Calculation of intravoxel incoherent motion parameter maps using a kernelized total difference-based method.

Quantitative analysis of diffusion-weighted magnetic resonance imaging (DW-MRI) has been explored for many clinical applications since its development. In particular, the intravoxel incoherent motion (IVIM) model for DW-MRI has been commonly utilized in various organs. However, because of the presence of excessive noise, the IVIM parameter maps obtained from pixel-wise fitting are often unreliable. In this study, we propose a kernelized total difference-based curve-fitting method to estimate the IVIM parameters. Simulated DW-MRI data at five signal-to-noise ratios (i.e., 10, 20, 30, 50, and 100) and real abdominal DW-MRI data acquired on a 1.5-T MRI scanner with nine b-values (i.e., 0, 10, 25, 50, 100, 200, 300, 400, and 500 s/mm2) and six diffusion-encoding gradient directions were used to evaluate the performance of the proposed method. The results were compared with those obtained by three existing methods: trust-region reflective (TRR) algorithm, Bayesian probability (BP), and deep neural network (DNN). Our simulation results showed that the proposed method outperformed the other three comparing methods in terms of root-mean-square error. Moreover, the proposed method could preserve small details in the estimated IVIM parameter maps. The experimental results showed that, compared with the TRR method, the proposed method as well as the BP (and DNN) method could reduce the overestimation of the pseudodiffusion coefficient and improve the quality of IVIM parameter maps. For all studied abdominal organs except the pancreas, both the proposed method and the BP method could provide IVIM parameter estimates close to the reference values; the former had higher precision. The kernelized total difference-based curve-fitting method has the potential to improve the reliability of IVIM parametric imaging.

Material decomposition using dual-energy CT with unsupervised learning.

Material decomposition (MD) is an application of dual-energy computed tomography (DECT) that decomposes DECT images into specific material images. However, the direct inversion method used in MD often amplifies noise in the decomposed material images, resulting in lower image quality. To address this issue, we propose an image-domain MD method based on the concept of deep image prior (DIP). DIP is an unsupervised learning method that can perform different tasks without using a large training dataset with known targets (i.e., basis material images). We retrospectively recruited patients who underwent non-contrast brain DECT scans and investigated the feasibility of using the proposed DIP-based method to decompose DECT images into two (i.e., bone and soft tissue) and three (i.e., bone, soft tissue, and fat) basis materials. We evaluated the decomposed material images in terms of signal-to-noise ratio (SNR) and modulation transfer function (MTF). The proposed DIP-based method showed greater improvement in SNR in the decomposed soft-tissue images compared to the direct inversion method and the iterative method. Moreover, the proposed method produced similar MTF curves in both two- and three-material decompositions. Additionally, the proposed DIP-based method demonstrated better separation ability than the other two studied methods in the case of three-material decomposition. Our results suggest that the proposed DIP-based method is capable of unsupervisedly generating high-quality basis material images from DECT images.

Single-shot quantitative phase contrast imaging based on deep learning.

Quantitative differential phase-contrast (DPC) imaging is one of the commonly used methods for phase retrieval. However, quantitative DPC imaging requires several pairwise intensity measurements, which makes it difficult to monitor living cells in real-time. In this study, we present a single-shot quantitative DPC imaging method based on the combination of deep learning (DL) and color-encoded illumination. Our goal is to train a model that can generate an isotropic quantitative phase image (i.e., target) directly from a single-shot intensity measurement (i.e., input). The target phase image was reconstructed using a linear-gradient pupil with two-axis measurements, and the model input was the measured color intensities obtained from a radially asymmetric color-encoded illumination pattern. The DL-based model was trained, validated, and tested using thirteen different cell lines. The total number of training, validation, and testing images was 264 (10 cells), 10 (1 cell), and 40 (2 cells), respectively. Our results show that the DL-based phase images are visually similar to the ground-truth phase images and have a high structural similarity index (>0.98). Moreover, the phase difference between the ground-truth and DL-based phase images was smaller than 13%. Our study shows the feasibility of using DL to generate quantitative phase imaging from a single-shot intensity measurement.

Self-supervised neural network for phase retrieval in QDPC microscopy.

Quantitative differential phase contrast (QDPC) microscope plays an important role in biomedical research since it can provide high-resolution images and quantitative phase information for thin transparent objects without staining. With weak phase assumption, the retrieval of phase information in QDPC can be treated as a linearly inverse problem which can be solved by Tikhonov regularization. However, the weak phase assumption is limited to thin objects, and tuning the regularization parameter manually is inconvenient. A self-supervised learning method based on deep image prior (DIP) is proposed to retrieve phase information from intensity measurements. The DIP model that takes intensity measurements as input is trained to output phase image. To achieve this goal, a physical layer that synthesizes the intensity measurements from the predicted phase is used. By minimizing the difference between the measured and predicted intensities, the trained DIP model is expected to reconstruct the phase image from its intensity measurements. To evaluate the performance of the proposed method, we conducted two phantom studies and reconstructed the micro-lens array and standard phase targets with different phase values. In the experimental results, the deviation of the reconstructed phase values obtained from the proposed method was less than 10% of the theoretical values. Our results show the feasibility of the proposed methods to predict quantitative phase with high accuracy, and no use of ground truth phase.

Classification of peripheral blood neutrophils using deep learning.

Deep learning has been used to classify the while blood cells in peripheral blood smears. However, the classification of developing neutrophils is rarely studied. Moreover, it is still unknown whether deep learning can work well on the data coming from different sources. In this study, we therefore investigate the classification performance of deep learning for immature and mature neutrophils. In particular, we used three open-access datasets obtained from different imaging systems: CellaVision DM 96, CellaVision DM 100, and iCELL ME-150. A total of 26,050 images identified by one laboratory technologist were randomly split into training, validation, and testing datasets. A total of 10 convolutional neural networks were trained to classify six blood cell types: myeloblast, promyelocyte, myelocyte, metamyelocyte, banded neutrophil, and segmented neutrophil. The experimental results showed that compared to any single model, the average ensemble model could achieve a better classification performance and provide a testing accuracy of 90.1%. The sensitivity and specificity of the average ensemble model for the six blood cell types were above 83.5% and 96.9%, respectively. Our results suggest that deep learning is a promising tool for the classification of developing neutrophils, but further improvement is required.

An unsupervised convolutional neural network method for estimation of intravoxel incoherent motion parameters.

Objective.Intravoxel incoherent motion (IVIM) imaging obtained by fitting a biexponential model to multipleb-value diffusion-weighted magnetic resonance imaging (DW-MRI) has been shown to be a promising tool for different clinical applications. Recently, several deep neural network (DNN) methods were proposed to generate IVIM imaging.Approach.In this study, we proposed an unsupervised convolutional neural network (CNN) method for estimation of IVIM parameters. We used both simulated and real abdominal DW-MRI data to evaluate the performance of the proposed CNN-based method, and compared the results with those obtained from a non-linear least-squares fit (TRR, trust-region reflective algorithm) and a feed-forward backward-propagation DNN-based method.Main results.The simulation results showed that both the DNN- and CNN-based methods had lower coefficients of variation than the TRR method, but the CNN-based method provided more accurate parameter estimates. The results obtained from real DW-MRI data showed that the TRR method produced many biased IVIM parameter estimates that hit the upper and lower parameter bounds. In contrast, both the DNN- and CNN-based methods yielded less biased IVIM parameter estimates. Overall, the perfusion fraction and diffusion coefficient obtained from the DNN- and CNN-based methods were close to literature values. However, compared with the CNN-based method, both the TRR and DNN-based methods tended to yield increased pseudodiffusion coefficients (55%-180%).Significance.Our preliminary results suggest that it is feasible to estimate IVIM parameters using CNN.

Simultaneous Denoising of Dynamic PET Images Based on Deep Image Prior.

Parametric imaging obtained from kinetic modeling analysis of dynamic positron emission tomography (PET) data is a useful tool for quantifying tracer kinetics. However, pixel-wise time-activity curves have high noise levels which lead to poor quality of parametric images. To solve this limitation, we proposed a new image denoising method based on deep image prior (DIP). Like the original DIP method, the proposed DIP method is an unsupervised method, in which no training dataset is required. However, the difference is that our method can simultaneously denoise all dynamic PET images. Moreover, we propose a modified version of the DIP method called double DIP (DDIP), which has two DIP architectures. The additional DIP model is used to generate high-quality input data for the second DIP model. Computer simulations were performed to evaluate the performance of the proposed DIP-based methods. Our simulation results showed that the DDIP method outperformed the single DIP method. In addition, the DDIP method combined with data augmentation could generate PET parametric images with superior image quality compared to the spatiotemporal-based non-local means filtering and high constrained backprojection. Our preliminary results show that our proposed DDIP method is a novel and effective unsupervised method for simultaneously denoising dynamic PET images.

Monte Carlo evaluation of a LYSO-based Compton camera using two origin ensemble algorithms with resolution recovery.

PURPOSE: Due to the lack of depth-of-interaction information, a Compton camera made of lutetium-yttrium orthosilicate (LYSO) crystals suffers from poor spatial resolution, which may lead to an unreliable range verification in proton therapy. The aim of this study is to evaluate the performance of a LYSO-based Compton camera using the origin ensemble algorithm with resolution recovery (OE-RR). We also proposed a regularized version of OE-RR called ROE-RR. METHODS: We simulated a two-layer LYSO-based Compton camera which was used to detect prompt gammas (PGs) produced by a proton beam irradiated on a water phantom. PG images reconstructed by the OE-RR algorithm were evaluated and compared with those reconstructed by the proposed ROE-RR algorithm. RESULTS: Our simulated results show that both the OE-RR and ROE-RR algorithms could provide an accurate estimate of the Bragg peak position, with a mean positioning error of 2.5 mm. Compared to the OE-RR algorithm, the proposed ROE-RR algorithm is less sensitive with respect to initial conditions and requires less iterations for converging to equilibrium. More importantly, the proposed ROE-RR algorithm could provide better image quality than the OE-RR algorithm, especially in low-count data. CONCLUSIONS: For LYSO-based Compton cameras, using a resolution-recovery image reconstruction algorithm is essential for reliable range verification.

Patch-Based U-Net Model for Isotropic Quantitative Differential Phase Contrast Imaging.

Quantitative differential phase-contrast (qDPC) imaging is a label-free phase retrieval method for weak phase objects using asymmetric illumination. However, qDPC imaging with fewer intensity measurements leads to anisotropic phase distribution in reconstructed images. In order to obtain isotropic phase transfer function, multiple measurements are required; thus, it is a time-consuming process. Here, we propose the feasibility of using deep learning (DL) method for isotropic qDPC microscopy from the least number of measurements. We utilize a commonly used convolutional neural network namely U-net architecture, trained to generate 12-axis isotropic reconstructed cell images (i.e. output) from 1-axis anisotropic cell images (i.e. input). To further extend the number of images for training, the U-net model is trained with a patch-wise approach. In this work, seven different types of living cell images were used for training, validation, and testing datasets. The results obtained from testing datasets show that our proposed DL-based method generates 1-axis qDPC images of similar accuracy to 12-axis measurements. The quantitative phase value in the region of interest is recovered from 66% up to 97%, compared to ground-truth values, providing solid evidence for improved phase uniformity, as well as retrieved missing spatial frequencies in 1-axis reconstructed images. In addition, results from our model are compared with paired and unpaired CycleGANs. Higher PSNR and SSIM values show the advantage of using the U-net model for isotropic qDPC microscopy. The proposed DL-based method may help in performing high-resolution quantitative studies for cell biology.

Generation of Brain Dual-Energy CT from Single-Energy CT Using Deep Learning.

Deep learning (DL) has shown great potential in conversions between various imaging modalities. Similarly, DL can be applied to synthesize a high-kV computed tomography (CT) image from its corresponding low-kV CT image. This indicates the feasibility of obtaining dual-energy CT (DECT) images without purchasing a DECT scanner. In this study, we investigated whether a low-to-high kV mapping was better than a high-to-low kV mapping. We used a U-Net model to perform conversions between different kV CT images. Moreover, we proposed a double U-Net model to improve the quality of original single-energy CT images. Ninety-eight patients who underwent brain DECT scans were used to train, validate, and test the proposed DL-based model. The results showed that the low-to-high kV conversion was better than the high-to-low kV conversion. In addition, the DL-based DECT images had better signal-to-noise ratios (SNRs) than the true (original) DECT images, but at the expense of a slight loss in spatial resolution. The mean CT number differences between the true and DL-based DECT images were within [Formula: see text] 1 HU. No statistically significant difference in CT number measurements was found between the true and DL-based DECT images (p > 0.05). The DL-based DECT images with improved SNR could produce low-noise virtual monoenergetic images. Our preliminary results indicate that DL has the potential to generate brain DECT images using single-energy brain CT images.

Kernel-based curve-fitting method with spatial regularization for generation of parametric images in dynamic PET.

Due to high levels of noise in pixel-wise time-activity curves, the indirect method that generates kinetic parametric images from dynamic positron emission tomography (PET) images often results in poor parametric image quality. We have demonstrated that the quality of parametric images can be improved by denoising dynamic PET images, using gradient-free curve-fitting and applying a kernel-based post-filtering to parametric images. However, many gradient-free curve-fitting methods are time-consuming. Moreover, some parameter estimates (e.g. k2 and k3) have large variability. To provide high-quality PET parametric images with low computational cost, we propose a curve-fitting method that incorporates the kernel-based denoising method and the highly constrained backprojection technique into the Levenberg-Marquardt (LM) algorithm. We conducted a simulation study to evaluate the performance of the proposed curve-fitting method. Dynamic PET images were reconstructed using the expectation-maximization (EM) algorithm and were denoised before parameter estimation. Compared to the LM algorithm with and without the kernel-based post-filtering, the proposed method achieved superior performance, offering a decrease in both bias and coefficient of variation (CV) on all parametric images. Overall, the proposed method exhibited lower bias and slightly higher CV than the gradient-free pattern search method with the kernel-based post-filtering (PatS-K). Moreover, the computation time of the proposed method was about 18 times lower than that of the PatS-K method. Finally, we show that the proposed method can further improve the quality of parametric images when dynamic PET images are reconstructed using the kernel-based EM algorithm.

Dose image prediction for range and width verifications from carbon ion-induced secondary electron bremsstrahlung x-rays using deep learning workflow.

PURPOSE: Imaging of the secondary electron bremsstrahlung (SEB) x rays emitted during particle-ion irradiation is a promising method for beam range estimation. However, the SEB x-ray images are not directly correlated to the dose images. In addition, limited spatial resolution of the x-ray camera and low-count situation may impede correctly estimating the beam range and width in SEB x-ray images. To overcome these limitations of the SEB x-ray images measured by the x-ray camera, a deep learning (DL) approach was proposed in this work to predict the dose images for estimating the range and width of the carbon ion beam on the measured SEB x-ray images. METHODS: To prepare enough data for the DL training efficiently, 10,000 simulated SEB x-ray and dose image pairs were generated by our in-house developed model function for different carbon ion beam energies and doses. The proposed DL neural network consists of two U-nets for SEB x ray to dose image conversion and super resolution. After the network being trained with these simulated x-ray and dose image pairs, the dose images were predicted from simulated and measured SEB x-ray testing images for performance evaluation. RESULTS: For the 500 simulated testing images, the average mean squared error (MSE) was 2.5 × 10-5 and average structural similarity index (SSIM) was 0.997 while the error of both beam range and width was within 1 mm FWHM. For the three measured SEB x-ray images, the MSE was no worse than 5.5 × 10-3 and SSIM was no worse than 0.980 while the error of the beam range and width was 2 mm and 5 mm FWHM, respectively. CONCLUSIONS: We have demonstrated the advantages of predicting dose images from not only simulated data but also measured data using our deep learning approach.

Denoising of multi b-value diffusion-weighted MR images using deep image prior.

The clinical value of multiple b-value diffusion-weighted (DW) magnetic resonance imaging (MRI) has been shown in many studies. However, DW-MRI often suffers from low signal-to-noise ratio, especially at high b-values. To address this limitation, we present an image denoising method based on the concept of deep image prior (DIP). In this method, high-quality prior images obtained from the same patient were used as the network input, and all noisy DW images were used as the network output. Our aim is to denoise all b-value DW images simultaneously. By using early stopping, we expect the DIP-based model to learn the content of images instead of the noise. The performance of the proposed DIP method was evaluated using both simulated and real DW-MRI data. We simulated a digital phantom and generated noise-free DW-MRI data according to the intravoxel incoherent motion model. Different levels of Rician noise were then simulated. The proposed DIP method was compared with the image denoising method using local principal component analysis (LPCA). The simulation results show that the proposed DIP method outperforms the LPCA method in terms of mean-squared error and parameter estimation. The results of real DW-MRI data show that the proposed DIP method can improve the quality of IVIM parametric images. DIP is a feasible method for denoising multiple b-value DW-MRI data.

Reliable estimation of brain intravoxel incoherent motion parameters using denoised diffusion-weighted MRI.

In this study, we evaluate whether diffusion-weighted magnetic resonance imaging (DW-MRI) data after denoising can provide a reliable estimation of brain intravoxel incoherent motion (IVIM) perfusion parameters. Brain DW-MRI was performed in five healthy volunteers on a 3 T clinical scanner with 12 different b-values ranging from 0 to 1000 s/mm2 . DW-MRI data denoised using the proposed method were fitted with a biexponential model to extract perfusion fraction (PF), diffusion coefficient (D) and pseudo-diffusion coefficient (D*). To further evaluate the accuracy and precision of parameter estimation, IVIM parametric images obtained from one volunteer were used to resimulate the DW-MRI data using the biexponential model with the same b-values. Rician noise was added to generate DW-MRI data with various signal-to-noise ratio (SNR) levels. The experimental results showed that the denoised DW-MRI data yielded precise estimates for all IVIM parameters. We also found that IVIM parameters were significantly different between gray matter and white matter (P < 0.05), except for D* (P = 0.6). Our simulation results show that the proposed image denoising method displays good performance in estimating IVIM parameters (both bias and coefficient of variation were <12% for PF, D and D*) in the presence of different levels of simulated Rician noise (SNRb=0 = 20-40). Simulations and experiments show that brain DW-MRI data after denoising can provide a reliable estimation of IVIM parameters.

A deep learning approach for converting prompt gamma images to proton dose distributions: A Monte Carlo simulation study.

PURPOSE: In proton therapy, imaging prompt gamma (PG) rays has the potential to verify proton dose (PD) distribution. Despite the fact that there is a strong correlation between the gamma-ray emission and PD, they are still different in terms of the distribution and the Bragg peak (BP) position. In this work, we investigated the feasibility of using a deep learning approach to convert PG images to PD distributions. METHODS: We designed the Monte Carlo simulations using 20 digital brain phantoms irradiated with a 100-MeV proton pencil beam. Each phantom was used to simulate 200 pairs of PG images and PD distributions. A convolutional neural network based on the U-net architecture was trained to predict PD distributions from PG images. RESULTS: Our simulation results show that the pseudo PD distributions derived from the corresponding PG images agree well with the simulated ground truths. The mean of the BP position errors from each phantom was less than 0.4 mm. We also found that 2000 pairs of PG images and dose distributions would be sufficient to train the U-net. Moreover, the trained network could be deployed on the unseen data (i.e. different beam sizes, proton energies and real patient CT data). CONCLUSIONS: Our simulation study has shown the feasibility of predicting PD distributions from PG images using a deep learning approach, but the reliable prediction of PD distributions requires high-quality PG images. Image-degrading factors such as low counts and limited spatial resolution need to be considered in order to obtain high-quality PG images.

Partial-ring PET image restoration using a deep learning based method.

PET scanners with partial-ring geometry have been proposed for various imaging purposes. The incomplete projection data obtained from this design cause undesirable artifacts in the reconstructed images. In this study, we investigated the performance of a deep learning (DL) based method for the recovery of partial-ring PET images. Twenty digital brain phantoms were used in the Monte Carlo simulation toolkit, SimSET, to simulate 15 min full-ring PET scans. Partial-ring PET data were generated from full-ring PET data by removing coincidence events that hit these specific detector blocks. A convolutional neural network based on the residual U-Net architecture was trained to predict full-ring data from partial-ring data in either the projection or image domain. The performance of the proposed DL-based method was evaluated by comparing with the PET images reconstructed using the full-ring projection data in terms of the mean squared error (MSE), structural similarity (SSIM) index and recovery coefficient (RC). The MSE results showed the superiority of the image-domain approach in reduction of 91.7% in contrast to 14.3% for the projection-domain approach. Therefore, the image-domain approach was used to study the influence of the number of detector block removal. The SSIM results were 0.998, 0.996 and 0.993 for 3, 5 and 7 detector block removals, respectively. The activity of gray and white matters could be fully recovered even with 7 detector block removal, while the RCs of two artificially inserted small lesions (3 pixels in diameter) in the testing data were 94%, 89% and 79% for 3, 5, and 7 detector block removals, respectively. Our simulation results suggest that DL has the potential to recover partial-ring PET images.

Noise reduction in dual-energy computed tomography virtual monoenergetic imaging.

PURPOSE: Virtual monoenergetic images (VMIs) derived from dual-energy computed tomography (DECT) have been explored for several clinical applications in recent years. However, VMIs at low and high keVs have high levels of noise. The aim of this study was to reduce image noise in VMIs by using a two-step noise reduction technique. METHODS: VMI was first denoised using a modified highly constrained backprojection (HYPR) method. After the first-step denoising, a general-threshold filtering method was performed. Two sets of anthropomorphic phantoms were scanned with a clinical dual-source DECT system. DECT data (80/140Sn kV) were reconstructed as VMI series at 12 different energy levels (range, 40-150 keV, interval, 10 keV). For comparison, the averaged VMIs obtained from 10 repeated DECT scans were used as the reference standard. The signal-to-noise ratio (SNR), contrast-to-noise ratio (CNR) and root-mean-square error (RMSE) were used to evaluate the quality of VMIs. RESULTS: Compared to the original HYPR method, the proposed two-step image denoising method could provide better performance in terms of SNR, CNR, and RMSE. In addition, the proposed method could achieve effective noise reduction while preserving edges and small structures, especially for low-keV VMIs. CONCLUSION: The proposed two-step image denoising method is a feasible method for reducing noise in VMIs obtained from a clinical DECT scanner. The proposed method can also reduce edge blurring and the loss of intensity in small lesions.

A kernel-based image denoising method for improving parametric image generation.

One of the main challenges in the pixel-wise modeling analysis is the presence of high noise levels. Wang and Qi proposed a kernel-based method for dynamic positron emission tomgraphy reconstruction. Inspired by this method, we propose a kernel-based image denoising method based on the minimization of a kernel-based lp-norm regularized problem. To solve the kernel-based image denoising problem, we used the general-threshold filtering algorithm in combination with total difference. In the present study, we investigated whether diffusion-weighted magnetic resonance imaging (DW-MRI) data denoised using the proposed method can provide improved intravoxel incoherent motion (IVIM) parametric images. We also compared the proposed method with the method using the local principal component analysis (LPCA). The simulated DW-MR magnitude images are assumed to have Rician distributed noise. Computer simulations show that the proposed image denoising method can achieve a better bias-variance trade-off than the LPCA method. Moreover, the proposed method can reduce variance while simultaneously preserving edges in the parametric images. We tested our image denoising method on in vivo DW-MRI data, and the result showed that the denoised DWI-MRI data obtained using the proposed method can substantially improve the quality of IVIM parametric images.

Indirect methods for improving parameter estimation of PET kinetic models.

PURPOSE: Parametric images obtained from kinetic modeling of dynamic positron emission tomography (PET) data provide a new way of visualizing quantitative parameters of the tracer kinetics. However, due to the high noise level in pixel-wise image-driven time-activity curves, parametric images often suffer from poor quality and accuracy. In this study, we propose an indirect parameter estimation framework which aims to improve the quality and quantitative accuracy of parametric images. METHODS: Three different approaches related to noise reduction and advanced curve fitting algorithm are used in the proposed framework. First, dynamic PET images are denoised using a kernel-based denoising method and the highly constrained backprojection technique. Second, gradient-free curve fitting algorithms are exploited to improve the accuracy and precision of parameter estimates. Third, a kernel-based post-filtering method is applied to parametric images to further improve the quality of parametric images. Computer simulations were performed to evaluate the performance of the proposed framework. RESULTS AND CONCLUSIONS: The simulation results showed that when compared to the Gaussian filtering, the proposed denoising method could provide better PET image quality, and consequentially improve the quality and quantitative accuracy of parametric images. In addition, gradient-free optimization algorithms (i.e., pattern search) can result in better parametric images than the gradient-based curve fitting algorithm (i.e., trust-region-reflective). Finally, our results showed that the proposed kernel-based post-filtering method could further improve the precision of parameter estimates while maintaining the accuracy of parameter estimates.

Accuracy of using high-energy prompt gamma to verify proton beam range with a Compton camera: A Monte Carlo simulation study.

Prompt gamma (PG) rays emitted during proton therapy has been used for proton range verification. Because high-energy PG emission is well correlated to the Bragg peak (BP), high-energy PG rays are well-suited for proton range verification. However, the low production and detection of high-energy PG rays often lead to inaccurate BP position estimates. The aim of this study is to improve the BP position estimates obtained from high-energy PG rays. We propose a BP position estimation method based on the local maximum closest to the distal fall-off region. We present the results of Monte Carlo simulations in which a water phantom was irradiated with a proton beam. Our results show that the BP position estimated from the 6.13 MeV PG rays can be improved using the proposed position estimation method. Moreover, the 6.92 and 7.12 MeV PG rays can be used for predicting the BP position. However, the accuracy of the BP position estimation decreases with decreasing tissue oxygen levels. We also found that the subtraction of the PG images of 6.13 MeV from those of 6.92 and 7.12 MeV can be used to predict the BP position with a mean accuracy of < 2 mm. The accurate estimation of the BP position can be achieved using different high-energy PG rays, but factors including position estimation, irradiated tissue and event selection should be carefully taken into account.