TWO LEVEL FUZZY LATTICE (2L-FL) SUPERVISED CLUSTERING : A NEW METHOD FOR SOFT TISSUE IDENTIFICATION IN SURGERY1 Vassilios Petridis(1), Vassilis G. Kaburlasos(1), Peter Brett (2), Tim Parker (2) (1)
Department of Electrical and Computer Engineering Aristotle University of Thessaloniki GR-54006 Thessaloniki, GREECE
(2)
Advanced Manufacturing and Automation Research Center AMARC, University of Bristol 1st Floor, Fanum House, 26-32 Park Row, Bristol BS1 5LY, U.K.
[email protected] [email protected] Abstract : A new technique is presented towards the intelligent control of a surgical tool to be used for the epidural puncture. Specifically the problem of the accurate soft tissue identification from laser-based spectral data was dealt with successfully. The proposed new technique, namely two level fuzzy lattice (2LFL) supervised clustering, employs tools and methods from the novel theory of fuzzy lattices and it might be helpful towards a mechatronic solution to the problem of accurate tool placement through flexible tissues in surgery. 1 - THE MEDICAL PROBLEM IN PERSPECTIVE The precision to which non-invasive measurements can be achieved in medicine through X-ray, high resolution computerized tomography and magnetic resonance imaging scans, has increased in many cases beyond the capabilities of the accurate deployment of surgical tools by manual methods. There is a need to be able to work with comparable accuracy in surgery, particularly as there is a move towards less invasive techniques. The more general problem of controlling precisely the position of a surgical tool in compliant tissues has been considered. Frequently in such situations the position of a tissue interface or of objects such as blood vessels and nerves will not be known in advance as most mediums are of unknown thickness and the position relative to the penetrating tool may change due to deformation of the medium, therefore the precise position will need to be determined in real-time. An approach which has been adopted by the authors, towards identifying accurately the location of a surgical tool, is to equip the penetrating tool with sensing components capable of identifying (or allowing the surgeon to identify) and reacting to the tissue type at the tool tip. The principal objective of this paper is to demonstrate a new and efficient tissue discrimination technique that might enable controlled (a) Lower back. penetration of soft tissues in surgery. A concrete soft tissue surgical procedure was indicates the approximate needle trajectory considered, that is the epidural puncture. The epidural puncture to the spinal cavity involves the insertion of a needle through different layers of visco-elastic tissue into a fluid cavity and the delivery of anesthesia [8] as shown in Figure F1. This is currently a manual procedure relying on the interpretation of tactile force data to determine when the process is complete. The epidural puncture has been considered as a simplified version of procedures involving biopsies to soft tissue organs of the abdomen or discectomy within the spinal cord. In such, more complex situations it (b) Lower back section. will be important to alert the user to the presence of nerve tissues and major blood vessels. An automated tool has been Figure F1 The anatomy of the epidural envisaged to control penetration through soft tissues. Hence, puncture site. there is a need to investigate appropriate sensing and decision making methods for this purpose. 1
This work was supported in part by Brite Euram Project BE7470
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2 - STATE OF THE ART AND CLAIMED INNOVATION The development of tools for automatic handling and cutting under variable working conditions is a topic of current research in a number of industrial application sectors such as textiles, and more recently to food and agriculture [1], [10]. Tools are deployed by a system embodying the integration of enabling technologies such as sensors, manipulators, automatic control schemes and expert systems, that is a mechatronic system. The principle difficulties that have been experienced typically include a great deal of processing sensory data, particularly vision, and the resolution that can be achieved from tactile data is often coarse, making for difficult control. The application of such new technology in surgery can offer many benefits. However, there are other hurdles, particularly safety [2], [4] and potential complexity of some working environments within the body that have reduced progress. Although various examples have demonstrated some of the benefits of applying these technologies, tool positioning has not been adaptive but has relied on predefined trajectory data. The need to perform a wider range of procedures with added safety requires tool reaction to sensory data and an ability to discriminate tissue types. The sensory method considered in this paper is laser-based spectroscopy and specifically the technique of Raman spectroscopy which is well established in the area of analytical chemistry [9]. When applied to biological soft tissues it was found that the degree of fluorescence as well as the detailed Raman structure could be useful in tissue identification [8]. To pursue an efficient sensory data interpretation we proceeded to the development of a novel and inherently hierarchical framework for global decision making that may handle with mathematical consistency disparate sensory data, for example images, waveforms, logic statements etc. Within this framework, namely the framework of fuzzy lattices, a specific technique was articulated called “two level fuzzy lattice supervised clustering” or 2L-FL Supervised Clustering in short, with the ability to capture the knowledge from previous operations, collect clinical and physiological data from the patient, and the potential to advise on the course of possible manipulator actions, perform early warning functions and invoke predefined safety actions. 3 - FUZZY LATTICES : A NOVEL FRAMEWORK FOR DECISION MAKING The novel framework for decision making, developed by Petridis, Kaburlasos, is based on fuzzy lattices. A mathematical lattice implies a relation of partial ordering among its elements. That is if x,y∈ then it might be x≤y, or y≤x, or x and y might not be related at all [5]. When x ≤y we say conventionally that x is included in y. Therefore the inclusion relation ≤ between x and y, x,y∈ is binary-valued since it is either 1 (it holds) or 0 (it does not hold). Note that x ∧y is the greatest lattice element included in both x and y, where ∧ denotes the lattice meet, whereas x∨y is the least lattice element that includes both x and y, where ∨ denotes the lattice join [3]. We are concerned only with complete lattices, that is lattices with a least and a greatest element denoted by O and I respectively. We proceeded by fuzzifying the conventionally binary-valued lattice-inclusion-relation ≤ which hence became applicable to any pair of lattice elements. To this end we have introduced the novel concept of inclusion measure σ(x,w), as a function from × to the interval [0,1] that indicates the degree of truth of the conventional lattice relation x ≤w. The function σ(x,w) ≡ σ(x≤w) should satisfy the following conditions (C1) σ(x≤O) = 0, x≠O, (C2) σ(x≤I) = 1, ∀x∈ , and (C3) u≤w ⇒ σ(x≤u) ≤ σ(x≤w), where u,w,x∈ . Equipped with an inclusion measure a conventional lattice becomes a fuzzy lattice. We have found a way to define systematically an inclusion measure σ(x≤w) in a lattice , provided there exists a positive valuation function v(.) in . For details on positive valuations refer to [3]. Notice also that a lattice with a positive valuation becomes a metric space with metric d( u,w)=v(u∨w)-v(u∧w). Our oldest known attempt to employing “fuzzy lattices” is noted in a neurocomputing application in [6]. However, the work in this research significantly systematizes and enhances it. A last word should be said about the lattice 1 whose elements are pairs of lattice elements. The significance of is that one of its elements defines an 1 interval of lattice elements and hence a finite set of lattice 1 elements can define an infinite set of lattice elements. It has been proven rigorously that a positive valuation function could be defined in 1 provided one in , and this occasions a metric and an inclusion measure in 1. Due to both the limited space and the emphasis of this paper on medical applications, the mathematical substantiation will be given elsewhere.
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The practical significance of the novel ma thematical approach of fuzzy lattices is that many sets are lattices that can be fuzzified and thus the degree of inclusion of a lattice element into another one can be specified, and consequently a sensible decision might be reached. A specific technique was developed along these lines which is applicable in the epidural procedure for tissue discrimination. This technique is presented in section 5. Note that disparate objects which are elements of other lattices can be treated jointly with mathematical consistency because it is known that the product of lattices is again a lattice [3]. Apparently this property fosters data fusion which might be of considerable value in a surgical operation where disparate data like images, waveforms, logic statements etc. might be fused to attain a sophisticated decision making capacity. 4 - PREPROCESSING, AND FEATURE EXTRACTION In this work 27 Raman spectra profiles were available that correspond to 4 distinct soft tissue types. The profiles have been provided by Peter Brett and Tim Parker. A typical profile from each tissue type is shown in Figure F2. The horizontal axis displays the wavenumbers away from the driving frequency, these are the frequencies where the spectra emission intensity levels were sampled, whereas the vertical axis displays the sampled spectra emission intensity levels themselves. For all the profiles the same range of frequencies was sampled but not at the same rate. Thus, for 24 profiles 2166 samples were available, whereas for 3 profiles 4320 samples were available. The 24 profiles with the 2166 samples were truncated at both ends by discarding the first 36 and the last 30 samples, resulting in 24 profiles of 2100 samples each. A truncation of this size was necessary in order to discard noisy measurements at both ends of the spectrum, but without distorting any distinctive features of the spectrums in question. The other 3 profiles were redefined by linear interpolation on exactly the same 2100 sample frequencies as the first 24 profiles. Hence, in conclusion 27 Raman spectra profiles p 1,…,p27 were produced each consisting of 2100 samples and defined at the same frequencies. Notice in Figure F2 that unique discrimination traits in these profiles include narrow/sharp “peaks” or “valleys” and the dc value of a profile. What is important for tissue identification is the sequence of features on its profile, the approximate location of the features along the spectrum of frequencies, as well as their approximate size. Apparently the tissue profile discrimination method should be tolerable to scaling and translation of the features. Hence, a need emerges for an “efficient” feature extraction method that meets the stated objectives. In addition, for the sake of “efficiency” the feature extraction scheme should be both rapid enough to be applicable on-line during a surgery and dependable by preserving, even enhancing, the discriminatory power of the original Raman profiles. A feature extraction technique tested in the lab and found to satisfy the requirements set above was the Fourier Transform, in particular its phase. The discriminatory power of the phase of a signal has been shown in [7] and it satisfies the above cited requirements. In the case of Raman spectra it was found that only the first 22 phase components were significantly greater than zero, therefore all the rest of the components were discarded. Such a feature extraction accounts for a significant data reduction from 2100 down to 22 real numbers. Notice that the set of the 22-numbers-long phase vectors is a subset of the lattice 22
of finite series of length 22. A positive valuation in this lattice is defined by the sum
∑ x , where x , i
i
i =1
i=1,…,22 is a vector entry, in our case that is a phase component. Implementation of these 22-numbers-long phase vectors by the intelligent algorithm to be presented in the following section showed a significant discriminatory capacity with occasionally a few misclassifications. To further enhance the discriminatory capacity of the intelligent algorithm we considered the magnitude of the Fourier Transform as well. Hence, the dc value of each profile was considered properly scaled by a constant in order to be comparable with its phase components. Then this scaled dc value of each profile replaced its first phase component. Notice that such a replacement implied no loss of useful information because the first phase component is always zero as it corresponds to the dc value of a Raman profile. 5 - THE TWO LEVEL FUZZY LATTICE (2L-FL) SUPERVISED CLUSTERING ALGORITHM The 2L-FL Supervised Clustering algorithm was applied on the data produced by the preprocessing procedure elaborated in the previous section. Recall that a positive valuation function has already been defined in the lattice of the data. Hence, both a distance function d(.,.) and an inclusion measure σ(.,.) may be defined in this lattice. In this specific soft tissue discrimination problem, the data profiles were given by an expert separated into four groups each group corresponding to a different type of soft tissue. Some of the data of each group were selected as training data and formed the training-set, whereas all the data of the group were used as testing data. Thus the testing-set was a super set of the training-set. In this way we were able, during the
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“Testing Phase”, to grade both the system’s ability to learn and its capacity to infer sensible conclusions with regards to new and hitherto unknown inputs.
Figure F2
Four Raman spectra profiles that correspond to four typical soft tissue representatives encountered in the epidural surgical procedure.
The “Training Phase” of the 2L-FL Supervised Clustering Algorithm Step-S1. A lower level clustering is conducted in the training-set by assuming that two training data are in the same cluster suffices their lattice distance is less than a threshold T d. A contradiction occurs when two data from different classes are put in the same cluster; then a decrease of T d is triggered followed by a new step-S1 cycle. Step-S1 keeps repeating until a contradiction-free cycle occurs which provides with homogeneous clusters, that is clusters that encode data from a single class. Step-S2. The lower level clustering is concluded in a procedure of experience extraction accomplished by the lattice join operation (∨) of lattice 1 data in the same cluster. By the conclusion of the lower level clustering one lattice 1 element has been defined for each lower level cluster. Step-S3. An upper level clustering is attained in a procedure of assigning each lower level cluster to a concrete tissue type. Such an assignment is administered by an external expert. Note that a set of clusters assigned to the same tissue type constitutes a class that corresponds to a single tissue type. The “Training Phase” of the 2L-FL Supervised Clustering scheme described above bases its decision making on the lattice metric d(.,.). In the sequel, to test the experience extracting efficacy of the “Training Phase” there follows the “Testing Phase” which bases its decision making on the fuzzy lattice inclusion measure σ(.,.). A description of the “Testing Phase” follows. The “Testing Phase” of the 2L-FL Supervised Clustering Algorithm All the data are fed one-by-one to the experience extracting and decision making mechanism, and their fuzzy degree of inclusion is calculated in all classes. The class that provides with the largest degree of lattice inclusion σmax is selected as the winner class the current input datum is classified in, provided that σmax is over a threshold T σ; otherwise no class is assigned to the current input datum, and we say that a class may not be defined for the input datum in question or in other words we say that the input datum is of an “undefined” class.
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In conclusion, the 2L-FL Supervised Clustering algorithm is a clustering with a teacher that occurs in two levels. In the lower level a clustering takes place based mainly on the lattice metric and a teacher regulates the rate of the distance threshold T d until homogeneous clusters are formed. Lower level learning results in lattice 1 by means of the lattice join (∨) of clustered data. The lower level clustering is in fact a mechanism that yields a lattice general representation of the training classes and, at the same time, it reduces potentially a set of multiple training elements to a few lattice intervals. On the other hand, the upper level clustering may be viewed again as an expert driven procedure, but one where labels are attached to the lower level clusters already identified and represented by lattice 1 elements. Both lower and upper level clustering is applied on the training-set. The worse case training scenario is to have data very near to each other, yet to belong in different classes. Then step-S1 of the algorithm will trigger repetitive decreases of Td until all the homogeneous learned clusters encode a single datum. This case might arise as a result of poor preprocessing, and never occurred in our experiments. Finally, testing occurs by correlating an input profile to already learned tissue types based on the affinity of the input profile and a tissue type as this affinity is determined by the fuzzy lattice inclusion measure. Experimental soft tissue discrimination results are provided in the next section. 6 - EXPERIMENTS AND RESULTS The 2L-FL Supervised Clustering algorithm was employed to process the set of 27 Raman spectra profiles p 1,…,p27; six data corresponded to connective-tissue, seven data to muscle-tissue, eight to fat-tissue, and another six data corresponded to skin-tissue. Each datum was a vector with 22 entries obtained as explained in section 4. Each Raman profile was treated by the 2L-FL Supervised Clustering algorithm as a whole individual object. Notice also that the original Raman profiles resulted in by actual measurements with all the consequent implications of noise interference. Two different training-sets selected randomly are shown in TABLE T1 under alternatives a1 and a2. For both alternatives of TABLE T1 as well as for all the other alternatives tested in the lab, it was found that the distance threshold T d=10.0 allows the formation of homogeneous clusters. Therefore this value for the Td was used in all subsequent experiments. Training Alternative a1 For the alternative a1, in the training-set the connective-tissue was represented by {p 1,p2,p3,p4}, the muscle-tissue was represented by {p 7,p8,p20}, the fat-tissue was represented by {p 12,p13,p16,p17}, and the skintissue was represented by{p 22,p23,p27}. Due to the efficacy of the suggested preprocessing technique, step-S1 of the “Training Phase” showed only six homogeneous clusters. Notice that three clusters were allocated to the connective-tissue group of data to accommodate the two outliers p 2 and p3. Each lower level cluster was specified in step-S2 by a single lattice 1 element. The “Training Phase” was concluded in step-S3 when an expert attached a label to each lower level cluster as shown in TABLE T1.
TABLE T1 Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5 Cluster 6
Two alternative training sets were used a1 and a2. The category formation results of the “Training Phase” are listed below. Training alternative a1 Training alternative a2 p / CONNECTIVE 6 p1∨p4 / CONNECTIVE p9 / MUSCLE p7∨p8∨p20 / MUSCLE p2 / CONNECTIVE p2 / CONNECTIVE p3 / CONNECTIVE p3 / CONNECTIVE p15 / FAT p12∨p13∨p16∨p17 / FAT p 24 / SKIN p22∨p23 ∨p27 / SKIN
“Testing Phase” results for the alternative a1 are shown in TABLE T2. Notice that the data used by the 2L-FL Supervised Clustering algorithm for training, and only those data yield a fuzzy degree of lattice inclusion equal to 1. Assuming that each one of the data p 1 through p 27 is classified to the cluster that provides with the largest rate of fuzzy lattice inclusion σ than in any other cluster, TABLE T2 implies an 100% correct tissue discrimination. Notice that for all profiles p 1 through p 27 the maximum degree of inclusion in a class is above the class’ decision threshold T σ=.75.
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Training Alternative a2 For the alternative a2, in the training-set the connective-tissue was represented by {p 2,p3,p6}, the muscle-tissue was represented by {p 9}, the fat-tissue was represented by {p 15}, and the skin-tissue was represented by{p 24}. The “Training Phase” revealed again six homogeneous clusters, each one encoding a single datum, and it was concluded when an expert attached a label to each cluster as shown in TABLE T1 under alternative a2. “Testing Phase” results for the alternative a2 are shown in TABLE T3. Note again tha t the data used by the 2L-FL Supervised Clustering algorithm for training, and only those data yield a fuzzy degree of lattice inclusion equal to 1. Assuming that each one of the data profiles p 1 through p 27 is classified to the cluster that provides with the largest rate of fuzzy lattice inclusion σ than in any other cluster, TABLE T3 implies an 100% correct soft tissue discrimination. For all profiles p 1 through p 27 the maximum degree of inclusion in a class is above the class’ decision threshold T σ=.55. It may be noted that the classification answers in TABLE T3 are not so “strongly asserted” as they are in TABLE T2. For instance, datum p 5 is correctly classified in cluster c-1 in both Tables. Nevertheless, the fuzzy degree of inclusion of p 5 in c-1 is .97 in TABLE T2, whereas it is .89 in TABLE T4. The same holds true for the rest of the data. This is explained by the fact that more data were used for training in alternative a1 than they were in alternative a2. In fact, in alternative a2 a cluster was defined by a single datum as shown in TABLE T1. It should also be noted that had we used only the 22 phase data components without a profile’s dc value, a more numerous than a six cluster partition would have resulted in, due to the fact that muscle-tissue and skin-tissue profiles tend to be put in the same cluster. Such an intermingling calls for a repetitive parameter T d fine tuning (repetitive decreases of T d) that leads to the definition of both the muscle and the skin classes by more than one clusters. To proceed in a more elegant solution and with respect to the system’s memory resources we introduced an additional feature, that is a profile’s dc value. As a result any muscle-tissues in the training-set were clustered together and apart from skin-tissues which were clustered among themselves in a single cluster in all the experiments (alternatives). Finally, note that had the outliers p 2 and p3 not been used for training then as implied by both Tables T2, T3 and the class decision thresholds T σ, the data p 2 and p3 would be of an “undefined” class. 7 - CONCLUSION AND FUTURE WORK A new intelligent scheme, that is the 2L-FL Supervised Clustering was presented for tissue discrimination during the epidural surgical procedure. The 2L-FL Supervised Clustering was able to extract knowledge from raw data in order to identify and discriminate soft tissues. The data were preprocessed by the conventional Fast Fourier Transform, phase and the dc component were used as features. The 2L-FL is fast as it requires a few passes through the training data to extract the knowledge, new tissue types may be identified on-the-fly, and the decisions are sensible to the user. The employment of the 2L-FL Supervised Clustering scheme via a surgical mechatronic tool was further advocated by the pointed experimental results reported in this paper, provided there are sufficient training data representing all the different cases expected to be encountered in practice. Future plans are mainly concerned with the optimization of the 2L-FL Supervised Clustering algorithm. Specifically, during training other clustering techniques could be sought; for example a clustering based on a non-linear function of both the distance and the inclusion measure. Moreover, an efficient law should be sought to decrease the threshold T d on-line; for such a small data set as the one used in this work the decrease of T d was performed manually. Finally, optimal methodologies should be investigated for defining a low level cluster by a set of lattice 1 elements rather than by a single element. Such a cluster definition by multiple lattice 1 elements is believed to optimize the classification decisions when the training-set is of considerable size. REFERENCES [ 1] [ 2] [ 3] [ 4] [ 5]
Allotta B., Buttazo G., Dario P., Quaglia F., “A force/torque sensor-based technique for robot harvesting of fruits and vegetables”, IEE International Workshop on Intelligent Robots and Systems, IROS’90. Benabid A.L., Cinquin P., Lavalle S., Le Bas J.F., Demongeot J. de Rougemont J., “Computer-driven robot for stereotactic surgery connected to CT scan and magnetic resonance imaging”, Proceedings American Society Stereotactic Functional Neurosurgery, Montreal 1987, Applied Neurophysiology; 50; 153-154, 1987. Birkhoff G. (1967), “Lattice Theory”, American Mathematical Society, Colloquium Publications, vol. 25. Edwards R., “Robotics in Medicine: Safety Aspects”, Conference on Robotics in Medicine, IMechE HQ, London, June 1990. Halmos P.R. (1960), “Naive Set Theory”, Van Nostrand, New York.
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[ 6] Kaburlasos V.G., “Adaptive Resonance Theory with Supervised Learning and Large Database Applications”, a University of Nevada Reno 1992 Dissertation, University Microfilms Inc., US Library of Congress-Copyright Office. [ 7] Oppenheim A.V., Lim J.S., “The Importance of Phase in Signals”, Proceedings of the IEEE, vol. 69, no. 5, May 1981, pp. 529-541. [ 8] Parker T.J., Harrison A.J., Brett P.N., Thomas T., “Towards a Mechatronic Tool for the Epidural Procedure”, Mechatronics and Machine Vision in Practice (M2ViP’95), Sep. 1995, Hong-Kong. [ 9] Straughan B.P., Spectroscopy, B.P. Straughan & S. Walker eds, vol. 2, pp. 138-265, 1976, Chapman & Hall, London. [10] Tillet R.D., Reed J.N., “Initial Development of a Mechatronic Mushroom Harvester”, Intl Conf. Mechatronics: Designing Intelligent Machines, IMechE, Cambridge, UK, 12-13 September 1990.
TABLE T2 - Classification results of the “Testing Phase” for alternative a1. Class decision threshold T σ=.75 c-1 c-2 c-3 c-4 c-5 c-6 p1 1 .74 .38 .36 .38 .62 p2 .47 .55 1 .24 .61 .43 p3 .49 .44 .24 1 .26 .66 p4 1 .67 .34 .41 .35 .58 p5 .97 .71 .36 .38 .36 .60 p6 .98 .72 .37 .38 .37 .62 p7 .74 1 .43 .31 .42 .59 p8 .71 1 .42 .31 .42 .57 p9 .71 .99 .42 .31 .42 .57 p10 .65 .82 .38 .34 .39 .64 p11 .62 .83 .48 .28 .45 .52 p12 .40 .46 .52 .22 1 .37 p13 .39 .47 .54 .21 1 .38 p14 .39 .47 .53 .21 .99 .38 p15 .42 .44 .46 .23 .84 .38 p16 .39 .47 .54 .21 1 .38 p17 .39 .48 .53 .22 1 .38 p18 .39 .48 .54 .22 .97 .38 p19 .42 .44 .46 .23 .83 .38 p20 .63 1 .38 .34 .39 .65 p21 .57 .81 .35 .38 .38 .76 p22 .61 .56 .29 .55 .31 1 p23 .53 .67 .33 .45 .34 1 p24 .59 .58 .30 .51 .32 .96 p25 .52 .67 .33 .44 .34 .95 p26 .61 .56 .29 .57 .31 .96 p27 .54 .65 .32 47 .33 1
TABLE T3 - Classification results of the “Testing Phase” for alternative a2 Class decision threshold T σ=.55 c-1 c-2 c-3 c-4 c-5 c-6 p1 .84 .60 .38 .36 .34 .49 p2 .37 .42 1 .24 .46 .30 p3 .38 .31 .24 1 .23 .51 p4 .81 .54 .34 41 .31 .44 p5 .89 .59 .36 .38 .32 .47 p6 1 .59 .37 .38 .33 .48 p7 .60 90 .43 .31 .32 .42 p8 .58 .96 .42 .31 .32 .41 p9 .59 1 .42 .31 .33 .40 p10 .50 .75 .38 .34 .30 .46 p11 .50 .74 .48 .28 .35 .37 p12 .31 .35 .52 .22 .74 .27 p13 .30 .36 .54 .21 .74 .26 p14 .30 .36 .53 .21 .75 .26 p15 .33 .33 .46 .23 1 .28 p16 .30 .36 .54 .21 .72 .26 p17 .30 .36 .53 .22 .73 .26 p18 .31 .36 .54 .22 .73 .26 p19 .33 .33 .46 .23 .92 .28 p20 .50 .69 .38 .34 .31 .46 p21 .45 .57 .35 .38 .29 .54 p22 .51 .39 .29 .55 .28 .83 p23 .43 .46 .33 .45 .26 .72 p24 .48 .40 .30 .51 .28 1 p25 .42 .47 .33 .44 .26 .71 p26 .52 .39 .29 .57 .27 .80 p27 .44 .45 .32 .47 .26 .69
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