Background Magnetically marked capsules serve for the analysis of peristalsis and throughput times within the intestinal tract. Moreover, they can be used for the targeted disposal of drugs. The capsules get localized in time by field measurements with a superconducting quantum interference device (SQUID) magnetometer array. Here it is important to ensure an online localization with high speed and high suppression of disturbing fields. In this article we use multipole expansions for the simultaneous localization and suppression of disturbing fields.
Background
Magnetically marked capsules serve for the analysis of peristalsis and throughput times within the intestinal tract. Moreover, they can be used for the targeted disposal of drugs. The capsules get localized in time by field measurements with a superconducting quantum interference device (SQUID) magnetometer array. Here it is important to ensure an online localization with high speed and high suppression of disturbing fields. In this article we use multipole expansions for the simultaneous localization and suppression of disturbing fields.Methods We expand the measurement data in terms of inner and outer multipoles. Thereby we obtain directly a separation of marker field and outer disturbing fields. From the inner dipoles and quadrupoles we compute the magnetization and position of the capsule. The outer multipoles get eliminated.
Methods
We expand the measurement data in terms of inner and outer multipoles. Thereby we obtain directly a separation of marker field and outer disturbing fields. From the inner dipoles and quadrupoles we compute the magnetization and position of the capsule. The outer multipoles get eliminated.Results The localization goodness has been analyzed depending on the order of the multipoles used and depending on the systems noise level. We found upper limits of the noise level for the usage of certain multipole moments. Given a signal to noise ratio of 40 and utilizing inner dipoles and quadrupoles and outer dipoles, the method enables an accuracy of 5 mm with a speed of 10 localizations per second.
Results
The localization goodness has been analyzed depending on the order of the multipoles used and depending on the systems noise level. We found upper limits of the noise level for the usage of certain multipole moments. Given a signal to noise ratio of 40 and utilizing inner dipoles and quadrupoles and outer dipoles, the method enables an accuracy of 5 mm with a speed of 10 localizations per second.Conclusion The multipole localization is an effective method and is capable of online-tracking magnetic markers.
Conclusion
The multipole localization is an effective method and is capable of online-tracking magnetic markers.

Background The transport of capsules in the alimentary tract underlies complex influencing factors like the patients peristalsis, the hydration and the form and size of the capsules. A procedure which allows the instantaneous localization of the capsules supports a number of patient examinations as well as examinations of new drug forms [ 1 - 4 ]. Capsules can be marked radioactively (scintigraphy) or magnetically. The scintigraphy [ 5 ] has a lower time resolution compared to the magnetic localization, and due to radiation it is not appropriate for examinations with healthy probands. The localization of magnetically marked capsules (magnetic markers) must be spatially accurate and with high temporal resolution. For the spatial localization the marker field must be separated from the external magnetic disturbing fields. This separation can be achieved by splitting the magnetic field in multipole moments [ 6 ]. The method proposed utilizes the multipole moments directly for the determination of the position and the magnetic moment of the marker. Thus, the separation of disturbing fields and the localization are integrated numerically effective into one procedure. This allows a fast online-localization of the marker capsules. Multipole expansions are used also to model spatially distributed biological sources such as brain currents [ 7 , 8 ]. The application of multipoles for the localization of magnetic dipoles is described in [ 9 , 10 ], and is used in other technical areas without disturbing field suppression [ 11 ]. Marking of capsules and pills takes place by partially filling them with black iron oxide (Fe 3 O 4 ) which is subsequently magnetized up to saturation. The magnetic field measurement is performed within magnetically shielded rooms by the use of highly sensitive SQUID arrays. For the investigation at hand we conduct simulation runs to determine the performance of the multipole localization.
Background
The transport of capsules in the alimentary tract underlies complex influencing factors like the patients peristalsis, the hydration and the form and size of the capsules. A procedure which allows the instantaneous localization of the capsules supports a number of patient examinations as well as examinations of new drug forms [ 1 - 4 ]. Capsules can be marked radioactively (scintigraphy) or magnetically. The scintigraphy [ 5 ] has a lower time resolution compared to the magnetic localization, and due to radiation it is not appropriate for examinations with healthy probands.The localization of magnetically marked capsules (magnetic markers) must be spatially accurate and with high temporal resolution. For the spatial localization the marker field must be separated from the external magnetic disturbing fields. This separation can be achieved by splitting the magnetic field in multipole moments [ 6 ]. The method proposed utilizes the multipole moments directly for the determination of the position and the magnetic moment of the marker. Thus, the separation of disturbing fields and the localization are integrated numerically effective into one procedure. This allows a fast online-localization of the marker capsules.Multipole expansions are used also to model spatially distributed biological sources such as brain currents [ 7 , 8 ].The application of multipoles for the localization of magnetic dipoles is described in [ 9 , 10 ], and is used in other technical areas without disturbing field suppression [ 11 ].Marking of capsules and pills takes place by partially filling them with black iron oxide (Fe 3 O 4 ) which is subsequently magnetized up to saturation. The magnetic field measurement is performed within magnetically shielded rooms by the use of highly sensitive SQUID arrays. For the investigation at hand we conduct simulation runs to determine the performance of the multipole localization.Methods Algorithm The field of a magnetic marker located adjacent to the point of origin can be expressed by a multipole expansion in Cartesian coordinates ( x 1 ,  x 2 ,  x 3 ). If the distance   between marker and origin is small compared to the distance   between a magnetic sensor and the origin, the field of the marker at the sensor position is given by the first elements of the multipole expansion. With the notation follows with  c m  being the dipole, quadrupole and octopole moments of the field expansion. The form functions   arise from a Taylor series expansion in the parameter   with   and  . It holds With the Kronecker delta   follows and Conversely, a Taylor series expansion of – compared to the sensor coordinates – far away located field sources in the parameter   with   yields a multipole expansion of external disturbing fields: We denote the multipole moments  c ex  of the expansion of fields of external sources as "outer moments" to distinguish them from the "inner moments"  c m . To get the same normalization and symmetry properties for the outer and inner form functions, we define the outer form functions   by It follows and The tensors of 3 rd  order   and of 4 th  order   own the following symmetry features which are identical for the inner and outer multipole expansion: We combine the resulting 3, 5 and 7 linearly independent components of the tensors of 2 nd , 3 rd  and 4 th  order to one vector for the marker field   and one vector for the external disturbing field  : The summation of equation (1) and equation (6) yields the field expansion for a magnetic marker with disturbing fields. We truncate this expansion after the octopole terms, and transcript it into a linear equation system for the determination of equivalent multipole moments  c  for a measurement  B meas : The structure of the vectors  B meas  and  c  and the matrix  F  is given below in the formulas (15...24). The residuals 0(·) are sufficiently small, if the coordinates of the marker are small compared to the coordinates of the field sensors  , and if the coordinates of the field sensors are small compared to the coordinates of the external disturbing field sources  . B meas  is a vector with the measurement values of the magnetometer sensor field in the positions   with the directions  : The matrix  F  is built from the linearly independent form functions for inner and outer field sources given in equation (13). Their scalar product with the sensor normal directions   yields one row for every sensor: The number of columns of  F  is the sum of the numbers of inner and outer field functions used. Each column describes the field of one specific magnetic moment with unit strength measured by the sensor system. The Matrix  F  is called the forward matrix of all moments considered. The Matrix  F  is structured into submatrices for different moments: Matrix   and   are the forward matrices for inner and outer dipoles: Matrix   and   are the forward matrices for inner and outer quadrupoles. The size of   is ( Nsen /5) with the rows belonging to quadrupole moments with indices (1,1; 3,3; 1,2; 2,3; 3,1). Matrix   and   are the forward matrices for inner and outer octopoles. The size of   is ( Nsen /7) with the rows belonging to octopole moments with indices (1,2,2; 2,3,3; 3,1,1; 1,3,3; 2,1,1; 3,2,2; 1,2,3). The vector of multipole moments  c  is composed of inner and outer dipole moments  c d , quadrupole moments  c q  and octopole moments  c o : The inner dipole moments   describe a dipole at the point of origin, the outer dipole moments   describe a homogeneous disturbing field: The   represent a quadrupole at the point of origin. The   describe an external gradient field, whose field strength vanishes at the origin and which has no spatial derivations of 2 nd  or higher order. This field can be measured by five ideal gradiometers at the origin and can be compared with the creation of software gradiometers. The   represent an octopole at the origin. The   describe an external gradient field of 2 nd  order, which has no spatial derivations of 3 rd  or higher order and whose field strength and spatial derivatives of 1 st  order vanish at the origin. This field could be measured by 7 ideal second order gradiometers at the origin, it can be compared with the creation of software gradiometers of 2 nd  order: Due to its small spatial extension, the magnetic marker can be described as a dipole of strength   at position   as a good approximation. The field of this dipole is With the Taylor series expansion follows in analogy to equation (1) A comparison of coefficients of (1) and (27) yields and The dipole strength   can be determined by the dipole moment  . An equation system for the adjacent calculation of the dipole position   from the dipole strength   and the quadrupole moment   follows from (29) and (23): with This equation system is named shift equation in analogy to [ 10 ]. It is overdetermined, and can be solved by means of the pseudo inverse of  m . We get the multipole moments  c , which are required for the localization of the marker dipole, from solving the overdetermined equation system (14) by means of the pseudo inverse of  F : c  = ( F T · F ) -1 · F T · B meas .     (33) Here, the matrix of form functions  F  must contain columns at least for the moments   and  . Iterative dipole localization for a fixed dipole (e.g. one time point) is achieved by using the localization position as a new point of origin. The step length of the last localization step serves as a stop criterion for the iterative localization procedure. This is justified by considering the residuals of equation (14) within the convergence range of the procedure, and will practically be shown by the results of the following simulations. The tracking of a moved dipole based on measurements at consecutive time steps works by updating both the point of origin and the measurement data set after each localization step (Fig.  1 ). The localization step must be monitored, since it contains information about the marker speed and the noise dependent and speed dependent localization errors. Figure 1 Flow chart of the algorithm for online localization.  This algorithm is meant for online localization, and therefore comprises only one iteration. A high signal to noise ratio and a high computing speed render 2–3 iterations per measurement cycle possible. Measurement system The simulations to determine the performance of the algorithm use the sensor geometry of the multi channel SQUID system Argos 200 from AtB (Advanced Technologies Biomagnetics, Pescara, Italy). The ARGOS 200 system contains fully integrated planar SQUID magnetometers produced using Nb technology with integrated pick-up loops. The sensing area is a square of 8 mm side length. The intrinsic noise level of the built in 195 SQUID sensors is below 5 fT Hz -1/2  at 10 Hz. Three sensors form one orthogonal triplet in each case. The measurement plane with a diameter of 23 cm consists of 56 of those triplets. The reference array consists of seven SQUID sensor triplets located in the second level in a plane which is positioned parallel to the measurement plane at a distance of 98 mm. The third (196 mm above the first plane) and fourth (254 mm above the first plane) levels contain one triplet each (Fig.  2 ). Figure 2 SQUID Array Argos 200.  The ATB SQUID Array Argos 200 consists of 195 magnetometers which are arranged in orthogonal sensor triplets in four levels. The measurement area of each sensor is a square of 8 mm edge length. The measurement system is positioned within a magnetically shielded room, consisting of 3 highly permeable shieldings and one eddy current shielding. The shielding performance is 38 db at 1 Hz, 55 db at Hz and 80 db at 20 Hz. The sensor arrangement in orthogonal triplets facilitates the measurement of all 3 spatial components of the magnetic field. Thus, the required field coverage for the localization of a magnetic marker with unknown dipole strength is achieved. The subdivision into 168 measurement and 27 reference sensors is meant for the creation of software gradiometers. We can use all sensors simultaneously for the multipole method which integrates the suppression of disturbing fields. With the above described measurement system we performed simulations with different signal to noise ratios.
Methods
Algorithm The field of a magnetic marker located adjacent to the point of origin can be expressed by a multipole expansion in Cartesian coordinates ( x 1 ,  x 2 ,  x 3 ). If the distance   between marker and origin is small compared to the distance   between a magnetic sensor and the origin, the field of the marker at the sensor position is given by the first elements of the multipole expansion. With the notation follows with  c m  being the dipole, quadrupole and octopole moments of the field expansion. The form functions   arise from a Taylor series expansion in the parameter   with   and  . It holds With the Kronecker delta   follows and Conversely, a Taylor series expansion of – compared to the sensor coordinates – far away located field sources in the parameter   with   yields a multipole expansion of external disturbing fields: We denote the multipole moments  c ex  of the expansion of fields of external sources as "outer moments" to distinguish them from the "inner moments"  c m . To get the same normalization and symmetry properties for the outer and inner form functions, we define the outer form functions   by It follows and The tensors of 3 rd  order   and of 4 th  order   own the following symmetry features which are identical for the inner and outer multipole expansion: We combine the resulting 3, 5 and 7 linearly independent components of the tensors of 2 nd , 3 rd  and 4 th  order to one vector for the marker field   and one vector for the external disturbing field  : The summation of equation (1) and equation (6) yields the field expansion for a magnetic marker with disturbing fields. We truncate this expansion after the octopole terms, and transcript it into a linear equation system for the determination of equivalent multipole moments  c  for a measurement  B meas : The structure of the vectors  B meas  and  c  and the matrix  F  is given below in the formulas (15...24). The residuals 0(·) are sufficiently small, if the coordinates of the marker are small compared to the coordinates of the field sensors  , and if the coordinates of the field sensors are small compared to the coordinates of the external disturbing field sources  . B meas  is a vector with the measurement values of the magnetometer sensor field in the positions   with the directions  : The matrix  F  is built from the linearly independent form functions for inner and outer field sources given in equation (13). Their scalar product with the sensor normal directions   yields one row for every sensor: The number of columns of  F  is the sum of the numbers of inner and outer field functions used. Each column describes the field of one specific magnetic moment with unit strength measured by the sensor system. The Matrix  F  is called the forward matrix of all moments considered. The Matrix  F  is structured into submatrices for different moments: Matrix   and   are the forward matrices for inner and outer dipoles: Matrix   and   are the forward matrices for inner and outer quadrupoles. The size of   is ( Nsen /5) with the rows belonging to quadrupole moments with indices (1,1; 3,3; 1,2; 2,3; 3,1). Matrix   and   are the forward matrices for inner and outer octopoles. The size of   is ( Nsen /7) with the rows belonging to octopole moments with indices (1,2,2; 2,3,3; 3,1,1; 1,3,3; 2,1,1; 3,2,2; 1,2,3). The vector of multipole moments  c  is composed of inner and outer dipole moments  c d , quadrupole moments  c q  and octopole moments  c o : The inner dipole moments   describe a dipole at the point of origin, the outer dipole moments   describe a homogeneous disturbing field: The   represent a quadrupole at the point of origin. The   describe an external gradient field, whose field strength vanishes at the origin and which has no spatial derivations of 2 nd  or higher order. This field can be measured by five ideal gradiometers at the origin and can be compared with the creation of software gradiometers. The   represent an octopole at the origin. The   describe an external gradient field of 2 nd  order, which has no spatial derivations of 3 rd  or higher order and whose field strength and spatial derivatives of 1 st  order vanish at the origin. This field could be measured by 7 ideal second order gradiometers at the origin, it can be compared with the creation of software gradiometers of 2 nd  order: Due to its small spatial extension, the magnetic marker can be described as a dipole of strength   at position   as a good approximation. The field of this dipole is With the Taylor series expansion follows in analogy to equation (1) A comparison of coefficients of (1) and (27) yields and The dipole strength   can be determined by the dipole moment  . An equation system for the adjacent calculation of the dipole position   from the dipole strength   and the quadrupole moment   follows from (29) and (23): with This equation system is named shift equation in analogy to [ 10 ]. It is overdetermined, and can be solved by means of the pseudo inverse of  m . We get the multipole moments  c , which are required for the localization of the marker dipole, from solving the overdetermined equation system (14) by means of the pseudo inverse of  F : c  = ( F T · F ) -1 · F T · B meas .     (33) Here, the matrix of form functions  F  must contain columns at least for the moments   and  . Iterative dipole localization for a fixed dipole (e.g. one time point) is achieved by using the localization position as a new point of origin. The step length of the last localization step serves as a stop criterion for the iterative localization procedure. This is justified by considering the residuals of equation (14) within the convergence range of the procedure, and will practically be shown by the results of the following simulations. The tracking of a moved dipole based on measurements at consecutive time steps works by updating both the point of origin and the measurement data set after each localization step (Fig.  1 ). The localization step must be monitored, since it contains information about the marker speed and the noise dependent and speed dependent localization errors. Figure 1 Flow chart of the algorithm for online localization.  This algorithm is meant for online localization, and therefore comprises only one iteration. A high signal to noise ratio and a high computing speed render 2–3 iterations per measurement cycle possible.
Algorithm
The field of a magnetic marker located adjacent to the point of origin can be expressed by a multipole expansion in Cartesian coordinates ( x 1 ,  x 2 ,  x 3 ). If the distance   between marker and origin is small compared to the distance   between a magnetic sensor and the origin, the field of the marker at the sensor position is given by the first elements of the multipole expansion. With the notationxxxfollowswith  c m  being the dipole, quadrupole and octopole moments of the field expansion.cThe form functions   arise from a Taylor series expansion in the parameter   with   and  . It holdsWith the Kronecker delta   followsandConversely, a Taylor series expansion of – compared to the sensor coordinates – far away located field sources in the parameter   with   yields a multipole expansion of external disturbing fields:We denote the multipole moments  c ex  of the expansion of fields of external sources as "outer moments" to distinguish them from the "inner moments"  c m .ccTo get the same normalization and symmetry properties for the outer and inner form functions, we define the outer form functions   byIt followsandThe tensors of 3 rd  order   and of 4 th  order   own the following symmetry features which are identical for the inner and outer multipole expansion:We combine the resulting 3, 5 and 7 linearly independent components of the tensors of 2 nd , 3 rd  and 4 th  order to one vector for the marker field   and one vector for the external disturbing field  :The summation of equation (1) and equation (6) yields the field expansion for a magnetic marker with disturbing fields. We truncate this expansion after the octopole terms, and transcript it into a linear equation system for the determination of equivalent multipole moments  c  for a measurement  B meas :cBThe structure of the vectors  B meas  and  c  and the matrix  F  is given below in the formulas (15...24). The residuals 0(·) are sufficiently small, if the coordinates of the marker are small compared to the coordinates of the field sensors  , and if the coordinates of the field sensors are small compared to the coordinates of the external disturbing field sources  .BcFB meas  is a vector with the measurement values of the magnetometer sensor field in the positions   with the directions  :BThe matrix  F  is built from the linearly independent form functions for inner and outer field sources given in equation (13). Their scalar product with the sensor normal directions   yields one row for every sensor:FThe number of columns of  F  is the sum of the numbers of inner and outer field functions used. Each column describes the field of one specific magnetic moment with unit strength measured by the sensor system. The Matrix  F  is called the forward matrix of all moments considered. The Matrix  F  is structured into submatrices for different moments:FFFMatrix   and   are the forward matrices for inner and outer dipoles:Matrix   and   are the forward matrices for inner and outer quadrupoles. The size of   is ( Nsen /5) with the rows belonging to quadrupole moments with indices (1,1; 3,3; 1,2; 2,3; 3,1).NsenMatrix   and   are the forward matrices for inner and outer octopoles. The size of   is ( Nsen /7) with the rows belonging to octopole moments with indices (1,2,2; 2,3,3; 3,1,1; 1,3,3; 2,1,1; 3,2,2; 1,2,3).NsenThe vector of multipole moments  c  is composed of inner and outer dipole moments  c d , quadrupole moments  c q  and octopole moments  c o :ccccThe inner dipole moments   describe a dipole at the point of origin, the outer dipole moments   describe a homogeneous disturbing field:The   represent a quadrupole at the point of origin. The   describe an external gradient field, whose field strength vanishes at the origin and which has no spatial derivations of 2 nd  or higher order. This field can be measured by five ideal gradiometers at the origin and can be compared with the creation of software gradiometers.The   represent an octopole at the origin. The   describe an external gradient field of 2 nd  order, which has no spatial derivations of 3 rd  or higher order and whose field strength and spatial derivatives of 1 st  order vanish at the origin. This field could be measured by 7 ideal second order gradiometers at the origin, it can be compared with the creation of software gradiometers of 2 nd  order:Due to its small spatial extension, the magnetic marker can be described as a dipole of strength   at position   as a good approximation. The field of this dipole isWith the Taylor series expansionfollows in analogy to equation (1)A comparison of coefficients of (1) and (27) yieldsandThe dipole strength   can be determined by the dipole moment  . An equation system for the adjacent calculation of the dipole position   from the dipole strength   and the quadrupole moment   follows from (29) and (23):withThis equation system is named shift equation in analogy to [ 10 ]. It is overdetermined, and can be solved by means of the pseudo inverse of  m .mWe get the multipole moments  c , which are required for the localization of the marker dipole, from solving the overdetermined equation system (14) by means of the pseudo inverse of  F :cFc  = ( F T · F ) -1 · F T · B meas .     (33)cFFFBHere, the matrix of form functions  F  must contain columns at least for the moments   and  .FIterative dipole localization for a fixed dipole (e.g. one time point) is achieved by using the localization position as a new point of origin. The step length of the last localization step serves as a stop criterion for the iterative localization procedure. This is justified by considering the residuals of equation (14) within the convergence range of the procedure, and will practically be shown by the results of the following simulations.The tracking of a moved dipole based on measurements at consecutive time steps works by updating both the point of origin and the measurement data set after each localization step (Fig.  1 ). The localization step must be monitored, since it contains information about the marker speed and the noise dependent and speed dependent localization errors.Measurement system The simulations to determine the performance of the algorithm use the sensor geometry of the multi channel SQUID system Argos 200 from AtB (Advanced Technologies Biomagnetics, Pescara, Italy). The ARGOS 200 system contains fully integrated planar SQUID magnetometers produced using Nb technology with integrated pick-up loops. The sensing area is a square of 8 mm side length. The intrinsic noise level of the built in 195 SQUID sensors is below 5 fT Hz -1/2  at 10 Hz. Three sensors form one orthogonal triplet in each case. The measurement plane with a diameter of 23 cm consists of 56 of those triplets. The reference array consists of seven SQUID sensor triplets located in the second level in a plane which is positioned parallel to the measurement plane at a distance of 98 mm. The third (196 mm above the first plane) and fourth (254 mm above the first plane) levels contain one triplet each (Fig.  2 ). Figure 2 SQUID Array Argos 200.  The ATB SQUID Array Argos 200 consists of 195 magnetometers which are arranged in orthogonal sensor triplets in four levels. The measurement area of each sensor is a square of 8 mm edge length. The measurement system is positioned within a magnetically shielded room, consisting of 3 highly permeable shieldings and one eddy current shielding. The shielding performance is 38 db at 1 Hz, 55 db at Hz and 80 db at 20 Hz. The sensor arrangement in orthogonal triplets facilitates the measurement of all 3 spatial components of the magnetic field. Thus, the required field coverage for the localization of a magnetic marker with unknown dipole strength is achieved. The subdivision into 168 measurement and 27 reference sensors is meant for the creation of software gradiometers. We can use all sensors simultaneously for the multipole method which integrates the suppression of disturbing fields. With the above described measurement system we performed simulations with different signal to noise ratios.
Measurement system
The simulations to determine the performance of the algorithm use the sensor geometry of the multi channel SQUID system Argos 200 from AtB (Advanced Technologies Biomagnetics, Pescara, Italy). The ARGOS 200 system contains fully integrated planar SQUID magnetometers produced using Nb technology with integrated pick-up loops. The sensing area is a square of 8 mm side length. The intrinsic noise level of the built in 195 SQUID sensors is below 5 fT Hz -1/2  at 10 Hz. Three sensors form one orthogonal triplet in each case. The measurement plane with a diameter of 23 cm consists of 56 of those triplets. The reference array consists of seven SQUID sensor triplets located in the second level in a plane which is positioned parallel to the measurement plane at a distance of 98 mm. The third (196 mm above the first plane) and fourth (254 mm above the first plane) levels contain one triplet each (Fig.  2 ).The measurement system is positioned within a magnetically shielded room, consisting of 3 highly permeable shieldings and one eddy current shielding. The shielding performance is 38 db at 1 Hz, 55 db at Hz and 80 db at 20 Hz.The sensor arrangement in orthogonal triplets facilitates the measurement of all 3 spatial components of the magnetic field. Thus, the required field coverage for the localization of a magnetic marker with unknown dipole strength is achieved.The subdivision into 168 measurement and 27 reference sensors is meant for the creation of software gradiometers. We can use all sensors simultaneously for the multipole method which integrates the suppression of disturbing fields.With the above described measurement system we performed simulations with different signal to noise ratios.Results We examined the localization characteristics of the multipole method by means of simulation runs at the sensor geometry of the measurement system Argos 200 (Fig.  2 ). All simulations performed are based on a dipole at position ( x ,  y ,  z ) = (0, 0, -300 mm), i.e. 30 cm below the measurement plane, with a dipole strength of 20 Amm 2 . This is a realistic dipole position for an examination within the digestive tract. The dipole field was superimposed by uncorrelated, Gauss distributed noise. The noise level in fT is also given as signal-to-noise-ratio (SNR), based on the channel with the strongest amplitude of the dipole field. The average localization accuracy over 100 simulations has been determined depending on the noise level and on the number of the multipole moments used in the vector  c  (21) (Fig.  3 ). Figure 3 Noise-dependent localization error.  The mean squared localization error e rr ( ) over 100 simulations has been determined depending on the noise level and on the different number of multipole moments used. The curves are plotted up to the noise level, where all simulations still produced a stable localization result. We used the inner moments up to the 3 rd  order  ,  , plotted in curves 3χ and the inner moments up to the 4 th  order  ,  ,  , plotted in curves 4χ. The outer moments which were used to model the disturbing fields were none (curves χ0), 2 nd  order moments (curves χ2: homogeneous fields), 2 nd  and 3 rd  order moments (curves χ3: homogeneous and gradient fields), and 2 nd  to 4 th  order moments (curves χ4: external fields up to 2 nd  order). The simplest disturbing field to model is a homogeneous field having index χ2. The dipole field of a dipole with a strength of 20 Amm 2  at position ( x ,  y ,  z ) = (0, 0, -300 mm) is superimposed by white, Gaussian distributed noise, which is given in fT and as the signal to noise ratio (SNR). The localization was run up to a stable point. We define the localization error as the mean quadratic error of the 100 stable points based on the true dipole position. The localization error increases if we use higher order multipole moments. This holds true for the inner moments  c m  and for the outer moments  c ex  as well. As a good approximation the interrelationship between noise level and localization error is linear, with raising proportionality factor for higher mode numbers. This corresponds to parallel translation of the curves in double logarithmic plotting. We examined the localization speed depending on the distance of the starting point to the dipole position. For any tested distance the starting point has been moved from the dipole position into 100 random directions. The remaining mean distance to the dipole position after one localization step is depicted in Fig.  4 . The localization speed turns to be significantly higher when using inner octopoles. It gets higher with a shorter starting distance in both an absolute and a relative manner based on the starting distance. Both effects are to be expected directly from the residuals of equation (14). The influence of the outer multipoles on the localization speed is low. The convergence radius at which the dipole was found from all 100 directions decreases slightly with the raising number of outer multipoles used, and increases slightly if inner octopoles are used (unequal right ends of the respective curves in Fig.  4 ). The convergence radius ranges between 6 and 10 cm. The maximum number of iterations for a target accuracy of 1 mm can be estimated from Fig.  4  as 3. Figure 4 Localization error depending on the starting point distance for one iteration.  For each distance  d s  the start position has been moved from the dipole into 100 random directions. The mean remaining distance  d r  after one localization step is shown. The curves are plotted until the starting distance  d r  where the localization was still stable from all 100 directions. To get the result after multiple iterative localization steps, the  d r -value has to be taken as the starting distance  d s  of the following step. We used the inner moments up to the 3 rd  order  ,  , plotted in curves 3χ and the inner moments up to the 4 th  order  ,  ,  , plotted in curves 4χ. The outer moments which were used to model the disturbing fields were none (curves χ0), 2 nd  order moments (curves χ2: homogeneous fields), 2 nd  and 3 rd  order moments (curves χ3: homogeneous and gradient fields), and 2 nd  to 4 th  order moments (curves χ4: external fields up to 2 nd  order). In the following we examined the interrelationship between the convergence distance at y-direction and the noise level. The maximum y-distance of the starting position to the dipole, at which the dipole could be found with 100 random noise distributions, is depicted in Fig.  5 . It shows that the convergence distance remains unchanged almost up to the point of critical noise level (see Fig.  3 ) at which localization becomes impossible. The convergence distance, also compare the maximum convergence radius from the curve ends of plot (Fig.  4 ), depends only marginally on the choice of the inner ansatz functions. It decreases slightly when using outer multipoles. Having a convergence radius of at least 6 cm for the dipole position tested, the choice of the starting position can be regarded as noncritical. Figure 5 Convergence distance depending on noise level.  The maximum y-distance  dy  between starting position and dipole position, at which for 100 random noise distributions the dipole could still get localized, is plotted. We used the inner moments up to the 3 rd  order  ,  , plotted in curves 3χ and the inner moments up to the 4 th  order  ,  ,  , plotted in curves 4χ. The outer moments which were used to model the disturbing fields were none (curves χ0), 2 nd  order moments (curves χ2: homogeneous fields), 2 nd  and 3 rd  order moments (curves χ3: homogeneous and gradient fields), and 2 nd  to 4 th  order moments (curves χ4: external fields up to 2 nd  order). The dipole field of a dipole with a strength of 20 Amm 2  at position ( x ,  y ,  z ) = (0, 0, -300 mm) is superimposed by white, Gaussian distributed noise given in fT and as the signal to noise ratio (SNR). The computing time used for one localization step is 5 ms with an implementation in Matlab at a standard Windows PC with a 2 GHz clock frequency. With maximum 3 iterations per localization step and additional computing time needed for data transfer and a basic visualization, 10 localizations per second are possible. This rate is normally sufficient for marker localizations.
Results
We examined the localization characteristics of the multipole method by means of simulation runs at the sensor geometry of the measurement system Argos 200 (Fig.  2 ). All simulations performed are based on a dipole at position ( x ,  y ,  z ) = (0, 0, -300 mm), i.e. 30 cm below the measurement plane, with a dipole strength of 20 Amm 2 . This is a realistic dipole position for an examination within the digestive tract. The dipole field was superimposed by uncorrelated, Gauss distributed noise. The noise level in fT is also given as signal-to-noise-ratio (SNR), based on the channel with the strongest amplitude of the dipole field.xyzThe average localization accuracy over 100 simulations has been determined depending on the noise level and on the number of the multipole moments used in the vector  c  (21) (Fig.  3 ).cThe localization was run up to a stable point. We define the localization error as the mean quadratic error of the 100 stable points based on the true dipole position. The localization error increases if we use higher order multipole moments. This holds true for the inner moments  c m  and for the outer moments  c ex  as well. As a good approximation the interrelationship between noise level and localization error is linear, with raising proportionality factor for higher mode numbers. This corresponds to parallel translation of the curves in double logarithmic plotting.ccWe examined the localization speed depending on the distance of the starting point to the dipole position. For any tested distance the starting point has been moved from the dipole position into 100 random directions. The remaining mean distance to the dipole position after one localization step is depicted in Fig.  4 . The localization speed turns to be significantly higher when using inner octopoles. It gets higher with a shorter starting distance in both an absolute and a relative manner based on the starting distance. Both effects are to be expected directly from the residuals of equation (14). The influence of the outer multipoles on the localization speed is low. The convergence radius at which the dipole was found from all 100 directions decreases slightly with the raising number of outer multipoles used, and increases slightly if inner octopoles are used (unequal right ends of the respective curves in Fig.  4 ). The convergence radius ranges between 6 and 10 cm. The maximum number of iterations for a target accuracy of 1 mm can be estimated from Fig.  4  as 3.In the following we examined the interrelationship between the convergence distance at y-direction and the noise level. The maximum y-distance of the starting position to the dipole, at which the dipole could be found with 100 random noise distributions, is depicted in Fig.  5 . It shows that the convergence distance remains unchanged almost up to the point of critical noise level (see Fig.  3 ) at which localization becomes impossible. The convergence distance, also compare the maximum convergence radius from the curve ends of plot (Fig.  4 ), depends only marginally on the choice of the inner ansatz functions. It decreases slightly when using outer multipoles. Having a convergence radius of at least 6 cm for the dipole position tested, the choice of the starting position can be regarded as noncritical.The computing time used for one localization step is 5 ms with an implementation in Matlab at a standard Windows PC with a 2 GHz clock frequency. With maximum 3 iterations per localization step and additional computing time needed for data transfer and a basic visualization, 10 localizations per second are possible. This rate is normally sufficient for marker localizations.Discussion The localization speed rises when using inner octopoles  , but this is associated with a higher localization error. At a signal to noise ratio lower than 10 3  inner octopoles cannot be used. An SNR of at least 10 2  is required for a source positioned 30 cm below the measurement plane. The outer moments  c ex  used enlarge the localization error depending on the uncorrelated sensor noise, as shown in Fig.  3 . Contrary, the localization error depending on the spatially correlated residual field within the measurement room lowers when using outer moments. Depending on the ratio between correlated and uncorrelated noise which has to be found with practical test series, noise suppression of homogeneous disturbing fields using   and possibly noise suppression of gradient fields using   are applicable. To ensure convergence, the starting point for the algorithm has to be within the convergence radius given in Fig.  4 . With typical measurement conditions, thus, a starting point 10 cm below the center of the measurement system will suffice.
Discussion
The localization speed rises when using inner octopoles  , but this is associated with a higher localization error. At a signal to noise ratio lower than 10 3  inner octopoles cannot be used. An SNR of at least 10 2  is required for a source positioned 30 cm below the measurement plane.The outer moments  c ex  used enlarge the localization error depending on the uncorrelated sensor noise, as shown in Fig.  3 . Contrary, the localization error depending on the spatially correlated residual field within the measurement room lowers when using outer moments. Depending on the ratio between correlated and uncorrelated noise which has to be found with practical test series, noise suppression of homogeneous disturbing fields using   and possibly noise suppression of gradient fields using   are applicable. To ensure convergence, the starting point for the algorithm has to be within the convergence radius given in Fig.  4 . With typical measurement conditions, thus, a starting point 10 cm below the center of the measurement system will suffice.cConclusions The multipole localization is an effective algorithm because it unites a method for the suppression of disturbing fields with a localization method. It can be used iteratively and online for the tracking of magnetic marker timelines within the intestinal tract.
Conclusions
The multipole localization is an effective algorithm because it unites a method for the suppression of disturbing fields with a localization method. It can be used iteratively and online for the tracking of magnetic marker timelines within the intestinal tract.


Prior to human settlement 700 years ago New Zealand had no terrestrial mammals—apart from three species of bats—instead, approximately 250 avian species dominated the ecosystem. At the top of the food chain was the extinct Haast's eagle,  Harpagornis moorei. H. moorei  (10–15 kg; 2–3 m wingspan) was 30%–40% heavier than the largest extant eagle (the harpy eagle,  Harpia harpyja ), and hunted moa up to 15 times its weight. In a dramatic example of morphological plasticity and rapid size increase, we show that the  H. moorei  was very closely related to one of the world's smallest extant eagles, which is one-tenth its mass. This spectacular evolutionary change illustrates the potential speed of size alteration within lineages of vertebrates, especially in island ecosystems.Harpagornis moorei. H. moorei

Introduction Since the discovery of the first fossil in 1872 the sheer size of  Harpagornis moorei  ( Figure 1 A) has fuelled speculation about its evolutionary history, ecology, and extinction, which like many other New Zealand bird species, is linked ultimately to human arrival in the 13th century. Even though morphology indicates that  H. moorei  was approaching the upper limit of body mass for powered flight [ 1 ], it was still an efficient predator. Consistent with other members of the family Accipitridae, it killed by piercing and crushing its prey with its large talons ( Figure 1 B). Rock art, Maori oral history, and bone artefacts prove early Polynesians co-existed with the eagle; however, there is no evidence that humans were targets for this aerial predator. An exploratory skeletal analysis, using representative genera within the Accipitridae (but lacking Australasian representatives of the genera  Hieraaetus ), placed  H. moorei  as a sister species to  Aquila audax,  the Australian wedge-tailed eagle (circa 4.5 kg; 2 m wingspan) [ 2 ]. However, shifts in body size are common in island ecosystems [ 3 ] and may distort skeletal characters used in phylogenetic reconstructions. Figure 1 Images and Phylogenetic Analysis of New Zealand's Extinct Giant Eagle,  H. moorei
 (A) An artist's impression of  H. moorei  attacking the extinct New Zealand moa. Evidence of eagle strikes are preserved on skeletons of moa weighing up to 200 kg. These skeletons show the eagle struck and gripped the moa's pelvic area, and then killed with a single strike by the other foot to the head or neck. (Artwork: John Megahan.) (B) Comparison of the huge claws of  H. moorei  with those of its close relative the  Hieraaetus morphnoides , the “little” eagle. The massive claws of  H. moorei  could pierce and crush bone up to 6 mm thick under 50 mm of skin and flesh. (C) Maximum-likelihood tree based on cyt  b  data (circa 1 kb), depicting phylogenetic relationships within the “booted eagle” group. Extraction numbers or GenBank accession numbers are shown along with taxa name.  Harpagornis moorei  (red) groups exclusively with the small  Hieraaetus  eagles, and genetic distances suggest a recent common ancestor about 0.7–1.8 million years ago (early to mid Pleistocene). The tree uses an HKY + Γ 4  + I likelihood model enforcing a molecular clock; maximum-likelihood bootstrap consensus values greater than 60% are shown.
Introduction
Since the discovery of the first fossil in 1872 the sheer size of  Harpagornis moorei  ( Figure 1 A) has fuelled speculation about its evolutionary history, ecology, and extinction, which like many other New Zealand bird species, is linked ultimately to human arrival in the 13th century. Even though morphology indicates that  H. moorei  was approaching the upper limit of body mass for powered flight [ 1 ], it was still an efficient predator. Consistent with other members of the family Accipitridae, it killed by piercing and crushing its prey with its large talons ( Figure 1 B). Rock art, Maori oral history, and bone artefacts prove early Polynesians co-existed with the eagle; however, there is no evidence that humans were targets for this aerial predator. An exploratory skeletal analysis, using representative genera within the Accipitridae (but lacking Australasian representatives of the genera  Hieraaetus ), placed  H. moorei  as a sister species to  Aquila audax,  the Australian wedge-tailed eagle (circa 4.5 kg; 2 m wingspan) [ 2 ]. However, shifts in body size are common in island ecosystems [ 3 ] and may distort skeletal characters used in phylogenetic reconstructions.HieraaetusAquila audax,Results/Discussion To further investigate the evolutionary history of this raptor we performed an ancient DNA study on the fossil remains of two extinct  H. moorei  specimens together with 16 extant eagles ( Table S1 ). Short, overlapping segments of the mitochondrial cytochrome  b  (cyt  b ) and ND2 genes were PCR-amplified, sequenced and analysed with data available on GenBank to build a maximum-likelihood tree ( Figures 1 C and S1). Appropriate ancient DNA controls were undertaken [ 4 ], including multiple, overlapping amplifications and independent replication (see  Materials and Methods ). Surprisingly, the resulting phylogeny firmly placed the  H. moorei  in a “clade” with a group of small eagles of the genus  Hieraaetus, H. morphnoides,  the little eagle, and  H. pennatus,  the booted eagle (both circa 1kg; 1.2m wingspan), and not with  A. audax . Moreover, the genetic distance separating  H. moorei  and the most recent common ancestor of the related  Hieraaetus  eagles is relatively small (1.25%). The lack of fossil calibration points for accipitrids precludes direct estimates of divergence times; however, when a molecular rate of 0.7%–1.7% per million years, as previously estimated for avian cyt  b  [ 5 ], is applied to the tree, a divergence estimate of approximately 0.7–1.8 million years ago is obtained. Although such indirect molecular dating estimates are error-prone, we believe that this range is the best available approximation of the “true” date when the lineages diverged (early to mid Pleistocene); however, additional molecular data and control region sequences may further clarify the topology and timing of the splits. The arrival of  H. moorei  into New Zealand's South Island appears to have been a recent event, probably involving a small bird-eating Asian/Australian  Hieraaetus  eagle that thereafter increased rapidly in size. An analysis of mean body mass using independent contrasts clearly demonstrated that the size of  H. moorei  is an anomaly in the context of the eagle phylogeny shown in  Figure 1 C (see  Materials and Methods ). Factors that may have influenced such rapid morphological evolution include the size of potential prey, competition with smaller harriers ( Circus  spp.), and a complete lack of terrestrial predatory mammals in New Zealand. The speed and magnitude of the increase in body mass seems unique within vertebrate lineages, and is more significant because it occurred in a species still capable of flight. The avian faunas of Islands such as Hawaii (Hawaiian goose), the Galápagos (Galápagos finches), Mauritius (dodo), and New Zealand (moa) are often cited as examples of rapid evolution of flightlessness, shifts in body size, and other specialisations in birds [ 1 , 6 , 7 ]. Isolated island faunas derived from vagile colonists often have vacant niches for large herbivores and predators. In New Zealand the large herbivore niches were occupied for millions of years by the moa. Clearly, in the absence of mammalian predators, selection for large body size in an avian predator with a relatively generalised body form was not limited by competition. Other large predatory birds have evolved on islands in the absence of mammalian competitors, notably giant eagles and owls on Cuba [ 8 ], but the magnitude of the size increase in  H. moorei  over its sister taxa is unrivalled.  H. moorei  therefore represents an extreme example of how freedom from competition on island ecosystems can rapidly influence morphological adaptation and speciation. The phylogeny in  Figure 1 C also reveals considerable problems with the current classification of the “booted eagle” group (eagles with feathered tarsi), especially the genera  Aquila, Hieraaetus,  and  Spizaetus,  which are clearly paraphyletic. Assignment of species within these genera has traditionally been problematic (see [ 9 ]), so this observation was not unexpected. However, it is apparent the name for the extinct New Zealand eagle should be amended to  Hieraaetus moorei  (Haast, 1872) (see  Materials and Methods ). The inclusion of  H. moorei  with the small  Hieraaetus  eagles implies the body size of the New Zealand species has changed by almost an order of magnitude since these lineages diverged. This spectacular evolutionary change illustrates the potential speed of size alteration within lineages of vertebrates and represents yet another example of the remarkable evolutionary processes that occur within island ecosystems.
Results/Discussion
To further investigate the evolutionary history of this raptor we performed an ancient DNA study on the fossil remains of two extinct  H. moorei  specimens together with 16 extant eagles ( Table S1 ). Short, overlapping segments of the mitochondrial cytochrome  b  (cyt  b ) and ND2 genes were PCR-amplified, sequenced and analysed with data available on GenBank to build a maximum-likelihood tree ( Figures 1 C and S1). Appropriate ancient DNA controls were undertaken [ 4 ], including multiple, overlapping amplifications and independent replication (see  Materials and Methods ). Surprisingly, the resulting phylogeny firmly placed the  H. moorei  in a “clade” with a group of small eagles of the genus  Hieraaetus, H. morphnoides,  the little eagle, and  H. pennatus,  the booted eagle (both circa 1kg; 1.2m wingspan), and not with  A. audax . Moreover, the genetic distance separating  H. moorei  and the most recent common ancestor of the related  Hieraaetus  eagles is relatively small (1.25%). The lack of fossil calibration points for accipitrids precludes direct estimates of divergence times; however, when a molecular rate of 0.7%–1.7% per million years, as previously estimated for avian cyt  b  [ 5 ], is applied to the tree, a divergence estimate of approximately 0.7–1.8 million years ago is obtained. Although such indirect molecular dating estimates are error-prone, we believe that this range is the best available approximation of the “true” date when the lineages diverged (early to mid Pleistocene); however, additional molecular data and control region sequences may further clarify the topology and timing of the splits. The arrival of  H. moorei  into New Zealand's South Island appears to have been a recent event, probably involving a small bird-eating Asian/Australian  Hieraaetus  eagle that thereafter increased rapidly in size.bbHieraaetus, H. morphnoides,H. pennatus,HieraaetusbHieraaetusAn analysis of mean body mass using independent contrasts clearly demonstrated that the size of  H. moorei  is an anomaly in the context of the eagle phylogeny shown in  Figure 1 C (see  Materials and Methods ). Factors that may have influenced such rapid morphological evolution include the size of potential prey, competition with smaller harriers ( Circus  spp.), and a complete lack of terrestrial predatory mammals in New Zealand. The speed and magnitude of the increase in body mass seems unique within vertebrate lineages, and is more significant because it occurred in a species still capable of flight. The avian faunas of Islands such as Hawaii (Hawaiian goose), the Galápagos (Galápagos finches), Mauritius (dodo), and New Zealand (moa) are often cited as examples of rapid evolution of flightlessness, shifts in body size, and other specialisations in birds [ 1 , 6 , 7 ]. Isolated island faunas derived from vagile colonists often have vacant niches for large herbivores and predators. In New Zealand the large herbivore niches were occupied for millions of years by the moa. Clearly, in the absence of mammalian predators, selection for large body size in an avian predator with a relatively generalised body form was not limited by competition. Other large predatory birds have evolved on islands in the absence of mammalian competitors, notably giant eagles and owls on Cuba [ 8 ], but the magnitude of the size increase in  H. moorei  over its sister taxa is unrivalled.  H. moorei  therefore represents an extreme example of how freedom from competition on island ecosystems can rapidly influence morphological adaptation and speciation.CircusThe phylogeny in  Figure 1 C also reveals considerable problems with the current classification of the “booted eagle” group (eagles with feathered tarsi), especially the genera  Aquila, Hieraaetus,  and  Spizaetus,  which are clearly paraphyletic. Assignment of species within these genera has traditionally been problematic (see [ 9 ]), so this observation was not unexpected. However, it is apparent the name for the extinct New Zealand eagle should be amended to  Hieraaetus moorei  (Haast, 1872) (see  Materials and Methods ). The inclusion of  H. moorei  with the small  Hieraaetus  eagles implies the body size of the New Zealand species has changed by almost an order of magnitude since these lineages diverged. This spectacular evolutionary change illustrates the potential speed of size alteration within lineages of vertebrates and represents yet another example of the remarkable evolutionary processes that occur within island ecosystems.Aquila, Hieraaetus,Spizaetus,HieraaetusMaterials and Methods DNA extraction and amplification DNA was extracted and amplified from two  H. moorei  bones ( Table S1 ) as previously described [ 10 ] using appropriate ancient DNA techniques. “Modern” toepad tissue (museum specimens) was extracted using Qiagen (Valencia, California, United States) DNeasy tissue extraction kits. Multiple negative extraction and amplification controls were included, to detect contamination. All PCR reactions were conducted as described in [ 10 ] using Platinum Taq HiFi (Invitrogen, Carlsbad, California, United States) together with the cyt  b  and ND2 primers listed in  Table S2 . Thermal cycling conditions were typically 40 cycles of 95 °C/55–60 °C/68 °C (30–45 s each). Sequences were determined using ABI Big Dye (v.3.1) on an ABI 3100 or 3730 (Applied Biosystems, Foster City, California, United States), according to manufacturer's instructions. Modern samples, and ancient samples subsequent to PCR amplification, were analysed in the Zoology Department, Oxford University. A single  Harpagornis  bone was sent to an ancient DNA facility at University College London (I. Barnes) for independent replication, where identical sequences were obtained for two cyt  b  amplifications. Similar cyt  b  and ND2 tree topologies, in addition to multiple overlapping sequences, make it unlikely that we are detecting a nuclear pseudogene. Phylogenetic methods Maximum-likelihood trees for cyt  b  and ND2 were selected using a heuristic search as implemented in PAUP*4.0b10 [ 11 ] under the HKY + Γ 4  + I substitution model. The assumption of a molecular clock was tested using a likelihood ratio test in which the χ 2  test statistic was two times the log likelihood difference between clock and non-clock models. For the cyt  b  tree the assumption of rate constancy was not rejected. Node support was evaluated for 1,000 bootstrap replicates. Bayesian Markov Chain Monte Carlo phylogenies were also generated on the cyt  b  dataset using BEAST [ 12 ] and MrBayes [ 13 ] under similar substitution models—the topology of these trees was consistent with  Figure 1 C and generated posterior support values higher than the bootstrap values. Using the maximum-likelihood tree in  Figure 1 C, an independent-contrasts analysis was employed to determine whether correlations existed between phylogenetic position and body mass. Mean live weight estimates were obtained from the literature, and the average mass of  H. moorei  was estimated from femur length [ 2 ]. A test to measure the index of phylogenetic dependence was conducted; this measures the degree to which traits vary across taxa (in a phylogeny) in accordance with predictions of a neutral Brownian model according to [ 14 ]. The results (not shown) clearly demonstrate that the mass of  H. moorei  is clearly an “outlier” in the context of the phylogeny presented here.
 Hieraaetus  systematics The type species for the genus  Hieraaetus  is  H. pennatus  (Gmelin, 1788); therefore, the taxa grouping strongly with  H. pennatus  must remain in that genus. The close genetic relationship of  H. morphnoides  with  H. pennatus  firmly embeds this species in  Hieraaetus . However, the New Guinea subspecies presently recognised as  H. morphnoides weiskei  is genetically, geographically, and morphologically distinct and warrants species status, which necessitates the new combination  Hieraaetus weiskei  (Reichenow, 1900).  Harpagornis moorei  is included in the clade with  H. pennatus  and  H. morphnoides,  and hence its generic assignment must reflect that. The name for the extinct  Harpagornis moorei  of New Zealand should therefore be amended to  Hieraaetus moorei  (Haast, 1872).
Materials and Methods
DNA extraction and amplification DNA was extracted and amplified from two  H. moorei  bones ( Table S1 ) as previously described [ 10 ] using appropriate ancient DNA techniques. “Modern” toepad tissue (museum specimens) was extracted using Qiagen (Valencia, California, United States) DNeasy tissue extraction kits. Multiple negative extraction and amplification controls were included, to detect contamination. All PCR reactions were conducted as described in [ 10 ] using Platinum Taq HiFi (Invitrogen, Carlsbad, California, United States) together with the cyt  b  and ND2 primers listed in  Table S2 . Thermal cycling conditions were typically 40 cycles of 95 °C/55–60 °C/68 °C (30–45 s each). Sequences were determined using ABI Big Dye (v.3.1) on an ABI 3100 or 3730 (Applied Biosystems, Foster City, California, United States), according to manufacturer's instructions. Modern samples, and ancient samples subsequent to PCR amplification, were analysed in the Zoology Department, Oxford University. A single  Harpagornis  bone was sent to an ancient DNA facility at University College London (I. Barnes) for independent replication, where identical sequences were obtained for two cyt  b  amplifications. Similar cyt  b  and ND2 tree topologies, in addition to multiple overlapping sequences, make it unlikely that we are detecting a nuclear pseudogene. Phylogenetic methods Maximum-likelihood trees for cyt  b  and ND2 were selected using a heuristic search as implemented in PAUP*4.0b10 [ 11 ] under the HKY + Γ 4  + I substitution model. The assumption of a molecular clock was tested using a likelihood ratio test in which the χ 2  test statistic was two times the log likelihood difference between clock and non-clock models. For the cyt  b  tree the assumption of rate constancy was not rejected. Node support was evaluated for 1,000 bootstrap replicates. Bayesian Markov Chain Monte Carlo phylogenies were also generated on the cyt  b  dataset using BEAST [ 12 ] and MrBayes [ 13 ] under similar substitution models—the topology of these trees was consistent with  Figure 1 C and generated posterior support values higher than the bootstrap values. Using the maximum-likelihood tree in  Figure 1 C, an independent-contrasts analysis was employed to determine whether correlations existed between phylogenetic position and body mass. Mean live weight estimates were obtained from the literature, and the average mass of  H. moorei  was estimated from femur length [ 2 ]. A test to measure the index of phylogenetic dependence was conducted; this measures the degree to which traits vary across taxa (in a phylogeny) in accordance with predictions of a neutral Brownian model according to [ 14 ]. The results (not shown) clearly demonstrate that the mass of  H. moorei  is clearly an “outlier” in the context of the phylogeny presented here.
 Hieraaetus  systematics The type species for the genus  Hieraaetus  is  H. pennatus  (Gmelin, 1788); therefore, the taxa grouping strongly with  H. pennatus  must remain in that genus. The close genetic relationship of  H. morphnoides  with  H. pennatus  firmly embeds this species in  Hieraaetus . However, the New Guinea subspecies presently recognised as  H. morphnoides weiskei  is genetically, geographically, and morphologically distinct and warrants species status, which necessitates the new combination  Hieraaetus weiskei  (Reichenow, 1900).  Harpagornis moorei  is included in the clade with  H. pennatus  and  H. morphnoides,  and hence its generic assignment must reflect that. The name for the extinct  Harpagornis moorei  of New Zealand should therefore be amended to  Hieraaetus moorei  (Haast, 1872).

DNA extraction and amplification DNA was extracted and amplified from two  H. moorei  bones ( Table S1 ) as previously described [ 10 ] using appropriate ancient DNA techniques. “Modern” toepad tissue (museum specimens) was extracted using Qiagen (Valencia, California, United States) DNeasy tissue extraction kits. Multiple negative extraction and amplification controls were included, to detect contamination. All PCR reactions were conducted as described in [ 10 ] using Platinum Taq HiFi (Invitrogen, Carlsbad, California, United States) together with the cyt  b  and ND2 primers listed in  Table S2 . Thermal cycling conditions were typically 40 cycles of 95 °C/55–60 °C/68 °C (30–45 s each). Sequences were determined using ABI Big Dye (v.3.1) on an ABI 3100 or 3730 (Applied Biosystems, Foster City, California, United States), according to manufacturer's instructions. Modern samples, and ancient samples subsequent to PCR amplification, were analysed in the Zoology Department, Oxford University. A single  Harpagornis  bone was sent to an ancient DNA facility at University College London (I. Barnes) for independent replication, where identical sequences were obtained for two cyt  b  amplifications. Similar cyt  b  and ND2 tree topologies, in addition to multiple overlapping sequences, make it unlikely that we are detecting a nuclear pseudogene.
DNA extraction and amplification
DNA was extracted and amplified from two  H. moorei  bones ( Table S1 ) as previously described [ 10 ] using appropriate ancient DNA techniques. “Modern” toepad tissue (museum specimens) was extracted using Qiagen (Valencia, California, United States) DNeasy tissue extraction kits. Multiple negative extraction and amplification controls were included, to detect contamination. All PCR reactions were conducted as described in [ 10 ] using Platinum Taq HiFi (Invitrogen, Carlsbad, California, United States) together with the cyt  b  and ND2 primers listed in  Table S2 . Thermal cycling conditions were typically 40 cycles of 95 °C/55–60 °C/68 °C (30–45 s each). Sequences were determined using ABI Big Dye (v.3.1) on an ABI 3100 or 3730 (Applied Biosystems, Foster City, California, United States), according to manufacturer's instructions. Modern samples, and ancient samples subsequent to PCR amplification, were analysed in the Zoology Department, Oxford University. A single  Harpagornis  bone was sent to an ancient DNA facility at University College London (I. Barnes) for independent replication, where identical sequences were obtained for two cyt  b  amplifications. Similar cyt  b  and ND2 tree topologies, in addition to multiple overlapping sequences, make it unlikely that we are detecting a nuclear pseudogene.bHarpagornisbbPhylogenetic methods Maximum-likelihood trees for cyt  b  and ND2 were selected using a heuristic search as implemented in PAUP*4.0b10 [ 11 ] under the HKY + Γ 4  + I substitution model. The assumption of a molecular clock was tested using a likelihood ratio test in which the χ 2  test statistic was two times the log likelihood difference between clock and non-clock models. For the cyt  b  tree the assumption of rate constancy was not rejected. Node support was evaluated for 1,000 bootstrap replicates. Bayesian Markov Chain Monte Carlo phylogenies were also generated on the cyt  b  dataset using BEAST [ 12 ] and MrBayes [ 13 ] under similar substitution models—the topology of these trees was consistent with  Figure 1 C and generated posterior support values higher than the bootstrap values. Using the maximum-likelihood tree in  Figure 1 C, an independent-contrasts analysis was employed to determine whether correlations existed between phylogenetic position and body mass. Mean live weight estimates were obtained from the literature, and the average mass of  H. moorei  was estimated from femur length [ 2 ]. A test to measure the index of phylogenetic dependence was conducted; this measures the degree to which traits vary across taxa (in a phylogeny) in accordance with predictions of a neutral Brownian model according to [ 14 ]. The results (not shown) clearly demonstrate that the mass of  H. moorei  is clearly an “outlier” in the context of the phylogeny presented here.
Phylogenetic methods
Maximum-likelihood trees for cyt  b  and ND2 were selected using a heuristic search as implemented in PAUP*4.0b10 [ 11 ] under the HKY + Γ 4  + I substitution model. The assumption of a molecular clock was tested using a likelihood ratio test in which the χ 2  test statistic was two times the log likelihood difference between clock and non-clock models. For the cyt  b  tree the assumption of rate constancy was not rejected. Node support was evaluated for 1,000 bootstrap replicates. Bayesian Markov Chain Monte Carlo phylogenies were also generated on the cyt  b  dataset using BEAST [ 12 ] and MrBayes [ 13 ] under similar substitution models—the topology of these trees was consistent with  Figure 1 C and generated posterior support values higher than the bootstrap values.bbbUsing the maximum-likelihood tree in  Figure 1 C, an independent-contrasts analysis was employed to determine whether correlations existed between phylogenetic position and body mass. Mean live weight estimates were obtained from the literature, and the average mass of  H. moorei  was estimated from femur length [ 2 ]. A test to measure the index of phylogenetic dependence was conducted; this measures the degree to which traits vary across taxa (in a phylogeny) in accordance with predictions of a neutral Brownian model according to [ 14 ]. The results (not shown) clearly demonstrate that the mass of  H. moorei  is clearly an “outlier” in the context of the phylogeny presented here.Hieraaetus  systematics The type species for the genus  Hieraaetus  is  H. pennatus  (Gmelin, 1788); therefore, the taxa grouping strongly with  H. pennatus  must remain in that genus. The close genetic relationship of  H. morphnoides  with  H. pennatus  firmly embeds this species in  Hieraaetus . However, the New Guinea subspecies presently recognised as  H. morphnoides weiskei  is genetically, geographically, and morphologically distinct and warrants species status, which necessitates the new combination  Hieraaetus weiskei  (Reichenow, 1900).  Harpagornis moorei  is included in the clade with  H. pennatus  and  H. morphnoides,  and hence its generic assignment must reflect that. The name for the extinct  Harpagornis moorei  of New Zealand should therefore be amended to  Hieraaetus moorei  (Haast, 1872).
Hieraaetus  systematics
HieraaetusThe type species for the genus  Hieraaetus  is  H. pennatus  (Gmelin, 1788); therefore, the taxa grouping strongly with  H. pennatus  must remain in that genus. The close genetic relationship of  H. morphnoides  with  H. pennatus  firmly embeds this species in  Hieraaetus . However, the New Guinea subspecies presently recognised as  H. morphnoides weiskei  is genetically, geographically, and morphologically distinct and warrants species status, which necessitates the new combination  Hieraaetus weiskei  (Reichenow, 1900).  Harpagornis moorei  is included in the clade with  H. pennatus  and  H. morphnoides,  and hence its generic assignment must reflect that. The name for the extinct  Harpagornis moorei  of New Zealand should therefore be amended to  Hieraaetus moorei  (Haast, 1872).HieraaetusHieraaetusH. morphnoides weiskeiH. morphnoides,Supporting Information Figure S1 Maximum-likelihood tree generated using 434 bp of ND2 data from a subset of eagle taxa. The tree topology seen here is identical to that seen in  Figure 1 C and is an independent verification of the tree topology (89 KB PDF). Click here for additional data file. Table S1 List of Eagle Taxa Used in This Study along with Museum Accession Numbers and Sample Provenance (64 KB PDF). Click here for additional data file. Table S2 List of Avian cyt  b  and ND2 Primers Used in This Study (54 KB PDF). Click here for additional data file. Accession Numbers Sequences have been deposited in GenBank ( http://www.ncbi.nlm.nih.gov/Genbank/index.html ) under accession numbers AY754044 to AY754056.
Supporting Information
Maximum-likelihood tree generated using 434 bp of ND2 data from a subset of eagle taxa. The tree topology seen here is identical to that seen in  Figure 1 C and is an independent verification of the tree topology
(89 KB PDF).Click here for additional data file.List of Eagle Taxa Used in This Study along with Museum Accession Numbers and Sample Provenance
(64 KB PDF).Click here for additional data file.List of Avian cyt  b  and ND2 Primers Used in This Study
b(54 KB PDF).Click here for additional data file.Accession Numbers Sequences have been deposited in GenBank ( http://www.ncbi.nlm.nih.gov/Genbank/index.html ) under accession numbers AY754044 to AY754056.
Accession Numbers
Sequences have been deposited in GenBank ( http://www.ncbi.nlm.nih.gov/Genbank/index.html ) under accession numbers AY754044 to AY754056.