Minimally invasive ionizing radiation detector
By arranging the recesses of the scintillator layer in the optical fiber in the ionizing radiation detector, the problems of large size and low sensitivity of the existing detector are solved, and the minimally invasive detection effect with higher sensitivity and smaller volume are achieved.
Patent Information
- Application Number
- CN201980078723.4
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Priority Date
- 2018-11-30
- Filing Date
- 2019-09-27
- Publication Date
- 2025-04-18
- Estimated Expiration
- 2039-09-27
AI Technical Summary
The existing ionizing radiation detectors are huge in size and have low sensitivity, making it difficult to meet the detection needs of minimally invasive or mildly invasive.
A detector including an optical fiber and a scintillator layer is designed, arranged in the recess of the optical fiber, in direct contact with the useful cladding and the optical fiber core, improving optical coupling.
By improving the optical coupling between the optical fiber and the scintillator, the sensitivity of the detector is improved, and the small, minimally invasive ionizing radiation detection effect is achieved.
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Figure CN113167915B_ABST
Abstract
Description
[0001] The present invention relates to a detector for high energy ionizing radiation greater than 100 eV. Technical Field
[0002] The field of the invention is that of detection of ionizing radiation for medical or industrial imaging, and more particularly that of detectors that are minimally or mildly invasive or have a negligible traumatic footprint. Background Art
[0003] Known ionizing radiation detectors include at least one optical fiber and a scintillator that is sensitive to radiation and is arranged at the end of the optical fiber. When subjected to ionizing radiation, the scintillator emits light. The purpose of these detectors is to capture part of this light and direct it, for example, through the core of the optical fiber to a photodiode. However, due to their inherently low sensitivity, these detectors are still bulky and, in fact, are of large size, thereby increasing the overall sensitivity.
[0004] The object of the present invention is to provide an ionizing radiation detector with a small volume.
[0005] Another object of the invention is to improve the sensitivity of such a detector by proposing a novel arrangement between the optical fiber and the scintillator. Summary of the invention
[0006] At least one of the above objects is achieved by a detector of high energy ionizing radiation greater than 100 eV, the detector comprising:
[0007] - an optical fiber having an outer diameter, called fiber diameter, of less than 250 micrometers and comprising an optical fiber core, a first cladding, called useful cladding, surrounding said optical fiber core and a second cladding, called protective cladding, surrounding said useful cladding, and
[0008] - a scintillator layer provided for converting said ionizing radiation into light.
[0009] The detector is characterized in that it comprises a portion, referred to as the detection portion, which is arranged on the length of the optical fiber and comprises a recess formed in the protective cladding, in the useful cladding and / or in the optical fiber core; the scintillator layer is arranged in the recess in contact with the useful cladding and the optical fiber core.
[0010] With the detector according to the invention, the scintillator layer is arranged in direct contact with the useful cladding and the optical fiber core over a wide contact surface. The optical coupling between the scintillator layer and the optical fiber is thus improved. As a result, the detector is more sensitive.
[0011] It is also conceivable that only the recess is produced in the optical fiber core.
[0012] The arrangement of the detector according to the invention allows a strong coupling between the optical fiber and the scintillator and therefore a larger luminescence signal at the output of the optical fiber, and an increased sensitivity of the detector. This special structure makes it possible to use optical fibers of very small dimensions, thus:
[0013] - having sensors that do not attenuate the therapeutic radiation in biological media and do not interfere with the treatment process (invisible concept), and
[0014] - Ultra-compact sensor with very limited dimensions.
[0015] The present invention makes it possible to obtain ultra-compact detectors on thin optical fibers, the effective diameter of which can be less than 125 μm, 80 μm, or even 60 μm. Therefore, regardless of the type of scintillator and optical fiber envisaged (organic or inorganic), the detector according to the present invention has a negligible invasive footprint for the patient and is invisible during the envisaged treatment process. Due to the reduced size of the detector according to the present invention compared to the devices of the prior art, the detector is considered to be minimally invasive. Therefore, it is described as minimally invasive because its use in the medical field, for example, requires a lesion of very small size compared to the use of existing detectors.
[0016] The detector according to the invention can be arranged to detect ionizing radiation having an energy greater than 100 eV, preferably 0.1 MeV to 500 MeV.
[0017] In an embodiment version of the detector according to the invention, the scintillator layer is completely contained in the recess and is flush with the outer surface of the protective envelope.
[0018] The scintillator layer does not increase the size of the detector. Therefore, the detector according to the invention is smaller than the detectors of the prior art. The scintillator layer can fill the recess and is flush with the outer surface of the useful envelope.
[0019] Advantageously, the detector according to the invention can comprise a metal layer surrounding the scintillator layer in the detection part.
[0020] The metal layer can be a thin enough layer so that ionizing radiation can pass through and light from the scintillator is reflected and redirected to the optical fiber.
[0021] In particular, the metal layer can be a deposit of aluminum and / or gold with a thickness less than or equal to 10 micrometers, in particular less than or equal to 300 nm.
[0022] In a special embodiment of the detector according to the invention, the detection portion can be arranged at the end of an optical fiber.
[0023] Preferably, the scintillator layer can extend over the entire end of the optical fiber to its cross section. In other words, the scintillator is arranged on the side of the tapered optical fiber and possibly on the cross section at the very end of the optical fiber.
[0024] The metal layer can also extend around the scintillator layer at the end of the optical fiber.
[0025] In a special embodiment of the detector according to the invention, the detection portion can be arranged at a non-zero distance from the end of the optical fiber.
[0026] In particular, the detection portion can be arranged at a distance of 0.1 cm to 1 cm from the end of the optical fiber.
[0027] In particular, the recess can extend inside the optical fiber core such that the scintillator layer is in direct contact with the interior of the optical fiber core.
[0028] In an advantageous embodiment of the detector according to the invention, the recess can extend around the axis of rotation of the optical fiber.
[0029] With such a recess produced around the entire cross section of the useful cladding, the coupling between the scintillator layer and the optical fiber is thus improved and, in case the recess is covered by a uniform scintillator layer, the detector becomes isotropic. The isotropic property means that the same signal is detected under the irradiation of the ionizing beam when the optical fiber is rotated along its axis of rotation. This feature is of particular interest in the medical field.
[0030] In a preferred embodiment of the detector according to the invention, the recess can form a cone in the detection portion.
[0031] In particular, the cone can be arranged at the end of the optical fiber and extend into the useful cladding and the optical fiber core. This arrangement improves the light coupling between the scintillator layer and the optical fiber core. The luminescence emitted by the scintillator and directed to the optical fiber is thus increased. Therefore, the detector according to the invention is more sensitive.
[0032] Furthermore, the recess can form a frustoconical portion in the detection portion, the frustoconical portion being arranged at a distance from the end of the optical fiber or at the end of the optical fiber.
[0033] In an embodiment of the detector according to the invention, the recess can form a cylindrical portion in the detection portion having a diameter smaller than the diameter of the optical fiber.
[0034] The diameter of the cylindrical portion can be less than or equal to 80% of the diameter of the optical fiber.
[0035] In particular, the recess can be produced by removing material from the useful cladding and / or the optical fiber core.
[0036] Advantageously, the diameter of the optical fiber can be less than or equal to 250 microns, in particular less than or equal to 95 microns, more particularly less than or equal to 80 microns, even more particularly less than or equal to 75 microns.
[0037] Preferably, the diameter of the optical fiber core is less than or equal to 110 micrometers, in particular less than or equal to 70 micrometers.
[0038] In an advantageous embodiment, the detector according to the invention can comprise a photon counter connected to an optical fiber.
[0039] A photon counter can be connected to the end of the optical fiber opposite the detection portion.
[0040] The scintillator layer can consist of a single scintillator bonded directly to the useful cladding and / or the fiber core. Bonding or adhesion by electrostatic treatment of the surfaces of the fiber and the scintillator is also conceivable.
[0041] The scintillator layer can also be a mixture of a polymer, for example of the photosensitive type, and an inorganic scintillator and / or an organic scintillator, for example in powder form.
[0042] Advantageously, the scintillator layer can be assembled to the useful envelope by bonding, in particular by means of an adhesive which is transparent to the luminescence.
[0043] In particular, the detection section can have a length of 0.1 mm to 5 mm.
[0044] Preferably, the thickness of the scintillator layer can be less than or equal to 100 micrometers, in particular less than or equal to 50 micrometers.
[0045] According to one aspect of the present invention, a method for manufacturing a detector of high-energy ionizing radiation according to the present invention is provided, the method comprising at least the following steps:
[0046] - tapering of the optical fiber over the detection section by removing the protective cladding and forming a recess in the useful cladding and / or in the fiber core, and
[0047] - depositing a scintillator layer in the recesses by photopolymerization.
[0048] Compared to the methods of the prior art, the thinning of the optical fiber by removing material in the useful cladding and optionally in the core makes it possible to maintain the good rigidity necessary for the optical fiber when in use. The protective cladding can be removed in a preliminary step using an acid different from that used for the recesses in the useful cladding and optionally in the core.
[0049] Advantageously, the step of depositing a scintillator layer comprises the step of irradiating the optical fiber with UV or visible light.
[0050] In particular, the thinning step can be performed by chemical etching and can comprise the following steps:
[0051] - immersing the optical fiber over a predetermined length in a solution provided to dissolve the protective cladding,
[0052] - immersing the unclad optical fiber in a solution provided for dissolving the useful cladding and / or the optical fiber core,
[0053] - moving the optical fiber vertically (along its axis of rotation) at a predetermined speed or at a continuous predetermined speed.
[0054] The solution used to form the recess in the useful cladding and / or the core can be hydrofluoric acid.
[0055] This solution can be covered with a liquid film (oil, silicone, etc.) so that the process of forming the recesses can be stabilized at the desired size and geometry of the structured optical fiber.
[0056] The predetermined length can be equal to the length of the detection portion.
[0057] Such an embodiment makes it possible to obtain a cone shape of variable frustoconical extension, or another shape depending on the applied (fixed or variable) speed, or a combination of both.
[0058] In order to obtain a cylindrical shape (a right circular cylinder), the detection part of the optical fiber is statically immersed in a tank containing the solution for a predetermined duration. This duration depends on the depth of the recess to be made. When the optical fiber core is etched, especially at its ends, a longer duration enables a short conical shape to be obtained.
[0059] In a particular embodiment, the thinning step can be performed by polishing the useful cladding and optionally the optical fiber core.The protective cladding is previously removed at least on the detection portion.
[0060] In a particular embodiment, the method according to the invention further comprises at least one step of depositing a metal layer after the step of depositing the scintillator layer.
[0061] According to one aspect of the invention, an endoscopic probe for detecting high-energy ionizing radiation is proposed, the endoscopic probe comprising at least one detector according to the invention.
[0062] In one embodiment, the probe according to the present invention can include a plurality of detectors, and ends of the plurality of detectors are arranged in the same plane.
[0063] In one embodiment, a probe according to the present invention can include a plurality of detectors whose ends are inscribed in parallel planes spaced apart by a non-zero distance (502).
[0064] The probe can include from 2 to 20 detectors.
[0065] Advantageously, the probe according to the invention can comprise a CMOS camera or a CCD camera connected to at least one detector, in particular to all detectors. BRIEF DESCRIPTION OF THE DRAWINGS
[0066] Other advantages and features will become apparent by studying the detailed description of non-limiting embodiments and in light of the accompanying drawings, in which:
[0067] Figure 1a is a schematic diagram of a side cross-sectional view of a first example of a detector having a first recess shape according to the present invention;
[0068] Figure 1b is a schematic diagram of a side cross-sectional view of a second example of a detector having a second recess shape according to the present invention;
[0069] Figure 1c is a schematic diagram of a side cross-sectional view of a third example of a detector having a third recess shape according to the present invention;
[0070] Figure 1d is a schematic diagram of a side cross-sectional view of a fourth example of a detector having a fourth recess shape according to the present invention;
[0071] Figure 1e is a schematic diagram of a side cross-sectional view of a fifth example of a detector having a fifth recess shape according to the present invention;
[0072] Figure 2 is a schematic diagram of an example of a probe according to the present invention; and
[0073] Figure 3 is a schematic diagram of an example of a method according to the present invention. DETAILED DESCRIPTION
[0074] It is understood that the embodiments to be described below are by no means limiting. Variants of the invention can be envisaged, in particular comprising only a selection of the features described below, separated from the other features described, if such selection of features is sufficient to confer a technical advantage or to distinguish the invention from the state of the prior art. The selection comprises at least one, preferably functional, feature without structural details, or with only part of the structural details, if this part alone is sufficient to confer a technical advantage or to distinguish the invention from the state of the prior art.
[0075] Figure 1a , 1b , 1c and 1d are schematic diagrams of side cross-sectional views of examples of detectors having several recess shapes according to the present invention.
[0076] Figure 1aThe detector 100 in the embodiment is configured to detect radiation having an energy of 100 eV to 500 MeV, preferably 0.1 MeV to 500 MeV.
[0077] The detector 100 comprises an optical fiber 102 comprising an optical fiber core 104 surrounded by a first cladding 106, called useful cladding, and a second cladding 108, called protective cladding, surrounding the useful cladding 106. The outer diameter of the optical fiber is 75 microns to 250 microns, in particular equal to 95 microns. The length of the optical fiber 102 can be greater than 1 m.
[0078] The optical fiber 102 comprises a portion 110, called the detection portion, arranged at its end. The detection portion 110 extends over a length of 0.1 mm to 5 mm, in particular equal to 5 mm, and comprises a recess so that the end of the optical fiber 102 forms a cone 114 pointing outwards. The recess 120 corresponds to the portion of the optical fiber that has been removed. The cone 114 is formed by removing material from the protective cladding 108, from the useful cladding 106 and from the optical fiber core 104.
[0079] In a previous step the protective cladding 108 is removed. For example, it can be removed using sulfuric acid which will etch it without degrading the rest of the optical fiber, ie the core 104 and the useful cladding 106. Other solutions exist, either by chemical etching or by mechanical removal.
[0080] The detector 100 further comprises, in the detection portion 110, a scintillator layer 116 arranged in contact with the useful cladding 106 and the optical fiber core 104. The scintillator layer 116 has a thickness of 20 to 100 microns, in particular equal to 50 microns.
[0081] The scintillator layer is molded into a pointed cone shape by completely covering the outer surface of the optical fiber core 104, which is exposed by creating a recess 120. The scintillator layer also extends until it partially covers the outer surface of the useful cladding 106, which is exposed by creating a recess 120.
[0082] The scintillator layer 116 is formed by dipping the detection portion of the optical fiber into a polymer / scintillator mixture during polymerization. The polymer allows the scintillator to aggregate and acts as a binder.
[0083] The detection section 110 in the shape of a cone 114 provides a large contact surface between the scintillator layer 116 and the useful cladding 106 on the one hand and between the scintillator layer and the optical fiber core 104 on the other hand. This greatly improves the optical coupling. Thus, the luminescence emitted by the scintillator layer 116 and directed to the optical fiber is increased compared to a device in which the scintillator is arranged at the end of an optical fiber cut orthogonally to the axis of rotation.
[0084] In addition, the cone 114 formed in the detection portion 110 makes it possible to maintain good rigidity required for the use of optical fibers.
[0085] The detector 100 also includes a photon counter 118 connected to the optical fiber 102 and configured to measure the amount of light emitted by the scintillator layer 116 and guided through the optical fiber core 104 .
[0086] Figure 1b is a schematic diagram of a cross section of the detector 200 viewed from the side. The detector 200 includes Figure 1a The detector 100 in FIG. 1 has the same elements as those in FIG. 1 , but has a different shape. On a portion of the end of the optical fiber 102, the protective cladding is removed, leaving the useful cladding 106 bare. On the detection portion 110, the useful cladding 106 has been chemically etched so that a first truncated cone 202 without a tip can be formed. At this position, the optical fiber core is not exposed. Then, a second truncated cone 204 is formed after the first truncated cone, which is longer than the first truncated cone and has a solid angle smaller than the solid angle of the first truncated cone. Therefore, the thickness of the useful cladding is tapered to zero at one end 123 where the optical fiber core is cut at a right angle. Therefore, the detection portion 110 is generated by creating a concave portion with a complex shape extending from the protective cladding 108 to the useful cladding 106. In this arrangement example, the scintillator layer covers the end of the optical fiber core and is in direct contact with the end of the optical fiber core over its entire radial cross section.
[0087] The scintillator layer 116 is arranged around the useful cladding 106, in contact with the outer surfaces of the first and second frustums 202 and 204. The scintillator layer is flush with the outer surface of the unetched portion of the useful cladding.
[0088] The detector 200 also includes a thin metal layer 206 produced around the scintillator layer 106. The metal layer 206 is a deposit of aluminum (Al) or gold (Au) with a thickness of less than 200 nm. The metal layer 206 makes it possible to reflect the light emitted by the scintillator layer 116 and direct it to the optical fiber core 104. Therefore, the detector 200 is more sensitive.
[0089] Figure 1c Describes the use of Figure 1b The detector 200 in FIG. 1 is a detector of the same configuration, the only difference being that the end of the optical fiber core is not cut at a right angle but extends to point 124 .
[0090] Figure 1d 3 is a schematic diagram of a cross section of the detector 300 as viewed from the side. The detector 300 includes Figure 1aThe detector 100 of the embodiment of the present invention is the same element as the detector 100 in the embodiment of the present invention. The difference is that the detection part 110 of the detector 300 is arranged at a distance of 2 mm to 10 cm, in particular equal to 1 cm, from the end of the optical fiber 102. Although recesses are created in the protective cladding 108, the useful cladding 106 and the optical fiber core 104 and a scintillator layer is deposited, this arrangement can still maintain good rigidity of the optical fiber 102.
[0091] Furthermore, the recess in the detector 300 forms two frustoconical portions 302 and 304 facing each other end to end, thereby creating a recess in the fiber core. The scintillator layer 116 completely fills the created recess and is flush with the outer surface of the protective cladding 108.
[0092] Figure 1e 4 is a schematic diagram of a cross section of the detector 400 as viewed from the side. The detector 400 includes Figure 1a The detector 100 in FIG. 1 is identical to the detector 100 in FIG. 1 . The difference is that the recess is in the shape of a flat spot 402 arranged at the end of the optical fiber 102 and only on one section. On the detection section 110, the recess is hollowed out from the protective cladding 108, through the useful cladding 106, to a part of the optical fiber core. The scintillator layer 116 completely fills the resulting recess.
[0093] Figure 2 is a schematic diagram of an example of a probe according to the present invention.
[0094] The probe 500 includes a plurality of detectors 100 arranged in a bundle. As an example, the probe 500 can include an arrangement of detectors of the same type or detectors of different types among the detectors 100, 200, 300, and 400.
[0095] exist Figure 2 In the embodiment, the ends of the detector 100 are separated from each other by a constant or variable distance 502, which is between 0.5 cm and 10 cm.
[0096] When the probe 500 is exposed to ionizing radiation 504, the scintillator layer 116 of the detector 100 emits light 506. The light 506 is captured by the fiber core 104 and directed to a camera (CCD, CMOS or other) to which the detector 100 is connected.
[0097] In other words, under illumination, the thin scintillating layer 116 emits light, which is collected and transmitted through the optical fiber toward an optical detector placed at a distance at the other end of the optical fiber.
[0098] This embodiment can be advantageously applied to the case of a probe constituted by a plurality of detectors of the same type or different types among the detectors 100 , 200 , 300 , and 400 .
[0099] Figure 3is a schematic diagram of an example of a method according to the present invention.
[0100] Method 600 is a method for manufacturing detector 100 , 200 , 300 , and / or 400 .
[0101] The method 600 includes a step 602 of tapering the optical fiber 102 above a detection portion by removing material from the protective cladding 108 and / or the useful cladding 106 and / or the optical fiber core 102 so as to form a recess in the optical fiber 102 .
[0102] The tapering step 602 comprises at least one step of immersing the optical fiber 102 at its detection portion 110 in a solution provided for dissolving the protective cladding 108 and / or the useful cladding 106 and / or the optical fiber core 102 .
[0103] The method 600 further comprises a step 604 of depositing a scintillator layer 116 in the recess by photopolymerization. The depositing step 604 comprises the step of irradiating the optical fiber 102 with UV or visible light.
[0104] The method 600 can also include a step 608 of depositing the metal layer 206 , for example by spraying or evaporating a metal such as aluminum or gold.
[0105] The structure at the end of the optical fiber according to the invention makes it possible to collect a maximum of the luminescence and transmit it toward the optical fiber.
[0106] The tapered shape of the optical fiber (called the "taper") facilitates the collection of the light emitted. This allows:
[0107] - satisfactory attachment of the scintillator to the optical fiber,
[0108] - Homogenization of the collection efficiency of the luminescence over the entire length of the optical fiber covered by the scintillating layer, and therefore optimization of the sensitivity of the probe given the low emissivity of the scintillator under illumination.
[0109] Miniaturization makes it possible to obtain ultracompact and invisible in vivo sensors, i.e., limiting the radiation attenuation in biological media or water equivalents to less than 1%.
[0110] Due to the improvement of the fiber / scintillator coupling, the diameter of the optical fiber can be reduced. This therefore enables the production of multisensors that are narrow enough to be compatible in particular with endoscopy.
[0111] The invention is advantageously applied to the production of sensors consisting of one or more optical fiber probes according to the invention, in particular for real-time in vivo dosimetry of ionizing radiation. The invention can be applied in the field of medical therapy: external beam radiation therapy, proton therapy, hadron therapy, brachytherapy (or Curie therapy), etc.
[0112] Of course, the invention is not limited to the examples that have just been described and many modifications may be made to these examples without exceeding the scope of the invention.
Claims
1. A detector for high energy ionizing radiation (504) greater than 100 eV, the detector comprising: - an optical fiber (102) having an outer diameter, referred to as the fiber diameter, of less than 250 micrometers and comprising an optical fiber core (104), a first cladding, referred to as the useful cladding (106), surrounding the optical fiber core (104) and a second cladding, referred to as the protective cladding (108), surrounding the useful cladding (106), and - a scintillator layer (116) provided for converting the ionizing radiation (504) into light (506); - a portion called the detection portion (110), which is arranged on the length of the optical fiber (102) and comprises a recess (120) formed in the protective cladding (108), in the useful cladding (106) and in the optical fiber core; the scintillator layer (116) is arranged in the recess, in contact with the useful cladding (106) and the optical fiber core (104), The invention is characterized in that the recess extends inside the optical fiber core (104), so that the scintillator layer is in direct contact with the inside of the optical fiber core and with the protective cladding (108).
2. The detector according to claim 1, characterized in that The scintillator layer (116) is completely contained in the recess and is flush with the outer surface of the protective envelope.
3. The detector according to claim 1 or 2, characterized in that: The detector includes a metal layer (206) surrounding the scintillator layer (116) in a detection portion (110).
4. The detector according to claim 1 or 2, characterized in that: The detection portion is arranged at the end of the optical fiber.
5. The detector according to claim 4, characterized in that: The scintillator layer (116) extends over the entire end of the optical fiber (102) to a cross section of the optical fiber.
6. The detector according to claim 1 or 2, characterized in that: The detection portion (110) is arranged at a non-zero distance from the end of the optical fiber (102).
7. The detector according to claim 1 or 2, characterized in that: The recess extends around the rotation axis of the optical fiber (102).
8. The detector according to claim 1, characterized in that The recess forms a cone in the detection portion (110).
9. The detector according to claim 1 or 2, characterized in that: The detector includes a photon counter (118) connected to the optical fiber (102).
10. A method (600) for manufacturing a detector for high energy ionizing radiation (504) greater than 100 eV according to any one of claims 1 to 9, the method comprising at least the following steps: - a step of tapering the optical fiber (102) on the detection portion (110) by removing the protective cladding (108) and creating a recess in the useful cladding (106) and / or in the optical fiber core (104), and - A step of depositing a scintillator layer (116) in the recess by photopolymerization so that the scintillator layer is in direct contact with the interior of the optical fiber core and with the protective cladding (108).
11. The method (600) according to claim 10, characterized in that The thinning step (602) is performed by chemical etching and comprises the following steps: - immersing the optical fiber (102) over a predetermined length in a solution provided to dissolve the protective cladding (108), - immersing the unclad optical fiber in a solution provided to dissolve said useful cladding and / or the optical fiber core, - vertically moving the optical fiber (102) at a predetermined speed or at a continuous predetermined speed.
12. An endoscopic probe for detecting high-energy ionizing radiation, the endoscopic probe comprising at least one detector according to any one of claims 1 to 9.
13. The endoscopic probe according to claim 12, characterized in that: The endoscope probe includes a plurality of detectors, and ends of the plurality of detectors are arranged in the same plane.
14. The endoscopic probe according to claim 12, characterized in that: The endoscopic probe includes a plurality of detectors whose ends are inscribed in parallel planes spaced apart by a non-zero distance (502).
Citation Information
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