values of different methods in the PZ, the TZ, and the whole prostate. The sensitivity and specificity values are calculated by maximizing Youden’s index.

Method | Ottens [38] | NLLS+Tofts+Parker | MRDI | fMRDI (Ours) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Zone | PZ | TZ | PZ+TZ | PZ | TZ | PZ+TZ | PZ | TZ | PZ+TZ | PZ | TZ | PZ+TZ |

AUC | 0.784 | 0.754 | 0.753 | 0.852 | 0.826 | 0.830 | 0.934 | 0.823 | 0.892 | 0.939 | 0.871 | 0.904 |

1 - specificity | 0.236 | 0.288 | 0.179 | 0.109 | 0.152 | 0.183 | 0.109 | 0.227 | 0.204 | 0.069 | 0.167 | 0.167 |

Sensitivity | 0.690 | 0.773 | 0.583 | 0.730 | 0.697 | 0.742 | 0.862 | 0.788 | 0.858 | 0.822 | 0.788 | 0.842 |

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## Share and Cite

**MDPI and ACS Style**

Zhao, K.; Pang, K.; Hung, A.L.; Zheng, H.; Yan, R.; Sung, K.
A Deep Learning-Based Framework for Highly Accelerated Prostate MR Dispersion Imaging. *Cancers* **2024**, *16*, 2983.
https://doi.org/10.3390/cancers16172983

**AMA Style**

Zhao K, Pang K, Hung AL, Zheng H, Yan R, Sung K.
A Deep Learning-Based Framework for Highly Accelerated Prostate MR Dispersion Imaging. *Cancers*. 2024; 16(17):2983.
https://doi.org/10.3390/cancers16172983

**Chicago/Turabian Style**

Zhao, Kai, Kaifeng Pang, Alex LingYu Hung, Haoxin Zheng, Ran Yan, and Kyunghyun Sung.
2024. "A Deep Learning-Based Framework for Highly Accelerated Prostate MR Dispersion Imaging" *Cancers* 16, no. 17: 2983.
https://doi.org/10.3390/cancers16172983