A model for segmentation of CT images of liver lesions based on regional fitting and gradient information

： Background: The Computed Tomography (CT) images of liver have such disadvantages as uneven gray scale, fuzzy boundary and missing, so the commonly used image segmentation model of liver lesions has low segmentation accuracy. Methods: We propose a new hybrid active contour model based on regional fitting and gradient information for segmenting CT images of liver lesions. Firstly, the problem of uneven gray scale of liver lesions image was solved by local area fitting method, and the gradient information of liver lesions image was integrated to enhance the detection ability of the model on the edge of liver lesions. Secondly, we introduce the region area term, which can keep the image segmentation curve smooth in the process of segmentation, and effectively control the direction and speed of curve evolution. Finally, the performance of the DRLSE model, RSF model and the present model was compared in the segmentation of liver lesions. Results: It can be concluded from the experimental results that: compared with DRLSE model and RSF model ， the average Dice similarity coefficient reached 97.7%, ncreased by 12.7% and 11.7% respectively; the under segmentation rate was 2%, 9% and 17% lower, and the over segmentation rate was 1.6%. Conclusion: Therefore, the segmentation model proposed in this paper has excellent segmentation performance and greatly improves the segmentation accuracy of liver lesions.


Background
The development of medical image segmentation technology [1,2] has passed the stage of manual segmentation, and has entered the stage of semi-automatic and full-automatic segmentation [3,4,5,6,7,8] .
However, when contrast agent flows into the hepatic artery, the image information in the tumor area is unevenly enhanced, resulting in blurry boundary, partial missing and uneven gray scale in CT images, so there will be a large error in the segmentation results of liver lesions. The commonly used medical image segmentation models of liver [9] can be divided into several categories according to its principle, such as region-based feature method, active contour model method, statistical shape method, clustering method and graph theory method. Where the active contour model method, due to its simple and efficient computing performance, and can well extract the deformation contour and other advantages, it is widely used in the medical image processing of liver with complex features. The active contour model method has developed rapidly since it was proposed. Caselles et al. [10] proposed the geodesic active contour model (GAC model), which abandoned the dependence of Snake model [11] on parameters and added the level set, making the contour curve closer to the topological structure of the target object. However, the results of CT images of liver lesions with blurred boundaries, partial missing and uneven gray scales are not satisfactory. Li et al. [12] proposed a level set model (DRLSE model) for distance regularization based on GAC model, which improved the evolution speed of segmentation curve. Zhang [13] , Wang [14] et al.
constrained the evolution of active contour by introducing local information, to some extent, it improves the ability of processing CT images of the liver.
Wei et al. [15] defined the edge stop function and weight coefficient, so as to improve the model's ability of extracting liver boundary information. However, these methods still can not achieve the cross region jump evolution of segmentation curve, and can not complete the task of segmentation of liver lesions with fuzzy boundary or chaotic gradient information. Chan et al. [16] proposed a region-based level set method (Chan-Vese model,CV model), it has a good effect on the segmentation of images with significantly different pixel values between the foreground and the background of liver focus, this model has poor ability of segmentation for CT images of liver with complex features such as blurred boundary, partial absence and uneven gray scale.
Li et al. [17] proposed an active contour model (RSF model) based on fitting energy of contractible region.
However, it is easy to fall into the local minimum when solving the energy functional, which is more suitable for global segmentation tasks. Akram et al. [18] proposed a locally and globally fitted image segmentation model (LGFI model). ZHANG [19] , ZHAO [20] et al., and Soomro et al. [21] proposed a region-based active contour model weighted by edge stop function. Although these models can process CT images of liver with complex features such as gray inequality, but it leads to a decrease in the evolutionary stability of the segmentation curve, it is also difficult to complete the task of segmenting the local area of the liver.
In recent years, the mainstream deep learning algorithm convolutional neural network [22] (CNN) has made great progress in medical image segmentation and other tasks.
In the final analysis, the methods based on convolutional neural network are to establish the mapping of pixels and pixels in a certain range to instances or categories through known sample data. However, the mechanical mapping model has obvious defects, almost no migration ability and generalization ability, and due to the complexity of the mapping model, slight changes in the input data will lead to unexpected errors. In addition, medical images have the characteristics of less semantic content, low imaging quality, less data [23] , and multiple modes [24] . In 2016, Ben Cohen et al. [25] used FCN for the first time to solve the problem of liver and liver tumor segmentation. UNET [26] took FCN as the backbone to fully mine the multi-scale features of the image. In the segmentation process, the above neural network realizes the automatic segmentation of liver and liver tumor through the independent classification of pixels. However, on the one hand, it is still urgent to solve the problem of too few data samples, and the segmentation labels in the training data need to be manually made by professional doctors, which is costly; on the other hand, other pixel categories in the neighborhood are not used in the classification process of each pixel Information is easy to appear small target area missing segmentation and fuzzy target boundary segmentation.
Based on the above situation, this paper proposes a model for segmentation of CT images of liver focus based on regional fitting and gradient information. It solves the problem that the common segmentation model of liver lesions image has a low accuracy in the segmentation of liver CT image with uneven gray scale, fuzzy boundary or absence. By combining local area fitting of liver lesions with image gradient information, it not only overcomes the uneven image gray level, but also enhances the edge detection ability of the model.
And then by introducing the area term, so that the segmentation curve in the evolution process to keep smooth, and effectively control the evolution direction and speed. These processing can effectively improve the accuracy of liver focus CT image segmentation, so that the model in this paper has excellent performance of liver focus segmentation.

Experimental materials and parameter Settings
In this study, the CT images of liver tumors were taken as the experimental objects, and the CT images were obtained from the public challenge dataset of the MICCAI liver tumor segmentation challenge. The experimental environment is Windows10 PC, the processor is inter (R) core i5-3230m 2.6GHz, the memory is 8GB, through C++ and Matlab2015 hybrid programming. In order to verify the effectiveness of the model in this paper, the DRLSE model, RSF model and the model in this paper were applied to the segmentation task of liver lesions in arterial phase images, and the segmentation performance was compared. can be clearly seen that the segmentation curve stops at the edge of the tumor more accurately, and the adjacent sites of the tumor and the blood vessels were also well differentiated.

Energy functional curve
In the experiment of image segmentation of three kinds of liver tumors mentioned above, we recorded the energy value of the model in this paper after each iteration, and made corresponding energy curves respectively. As shown in figure 4, the energy curve shows a downward trend in the evolution process and finally tends to be stable, that is, the energy of the model reaches the

Quantitative evaluation of segmentation results
In order to quantitatively evaluate the segmentation results, this paper adopts three commonly used

Conclusion
The segmentation accuracy of the liver lesion  Therefore, the model proposed in this paper, which is used to segment the CT images of liver lesions, can obtain higher accuracy and lower undersegmentation rate and oversegmentation rate,the segmentation accuracy of liver lesions was greatly improved.

Boundary based active contour model
The boundary-based active contour model controls the movement speed of the curve through the boundary information of the liver image. The Geodesic Active Model (GAC) proposed by Caselles and Kimmel et al. [10] is one of the most classical boundary-based Active contour models. Then Li et al. [19] proposed a level set model (DRLSE) for distance regularization based on GAC model. By introducing a distance regularization term to keep the evolving surface regular, the distance regularization term is introduced to keep the evolving surface regular and reduce the reinitialization step in the iterative process, thus the evolution of curve for segmenting liver focus is accelerated. The GAC model and its subsequent development are often applied to liver segmentation tasks [12,27,28,29] . These models,

Area based active contour model
The borderless active contour model proposed by Chan and Vese [16] is the most classical model based on the segmentation of CT images of the liver in the region, known as the Chan-Vese model. In various tasks related to liver segmentation [30,31] , The CV model uses an arbitrary curve C to divide the images of liver focus into two parts: the inside part and the outside part. Then the energy function is defined and minimized. When the value of the energy function is minimum, the image is divided into foreground and background.
Later, Li et al [17] proposed an active contour model (RSF) They are generally only suitable for global segmentation, and cannot be directly applied to such local segmentation tasks as liver lesions.

Segmentation model of liver focus image based on region fitting and gradient information
We propose a hybrid active contour model based on region fitting and gradient information, which can well solve the shortcomings of the above models, and has excellent segmentation ability in the face of the CT image of liver lesions with blurred boundary, partial missing and uneven gray level. We define the energy functional of the segmentation model as formula (1): In the formula, C represents any curve, which divides the CT image of liver focus into curve inside and curve outside.
() Fit C is the local area fitting term of liver CT image, () Grad C is the gradient information item of liver image. Different from the RSF model, the local area fitting term () Fit C only calculates the sum of the fitting values of all points on the segmentation curve C of liver lesions. The complete formula of local area fitting term is defined as formula (2): In this model, the gradient information item () Grad C of liver focus CT image is added, that is, the gradient energy of the pixels on the segmentation curve C is calculated by the liver focus boundary detection equation g , it overcomes the problem of easily falling into local minimum value when solving energy functional. The formula for defining the gradient information item is as follows (3): Because the segmentation curve C of liver lesions is unknown, it is difficult to solve the evolution equation directly. According to the thought of level set [32] , We use the level set function  to replace the arbitrary partition curve C, and the evolution of the plane curve is also replaced by the evolution of the three-dimensional surface. Then, the points on the three-dimensional surface and the points on the surface plane are mapped by introducing the heviside function () H  . Finally, Dirac function ()  is added to obtain the set of points on the zero horizontal plane of the curved surface. Thus, the energy functional can be expressed as formula (4) and (5): In order to keep the surface  stable as it evolves, it is necessary to reinitialize the level set surface  before each iteration, but it has also led to a large increase in the amount of computation. By adding distance regularization items p R into the model in this paper, the regularization term p R calculates the sum of the corresponding energy density values at the zero level set, so that the gradient of the three-dimensional surface is always 1 during the evolution process, that is, the value of the energy density function is minimum, so as to realize the acceleration of the evolution of the surface.
The definition formula is as follows (6): In order to keep the segmentation curve of liver focus smooth in the evolution process, we add the regional area term in this model as formula (7): In the area term, the detection equation g of the boundary of liver lesions is used as the dynamic coefficient, which can accelerate the evolution of the segmentation curve in the homogeneous region of liver lesions. However, in regions with a larger gradient, a smaller g value will slow down the evolution of the segmentation curve. Since we also defined the level set surface  , when the initial curve is defined outside the contour of the liver focus, the segmentation curve should be shrunk inward, that is, the area term coefficient should be less than zero. Inside the contour of the liver focus, the opposite is true. Therefore, the area term also has the effect of controlling the direction and speed of the movement of the segmentation curve.
In the actual segmentation of liver focus, the values on the three-dimensional surface function x M x y y I y u t