Introduction

The segmentation of anatomical structures of the human brain, present in images acquired by any imaging modality, constitutes the starting point for the diagnosis of a high number of diseases or pathologies that affect the brain, among these are intracranial hematomas which can be classified in various ways^1,2. On the other hand, digital neural images are accompanied by various imperfections such as noise and artifacts^3,4,5. These imperfections become real challenges, when computational strategies are implemented to generate the morphology (normal or abnormal) of the mentioned structures.

Additionally, when reviewing the state of the art regarding the segmentation of EDH, the works described below were found. The work described by Liao et al.⁶ shows an automatic computational technique for the segmentation of cerebral hematomas, considering the modality of multiscale computer tomography (MSCT). The aforementioned technique consists of: a) Application of a filtering stage, based on the maximum filter (MaxF) and the mean filter (MF). b) Use of multiresolution Gaussian pyramids (MRGP) generated from the filtered images. c) Implementation of multiresolution level sets (MRLS) to obtain the EDH segmentations. They report an average Dice coefficient⁷ of 0.9140 for the segmented EDHs.

Recently, Kamnitsas et al.8, reported an automatic technique based on convolutional neural networks, that apply deep learning, for the segmentation of space-occupying lesions that include EDHs present in multimodality medical images. They report a Dice Coefficient (Dc) greater than 0.90 in the segmentation of this type of injury.

On the other hand, the present work presents a non-linear computational technique (NLCT) for the segmentation of epidural hematomas, present in 7 databases conformed by three-dimensional brain images of MSCT. The aforementioned technique considers the stages of pre-processing, segmentation, post-processing and quantification of EDH, via volume occupied by the hematoma in the skull.

Materials and methods

A. Database: The databases (DB) used were supplied by the Central Hospital of San Cristóbal-Táchira-Venezuela. They were acquired through the MSCT modality and are constituted by three-dimensional images (3D), corresponding to the anatomical structures present in the head of 7 male patients.

B. Proposed computational technique for the automatic segmentation of epidural hematomas

In figure 1, a schematic diagram synthesizes the methods that make up the proposed technique, in the present investigation, to segment the aforementioned hematomas.

Figure 1.
Block diagram of the NLCT proposed for epidural hematomas segmentation.

C. Pre-processing stage:

Defining a volume of interest (VOI) which contains each hematoma to be considered: In figure 1, the phase which establishes a volume of interest corresponds to the block called Thresholding. Thresholding algorithms are, generally, simple structures and allow for efficient classification of the elements of an image considering one or more thresholds.. In the present work simple thresholding was used, allowing for discrimination between the anatomical structure of interest and the rest of the structures present in an image^9,10.

Filtering phase: In the block diagram, presented in figure 1, this phase corresponds to the morphological erosion and median filters. A brief explanation of both filters follows.

- Morphological Erosion Filter (MEF):

Mathematical morphology^11,12 is implemented in practice, through various morphological filters whose basic operators are erosion and dilation¹³. These operators are non-linear spatial filters that can be applied to binary, grayscale or color images. In particular, the erosion (⊖) of a two-dimensional image (I), composed of gray levels, using a two-dimensional structuring element (SE), as shown in Equation 1^14,15.

Where min is the minimum gray level contained in SE, (s, t) and defines the size of the SE and (x, y) represents the position of the pixel under study.

Median Filter (MF):

The median filter (MF) is also non-linear and normally, it is used to minimize the impulsive type noise present in the gray levels of the voxels neighboring the voxel object of study¹⁶. This type of filter is characterized by the conservation of the edges of the objects present in the image and it has the advantage that the final value, of the voxel, is a real value present in the image and not an average. In addition, the median filter is less sensitive to extreme values. One of the main drawbacks is that the computation time increases, substantially, as the size of the neighborhood increases¹⁷.

D. Segmentation stage

Computer intelligence operators: Support vector machines (SVM).

Support vector machines (SVM) are paradigms that undergo training and detection processes, and are based on both the Vapnik-Chervonenkis learning theory and the minimization principle that considers structural risk¹⁸. SVMs can be considered as classification and functional approach tools^19,20.

A variant of the SVM, called the least squares support vector machine (LSSVM), can be obtained using robust statistics, Fisher discriminant analysis and replacing the inequation system that governs the SVM, by an equivalent system of linear equations, which can be solved more efficiently^21,22,23.

In this work, the location of the seed voxel, to initialize the segmentation technique called Region Growing¹⁶ (RG), is calculated using LSSVM. For the purposes of this work, a radial base function (RBF) is considered in the LSSVM as the decision surface and, therefore, a formulation is obtained that depends on the hyperparameters identified as: a) Parameter for the error penalty (γ). b) To control the selectivity (σ²) of the LSSVM.

To automatically identify the coordinates of the seed voxel, the following procedure was implemented:

i) A size reduction technique is applied, based on bicubic interpolation, optimal reduction factor, is matched with that obtained in [3]. This allows to generate sub-sampled images of 64x64 pixels from filtered images of 512x512.

ii) A neurosurgeon selects, on the sub-sampled image, a reference point (P1) given by the centroid of the layer containing the maximum blood pool occupied by the EDH. For this point, the manual coordinates that unambiguously establish their spatial location in each considered image are identified.

iii) A LSSVM is implemented to recognize and detect point P1. For this, the processes of:

Training. Training circle circular neighborhoods of 10 pixels are selected, manually traced by a neurosurgeon, containing both point P1 (markers) and regions not containing P1 (no markers) are selected as a training set.

Then, each neighborhood is vectorized and, considering its gray levels, the attributes are calculated: mean, variance, standard deviation and median. Thus, both markers and non-markers are described by vectors (Va) of statistical attributes, given by: Va = [mean, variance, standard deviation and median]. Additionally, the LSSVM is trained considering the vectors Va as a training pattern and intoning the values of the parameters that control its performance, γ and σ².

The training set is constructed with a ratio of 1:10, which means that 10 non-markers are included for each marker. During training, a classifier with a decision boundary is generated to detect LSSVM entry patterns as markers or non-markers. Subsequently, due to the presence of false positives and negatives, a process is applied that allows incorporating into the training set the patterns that the LSSVM initially classifies inappropriately.

In this sense, we considered a toolbox called LS-SVMLAB and the Matlab15 application to implement an LSSVM classifier based on a radial base gaussian kernel with parameters σ² and γ.

Validation. The trained LSSVMs are used to detect P1, in images not used during training. To do this, a trained LSSVM looks for this reference point, in the axial view, from the first to the last image that makes up each of the 7 databases considered.

Finally, as a synthesis, figure 2 illustrates the process followed to locate the seed voxel in the databases considered.

Figure 2.
Synthetic diagram that illustrates the operation of the LSSVM for the detection of seed voxel coordinates

Unsupervised clustering: Region Growing (RG).

Region Growing is an unsupervised grouping technique, which performs an iterative process that tries to characterize each of the classes, according to the similarity between the voxels that integrate each of them and thus perform the segmentation¹⁶. The RG requires a "seed" point which can be selected, manually or automatically, to extract all the pixels connected to said seed16. To apply the RG, to the pre-processed images, the following considerations were made: a) The initial neighborhood, which is constructed from the seed, is assigned a cubic shape whose side depends on an arbitrary scalar r. The r parameter requires an intonation process. b) As a pre-defined criterion, modeling is chosen through Equation 2.

|I(x) − µ| < mσ (2)

Where I(x) is the intensity of the seed voxel, μ and σ the arithmetic mean and the standard deviation of the gray levels of the initial neighborhood and m a parameter that requires intonation.

Binary dilation filter

In order to compensate the effect produced by the morphological erosion filter, the application of a morphological dilation filter (MDF) is considered, taking into account the binary image obtained by RG. The effect of morphological dilation is to enlarge the regions of the maximum intensity image. In particular, the dilation (⊕) of a two-dimensional binary image (Ib), using a bidimensional structuring element (B), is defined as the result of operating the Ib with the values of the SE under the logical operation OR¹⁴. For the purposes of this work, a cubic structuring element was considered and the size of said SE is left as a parameter to control the performance of the dilatation process.

Tuning parameters: Obtaining optimal parameters

The adequate performance of the proposed technique requires obtaining optimal parameters for each of the algorithms that comprise it. To do this, Database 1 (DB1) was used as a reference, the parameters associated with the technique are modified which you wish to intone by systematically going through the values belonging to certain ranges, as described below.

The erosion, median and expansion filters have as parameter the size of the observation window. In order to reduce the number of possible combinations, an isotropic approach was considered to establish the range of values, which control the size of the aforementioned window, which is given by the odd combinations, given by the following ordered lists: (1,1,1), (3,3,3), (5,5,5), (7,7,7) and (9,9,9).

The parameters of the LSSVM, σ² and γ, are toned assuming that the cost function is convex and developing tests based on the following steps:

In order to tone the parameter γ, the value of σ² is arbitrarily set and values are systematically assigned to the parameter γ. The value of σ² is initially set at 2,5. Now, γ is varied considering the range [0,100] and a step size of 0.25. An analogous process is applied to tone the parameter σ², that is, it is assigned to γ the optimal value obtained in the previous step and a step size of 0.25 is considered to assign to σ the range of values contained in the interval [0.50].

During the intonation of the parameters of the RG, each one of the automatic segmentations of the EDH corresponding to the DB1 described, are compared with the manual segmentations of the EDH generated by a neurosurgeon, considering the Dc. The optimal values for the parameters of the RG (r and m), are matched to that experiment that generates the highest value for the Dc

The Dc is a metric that allows to compare segmentations of the same 2D or 3D image, obtained by different methodologies. In the medical context, usually, the Dc is considered to establish how similar are, spatially, manual segmentation (RD) and automatic segmentation (RP) that generates the morphology of any anatomical structure. Additionally, the Dc is maximum when a perfect overlap between RD and RP is reached but it is minimal when RD and RP do not overlap at all. In addition, the expected values for the Dc are real numbers between 0 and 1. The mathematical model that defines the Dc is given by Equation 3.

On the other hand, table 1 presents the information related to the parameters obtained after applying the tuning stage.

Obtaining the volumes related to manual segmentations: The volume of the hematomas, present in the original images, can be estimated manually by applying the following protocol, applicable only to epidural or intraparenchymal hematomas:

It is assumed that the hematoma, candidate to be characterized, has an ellipsoidal shape. The volume of the hematoma is estimated, considering the axial layers, dividing by 2 the multiplication of the lengths A, B and C; where A and B are the lengths of the major and minor axis of the hematoma, respectively. Such lengths are measured in the image with the "pool" of blood, linked to the hematoma, of greater area. The length C is obtained by multiplying the number of layers, in which the hematoma is present, by the thickness (E) of each layer.

Obtaining the volumes related to the automatic segmentations: The proposed technique generates the automatic segmentations of the EDH present in each of the 7 databases described. From such segmentations, the volume of the hematoma, candidate to be characterized, is calculated by multiplying the voxel dimensions by the number of voxels that make up the automatically segmented EDH.

Clinical utility of the volumes occupied by the hematomas: The main clinical utility of the characterization of the hematomas by obtaining the volume lies in the decision making that is made to establish the behavior to be followed to address the presence of the hematomas in a patient^24,25.

Results

Quantitative results: Regarding the trained LSSVM, the values of 2.50 and 0.25, respectively, were obtained as optimal parameters for γ and σ². The maximum value of the Dc obtained for the segmentation of the EDH is comparable with that reported in references^6,8, as shown in table 2.

Qualitative results: Figure 3, shows a 2-D view of both the original image and the processed versions after applying the NLCT technique

Figure 3
Axial view of images belonging to DB1: a) Original, b) Thresholdized, c) Eroded, d) Median, e) Segmented, f) Postprocessed with the binary morphological dilation filter.

Additionally, table 3 shows the values for the volume calculated considering the automatic segmentations of the EDHs.

It is important to note that one of the main clinical utilities of these numeric values, for the volumes, is that they define (in a high percentage) the behavior that is to be followed regarding the patient. Additionally, if only volume is considered, hematomas that exceed the threshold of 30 cm³ are susceptible to surgery. In this case, the patients represented by the databases from 2 to 6, according to the calculated volume should be taken to the operating room.

Conclusions

We have presented a non-linear computational technique whose intonation allows a precise segmentation of cerebral epidural hematomas, present in computed tomography images, since the Dc obtained is comparable with that reported in the literature.

The use of intelligent operators, represented by the least squares support vector machines, allowed for the automatic identification of the coordinates corresponding to the seed voxels that play a crucial role in the adequate initialization of the unsupervised grouping algorithm based on growth of regions.

The segmentations generated, automatically, by the proposed computational technique allows for the calculation of the volume of each hematoma considered. This volume is vital when deciding if a patient is surgically treated or not to address the hematoma that affects their health status.

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Author notes

m.avera@unisimonbolivar.edu.co