SYSTEM AND METHOD FOR TRACKING MIGRATION OF A STRUCTURE

Abstract

A system for tracking relative displacement between a first element and a second element in a structure, the system includes: a permanent magnet fixed relative to the first element; a magnetic sensor fixed relative to the second element, the magnetic sensor configured to measure a magnetic field strength; and a processor configured to determine the relative displacement between the first element and the second element, based on the magnetic field strength measured by the magnetic sensor. This invention has particular applications in e.g. orthopaedic prostheses. A related method includes the steps of measuring, using the magnetic sensor, a magnetic field strength; and determining the relative displacement between the relative displacement between the first element and the second element, based on the magnetic field strength measured by the magnetic sensor.

Claims

1. A system for tracking relative displacement between a first element and a second element in a structure, the system including: a permanent magnet fixed relative to the first element; a magnetic sensor fixed relative to the second element, the magnetic sensor configured to measure a magnetic field strength; and a processor configured to determine the relative displacement between the first element and the second element, based on the magnetic field strength measured by the magnetic sensor.

2. The system according to claim 1, wherein: the magnetic sensor is configured to generate magnetic field strength data based on the measured magnetic field strength, and to transmit the generated magnetic field strength data to the processor.

3. The system according to claim 2, wherein: the magnetic sensor is configured to measure the magnetic field strength, and thereby to generate the magnetic field strength data at predetermined intervals.

4. The system according to claim 1, wherein: the magnetic sensor includes three one-dimensional sub-sensors arranged mutually orthogonally, each of the three one-dimensional sensors configured to measure a magnitude of a respective one of the three orthogonal components of the magnetic field at the magnetic sensor; and the direction of the component of the magnetic field at the magnetic sensor of which a given sub-sensor is configured to measure or determine the magnitude is referred to the “orientation” of that sub-sensor.

5. The system according to claim 4, wherein: the orientations of the three sub-sensors correspond to the axes of the magnetic sensor.

6. The system according to claim 5, wherein: the permanent magnet is a cylindrical magnet, the z-axis of which is initially aligned with and is parallel to the z-axis of the magnetic sensor.

7. The system according to claim 4, wherein: the magnetic sensor is configured to determine the value of the magnitude of the component of the magnetic field strength of the permanent magnet which is along the z-axis of the permanent magnet; and the processor is configured to determine the z-distance between the permanent magnet and the magnetic sensor using the value of the magnitude of the component of the magnetic field strength of the permanent magnet which is along the z-axis of the permanent magnet.

8. The system according to claim 7, wherein: the processor is configured to determine the x-distance based on at least the z-distance, and the values of the magnitudes of the x- and z-components of the magnetic field strength at the magnetic sensor, measured by the magnetic sensor; and/or the processor is configured to determine the y-distance based on at least the z-distance, and the values of the magnitudes of the y- and z-components of the magnetic field strength at the magnetic sensor, measured by the magnetic sensor.

9. The system according to claim 7, wherein: the processor is configured periodically to update the value of the z-distance used to calculate the x-distance and the y-distance.

10. The system according to claim 9, wherein: after a predetermined amount of time or a predetermined number of measurements have been taken by the magnetic sensor, the process is configured to: calculate a new z-distance; compare the new z-distance with the old z-distance; and if the new z-distance differs from the old-distance by more than or equal to a predetermined distance threshold, the processor is configured to adopt the new z-distance; and if the new z-distance differs from the old z-distance by less than the predetermined difference threshold, the processor is configured to reject the new z-distance, and to continue calculating the x-distance and the y-distance using the old z-distance.

11. The system according to claim 1, wherein: the system includes a plurality of magnetic sensors.

12. The system according to claim 11, wherein: the plurality of magnetic sensors includes a first pair of sensors consisting of a first sensor and a second sensor, and a second pair of sensors consisting of a third sensor and a fourth sensor; the plurality of magnetic sensors are arranged in a cross formation, in which the first pair of sensors are spaced from each other in a first direction, and the second pair of sensors are spaced from each other in a second direction which is perpendicular or substantially perpendicular to the first direction.

13. The system according to claim 12, wherein: the plurality of sensors are arranged in the x-y plane; the value of the magnitude of the z-component of the magnetic field strength at the plurality of sensors is calculated by taking an average of the value of the magnitude of the z-component of the magnetic field strength measured by each of the plurality of the magnetic sensors.

14. The system according to claim 13, wherein: the values of the magnitudes of the x-component and/or y-component of the magnetic field strength at the plurality of sensors are calculated as a linear superposition of the values of the magnitudes of the x-component and/or y-component of the magnetic field strength measured by each of the plurality of magnetic sensors.

15. The system according to claim 1, wherein: the processor is configured to apply Savitzky-Golay filter or a modified Savitzky-Golay filter to the magnetic field strength data or the relative displacement data in order to smooth it and to remove high frequency noise.

16. The system according to claim 1, wherein: the processor is configured to remove noise from the magnetic field strength data or relative displacement data by using a discrete wavelet transform (DWT) denoising technique.

17. The system according to claim 16, wherein: the DWT denoising technique includes the following steps: transforming the data into the wavelet domain; defining a decomposition level; reducing selected components of the coefficients of the wavelet transform; rescaling the reduced coefficients; and inversely transforming the wavelet transform, to give the denoised signal.

18. The system according to claim 1, wherein: the first element includes an orthopaedic prosthesis; and the second element includes a component which is fixable to a bone, such that the permanent magnet is fixed relative to the bone.

19. The system according to claim 18, wherein: the system further includes: bone cement filling a space between the prosthesis and bone; and a cement restrictor configures to prevent the bone cement from diffusing into the bone.

20. The system according to claim 19, wherein: the orthopaedic prosthesis is an artificial elbow joint.

21. The system according to claim 18, wherein: the magnetic sensor is attached to or embedded into the orthopaedic prosthesis.

22. The system according to claim 18, wherein: the magnet is either: fixable to the bone, fixed to the cement restrictor, or embedded within the bone cement.

23. A method of tracking relative displacement between a first element and a second element in a structure, the structure including a magnet fixed to the first element and a magnetic sensor fixed to the second element, and the method including the steps of: measuring, using the magnetic sensor, a magnetic field strength; and determining the relative displacement between the relative displacement between the first element and the second element, based on the magnetic field strength measured by the magnetic sensor.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Embodiments of the present invention, and experimental results will now be described with reference to the drawings, in which:

[0053] FIG. 1 shows a schematic diagram of a system for tracking migration of an elbow prosthesis.

[0054] FIG. 2 shows a cylindrical magnet and a Hardinge cement restrictor which may be used in the system of FIG. 1.

[0055] FIG. 3 shows a diagram of a cylindrical magnet illustrating its dimensions.

[0056] FIG. 4 shows the magnetic field vector distribution in the z-r plane around the cylindrical magnet.

[0057] FIG. 5 shows a plane contour plot of the radial component of the magnetic field strength (B.sub.r) around a cylindrical magnet.

[0058] FIG. 6 shows a plane contour plot of the z-component of the magnetic field strength (B.sub.z) around a cylindrical magnet.

[0059] FIG. 7 shows a block diagram of the sensor open drain configuration.

[0060] FIG. 8 is a flowchart showing the algorithm which is used to determine the z-distance.

[0061] FIG. 9 is a set of diagrams illustrating movement in the x- and y-directions, along with the associated angles.

[0062] FIG. 10A shows the arrangement of four sensors S1 to S4 which are used in the “quad sesor configuration” described in the “experimental results” section below.

[0063] FIG. 10B shows a process by which a 2-bit address can be configured and allocated to each of sensors S1 to S4.

[0064] FIG. 11 is a flowchart illustrating a DWT denoising process.

[0065] FIG. 12 is a schematic diagram of the setup used to test the system of the present invention.

[0066] FIG. 13 is a plot of estimated vs. actual distance measured using a single sensor configuration.

[0067] FIG. 14 is a plot of displacement in the x-, y-, and z-directions over time for both filtered and unfiltered signals.

[0068] FIG. 15 is a plot demonstrating angular movement around the sensor.

[0069] FIG. 16 is a plot demonstrating the reduction in noise and cross-talk achieved by the present invention, for movement in the y-direction.

[0070] FIG. 17 is a plot demonstrating the reduction in noise and cross-talk achieved by the present invention, for movement in the z-direction.

[0071] FIG. 18 is a plot demonstrating the response of the system to movement in the y-direction with different materials located between the sensor and the magnet.

[0072] FIG. 19 includes four plots demonstrating the response of the system to movement in the y-direction at different, fixed z-distances.

[0073] FIG. 20 demonstrates the sensitivity of the system at different z-values.

[0074] FIG. 21 demonstrates the ability of the system to measure linear and angular displacement simultaneously.

[0075] FIG. 22 shows the response to movement in the y-direction of all four sensors in the quad sensor configuration.

[0076] FIG. 23 demonstrates the reduction in noise in the output of the quad sensors configuration as compared to the single sensor configuration.

[0077] FIG. 24 is a plot of estimated vs. actual z-distance for the quad sensor configuration.

[0078] FIG. 25 shows plots of quasi-static linear movement of the magnet in the y-direction, at constant x- and z-distances, showing both filtered and unfiltered plots.

[0079] FIG. 26 shows a comparison of linear movement detection between a single sensor configuration and a quad sensor configuration.

[0080] FIG. 27 shows filtered plots of quasi-static linear movement of the magnet in the x-direction, at constant y- and z-distances.

[0081] FIG. 28 shows a comparison of angular displacement detection between single sensor and quad sensor configuration.

[0082] FIG. 29 shows a comparison of raw magnetic data and calibrated quad sensor data for movement in the z-direction at constant y-distance, to demonstrate the improvements achieved by the present invention in terms of noise-reduction and cross-talk.

[0083] FIG. 30 shows a comparison of raw magnetic data and calibrated quad sensor data for movement in the y-direction at constant z-distance, to demonstrate the improvements achieved by the present invention in terms of noise-reduction and cross-talk.

[0084] FIG. 31 shows dynamic and quasi-static movement of the magnet in the y-direction at different z-distances.

[0085] FIG. 32 shows a comparison of the single sensor configuration with the quad sensor configuration at different z-values.

DETAILED DESCRIPTION OF THE DRAWINGS

[0086] FIG. 1 shows an example of a system 100 of the present invention in use with a prosthetic elbow joint 102. The drawing shows a user's arm 104, and shows the humerus H, radius R, and ulna U bones. The elbow E is the joint between the humerus H on one side, and the radius R and ulna U on the other side. In FIG. 1, the user has had a prosthetic elbow joint 102 inserted. The prosthetic elbow joint 102 includes an ulnar component 106, a pivot 108, and a humeral component 110. The ulnar component 106 includes an elongate ulnar shaft 112, having a distal end 114, and a proximal end 116. The proximal end 116 is broad, and includes two lateral projections 118a, 118b. The pivot 108 includes a humeral pivot component 120 and an ulnar pivot component 122. The humeral pivot component 120 is rotatable relative to the ulnar pivot component 122. Specifically, the humeral pivot component 120 is rotatable relative to the ulnar pivot component 122 in a manner which mimics the rotation of a human elbow joint, i.e. it operates as a simple hinge joint, which is not able to hyperextend, and with very limited rotation. The ulnar pivot component 122 includes a pair of recesses 124a, 124b, each of which receives a respective lateral projection 118a, 118b of the proximal end 116 of the ulnar shaft 112. Preferably the lateral projections of 118a, 118b are not able to rotate within the recesses 124a, 124b, i.e. there is a tight or snug fit between the lateral projections 118a, 118b and their respective recesses 124a, 124b. The humeral component 110 preferably includes a humeral shaft 126 having a distal end 128 and a proximal end 130, the proximal end 130 being connected to the humeral pivot component 120. The humeral component 110 extends into a cavity 132 inside the humerus H. This cavity 132 is filled with bone cement 134. Preferably, the various components of the elbow prosthesis, specifically the ulnar component 106, the pivot 108 and the humeral component 110 are made of a cobalt-chromium or titanium alloy, since these alloys will have a negligible effect of the magnetic field, and the magnetic sensor 136 will be able to detect the magnetic field without any attenuation.

[0087] The bone cement 134 (either in the configuration shown in FIG. 1, or in any other conceivable configuration) is preferably an inert material which is able to withstand forces to which the human elbow joint is usually subject. It is preferably a polymer, and specifically it is preferably polymethylmethacrylate (PMMA). However, there are many types of bone cement that vary by their viscosity and antibiotic content. In total elbow arthroplasty the low viscosity cement is commonly used with or without antibiotics depending upon the surgeon's preference. PALACOS (Heraeus Noblelight Ltd, UK) low viscosity (PMMA) bone cement with and without antibiotic (Gentamicin) was used in the specific example which was used to obtain the results set out later in this application. The bone cement comes in two components made up of a powder (copolymer) and liquid (monomer). The two components are preferably mixed together at a ratio of 2:1 in a bowl in a fume cupboard (Airone FC 750 model, Safelab systems Ltd. Somerset, UK) for 2 to 3 minutes to form PMMA cement. While in a semi-solid form, the cement was poured into a pre-made mould to make PMMA slabs with thicknesses of 2.5 mm, 5 mm and 7.5 mm. The PMMA cement was left in the mould for 5 minutes to completely polymerize and harden before removing from the mould.

[0088] At the distal end 128 of the humeral shaft 126 is a magnetic sensor 136. The magnetic sensor 136 may take any of the forms suggested earlier in this application, as well as any other of which the skilled person may be aware. In preferred implementations, the magnetic sensor 136 is a magnetoresistive sensor. In the example which was used to obtain the results set out later in this application, the Infineon TLV493D magnetic sensor was used to detect magnetic field intensity in 3 orthogonal directions and from this, the prosthesis position can be determined. The PCB was designed for the sensor and then the sensor was enclosed in 2 mm thick titanium alloy (Ti-6AL-4V). Before calibrating the sensor according to the appropriate working envelope, the sensor 136 must be configured with the data acquisition device, since the sensor 136 is a digital sensor utilizing 120 communication protocol. NI MyRio was used as a data acquisition device to retrieve data from the magnetic sensor via its serial data pin (SDA) serial clock pin (SCL). As the magnetic sensor has the output via 120 protocol, two pull up resistors were required on the 120 line (SDA and SCL) as shown in FIG. 7. These resistors are necessary because the device has an open collector configuration. In open collector configuration, the system can only connect to the clock line (SCL) or signal data line (SDA) to ground but it cannot drive the lines to high. For the line to be able to go to the high voltage the pull up resistor must be inserted because we need a stable voltage state to define the two-binary state of bits i.e. 0 V as 0 bit and 3.3V as 1 bit. The value of the pull up resistor is important for the design configuration because an incorrect value of the resistor can lead to signal loss. By using the following equations the values of the pull up resistor can be calculated:

[00005]$R_{\min }=\frac{V_{cc}-V_{\mathrm{OL}\left(\mathrm{Max}\right)}}{I_{OL}}$

[0089] In which R.sub.min is the minimum pull-up resistor value, V.sub.cc is the supply voltage, V.sub.CL(Max) and I.sub.CL are low level output voltage and current respectively.

[00006]$R_{\max }=\frac{t_r}{0.8473*C_b}$

[0090] In which R.sub.max is the maximum pull-up resistor value, t.sub.r is the rise time, and C.sub.b is the bus capacitance. The pull-up resistor value can then be selected as any value between R.sub.min and R.sub.max.

[0091] At the end of the matrix of bone cement 134 which is furthest from the pivot component 108, there is a permanent magnet 138 and a cement restrictor 140. In the present invention, the cement restrictor 140 is in the form of a disc of material including a number of slits cut into it, which is placed in the cavity 132 at its distal end, in order to prevent cement migration towards the shoulder S. The particular cement restrictor which was used to obtain the experimental results below was a Hardinge Cement Restrictor (see FIG. 2, which shows the cement restrictor 140 and the magnet 138). As discussed, in order to locate the position of the magnet, it is important to know its magnetic field strength. In the setup which was used to obtain the results set out below, an axis-magnetized cylindrical magnet was selected as a suitable source. A permanent magnet such as the example used here is a reliable source of steady magnetic field. The range and accuracy of magnetic field localization depends upon the magnetic remanence (which is the residual magnetization left behind in a ferromagnetic material after an external magnetic field is removed). The higher the magnetic remanence, the higher the range and accuracy. The remanence of the magnet depends on the size of the magnet: the bigger the magnet, the larger the magnetic remanence. In preferred cases, including the setup which was used to obtain the results below, a rare earth neodymium magnet (NdFeB, preferably Nd.sub.2Fe.sub.14B) was used, as it provides the highest available remanence to size ratio. An example of the magnet 138, is shown in FIGS. 2 and 3, in which the magnet 138 has a radius of 7 mm, and a height of 3 mm. In order to investigate the magnetic field of the permanent magnet 138, the inventors carried out an axisymmetric finite element model. FIG. 4 shows the magnetic field vector distribution in the z-r plane (because the simulation is axisymmetric, the magnetic field is axisymmetric around the z-axis of the magnet, and so the x-y plane can be considered as the r-plane). The magnetic field strength at any point P, and its specific coordinate point can be obtained by using the following equations:

[00007]$\frac{B_x}{B_y}=\frac{x}{y}{B}_r=\sqrt{B_x^2+B_y^2}r=\sqrt{x^2+y^2}$

[0092] Where B.sub.x and B.sub.y are the components of the magnetic field at point P in the x- and y-directions respectively, and x and y are the coordinate positions of point P on the x- and y-axes. B.sub.r is the magnetic field in the component of the magnetic field in the radial direction (i.e. the component of the magnetic field in the x-y plane), and r is the distance from the z-axis of the magnet in a direction parallel to the x-y plane.

[0093] Thus, the coordinate of the magnetic field at point P can be calculated by using the following equation.

[00008]$x=r\frac{B_x}{B_r}y=r\frac{B_y}{B_r}$

[0094] So, to obtain the three-axis displacement of the sensor from the three-axis magnetic field the relationship between (B.sub.r, B.sub.r) and (z, r) is enough where B.sub.z is the magnetic field along the z axis of the sensor. FIG. 4 shows the contour plot of the magnetic field density of the permanent magnet (B.sub.r and B.sub.z) in the z-r plane. FIGS. 5 and 6 indicate that the value of Br increases when r is increased, it reaches its maximum value when r=R, the radius of the magnet then Br decreases as r is increased and eventually goes to zero at longer distances. Also FIG. 5 indicates that the value of B.sub.z decreases when z is increased and at distance z>16 mm the value of B.sub.z is zero. To avoid multiple results when determining r and z from the magnetic field the movement of the magnet in r plane should preferably not exceed the specified regions as shown in plots and all the movement should be restricted between these regions. FIG. 6 shows a plot of the radial and z-direction components of the magnetic field of a permanent cylindrical magnet in the z-r plane. From this plot, the following features are apparent: [0095] Nonlinearity: both B.sub.z and B.sub.r in the z-r plane vary nonlinearly when the observation point moves away from the magnet. [0096] Crosstalk effect: the value of B.sub.r varies according to the value of z as well as the value of r, and the value of B.sub.z varies according to the value of r, as well as the value of z.

[0097] For these reasons, and as discussed, it is also necessary to calibrate the magnetic sensor 136. For example, to determine the distance between the magnet and sensor in one direction requires that we understand the interdependency of the output data with movement in the other axes. So, in this section a preferred method is described for the calibration of the magnetic sensor 136 which can be used to determine the change in relative displacement of the second element (i.e. the magnet 138) and the first element (i.e. the magnetic sensor 136). As the magnetic sensor has the capability of measuring magnetic field along three orthogonal axes simultaneously, the sensor can be used to measure field direction in two different planes. By using this concept, we can calibrate the sensor and identify its starting position. For the axis magnetised cylindrical permanent magnet the magnetic field along its axis (i.e. at a radius of zero) can be derived from:

[00009]$\begin{array}{cc}{B}_{Z\_T}\left(z\right)=\frac{{\mu }_{0}M}{2}\left(\frac{z+H_m}{\sqrt{{\left(z+H_m\right)}^2+{\left(\frac{D_m}{2}\right)}^2}}-\frac{z}{\sqrt{z^2+{\left(\frac{D_m}{2}\right)}^2}}\right)& \mathrm{Eqn}.8\end{array}$

[0098] In which M is the axial magnetization of the magnetic, μ.sub.0 is the relative magnetic permeability, and z is the distance from the pole of the magnet in the z-direction. D.sub.m is the diameter of the magnet. The theoretical localization of the permanent magnet can be found by using the equation:

[00010]$\begin{array}{cc}B=B_x+B_y+B_z=\frac{{\mu }_{0}M}{2}\left(\frac{z+H_m}{\sqrt{{\left(z+H_m\right)}^2+{\left(\frac{D_m}{2}\right)}^2}}-\frac{z}{\sqrt{z^2+{\left(\frac{D_m}{2}\right)}^2}}\right)& \mathrm{Eqn}.9\end{array}$

[0099] Where B.sub.x, B.sub.y, B.sub.z are the three components of the magnetic flux density measured by the magnetic sensor 136 along its x-, y- and z-axes respectively. The overall method which is used to determine the z-distance from the measured magnetic field values, the algorithm shown in FIG. 8 may be used.

[0100] In some configurations, a plurality of magnetic sensors may be used, as illustrated in FIG. 10A. In this case, the calculations are slightly different. Here, in order to measure the radiograph-free migration of the elbow prosthesis, the inventors used a magnetic measuring system consisting of 2×2 magnetic sensing array (“quad sensor”) hermetically sealed in titanium metal, a permanent magnet embedded in the cement restrictor and a layer of PMMA cement between them to mimic the original implant position. The selection of the magnetic system was based on the human body transparency towards the magnetic field and the negligible effect of prosthesis material on magnetic flow. In this configuration, a compact 3D magnetic sensor (Infineon TLV493D) with digital output via the 120 bus was used to detect the position of the implant via measuring the magnetic field intensity emitted from the magnet in 3 orthogonal directions. The sensors were mounted on a rigid base (PCB) with a distance of 6 mm between sensors S1 and S2 in the y-direction and 6 mm between sensors S3 and S4 in the x-direction as shown in the FIG. 10A. The size of the rigid base (PCB) was designed based on the dimensions of a commercially available humeral stem diameter. According to Zimmer humeral stem, the diameter of the humeral stem varies from 6 mm to 20 mm. As the quad sensor configuration requires an area of 10 mm, an extra 6 mm area was added in the final PCB design for the auxiliary components. National Instrument (My DAQ RIO 1900) was used as a data acquisition device for configuring and retrieving data from the quad sensor. In the quad sensor configuration, each sensor needs to be configured via the 120 bus and should be allocated with a specific 2-bit address. FIG. 10B describes the configuration process for how to achieve the specific bit address for sensor S1, S2, S3, and S4. In configurations employing four sensors arranged as shown in FIG. 10A, the following equations may be used to calculate the values of B.sub.x, B.sub.y, and B.sub.z:

[00011]$\mathrm{Bxc}=\mathrm{Bx}1-\mathrm{Bx}2+\mathrm{Bx}3-\mathrm{Bx}4\mathrm{Byc}=\mathrm{By}1-\mathrm{By}2+\mathrm{By}3-\mathrm{By}4\mathrm{Bzc}=\frac{{.Math.}_{i=1}^n{B}_{z}i}{n}$

[0101] Thereafter, the same relations as outlined above apply. In order to determine the z-value, the method shown in FIG. 8 may be used. In a first step S01, the magnetic sensor 136 is initialized, and the magnet parameters are specified. In the present case, the magnet parameters may include (but are not limited to) the height of the magnet, the radius or diameter of the magnet, the axial magnetization of the magnet, the material from which the magnet is made. The following additional parameters may also be specified: the material properties (e.g. relative magnetic permeability) of the bone cement, or whichever material is located between the magnetic sensor and the magnet. Then in step S02, a theoretical magnetic field B.sub.z_T is calculated or determined using equation 8 above, using a first z-value. In parallel with this calculation, in step S03, a three-axis magnetic field strength reading is taken from the magnetic sensor, and the three components of the magnetic field strength at the magnetic sensor are summed together. Then, in step S04, a length of the data set is specified. At step S05, it is determined whether the number of readings from the magnetic sensor is equal to the specified data set. If so, the method proceeds to step S06, in which an average of the sum of the magnetic field strength values is taken to give an average B.sub.0. If not, then the method returns to step S03, and the process is repeated until the required number of readings has been taken.

[0102] In step S07, the values of B.sub.C and B.sub.z_T are compared, and in step S08 it is determined whether the difference between the two meets a certain criterion. If so, the z-value used to calculate B.sub.z_T is deemed to be the “correct” value for the z-distance. In other words, if B.sub.C and B.sub.z_T differ by less than or equal to a predetermined threshold, the z-value which gives that value of B.sub.z_T is determined to be the “correct” z-value in step S09, and the corresponding value of z, is noted. If the difference between the two exceeds the threshold, then the process returns to step S02, taking the next value of z, until a suitable value of B.sub.z_T is arrived at. In some cases, the successive z-values which are input into the equations to determine B.sub.T are at 0.01 mm intervals, but other intervals are also envisaged (e.g. 0.05 mm, 0.1 mm, 0.5 mm).

[0103] As discussed, the focus of the present invention is to track the change in relative displacement between the magnet and magnetic sensor. So, in step S10, it is determined whether the value of z is the same as the previous value. In preferred cases, the requirement is not that the z-value is exactly the same; rather, if the “new” z-value differs from the previous z-value by an amount which is less than a predetermined difference threshold, then the previous z-value is maintained, in step S11. And, if not, in step S11, the value of z is updated. Then, after a fixed interval (i.e. the sampling interval) the process is repeated. Now that the z distance is determined, it is possible to determine the linear displacements in the x- and y-axes as follows.

[0104] As shown in FIG. 9, when the magnet is displaced in the y-direction (with the z-distance remaining constant), it makes an angle of α with the z-axis, in the y-z plane. Similarly, when the magnet is displaced in the x-direction (with the z-distance remaining constant), it makes an angle of β with the z-axis in the x-z plane. The values of α and β may be found as follows:

[00012]$\alpha ={\tan }^{-1}\left(\frac{B_y}{B_z}\right)\beta ={\tan }^{-1}\left(\frac{B_x}{B_z}\right)$

[0105] Then, the x- and y-distances may be found as follows:

x=z.Math.tan β

y=z.Math.tan α

[0106] Filtering the Results (i): Savitzky-Golay Filter

[0107] As discussed earlier in this application, the results can be improved by filtering the received data. The data received from the sensor need to be smoothened as it contain high frequency content that cannot be removed by plain FIR average filter. To achieve the high degree of noise removal from the desired signal, the length (N) of the signal has to be larger so that the signal bandwidth become greater than the filter pass band frequency.

[00013]${\omega }_c=\frac{\pi }{N}$

[0108] In the current study, Savitzky-Golay (SG) filter also known as least-square or polynomial smoothing filter is used to smooth the desired signal. The SG filter is basically a low pass filter or it can be also consider as a type of finite impulse response FIR digital filter that can preserve the high-frequency content of the desired signal. The output signal from the sensor can be represented as:

y(n)=x(n)+w(n)

[0109] Where x(n) represents the magnetic field signal with high frequency content while w(n) is the associated noise with magneto resistive sensor i.e. Johnson (thermal noise), shot noise, 1/f (flicker) noise. The SG filter can be defined by two parameters that are denoted as K for the polynomial degree and M for sequence. The following assumptions are made in the SG filter:

[0110] I. All data points of the signal should be natural numbers.

[0111] II. The length of the signal should be N=2M+1 and is odd for the sequence of M.

[0112] III. Data points should be positioned symmetrically about the origin x.sub.o as follow:

x.sub.N=[x.sub.−M, . . . ,x.sub.−1,x.sub.0,x.sub.1, . . . ,x.sub.M]

[0113] Polynomial smoothing of N samples of data is equivalent to replacing them by the values that lie on smooth polynomial curves drawn between the noisy samples.

{circumflex over (x)}.sub.m=c.sub.0+c.sub.1m+ . . . +c.sub.km.sup.k,−M≤m≤M

[0114] Where {circumflex over (x)}.sub.m represent the m.sup.th smooth data point. The coefficients c.sub.i are determined optimally by least square fit that minimize the least square error and also fits the given data on corresponding polynomial curve. For N data samples the performance index can be minimized as follows:

[00014]$E\underset{m=-k}{\overset{k}{.Math.}}{e}_m^2=\underset{m=-k}{\overset{k}{.Math.}}({x_{m-}\left(c_0+c_{1}m+.Math.+c_k{m}^k\right)}^2$

[0115] Similarly, we define K+1 polynomial basis vectors as follows:

S=[s.sub.0,s.sub.1, . . . ,s.sub.k]

[0116] Hence, the smooth data can be represented in vector form as:

[00015]$\hat{x}=\underset{i=0}{\overset{k}{.Math.}}{c}_i{s}_i=B_x$

[0117] As the data points should be symmetrically about the origin, the middle smoothed value y.sub.0={circumflex over (x)}.sub.0 is given in terms of middle SG filter b.sub.0.

[00016]$y_0=b_0^{T}x=\underset{m=-M}{\overset{M}{.Math.}}{b}_0\left(m\right)x_m$

[0118] The N-dimensional vector x can be shifted to n instants of time as follows:

x.fwdarw.[x.sub.n−M, . . . ,x.sub.n−1,x.sub.n,x.sub.n+1, . . . ,x.sub.n+M]

[0119] The resulting length N, order K, SG filter for smoothing a noisy sequence x(n) will be, in its steady-state form as:

[00017]$y\left(n\right)=\underset{m=-M}{\overset{M}{.Math.}}{b}_0\left(-m\right)x\left(n-m\right)$

[0120] Filtering the Results (ii): DWT Denoising.

[0121] As discussed, the data received from the sensors can be difficult to analyse because it may contain high content of noise. Therefore the data may need to be smooth. In an alternative to Savitzky-Golay filtering, discrete wavelet transform (DWT) denoising techniques may be used to estimate the signal from the sensor and remove noise component. DWT provides an effective denoising with minimal computational complexity. The DWT denoising technique consists of three main steps with five parameters as shown in the flowchart in FIG. 11.

[0122] First, is the data from the sensor needs to be transformed into a wavelet domain with the length of the signal power of 2. This transformation can be done by selecting a mother wavelet function (ø) from the wavelet family. Selecting a suitable type of wavelet is extremely important for denoising the data because something two similar wavelets may give different denoising data. After selection of the wavelet function, the decomposition level (k) needs to be defined. Then, a criterion is selected to reduce or shrink the coefficient of the wavelet transform. The coefficient of the wavelet transform can be reduced by selecting a specific thresholding function (β). There are four types of thresholding functions that are commonly used. Among them, Stein's Unbiased Risk Estimate (SURE) threshold is mostly used because of its state of the art decomposition of noise and better performance. The SURE thresholding can be define as:

[00018]$\mathrm{SURE}\left(t,x\right)=N-2\times M_{\left(i:.Math.x_i.Math.\le t\right)}+\underset{i=1}{\overset{N}{.Math.}}{\left(.Math.x_i.Math.\wedge t\right)}^2$

[0123] Where x, is the detailed wavelet coefficient, t is the candidate threshold, N is the length of data and M is the number of the data points less than t. SURE thresholding is generally used to obtain the unbiased variance between the unfiltered and filtered data. After defining the function, the thresholding selection rule (γ) is selected. In DWT denoising, denoising of the signal depends greatly on the selection of the noise threshold. The wrong choice can result in lowering down the signal strength. Traditional thresholding transforms the coefficient whose magnitude is below the specified value. In the DWT denoising technique, there are different thresholding selection parameters but the most commonly used threshold is the global thresholding. In global thresholding noise is assumed to have Gaussian distribution, having the same amplitude and frequency distribution that span the same data length. Global thresholding can be further divided into soft and hard thresholding which can be defined respectively in the equations below:

[00019]$x_{j,i}^{\prime }=\left\{\begin{array}{ccc}{x}_{j,i}& :& .Math.x_{j,i}.Math.\ge t\\ 0& :& .Math.x_{j,i}.Math.<t\end{array}\right\}{x}_{j,i}^{\prime }=\left\{\begin{array}{ccc}\mathrm{sign}\left(x_{j,i}\right)\left(.Math.x_{j,i}.Math.-t\right)& :& .Math.x_{j,i}.Math.\ge t\\ 0& :& .Math.x_{j,i}.Math.<t\end{array}\right\}$

[0124] Where and x.sub.j,i are x′.sub.j,i the noise and denoise coefficients of the wavelet at a j.sup.th decomposing level and i location. Finally, the shrunk coefficients are first rescaled (p) and then inversely transform to the original domain which is the denoised signal.

[0125] To check the performance of the filtered data signal to noise ratio (SNR) and its root mean square error (RMSE) can be determined as below:

[00020]$\mathrm{RMSE}=\sqrt{\frac{1}{N}\underset{n=1}{\overset{N}{.Math.}}{\left[x\left(n\right)-x^{\prime }\left(n\right)\right]}^2}\mathrm{SNR}=10{\log }_{10}\left\{\frac{{.Math.}_{n=1}^N{\left[x\left(n\right)\right]}^2}{{.Math.}_{n=1}^N{\left[x\left(n\right)-x^{\prime }\left(n\right)\right]}^2}\right\}$

EXPERIMENTAL RESULTS

[0126] The section below includes experimental data from two sets of experiments. In the first, a single magnetic sensor was used. In the second set of data, a quad sensor (i.e. a sensing arrangement including four magnetic sensors, such as the one shown in FIG. 10) was utilized.

[0127] A. Single Sensor Configuration

[0128] To evaluate the performance of the sensor and to obtain the correlation between the magnetic field and displacement, a mechanical testing system, shown in FIG. 12, (TA, Electro Force 3300, Boston, USA) was used to provide input migration of the implant via its two motorized stages i.e. linear and rotational with the resolution of 0.5 μm linearly, 0.01 degree angularly and 0.01 Hz frequency. To provide a repeatable simulation of implant migration, the inventors designed and fabricated an adjustable fixture/bracket for holding the sensor and magnet embedded in a cement restrictor. The fixture was attached with the Electro-force machine. One of the motorized motors moved the sensor bracket linearly in the y axis while the other can move the magnet bracket rotationally in the x/z plane. In order to move the magnet bracket in z-axis a linear actuator was attached with the resolution of 1 mm and this was used to adjust the distance between the sensor and magnet. The Electro-Force machine was programmed to move in the y axis quasi-statically (3 minute intervals) at amplitudes of 0.3 to 4 mm using square waveforms and dynamically using sine waveform at 0.1 Hz. Similarly the machine was programmed to move in the x/z plane (rotation) quasi-statically. The z direction was controlled via the linear actuator, which was programmed using an external DAQ card (NI LabView) the DAQ card was also used to communicate and acquire data with magnetic sensor and to record data into a measurement file.

[0129] Sensor Calibration

[0130] The sensor was first calibrated in the z-axis only in order to estimate the distance between the sensor and magnet. The permanent magnet was placed perpendicular to the z-axis of the sensor (Note that North Pole was facing towards the sensor, if poles changes the magnetic field sign change from positive to negative) and was moved linearly with the range of 18 mm at a step size of 1 mm. The resulted data set was processed with the algorithm as shown in FIG. 8 to determine the z-distances. FIG. 13 shows the comparison of the estimated distance with actual distance. The result showed how well the algorithm fitted the actual value with R.sup.2 close to 0.9991.

[0131] After estimating the distance between sensor and magnet, a similar set of experiments were conducted to analyse how the system detected the displacement of the sensor/magnet if they moved in a non-z direction. To evaluate this, the sensor was first moved linearly in y-axis ranging from 0.1 mm to 4 mm with the step size of 0.5 mm (at z=15 mm). It was and also moved angularly around the y-axis ranging from 0 to 4.0 degrees. FIG. 14 shows that the system was able to detect the displacement of magnet in the y-axis with no change in the x-axis showing a resolution of up to 0.3 mm. Displacement detection at different movement are provided in Table 1 for both filtered and unfiltered signals. The filtered version shows a considerably low standard deviation at 0.3 mm movement of 0.079 compared to the unfiltered standard deviation of 0.39. Similar results were seen during x-axis linear movement where the y-distance remained constant.

TABLE-US-00001 TABLE 1 Real and estimated values of movement in the y-direction. Actual Estimated Movement Estimated Movement movement (mm) without filter (mm) with filter (mm) 0.3 0.321 ± 0.390 0.328 ± 0.079 0.5 0.527 ± 0.393 0.530 ± 0.075 1.0 1.078 ± 0.386 1.082 ± 0.089 1.5 1.627 ± 0.384 1.632 ± 0.074 2.0 2.126 ± 0.372 2.133 ± 0.079 2.5 2.660 ± 0.371 2.665 ± 0.080 3.0 3.169 ± 0.380 3.173 ± 0.079 3.5 3.701 ± 0.371 3.705 ± 0.078 4.0 4.191 ± 0.369 4.195 ± 0.085

[0132] FIG. 15 shows the angular movement of the magnet around the sensor. The system was able to detect angular displacement until 3.0 degrees (approximately 2 mm). Also, from FIG. 15 it can be observed that up to 1 degree rotation (1.2 mm angular displacement) there is no change in the y movement but beyond 1 degree the y displacement start to increase. Also, it was observe that beyond 3.0 mm the sensor was able to detect the magnetic field but it introduced error in the tracking algorithm this error was due to the tilting effecting of the sensor.

[0133] The graphs in FIG. 16 show the comparison of magnetic field (B.sub.y) with the calibrated sensor output during the y-axis displacement at different z values. It was observed that during the y-axis displacement at different z-distances the magnetic field (B.sub.y) changes, demonstrating a strong cross-talk effect, whereas the output from the calibrated sensor demonstrated a strong correlation with the actual y-distance across different z values. In other words, the undesirable cross-talk effect can be eliminated using the system of the present invention.

[0134] Also, the graphs in FIG. 17 show that during z-axis displacement at different y-distances, similar effects were observed with strong cross-talk in magnetic field (Bz) which is eliminated when using the calibrated sensor.

[0135] Sensor Performance for Different Movements and Materials

[0136] The calibration steps described above, and whose results were shown in FIGS. 13 to 17, were performed without introducing any material between the sensor and magnet. To further investigate the performance of the sensor, the system was tested with the biomaterials that resembles the environment, that are present in the elbow prosthesis, namely, PMMA cement, UHMWPE and Titanium alloy (Ti-6Al-4V). FIG. 18 shows that these materials have negligible effect on the measuring system, as do the results in Table 2.

TABLE-US-00002 TABLE 2 Mean and standard deviations of static and dynamic distances for a range of different materials. Static Movement Dynamic Movement Materials (0.3 mm) (0.3 mm) No Material 0.3091 ±0.023 0.3133 ± 0.1012 Titanium 0.3191 ± 0.023 0.3231 ± 0.1040 Titanium and Bone cement 0.3085 ± 0.033 0.3305 ± 0.1212 Titanium and UHMWPE 0.3043 ± 0.028  0.3323 ± 0.11175 Titanium, Bone cement, 0.3085 ± 0.033 0.3317 ± 0.1221 and UHMWPE

[0137] Sensitivity

[0138] In the literature, there is no specific information about the minimum distance between the humeral tip and the cement restrictor in total elbow arthroplasty. However, in shoulder arthroplasty the minimum distance between cement restrictor and humeral component is around 10 mm. In order to check the sensitivity of the system of the present invention, the inventors placed the permanent magnet in the cement restrictor at a distance of greater than 10 mm from the magnetic sensor and showed the system performs under static and dynamic linear movement in they direction ranging from 0.15 mm to 1 mm. FIGS. 19 and 20 show that as the distance between sensor and magnet is increased the resolution of the system decreases and more noise content are introduce to the signal. At z=15 mm, the system was able accurately to detect static and dynamic movement. Beyond this distance 0.5 mm static movement was detected but with higher noise content. Specifically, FIG. 20 shows the sensitivity of the single sensor configuration with respect to various z-distance values when detecting linear movement in the y-direct. Also, the bars in the graph represent different implant movement e.g. the left-most bar shows the results when the implant is moved 0.15 mm, the next bar 0.20 mm, the next bar 0.30 mm, the next bar 0.50 mm and the right-most bar 1.00 mm.

[0139] The results show that, ideally, the distance between sensor and magnet for static and dynamic displacement should be below 16 mm.

[0140] Linear and Angular Movement

[0141] FIG. 21 shows that the system was able to detect linear and angular displacement simultaneously. As describe previously, there was a small increment in the y axis movement when angular displacement exceed 1.2 mm (1 degree).

[0142] B. Quad Sensor Configuration

[0143] A similar experimental setup was used to analyse the quad sensor arrangement, as for the single sensor arrangement. First, the effect of the magnetic field on the quad sensor configuration was analysed. As shown in FIG. 10, the sensors are very closely placed to each other. This means that if there is any type of magnetic variation all the sensors should read the same effect. FIG. 22 shows the variation in the y-component of the magnetic field strength, when the permanent magnet was randomly moved.

[0144] It can be seen that the y-components of the magnetic field strength of all four sensors change in unison as the permanent magnet is moved in the y-direction, but with different magnitude because of their position. The magnetic field of sensors 3 and 4 were almost the same because their y-distance values are the same. In contrast, because sensors 1 and 2 are 6 mm apart from each other in the y-direction, they have a magnetic field magnitude difference. Similarly, when the magnet was moved in x-axis all the sensors showed the same magnetic field variation response but with different magnitude.

[0145] As shown in FIG. 23, the magnetic field strengths measured by the quad sensor configuration show a reduction in the noise content as compared with the single sensor configuration. This reduction was observed even without the application of any kind of filtering technique to it. FIG. 23 shows the raw magnetic field strength data measured by the single sensor configuration and quad sensor configuration, for each of the x-, y-, and z-components of the magnetic field strength. This reduction in noise which is exhibited in the data from the the quad sensor configuration arises due to the cancellation of external magnetic fields: the sensors each measure the same amount of external magnetic field so by combining them external magnetic field variation is cancelled.

[0146] Z-Distance Estimation

[0147] As described in the previous section the quad sensor needs to be calibrated in order to determine the localization of the magnet. The first step of the detection algorithm is to determine the z-distance between the sensor and magnet. In order to check the effect of the quad sensor in estimating the z-distance, the permanent magnet enclosed in cement restrictor was placed perpendicularly, pointing towards to the quad sensor that was hermetically sealed in 2 mm thick titanium alloy. By using the linear actuator the distance between the quad sensor and magnet was increased with a step size of 1 mm. FIG. 24 shows that the estimated distance result from the quad sensor configuration matches well with the actual distance with a R.sup.2 values of 0.9993.

[0148] Linear Movement in y-Direction

[0149] As discussed in the calibration section, displacement detection in the x and y-axis can be determined by using the equations given earlier in this application. In order to analyse the performance of the quad sensor in detecting movement in the x- and y-directions. The sensor was first moved linearly in the y-direction ranging from 0.15 to 4.0 mm with a step size of 0.5 mm, keeping the x-distance constant and with a z-Distance of 15 mm. Compared with the single sensor configuration, the noise content in the quad sensor is low as previously described. However, additional filtering is carried out here, in order to smooth the data. Table 3 below shows the selected type of DWT filter along with its parameters. The selection of mother wavelet and decomposition was based on the signal reconstruction and SNR value.

TABLE-US-00003 TABLE 3 Selected parameters for DWT 1.sup.st Step 2.sup.nd Step 3.sup.rd Step Mother Decomposition Threshold Threshold Threshold Parameters Wavelet level Function Selection rescaling Selected Sym 6 6 SURE Hard one Parameter

[0150] FIG. 25 shows both filtered and unfiltered x-, y-distances along with the z-Distance estimation value. The result from the quad sensor configuration showed a well-matched linear displacement detection with a resolution of 0.15 mm.

[0151] FIG. 26 shows a comparison of a single sensor configuration with a quad sensor configuration, keeping the z-distance 15 mm. Single sensor configuration has a resolution of 0.3 mm while the quad sensor has a 0.15 mm. In the quad sensor configuration, the standard deviation error and mean error remained minimal as compared to single sensor configuration, the standard deviation error and mean error of which increased with the increase of displacement. The bars in the graph are used to differentiate the different linear movement in y-axis. Starting from the left: 0.15 mm, 0.30 mm, 0.50 mm, 1.0 mm, 1.5 mm, 2.0 mm, 2.5 mm, 3.0 mm, 3.5 mm, 4.0 m.

TABLE-US-00004 TABLE 4 Comparison of RMSE and SNR for single and quad sensor configurations. Displacement Single Sensor Configuration Quad Sensor Configuration (mm) RMSE SNR RMSR SNR 0.15 0.063 33.27 0.031 39.34 0.30 0.051 41.14 0.032 45.30 0.50 0.066 43.36 0.021 53.51 1.00 0.137 43.03 0.026 57.59 1.50 0.157 45.40 0.035 58.47 2.00 0.222 44.87 0.051 57.72 2.50 0.288 44.55 0.047 60.31 3.00 0.355 44.32 0.033 64.92 3.50 0.390 44.64 0.026 68.41 4.00 0.479 44.21 0.020 71.83

[0152] Table 4 describes the RMSE and SNR between both configurations. It was observed that in the single sensor configuration the maximum SNR value was 45 dB at 1.50 mm displacement and with increase in distance the RMSE increased. While, in quad senor configuration with increase in distance the SNR value increased and this configuration showed a maximum RMSE of 0.051.

[0153] Angular Movement in x-Direction

[0154] FIG. 27 shows the results obtained in the quad sensor configuration, when the magnet was moved angularly about the x-axis ranging from 0.5 degrees to 4.0 degrees with a step size of 0.5 degrees keeping the position with respect to the y-axis constant and at Z-Distance of 15 mm. Up to 1 degree of movement (approximately 0.60 mm) the z-distance remains same, but beyond 1 degree there was a change in the z-distance value. This change in value was only observed when the magnet was angular moved in a negative direction up to 1.5 degrees. After 2 degrees the same change in the z-distance was observed in both negative and positive axis. Also, there was a slight change in the y-distance when the magnet was angularly moved beyond 2 degrees.

[0155] The change in z-distance arises because as the magnet angularly moves it changes the distance between the sensor and magnet, which our algorithm was able to detect. Also, the in z-distance can differentiate between linear and angular movement. The change in the y-distance is due to the orientation of the magnet.

[0156] FIG. 27 shows a comparison of a single sensor configuration with a quad sensor configuration during angular motion. As described previously, the main drawback of single sensor configuration was the effect of tilt. Due to this effect, the single sensor configuration was only able to detect movement within 3 degrees depending upon its orientation position. In the quad sensor configuration, the tilting effect is resolved. This was achieved by introducing specific sensor selection criteria.

[0157] FIG. 28 shows the position of the magnet observed by all four sensors. In the drawing, S1, S2, S3, S4 show the position of the sensors, while SP-S1 to SP-S4 show the starting position of the magnet when there was no angular movement and EP-S1 to EP-S4 show the end position of the magnet when it was angularly moved 4 degrees (approximately 2.3 mm) in both positive and negative direction.

[0158] This shows that the quad sensor configuration of the present invention clearly specifies the position of the magnet. During the angular movement experiment, the starting position of the magnet was at the centre of sensor S1, then moved 4 degrees in both the positive and negative directions. As discussed, the single sensor can detect angular displacement up to 3 degrees depending upon the orientation. In FIG. 28 during the positive x-direction displacement, S1 was only able to measure the till 1.5 degrees beyond that the mean error and standard deviation error started to increase. Similarly, the quad configuration also showed the same response. By introducing the sensor selection criteria this tilting effect was compensated.

[0159] The sensor selection criteria depend upon the localization of the magnet. It can be observed that during positive x-direction displacement, the magnet is displaced between sensor S1 and S3. Therefore eliminating S2 and S4 value from the quad configuration gives a well-matched result. Similarly, during the negative x-direction displacement the single sensor and quad sensor configurations were able to detect the movement with minimum error, but as explained previously as the magnet lies between sensor S1 and S4. Therefore, by combining the values of these sensors and eliminating sensors S2 and S3 gives better, well-matched results.

[0160] Cross-Talk

[0161] The magnetic field density of a permanent cylindrical magnet displays two features: non-linearity and cross talk. Due to these, the magnetic field does not vary linearly with spatial location, and movement in a single can direction (e.g. the x-direction) can lead to a change in all three of the components of the magnetic field strength, which is experienced by e.g. the magnetic sensor. These features give rise to a correlation between displacement and magnetic field which is not-insignificant and challenging to resolve.

[0162] FIG. 29 shows the y-component of the magnetic field strength, as measured by the quad sensor, when the magnet was moved to 1 mm with a step size of 0.1 mm at different z-distance positions. FIG. 29 (LHS) shows that without calibration the y-component of the magnetic field strength, with varying z-distances shows a strong cross talk effect, along with non-linearity.

[0163] FIG. 29 (RHS) shows that the calibrated output from the quad sensor shows a close agreement with the actual displacement, along with elimination of the cross talk effect.

[0164] A similar set of experiments was conducted to analyse the effect of the correlation of magnetic field with respect to displacement by moving the magnet in the z-direction to 4 mm with the step size of 1 mm at different y-distances. FIG. 30 (LHS) demonstrates the same effect of strong cross-talk and non-linearity as the magnet was moved in z-axis at different y distance value, and FIG. 30 (RHS) shows the calibrated sensor output, clearly demonstrating elimination of the cross talk effect.

[0165] Resolution

[0166] To analyse the sensitivity of the quad sensor configuration in static and dynamic movement. 4 sets of experiments were conducted by moving the magnet quasi statically and dynamically in the y-direction to 1 mm with a set of 0.15, 0.20, 0.30, 0.50 and 1.00 mm at different z-distances. The selection of z-distances was based on the teaching of the literature, that the minimum distance between the humeral tip and cement restrictor is 10 mm. So, the starting z-distance was selected to be 10 mm and the end z-distance 20 mm. The results of these experiments are shown in FIGS. 30 and 31.