Abstract
Accurate prostate segmentation in ultrasound images is crucial for the clinical diagnosis of prostate cancer and for performing image-guided prostate surgery. However, it is challenging to accurately segment the prostate in ultrasound images due to their low signal-to-noise ratio, the low contrast between the prostate and neighboring tissues, and the diffuse or invisible boundaries of the prostate. In this paper, we develop a novel hybrid method for segmentation of the prostate in ultrasound images that generates accurate contours of the prostate from a range of datasets. Our method involves three key steps: (1) application of a principal curve-based method to obtain a data sequence comprising data coordinates and their corresponding projection index; (2) use of the projection index as training input for a fractional-order-based neural network that increases the accuracy of results; and (3) generation of a smooth mathematical map (expressed via the parameters of the neural network) that affords a smooth prostate boundary, which represents the output of the neural network (i.e., optimized vertices) and matches the ground truth contour. Experimental evaluation of our method and several other state-of-the-art segmentation methods on datasets of prostate ultrasound images generated at multiple institutions demonstrated that our method exhibited the best capability. Furthermore, our method is robust as it can be applied to segment prostate ultrasound images obtained at multiple institutions based on various evaluation metrics.
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Data will be made available on reasonable request.
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All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by Yiyun Wu, Jing Zhao, and Caishan Wang. The first draft of the manuscript was written by Tao Peng, and writing checking and review and supervision was performed by Jin Wang and Jing Cai. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
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Appendix
Appendix
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Used symbols in this work
Appendix table used symbols in this work
Description | Symbols | |
---|---|---|
Global variables | D-dimensional real number set space | RD |
Number of points in dataset | n | |
X-axis coordinate of data point | x | |
Y-axis coordinate of data point | y | |
ASPC | Principal curve | f |
Data point set/data point | Pn/pn | |
Vertex/segment subset of principal curve | Vi = {v1, v2, …, viv}/Si = {s1, s2, …, sis} | |
Vertex/segment of principal curve | v/s | |
Number of vertices/segments of principal curve | iv/is | |
Projection index | t | |
Mean shift vector | m(•) | |
Normalization factor | z | |
Kernel bandwidth | h | |
Radially symmetric kernel | K(·) | |
Kernel density estimator | S(·) | |
Derivative of the kernel profile | L(·) | |
Bright/indeterminate/non-bright pixel point sets | TC/IC /FC | |
Intensity value at data point p in the image | gd | |
Gradient value at data point p in the image | Gd | |
Kernel function of the indeterminacy filter | GIc | |
Standard deviation of kernel function | \(\sigma_{I}\) | |
Adjustment parameters in the linear function | ap1/ap2 | |
Average indeterminacy value of the current cluster point | Icavg | |
FBNNL | Number of neurons of input layer | I |
Number of neurons of hidden layer | H | |
Number of neurons of output layer | O | |
Weight from input layer to the hidden layer | w1 | |
Weight from hidden layer to the output layer | w2 | |
Threshold of the hidden neuron | a | |
Threshold of the output neuron | b | |
Learning rate from input layer to the hidden layer | μ1 | |
Learning rate from hidden layer to the output layer | μ2 | |
Adjustment parameter | ap | |
Fractional order | α | |
Iteration number | g | |
Output of output units | c(•) | |
Total error | E | |
Evaluation metrics | Dice similarity coefficient | DSC |
Jaccard similarity coefficient | OMG | |
Accuracy | ACC |
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Values of important neural parameters
After training, we can determine the optimal parameters of our model to determine the experimental result contour according to Eqs. (14) and (15). Table 5 represents the values of important parameters of our model for expressing the contours in three randomly selected cases, and the qualitative results of these cases are shown in Fig. 8.
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MSC method
To search the cluster of data, Cheng et al. [27] proposed the MSC method, whose details are summarized as follows:
Step 1: Compute the mean shift vector mh(pi), shown as follows.
To perform kernel density estimation, the kernel density estimator S(•) for pi is denoted as
where h is kernel bandwidth with a constant, K(·) is the radially symmetric kernel, and z is a normalization factor.
The density gradient estimator is obtained as the gradient of the density estimator based on Eq. (16), shown as below:
where L(p) = K' (p) shows the derivative of the selected kernel profile. The first term of Eq. (17) is a constant, and the second term of Eq. (17) is called the mean shift vector m(p), which points to the direction of the greatest increase in density, shown below:
Step 2: Update each initial point pi on the m(p) direction according to pi = + mh(pi).
Step 3: Iterate Step 1 and Step 2 until convergence.
Step 4: Determine the cluster point.
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NAMSC method
According to the traditional MSC method, Guo et al. [21] proposed the NAMSC method, which combined an NS-based filter into the MSC method. The details of the NAMSC method are shown as follows:
Step 1: To unify the dataset, the initial data points Pn are normalized into the range of {(− 1, − 1) ~ (1,1)}.
Step 2: Map the Pn to each channel of the neutrosophic domain, where Tc(Pn), Ic(Pn), and Fc(Pn) represent the bright, indeterminate, and non-bright pixel point sets, respectively.
where gd(x, y) and Gd(x, y) are the intensity value and gradient value at the position of (x, y) in the image, respectively.
Step 3: Convolute each channel using the indeterminacy filter.
where \(\sigma_{I}\) is the standard deviation, which is used to determine the shape of the kernel function. GIc is a kernel function of the indeterminacy filter. ap1 and ap2 are the adjustment parameters in the linear function, which is used to transform the indeterminate value to filter’s parameter value.
Step 4: Compute the indeterminate values of the channels in the neutrosophic domain.
where Tc' is the result after indeterminate filter on Tc and fc is the filter size.
Step 5: Select a point pi (x, y) and compute the bandwidth h according to the corresponding indeterminate value.
where Icavg is the average indeterminacy value of the current cluster point. \(Tc^{\prime}_{\max}\) and \(Tc^{\prime}_{\min}\) are the maximum and minimum of Tc at the current cluster among all the channels of the neutrosophic domain, respectively.
Step 6: Compute the mean shift vector m(p) (same with Step 2 of the MSC method).
Step 7: Transfer each initial point pi on the m(p) direction (same with Step 3 of the MSC method).
Step 8: Go to Step 6 until it meets the condition \(\nabla S\left(p\right)=0\) (same as Step 4 of the MSC method).
Step 9: Update the bandwidth h using the average value of the indeterminate values in the current cluster.
Step 10: Go to Step 6 until the mean of the current cluster point become unchanged.
Step 11: Go to Step 5 until all the data points are clustered.
Step 12: Exit the loop and output the cluster points.
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Peng, T., Wu, Y., Zhao, J. et al. Ultrasound Prostate Segmentation Using Adaptive Selection Principal Curve and Smooth Mathematical Model. J Digit Imaging 36, 947–963 (2023). https://doi.org/10.1007/s10278-023-00783-3
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DOI: https://doi.org/10.1007/s10278-023-00783-3