CN112712529A - Method and system for generating crystal position lookup table of PET (positron emission tomography) detector and PET equipment - Google Patents
Method and system for generating crystal position lookup table of PET (positron emission tomography) detector and PET equipment Download PDFInfo
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Abstract
The invention belongs to the technical field of medical imaging, and particularly relates to a method and a system for generating a crystal position lookup table of a PET (positron emission tomography) detector and PET equipment. The generation method of the PET detector crystal position lookup table comprises the following steps: and obtaining a target peak point coordinate value of the crystal position according to the theoretical crystal position model, obtaining the crystal position calculation peak point coordinate value after data acquisition and analysis by adopting a neural network, comparing the crystal position calculation peak point coordinate value with the target peak point coordinate value, and obtaining a final crystal position segmentation area so as to generate a crystal position lookup table. The invention solves the problem of the error segmentation of the crystal position point region, does not need manual intervention, can automatically segment all the crystal regions, and improves the generation efficiency of the crystal position lookup table.
Description
Technical Field
The invention belongs to the technical field of medical imaging, and particularly relates to a method and a system for generating a crystal position lookup table of a PET (positron emission tomography) detector and PET equipment.
Background
Positron Emission Tomography (PET) belongs to the more advanced imaging technology in the nuclear medicine imaging technology, and a component which directly influences the image resolution is a PET detector which is generally composed of a scintillation crystal array coupled photoelectric conversion device.
For a PET system, a crystal position lookup table is generally established to record a corresponding relationship between a gamma photon event position coordinate and a scintillation crystal, and in clinical application, the scintillation crystal which reacts with gamma photons can be determined through the crystal position lookup table, so that an actual physical position of the scintillation crystal is obtained for subsequent image reconstruction. Therefore, the accuracy of the crystal position lookup table directly affects the spatial resolution of the PET system.
In the prior art, there are two main methods for generating a crystal position lookup table of a PET detector:
firstly, based on a crystal position scatter diagram and a preset contour line threshold value, carrying out region segmentation on the crystal position scatter diagram, carrying out X-direction and Y-direction projection on the segmented region, and acquiring a lookup table corresponding to the crystal position;
secondly, extracting real response vertexes of the crystals in the crystal array based on the flood field image, numbering the real response vertexes according to the arrangement sequence of the crystals, and obtaining region segmentation with the crystal numbers, so that a crystal position lookup table is obtained.
Due to uncertainty of the contour line threshold, incomplete segmentation and mistaken segmentation of the position scatter diagram can be caused, so that the segmentation of the scatter diagram needs to be completed by means of manual operation, and updating of a position lookup table is not facilitated; moreover, the accuracy of the manual segmentation for the segmented regions is difficult to guarantee. In addition, due to factors such as environmental temperature change of the PET system, working voltage drift, gain change of the detector caused by long-term work and the like, a flood field image is easy to change, misjudgment is generated on the position of a scattered point, false scattered points are caused, accordingly, the segmentation boundary is blurred, and finally the accuracy of the segmentation region is difficult to guarantee.
Disclosure of Invention
Based on the above disadvantages and shortcomings of the prior art, it is an object of the present invention to at least solve one or more of the above problems of the prior art, in other words, to provide a method and system for generating a PET detector crystal position lookup table, and a PET apparatus, which satisfy one or more of the above requirements.
In order to achieve the purpose, the invention adopts the following technical scheme:
a method for generating a PET detector crystal position lookup table comprises the following steps:
s1, establishing a theoretical crystal position model: obtaining the distance D between adjacent crystal units and the central position P of each crystal unit according to the theoretical crystal array N x N and the position lookup table matrix M x M0i(ii) a Wherein D is M/N, N is an integer greater than 1, and M is an integer multiple of N;
s2, acquiring data based on the background radiation of the crystal, and acquiring a crystal position scatter diagram of each crystal array;
s3, inputting the crystal position scatter diagram into the neural network for training to obtain the position P of each crystal peak pointi；
S4, judging the position P of each crystal peak pointiWith the central position P of the corresponding crystal unit0iWhether the distance therebetween is satisfied
If yes, go to step S6; if not, go to step S5;
s5, obtaining the number K of crystal peak points of the row or column of the crystal array; if K is greater than N, increasing the training threshold of the neural network, and turning to the step S3; if K is less than N, reducing the training threshold of the neural network, and turning to the step S3;
s6, calculating distances between all points in the matrix and crystal peak points of corresponding areas and sequencing to obtain area dividing lines;
and S7, performing region division according to the region division line, and assigning crystal serial numbers to the regions after the region division to generate a crystal position lookup table.
Preferably, the following steps are further included between step S2 and step S3:
and S21, extracting the interested region of the crystal position scatter diagram of each crystal array to obtain a target crystal position scatter diagram of the crystal array.
Preferably, a binarization process is applied to the crystal position scattergram of the crystal array to extract the region of interest.
Preferably, the extracting of the region of interest includes:
collecting a gray scale maximum G in a crystal position scattergram of a crystal arraymaxAnd the minimum value of gray scale Gmin(ii) a Determining a threshold value
Setting pixel points of which the gray value is not less than a threshold T in a crystal position scatter diagram of the crystal array at 1, and setting pixel points less than the threshold T at 0;
and extracting an area with the pixel point of 1 in the crystal position scatter diagram of the crystal array as an interested area.
Preferably, the following steps are further included between step S2 and step S3:
and S22, performing distance transformation on the target crystal position scatter diagram of the crystal array to obtain a distance-transformed crystal position scatter diagram serving as the input of the neural network.
The invention also provides a system for generating the crystal position lookup table of the PET detector, which applies the method for generating the crystal position lookup table of the PET detector, and comprises the following steps:
the theoretical crystal position model comprises theoretical crystal array N and position lookup table matrix M, and adjacent crystal unitsDistance D and center position P of each crystal unit0i(ii) a Wherein, D is M/N;
the data acquisition module is used for acquiring data based on the background radiation of the crystal and acquiring a crystal position scatter diagram of each crystal array;
a neural network training module for inputting the crystal position scatter diagram into the neural network for training to obtain the position P of each crystal peak pointi；
A judging module for judging the position P of each crystal peak pointiWith the central position P of the corresponding crystal unit0iWhether the distance therebetween is satisfied
The method is also used for judging the relation between the number K of the crystal peak points of the rows or the columns of the crystal array and N;
the calculation module is used for calculating and sequencing the distances between all points in the matrix and the crystal peak points of the corresponding region to obtain region dividing lines;
and the generation module is used for carrying out region division according to the region division line and carrying out crystal serial number assignment on each region after the region division so as to generate a crystal position lookup table.
Preferably, the system for generating the PET detector crystal position lookup table further comprises:
and the extraction module is used for extracting the interested region of the crystal position scatter diagram of each crystal array so as to obtain a target crystal position scatter diagram of the crystal array.
Preferably, the extraction module extracts the region of interest by applying binarization processing to a crystal position scattergram of the crystal array.
Preferably, the system for generating the PET detector crystal position lookup table further comprises:
and the distance conversion module is used for performing distance conversion on the target crystal position scatter diagram of the crystal array to obtain a crystal position scatter diagram after the distance conversion, and the crystal position scatter diagram is used as the input of the neural network.
The invention also provides a PET device, which applies the method for generating the PET detector crystal position lookup table according to any one of the above aspects or configures the system for generating the PET detector crystal position lookup table according to any one of the above aspects.
Compared with the prior art, the invention has the beneficial effects that:
the method comprises the steps of obtaining a target peak point coordinate value of a crystal position based on a theoretical crystal position distribution model, obtaining a position coordinate value of a crystal peak point collected in real time by adopting a neural network, and comparing the position coordinate value of the crystal peak point with the target peak point coordinate value to obtain a final crystal position segmentation area; the problem of the regional mistake of crystal position is cut apart is solved to and need not artificial intervention, all crystal regional divisions can be realized automatically, improve work efficiency.
Drawings
FIG. 1 is a flow chart of a method for generating a crystal position lookup table of a PET detector according to embodiment 1 of the invention;
FIG. 2 is a schematic diagram of the construction of a PET detector of embodiment 1 of the invention;
FIG. 3 is a schematic view showing the distribution of the central positions of the crystal units in example 1 of the present invention;
FIG. 4 is a crystal position scattergram of a collected crystal array of example 1 of the present invention;
fig. 5 is a scattergram of target crystal positions for extracting a region of interest according to embodiment 1 of the present invention;
FIG. 6 is a crystal position scattergram after distance conversion of example 1 of the present invention;
FIG. 7 is a schematic diagram showing the positions of the crystal peak points of the neural network output in example 1 of the present invention;
FIG. 8 is a schematic diagram showing the division of the crystal according to example 1 of the present invention in which the peak point position is abnormal;
FIG. 9 is a schematic view of the crystal position dividing region of example 1 of the present invention;
FIG. 10 is a schematic diagram of a crystal position lookup table of example 1 of the present invention;
FIG. 11 is a block diagram of a system for generating a crystal position lookup table of a PET detector according to embodiment 1 of the present invention.
Detailed Description
In order to more clearly illustrate the embodiments of the present invention, the following description will explain the embodiments of the present invention with reference to the accompanying drawings. It is obvious that the drawings in the following description are only some examples of the invention, and that for a person skilled in the art, other drawings and embodiments can be derived from them without inventive effort.
Example 1:
as shown in fig. 1, the method for generating a PET detector crystal position lookup table of the present embodiment includes the following steps:
s1, obtaining the distance D between adjacent crystal units and the central position P of each crystal unit according to the theoretical crystal array N and the position lookup table matrix M0i(ii) a Wherein D is M/N, N is an integer greater than 1, and M is an integer multiple of N;
specifically, as shown in fig. 2, the PET system of the present embodiment has 36 large modules, each of which has 24 detector modules, and each of the detector modules is composed of an 8 × 8 crystal array, i.e., the PET system has 55296 pixels.
The theoretical crystal position model means that the actual crystal array is partitioned at equal intervals according to a certain matrix; as shown in fig. 3, the present embodiment is described in detail with the value of N being 8 and the value of M being 512; correspondingly, according to the position lookup table matrix, an equidistant partition is calculated, the distance D between adjacent crystal units is 512/8-64, and the central position of each crystal unit is defined as P0i。
And S2, acquiring data based on the background radiation of the crystal, and acquiring a crystal position scatter diagram of each crystal array, as shown in FIG. 4.
S3, extracting the interested region of the crystal position scatter diagram of each crystal array to obtain a target crystal position scatter diagram of the crystal array;
specifically, the method for extracting the region of interest by using binarization processing on a crystal position scatter diagram of a crystal array specifically comprises the following steps:
collecting maximum gray in a crystal position scattergram of a crystal arrayValue GmaxAnd the minimum value of gray scale Gmin(ii) a Determining a threshold value
Setting pixel points of which the gray value is not less than a threshold T in a crystal position scatter diagram of the crystal array at 1, and setting pixel points less than the threshold T at 0;
and extracting an area with a pixel point of 1 in the crystal position scatter diagram of the crystal array as an area of interest, namely obtaining a target crystal position scatter diagram of the crystal array, as shown in fig. 5.
S4, performing distance transformation on the target crystal position scatter diagram of the crystal array to obtain a crystal position scatter diagram after distance transformation, wherein the crystal position scatter diagram is used as input of a neural network, and the accuracy of subsequent region segmentation is improved;
specifically, the distance transformation formula is:
(xi，yi)、(xj，yj) The coordinates of two pixel points in the crystal position scattergram are respectively obtained, so as to obtain a crystal position scattergram after distance transformation, as shown in fig. 6.
S5, inputting the crystal position scatter diagram after distance conversion into a neural network for training to obtain the position P of each crystal peak pointi(ii) a As shown in fig. 7.
Specifically, the objective function model of the neural network is:
wherein, XiData (x) of a crystal position scattergram input for a neural networki，yi)，YiPosition P of crystal peak point output by neural networkiOf (i), i.e. (P)ix，Piy) (ii) a n is the total number of data of the crystal position scattergram, ωiAnd weighting factors corresponding to the data of the crystal position scatter diagram.
In addition, the neural network may be an existing commonly used neural network, for example, a convolutional neural network, and a corresponding neural network model is obtained through pre-training, so as to realize input of a crystal position scattergram and output of a crystal peak point position. By adopting the neural network algorithm, the artificial participation can be reduced, and the region boundary can be ensured to be clearer.
S6, judging the position P of each crystal peak pointiWith the central position P of the corresponding crystal unit0iWhether the distance therebetween is satisfied
To be a partition or not; if yes, i.e. the crystal peak point is located in the corresponding divided region, go to step S8; if not, namely the crystal peak point is outside the corresponding crystal unit dividing area, the step is switched to step S7;
s7, obtaining the number K of crystal peak points of the row or column of the crystal array; if K is greater than N, increasing the training threshold of the neural network, and turning to the step S5; if K is less than N, reducing the training threshold of the neural network, and turning to the step S5;
specifically, if the number of crystal peak points in a row or a column of the crystal array is greater than 8, the training threshold of the neuron corresponding to the neural network needs to be increased, and the position coordinates of the crystal peak points in the corresponding region are optimized; if the number of the crystal peak points of the rows or columns of the crystal array is less than 8, as shown in fig. 8, the training threshold of the neurons corresponding to the neural network needs to be reduced, and the position coordinates of the crystal peak points in the corresponding region need to be optimized; jumping to step S5, the corresponding step operation is continued.
S8, calculating distances between all points in the matrix and crystal peak points of corresponding areas and sequencing to obtain area dividing lines; specifically, the euclidean distances between all the points in the matrix and the crystal peak points of the corresponding region are calculated and sorted, that is, the region boundary surrounded by the set of points whose X-axis and Y-axis distances from all the points in the matrix to the crystal peak points of the corresponding region are 32 is calculated as the region dividing line, as shown in fig. 9.
S9, dividing the regions according to the division lines, assigning the crystal numbers to the regions, and generating a crystal position lookup table, as shown in fig. 10.
Corresponding to the above method for generating a crystal position lookup table of a PET detector, as shown in fig. 11, the embodiment further provides a system for generating a crystal position lookup table of a PET detector, including:
the theoretical crystal position model 100 comprises theoretical crystal array N and position lookup table matrix M, spacing D between adjacent crystal units and central position P of each crystal unit0i(ii) a Wherein, D is M/N; n is an integer greater than 1, and M is an integer multiple of N;
specifically, as shown in fig. 2, the PET system of the present embodiment has 36 large modules, each of which has 24 detector modules, and each of the detector modules is composed of an 8 × 8 crystal array, i.e., the PET system has 55296 pixels.
The theoretical crystal position model means that the actual crystal array is partitioned at equal intervals according to a certain matrix; as shown in fig. 3, the present embodiment is described in detail with the value of N being 8 and the value of M being 512; correspondingly, according to the position lookup table matrix, an equidistant partition is calculated, the distance D between adjacent crystal units is 512/8-64, and the central position of each crystal unit is defined as P0i。
And the data acquisition module 200 is configured to acquire data based on background radiation of the crystal itself and obtain a crystal position scattergram of each crystal array, as shown in fig. 4.
An extraction module 300, configured to extract an area of interest from the crystal position scattergram of each crystal array to obtain a target crystal position scattergram of the crystal array;
specifically, the method for extracting the region of interest by using binarization processing on a crystal position scatter diagram of a crystal array specifically comprises the following steps:
collecting a gray scale maximum G in a crystal position scattergram of a crystal arraymaxAnd the minimum value of gray scale Gmin(ii) a Determining a threshold value
Setting pixel points of which the gray value is not less than a threshold T in a crystal position scatter diagram of the crystal array at 1, and setting pixel points less than the threshold T at 0;
and extracting an area with a pixel point of 1 in the crystal position scatter diagram of the crystal array as an area of interest, namely obtaining a target crystal position scatter diagram of the crystal array, as shown in fig. 5.
A distance transformation module 400, configured to perform distance transformation on the target crystal position scattergram of the crystal array to obtain a distance-transformed crystal position scattergram, which is used as an input of the neural network;
specifically, the distance transformation formula is:
(xi，yi)、(xj，yj) The coordinates of two pixel points in the crystal position scattergram are respectively obtained, so as to obtain a crystal position scattergram after distance transformation, as shown in fig. 6.
A neural network training module 500, configured to input the crystal position scatter diagram into the neural network for training, so as to obtain a position P of each crystal peak pointiAs shown in fig. 7.
Specifically, the objective function model of the neural network is:
wherein, XiData (x) of a crystal position scattergram input for a neural networki，yi)，YiPosition P of crystal peak point output by neural networkiOf (i), i.e. (P)ix，Piy) (ii) a n is the total number of data of the crystal position scattergram, ωiAnd weighting factors corresponding to the data of the crystal position scatter diagram.
In addition, the neural network may be an existing commonly used neural network, for example, a convolutional neural network, and a corresponding neural network model is obtained through pre-training, so as to realize input of a crystal position scattergram and output of a crystal peak point position.
A judging module 600 for judging the position P of each crystal peak pointiWith the central position P of the corresponding crystal unit0iWhether the distance therebetween is satisfied
The method is also used for judging the relation between the number K of the crystal peak points of the rows or the columns of the crystal array and N;
specifically, the position P of each crystal peak point is judgediWith the central position P of the corresponding crystal unit0iWhether the distance therebetween is satisfied
To be a partition or not; if yes, the crystal peak point is located in the corresponding segmentation area; if not, namely the crystal peak point is outside the corresponding crystal unit partition area, the training threshold of the neurons of the neural network needs to be reset at this time, so as to perform the training of the position of the crystal peak point again.
In addition, the judging module is used for judging the relation between the number K of the crystal peak points of the rows or the columns of the crystal array and N; if K is larger than N, improving the training threshold of the neural network; and if K is smaller than N, reducing the training threshold of the neural network. Specifically, if the number of crystal peak points in a row or a column of the crystal array is greater than 8, the training threshold of the neuron corresponding to the neural network needs to be increased, and the position coordinates of the crystal peak points in the corresponding region are optimized; if the number of the crystal peak points in the row or column of the crystal array is less than 8, as shown in fig. 8, it is necessary to reduce the training threshold of the neuron corresponding to the neural network, and optimize the position coordinates of the crystal peak points in the corresponding region.
A calculation module 700, configured to calculate distances between all points in the matrix and crystal peak points in corresponding regions and sort the distances to obtain region dividing lines;
specifically, the euclidean distances between all the points in the matrix and the crystal peak points of the corresponding region are calculated and sorted, that is, the region boundary surrounded by the set of points whose X-axis and Y-axis distances from all the points in the matrix to the crystal peak points of the corresponding region are 32 is calculated as the region dividing line, as shown in fig. 9.
A generating module 800, configured to perform region segmentation according to the region segmentation line, and perform crystal number assignment on each region after the region segmentation to generate a crystal position lookup table, as shown in fig. 10.
In addition, the embodiment further provides a PET apparatus, which applies the method for generating the PET detector crystal position lookup table of the embodiment or configures the system for generating the PET detector crystal position lookup table of the embodiment, so as to improve the detection accuracy of PET.
Example 2:
the difference between the method for generating the crystal position lookup table of the PET detector of the present embodiment and the embodiment 1 is that:
the step S3 and the step S4 are omitted, namely, the crystal position scatter diagram of the crystal array is directly input into the neural network for training, the training data volume is increased, the training precision is reduced, and the requirements of different applications can be met.
Other procedures can be referred to example 1;
accordingly, the generation system of the PET detector crystal position lookup table of the present embodiment is different from that of embodiment 1 in that:
an extraction module and a distance conversion module are omitted, the framework of the system is simplified, and the requirements of different applications are met.
Other architectures can refer to example 1.
The PET apparatus of this embodiment applies the method for generating the PET detector crystal position lookup table of this embodiment or configures the system for generating the PET detector crystal position lookup table of this embodiment.
Example 3:
the difference between the method for generating the crystal position lookup table of the PET detector of the present embodiment and the embodiment 1 is that:
step S4, namely the step of distance transformation, is omitted, the flow of the generation method is simplified, and the requirements of different applications are met.
Other procedures can be referred to example 1;
accordingly, the generation system of the PET detector crystal position lookup table of the present embodiment is different from that of embodiment 1 in that:
and a distance conversion module is omitted, so that the framework of the system is simplified, and the requirements of different applications are met.
Other architectures can refer to example 1.
The PET apparatus of this embodiment applies the method for generating the PET detector crystal position lookup table of this embodiment or configures the system for generating the PET detector crystal position lookup table of this embodiment.
The foregoing has outlined rather broadly the preferred embodiments and principles of the present invention and it will be appreciated that those skilled in the art may devise variations of the present invention that are within the spirit and scope of the appended claims.
Claims (10)
1. A method for generating a PET detector crystal position lookup table is characterized by comprising the following steps:
s1, obtaining the distance D between adjacent crystal units and the central position P of each crystal unit according to the theoretical crystal array N and the position lookup table matrix M0i(ii) a Wherein D is M/N, N is an integer greater than 1, and M is an integer multiple of N;
s2, acquiring data based on the background radiation of the crystal, and acquiring a crystal position scatter diagram of each crystal array;
s3, inputting the crystal position scatter diagram into the neural network for training to obtain the position P of each crystal peak pointi；
S4, judging the position P of each crystal peak pointiWith the central position P of the corresponding crystal unit0iWhether the distance therebetween is satisfied
If yes, go to step S6; if not, go to step S5;
s5, obtaining the number K of crystal peak points of the row or column of the crystal array; if K is greater than N, increasing the training threshold of the neural network, and turning to the step S3; if K is less than N, reducing the training threshold of the neural network, and turning to the step S3;
s6, calculating distances between all points in the matrix and crystal peak points of corresponding areas and sequencing to obtain area dividing lines;
and S7, performing region division according to the region division line, and assigning crystal serial numbers to the regions after the region division to generate a crystal position lookup table.
2. The method as claimed in claim 1, wherein the step S2 and the step S3 further include the following steps:
and S21, extracting the interested region of the crystal position scatter diagram of each crystal array to obtain a target crystal position scatter diagram of the crystal array.
3. The method for generating the crystal position lookup table of the PET detector as claimed in claim 2, characterized in that a binarization process is applied to the crystal position scatter diagram of the crystal array to extract the region of interest.
4. The method as claimed in claim 3, wherein the extracting of the region of interest comprises:
collecting a gray scale maximum G in a crystal position scattergram of a crystal arraymaxAnd the minimum value of gray scale Gmin；
Determining a threshold value
Setting pixel points of which the gray value is not less than a threshold T in a crystal position scatter diagram of the crystal array at 1, and setting pixel points less than the threshold T at 0;
and extracting an area with the pixel point of 1 in the crystal position scatter diagram of the crystal array as an interested area.
5. The method for generating the PET detector crystal position lookup table according to any one of claims 2-4, wherein the step S2 and the step S3 further include the following steps:
and S22, performing distance transformation on the target crystal position scatter diagram of the crystal array to obtain a distance-transformed crystal position scatter diagram serving as the input of the neural network.
6. A generation system of a PET detector crystal position lookup table, which applies the generation method of the PET detector crystal position lookup table according to claim 1, and is characterized by comprising:
the theoretical crystal position model comprises theoretical crystal array N and position lookup table matrix M, spacing D between adjacent crystal units and central position P of each crystal unit0i(ii) a Wherein, D is M/N;
the data acquisition module is used for acquiring data based on the background radiation of the crystal and acquiring a crystal position scatter diagram of each crystal array;
a neural network training module for inputting the crystal position scatter diagram into the neural network for training to obtain the position P of each crystal peak pointi；
A judging module for judging the position P of each crystal peak pointiWith the central position P of the corresponding crystal unit0iWhether the distance therebetween is satisfied
The method is also used for judging the relation between the number K of the crystal peak points of the rows or the columns of the crystal array and N;
the calculation module is used for calculating and sequencing the distances between all points in the matrix and the crystal peak points of the corresponding region to obtain region dividing lines;
and the generation module is used for carrying out region division according to the region division line and carrying out crystal serial number assignment on each region after the region division so as to generate a crystal position lookup table.
7. The system for generating a crystal position lookup table for a PET detector as claimed in claim 6, further comprising:
and the extraction module is used for extracting the interested region of the crystal position scatter diagram of each crystal array so as to obtain a target crystal position scatter diagram of the crystal array.
8. The system for generating the crystal position lookup table of the PET detector as claimed in claim 7, wherein the extraction module applies binarization processing to the crystal position scatter diagram of the crystal array to extract the region of interest.
9. The system for generating a crystal position lookup table for a PET detector as claimed in claim 6, further comprising:
and the distance conversion module is used for performing distance conversion on the target crystal position scatter diagram of the crystal array to obtain a crystal position scatter diagram after the distance conversion, and the crystal position scatter diagram is used as the input of the neural network.
10. A PET apparatus, characterized by applying the method for generating a PET detector crystal position lookup table according to any one of claims 1 to 5 or configuring the system for generating a PET detector crystal position lookup table according to any one of claims 6 to 9.
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