referring to fig1 there is shown a hyperthermia annular phased - array system 100 having improved &# 34 ; hot spot &# 34 ; characteristics through utilizing the focused near - field adaptive nulling apparatus of this invention . an annular hyperthermia phased - array transmit antenna 102 , energized by a hyperthermia array controller 101 , has a plurality of dipole transmit antenna elements 104 is placed around a patient to be treated , or target body 106 , so that the dipole antenna elements are uniformly disposed around the patient . each dipole antenna element is oriented parallel to the other dipole antennas and parallel to a longitudinal axis a passing through the center of the cylinder . the patient is positioned within the hyperthermia phased - array such that the deep - seated tumor to be treated 107 is at the approximate center , or focus , of the array . a water - bolus 105 is maintained between the patient and the array to control the temperature of the patient &# 39 ; s skin . phased - array 102 therapeutically illuminates the target body 106 by focusing electric field energy radiated by dipole antenna elements 104 on tumor 107 deep within the body . an example of a deep - seated tumor is cancer of the prostate . the tumor volume often has a decreased blood flow which aids in heating the tumor , compared to normal tissue for which heat is carried away by normal blood flow . in practice , undesired high - temperature regions away from the focus can also occur on the skin and inside the volume of the target body . for example , scar tissue , which has a decreased blood flow rate , will tend to heat up more rapidly than normal tissue having normal blood flow . in the adaptive hyperthermia array of this invention , electric - field nulls are used to reduce the power delivered to potential hot spots . computer simulations , presented herein , show that non - invasive field probes , or sensors , 112 placed on the surface of the target can be used to eliminate hot spots interior to the target tissue . with the adaptive hyperthermia phased - array described herein , rf energy nulls are adaptively formed to reduce the electric field energy delivered to these potential hot spots . as shown , the energy nulls achieved by the adaptive nulling apparatus of this invention are both invasive to the target , i . e ., extend into the target body , and non - invasive to the target , i . e ., on the surface of the target . referring to fig2 there is shown a schematic cross - sectional representation of an embodiment of an eight - element hyperthermia phased - array 102 of fig1 . phased - array 102 has transmit antennas 104 ; through 104 8 , arranged symmetrically surrounding a human body target 106 &# 39 ; at the prostate level . an analytical model of the embodiment of fig2 is shown in fig3 . here , an elliptical phantom target 106 is used to model the prostate - level cross section of the human body 106 &# 39 ;. the center 107 of the elliptical phantom models the location of the prostate tumor to receive hyperthermia treatment , i . e ., the focus of rf energy for the phased array 102 . water bolus 105 is assumed to surround the target body 106 , and is treated as a homogeneous medium for analysis purposes . four auxiliary rf e - field probes , or sensors , 112 1 through 112 4 , i . e ., receiving antennas , are placed around the perimeter of the target to model non - invasive probes placed on the skin of the human body target . each auxiliary probe 112 1 through 112 4 has a corresponding null zone 120 1 through 120 4 , respectively , centered at each auxiliary probe and extending into the elliptical target region 106 . each null zone indicates an area in which undesired &# 34 ; hot spots &# 34 ; are reduced or eliminated . the width of each null zone is directly related to the depth of each null . the depth of each null ( sometimes referred to as the amount of cancellation ) is directly related to the signal - to - noise ratio at the probe position ( snr p ). a low snr p indicates a large amount of nulling ( strong null ), and a high snr p indicates a small amount of nulling ( weak null ). the resolution , or minimum spacing , between the focus 107 and any null position is normally equal to the half - power beamwidth of the transmit antenna . resolution may be enhanced somewhat by using weak nulls whenever the separation between the null and focus is closer than the half - power beamwidth . referring to fig4 there is shown the results of a simulation of the thermal distribution inside the target body 106 for the hyperthermia ring array 102 of the analytical model of fig3 without adaptive nulling , transmitting into the target body . for simulation purposes , target body 106 is assumed to be a homogeneous elliptical region , and the rf energy from the array is focused at the center of the ellipse 107 , simulating the tumor site . no adaptive nulling is used . the contour lines of the thermal distribution represent isotherms having the indicated temperature in degrees celcius (° c . ), and are spaced at 2 ° c . intervals . the simulation shows that the focus is expected to have a temperature of approximately 46 ° c ., while two undesired &# 34 ; hot spots &# 34 ; 122 and 124 to the left and right of the focus , respectively , are expected to have temperatures of approximately 42 ° c . fig5 shows a simulated thermal distribution for the model of fig3 where the adaptive nulling methods of this invention is applied . comparison of fig5 with fig4 show that the &# 34 ; hot spots &# 34 ; 122 and 124 are essentially eliminated , no new &# 34 ; hot spots &# 34 ; have been produced within the target body , and the peak temperature induced at the focus is still approximately 46 ° c . referring to fig6 a generalized schematic of the non - invasive adaptive - nulling hyperthermia system of fig3 includes hyperthermia transmitting antenna array 102 having a plurality of transmitting antenna elements 104 n , where n = 1 , . . ., n , surrounding target body 106 for focusing rf energy at focus 107 within the target body . antenna array 102 is energized by an rf energy source 108 which is distributed to and drives each transmit antenna element 104 n through a corresponding transmit weighting function 110 n , each having a corresponding weight w n . each weighting function w n may affect the gain and phase of the rf energy fed to its corresponding antenna 104 n in the array , i . e ., w n represents a complex weighting function . each weighting function 110 n may be implemented by a voltage controlled rf amplifier and a voltage controlled rf phase shifter . an amplitude control voltage representing the amplitude component of transmit weight w n is fed to the voltage controlled amplifier , and a phase control voltage representing the phase of transmit weight w n is fed to the voltage controlled phase shifter . target body 106 has a plurality of e - field auxiliary probes 112 m , where m = 1 , . . . , n aux , i . e ., receiving antennas , positioned at various locations on the surface of the body for sampling the e - field at each particular location . another receiving probe 115 may be placed at the desired focus 107 of the array . receiving probes 112 m and 115 each drive an input to an rf receiver 114 . the transmit amplitude and phase weights of each weighting function w n are fed to the receiver 114 through lines 103 n and are used to find the transmit level of each transmit element 104n . the outputs of receiver 114 represent the auxiliary probe - received complex voltages v 1 , v 2 , . . . , v naux , the focus probe - received complex voltage v f , and the transmit level of the phased array . the receiver outputs drive the inputs of a signal processor 116 , which applies a nulling algorithm to adjust the weighting functions w n and thereby null , or minimize , the rf signal received by each receiving probe 112 m , i . e ., minimize the snr p at each probe . to generate the desired field distribution in a clinical adaptive hyperthermia system , the receiving probes are positioned as close as possible to the focus ( tumor site ) and to where high temperatures are to be avoided ( such as near the spinal cord and scar tissue ). for an annular array configuration the receiving probes can be located non - invasively on the surface ( skin ) of the target . initially , the hyperthermia array is focused to produce the required field intensity at the tumor . an invasive probe may be used to achieve the optimum focus at depth . to avoid undesired hot spots , it is necessary to minimize the power received at the desired null positions and to constrain the array transmit weights w n to deliver a required amount of transmitted or focal region power . signal processor 116 performs either a sample matrix inversion ( smi ) algorithm or a gradient search algorithm on the signals output from receiver 114 and updates the adaptive array weights w n ( with gain g and phase φ ) to rapidly ( within seconds ) form the nulls at the auxiliary probes before a significant amount of target heating takes place . with this adaptive system , it is possible to avoid unintentional hot spots in the proximity of the auxiliary probes and maintain a therapeutic thermal dose distribution at the focus ( tumor ). signal processor 116 may also perform a maximizing algorithm to maximize energy at the focus 107 . the focus probe 115 is invasively placed at the desired focus 107 , and used to generate a maximum signal , or signal - to - noise ratio ( snr f ), at the tumor site . rf receiver 114 makes an amplitude and phase measurement on the output signal from invasive probe 115 for each transmit antenna element 104 n radiating one at a time . signal processor 116 processes these measurements and feeds back weight command signals to the transmit weighting functions 110 n to calibrate or phase align the transmit channels to thereby maximize the snr f , or rf power , at the invasive focal point probe . if receiver 114 makes amplitude - only measurements from invasive focus probe 115 , then a gradient search technique may be applied by the signal processor with all elements transmitting simultaneously to maximize the snr f at the invasive focal point probe . fig7 shows an analytical model of a hyperthermia phased - array antenna system 200 , paralleling the generalized hyperthermia phased - array antenna system 100 of fig6 illustrating the principles of the near - field adaptive nulling technique of this invention . the phased - array antenna system 200 includes a hyperthermia transmitting antenna array 202 having a plurality of transmitting antennas 204 n , where n = 1 , . . . , n for focusing rf energy at a desired focus 207 in the near field of the antenna . antenna array 202 is energized by an rf energy source 208 which drives a power divider 209 . power divider 209 has one output for driving each antenna 204 n a corresponding transmit weighting function 210 n , each having a corresponding transmit weight w n . it is assumed here that each weighting function w n may affect the phase of the rf energy fed to its corresponding antenna 204 n in the array . a calibration e - field probe 212 , or focus probe antenna , is positioned at focus 207 for sampling the e - field at that location . it is assumed that the hyperthermia phased - array antenna 200 is focused ( as it normally is ) in the near field and that a main beam 220 and possibly sidelobes 222 are formed in the target . in general , phase and amplitude focusing is possible . it is assumed that phase focusing alone is used to produce the desired quiescent main beam , i . e ., weighting functions w n affects only the phase of the rf signal driving each antenna . the signal received by the calibration probe can be maximized by adjusting the phase weighting functions w n so that the observed transmit antenna element - to - element phase variation is removed , i . e ., all transmit antennas appear to be in - phase when observed from the focus . one way to achieve phase coherence at the focus in a numerical simulation is to choose a reference path length as the distance from the focus to the phase center 224 of the array . this distance is denoted r f and the distance from the focus to the nth array transmit antenna element is denoted r n f . the voltage received at the calibration probe 212 ( located at focus 207 ) due to the nth array element may be computed using the &# 34 ; method of moments &# 34 ;, as described below . to maximize the received voltage at the calibration probe output , it is necessary to apply the phase conjugate of the signal observed at the calibration probe , due to each array transmit antenna element , to the corresponding element at the transmit array . the resulting near - field radiation pattern will have a main beam and sidelobes . the main beam will be pointed at the array focal point , and sidelobes will exist at angles away from the main beam . auxiliary probes can then be placed at the desired null positions in the quiescent sidelobe region . these sidelobes occur where tissue hot spots are likely to occur , and they are nulled by one of the adaptive nulling algorithms described below . considering again the hyperthermia array and probe geometry shown in fig7 the hyperthermia transmit antenna array 202 typically contains n identical transmit antenna elements 204 . the number of adaptive channels is denoted m , and for a fully adaptive array m = n . the ideal transmit weights w n ( a complex voltage gain vector ) are assumed in the computer simulation , with w =( w 1 , w 2 , . . . , w n ) t denoting the adaptive channel weight vector as shown in fig6 . ( superscript t means transpose ). to generate adaptive nulls , the transmit weights ( phase and gain ) are controlled by either the sample matrix inversion ( smi ) algorithm or a gradient search algorithm . the smi algorithm has the flexibility to operate in either open - or closed - loop feedback modes ; the gradient search algorithm operate only in a feedback mode . for the smi algorithm , the fundamental quantities required to fully characterize the incident field for adaptive nulling purposes are the adaptive channel cross correlations . to implement this algorithm it is necessary to know the complex received voltage at each of the auxiliary probes . for example , the moment - method formulation ( described below ) allows computation of complex - received voltage at each of the auxiliary probes . fig8 is a block diagram showing the smi algorithm applied to the adaptive hyperthermia phased - array of fig7 and the derivation of performance measures to quantify computer simulation results . four performance measures are used to quantify the computer simulations : electric - field distribution e ( x , y , z ), channel correlation matrix eigenvalues λ k , k = 1 , . . . , n , adaptive transmit weights w a , and interference cancellation c . the calculation of these performance measures is described in detail below . assuming a spherical wavefront is incident at an ith probe antenna 226 due to each of the n array transmit antenna elements 204 n ( radiating one at a time with a unity - amplitude reference signal ), the result is a set of probe - received complex voltages denoted v 1 i , v 2 i , . . . , v n i after a gain adjustment 250 . the cross correlation r mn i of the received voltages due to the mth and nth transmit antenna ( adaptive transmit channel ) at the ith probe is given by where * means complex conjugate and e (*) means mathematical expectation . ( note : for convenience , in equation ( 1 ) the superscript i in v m and in v n has been omitted .) because v m and v n represent voltages of the same waveform but at different times , r mn i is also referred to as an autocorrelation function . in the frequency domain , assuming the transmit waveform has a band - limited white noise power spectral density ( as commonly assumed in radar system analysis ), equation ( 1 ) can be expressed as the frequency average ## equ1 ## where b = f 2 - f 1 is the nulling bandwidth , or bandwidth of frequencies applied by the hyperthermia treatment , and f is the transmit frequency of the hyperthermia array . it should be noted that v m ( f ) takes into account the transmit wavefront shape , which is spherical for the hyperthermia application . for the special case of a continuous wave ( cw ) transmit waveform , as normally used in hyperthermia , the cross correlation reduces to where f o is the transmit frequency of the hyperthermia array . next , the channel correlation matrix , or interference covariance matrix , denoted r is determined 252 . ( note : in hyperthermia , interference is used to refer to the signals received at the auxiliary probes . the undesired &# 34 ; hot spots &# 34 ; can be thought of as interfering with the therapy .) if there are n aux independent desired null positions or auxiliary probes , the n aux - probe channel correlation matrix is the sum of the channel correlation matrices observed at the individual probes . that is , ## equ2 ## where r i is the sample channel correlation matrix observed at the ith probe and i is the identity matrix used to represent the thermal noise level of the receiver for simulation purposes . prior to generating an adaptive null , the adaptive channel weight vector , w , is chosen to synthesize a desired quiescent radiation pattern . when nulling is desired , the optimum set of transmit weights to form an adaptive null ( or nulls ), denoted w a , is computed 254 by where - 1 means inverse and w q is the quiescent weight vector . during array calibration , the normalized quiescent transmit weight vector , with transmit element 204 1 radiating , is chosen to be w q =( 1 , 0 , 0 , . . . , 0 ) t , i . e ., the transmit channel weight of element 204 1 is unity and the remaining transmit channel weights are zero . similar weight settings are used to calibrate the remaining transmit elements . for a fully adaptive annular array focused at the origin in homogeneous tissue , the normalized quiescent weight vector is simply w 4 =( 1 , 1 , 1 , . . . , 1 ) t . commonly , the weight vector is constrained to deliver a required amount of power to the hyperthermia array or to the tumor . for simplicity in the computer simulation used to analyze the hyperthermia array , the weights are constrained such that ## equ3 ## where w n is the transmit weight for the nth element . it should be noted that in the computer simulations , the electric field due to the normalized weight vector is scaled appropriately to deliver the required amount of power to the tissue so that a desired focal - region temperature level is achieved after t minutes . the summation of power received at the probes is given by where means complex conjugate transpose . the signal - plus - noise - to - noise ratio for the auxiliary probe array , denoted snr p , is computed as the ratio of the auxiliary probe array output power ( defined in equation ( 7 )) with the transmit signal present , to the probe array output power with only receiver noise present , that is , ## equ4 ## next , the adaptive array cancellation ratio , denoted c , is determined 255 . c is defined here as the ratio of the summation of probe - received power after adaptation to the summation of probe - received power before adaptation ( quiescent ); that is , ## equ5 ## substituting equation ( 7 ) into equation ( 9 ) yields ## equ6 ## next , the channel correlation matrix defined by the elements in equations ( 2 ) or ( 3 ) is hermitian ( that is , r = r . sup . ), which , by the spectral theorem , can be decomposed 256 in eigenspace as ## equ7 ## where λ k , k = 1 , 2 , . . . , m are the eigenvalues of r , and e k , k = 1 , 2 , . . . , m are the associated eigenvectors of r . the channel correlation matrix eigenvalues ( λ 1 , λ 2 , . . . , λ m ) are a convenient quantitative measure of the use of the adaptive array degrees of freedom . the amplitude spread between the largest and smallest eigenvalues is a quantitative measure of the dynamic range of the interference ( hot spot ) signals . fig9 is a block diagram of the sample matrix inversion algorithm implemented by the signal processor 116 of fig6 . receiver 114 generates probe - received complex voltage vector v 1 i , v 2 i , . . . , v n i for the ith auxiliary probe . the signal processor generates 280 the transmit channel correlations r mn i defined by equation ( 3 ), and sums 282 them to form the channel correlation matrix r defined by equation ( 4 ). next , the signal processor multiplies 284 the inverse of the channel correlation matrix r - 1 by the quiescent transmit weight vector w q to form the new adapted transmit weight vector w a containing the adapted transmit weights fed back to the transmit weight networks 110 n of fig6 . under conditions where only the probe received voltage amplitude is measured , it is appropriate to consider a gradient search algorithm to minimize the interference power at selected positions . the gradient search is used to control the transmit weights w n iteratively such that the rf signal received by the probe array is minimized . the transmit array weights ( gain and phase ) are adaptively changed in small increments and the probe array output power is monitored to determine weight settings that reduce the output power most rapidly to a null . consider j sets of n transmit weights that are applied to adaptive hyperthermia phased array antenna 102 of fig6 . in terms of adaptive nulling , the optimum transmit weight settings ( from the collection of j sets of n transmit weights ) occur when the snr p is minimized . equivalently , the total interference power received by the auxiliary probe array , denoted p rec , is to be minimized . for notational l convenience let a figure of merit f denote either the snr p or p rec and employ a gradient search to find the optimum transmit weights to minimize f , that is , the transmit weight settings for which f opt occurs yields the closest approximation to the optimal transmit weights determined by using the sample matrix inverse approach described above . fig1 shows an amplitude and phase scatter diagram for the n complex transmit weights w n at the jth configuration , i . e ., the jth set of weights tried . the nth transmit weight in the jth configuration of transmit weights is denoted where a nj is the transmit weight amplitude distributed over the range a min to a max and φ nj is the transmit weight phase distributed over the range - π to π radians . referring also to fig1 , it is desired to find the values of amplitude and phase for each of the n transmit weights such that the figure of merit f ( snr p or p rec ) is minimized . when the figure of merit is minimized , adaptive radiation pattern nulls will be formed at the auxiliary probe positions . assuming an initial setting of the n transmit weights such as those selected to focus the radiation pattern on a tumor , the weights are adjusted by dithering them until the optimum figure of merit is achieved . it is desired to find the collective search directions for the n transmit weights such that f decreases most rapidly . that is , weights are selected so that the directional derivative is minimized at ( a j , φ j ), where a j and φ j are vectors representing the transmit amplitude weights and transmit phase weights , respectively , for the jth configuration . the directional derivative of f j is expressed in terms of the amplitude and phase changes of the transmit weights as ## equ8 ## where ∂ means partial derivative , and r anj , r . sub . φnj are the ( a , φ ) directions for which f j is decreasing most rapidly . the directions r anj , r 101 nj are constrained by ## equ9 ## it is desired to minimize d ( f j ) subject to the above constraint equation . using lagrange multipliers it is possible to construct the lagrangian function ## equ10 ## where g is a constant to be determined . the requirement that l j be an extremum implies ## equ11 ## squaring equations ( 19 ) and ( 20 ) and invoking equation ( 15 ) ## equ12 ## substituting the expression for g in equations ( 19 ) and ( 20 ) gives ## equ13 ## the minus sign was chosen corresponding to the direction of maximum function decrease . this choice of minus sign in equation ( 22 ) enforces nulls in the hyperthermia array radiation pattern . alternatively , if the positive sign in equation ( 22 ) is selected , then the gradient directions can be used to maximize the figure of merit for the purposes of focusing at an invasive probe at the tumor site , i . e ., maximize the snr f . this may be used , for example , to determine the quiescent transmit weight vector w q . thus , two gradient searches may be performed to optimize the radiation pattern of the hyperthermia array . the first to produce a peak or focused radiation pattern at the tumor , and the second to form the desired nulls at the auxiliary probes . furthermore , these two gradient searches may be implemented as a single , combined gradient search constrained to maximize the radiation pattern at the focus and minimize the radiation pattern at the desired nulls . the combined gradient search is implemented by minimizing the figure of merit defined as the ratio of the power received at the auxiliary probes to the power received by the probe at the focus . the partial derivatives ## equ14 ## represent the gradient directions for maximum function decrease . since the figure of merit f cannot be expressed here in analytical form , the partial derivatives are numerically evaluated by using finite differences . thus , we write ## equ15 ## where as shown in fig2 the figure of merit differences are and δa nj and δφ nj are assumed to be small increments . we will assume that the increments δa nj and δφ nj are independent of the configuration number and element number , that is , substituting equations ( 26 ), ( 27 ), ( 30 ) and ( 31 ) in equations ( 23 ) and ( 24 ) gives the desired result for the search directions ## equ16 ## equations ( 32 ) and ( 33 ) are used to compute the new amplitude and phase settings of the ( j + 1 ) th transmit weight configuration according to in practice , it may be necessary to keep one of the transmit weights fixed ( in amplitude and in phase ) during the gradient search to guarantee convergence . fig1 is a block diagram of the gradient search algorithm implemented by the signal processor 116 of fig6 . each of the n transmit antennas 104 n of array 102 ( fig6 ) is driven through its corresponding weighting network 110 n which applies complex transmit weights w nj at the jth configuration of the weights . the transmit antennas induce a voltage across the ith probe antenna 112 i at the corresponding input to receiver 114 ( fig6 ). receiver 114 amplifies the signal received from the ith probe by gain a i to produce voltage amplitude vector | v 1j i |,| v 2j i |, . . . ,| v nj i | at the receiver output . the voltage amplitude vector is input to signal processor 116 which performs the gradient search . for any initial configuration ( j = 1 ) of the transmit weights w nj , the signal processor causes each weight to be dithered by a small amount in amplitude , δa nj , and phase , δφ nj . each transmit weight is dithered independent of the other transmit weights , which remain in their jth configuration state . received voltage vectors | v 1j i |,| v 2j i |, . . . ,| v nj i |, i . e ., are stored and used to calculate the resulting figure of merit f nj 300 for each dithered condition , the figure of merit being the power received by the auxiliary prove array . the figure of merit is a rectangular matrix of dimension n × 4 , where the dimensionality of four is due to the plus and minus dithering of both of the amplitude and phase . the figure of merit differences δf anj and δf . sub . φnj caused by dithering the amplitude and phase , respectively , are calculated according to equations ( 28 ) and ( 29 ). the gradient search directions r anj and r . sub . φnj , based upon minimizing the auxiliary probe array received power , are then determined 302 from the figure of merit differences according to equations ( 32 ) and ( 33 ), respectively . the resulting search directions are used to update 304 transmit weights w nj to the ( j + 1 ) th configuration transmit weights w n ,( j + 1 ) according to equations ( 34 ) and ( 35 ). the transmit weights w n ,( j + 1 ) are sent to update the transmit weighting networks 110 n , and the process is repeated . the final adaptive weight vector w a is achieved when the ( j + 1 ) th transmit weight configuration has converged . convergence is expected to occur within several hundred iterations depending on the dither step size δa and δφ . referring again to fig8 a method of moments formulation 258 is used to compute the probe - received voltages in equation ( 2 ) due to the transmitting hyperthermia phased - array antenna in an infinite homogeneous conducting medium . the medium is described by the three parameters μ , ε , and σ , which are discussed below . the formulation given here is analogous to that developed under array - receiving conditions for an adaptive radar . the software used to analyze a hyperthermia array is based on the receive - array analogy but the theory presented below is given in the context of a transmit array . an antenna analysis code ( wires ) originally developed by j . h . richmond is capable of analyzing antenna or radar cross section problems . see , j . h . richmond , &# 34 ; computer program for thin - wire structures in a homogeneous conducting medium &# 34 ;, ohio state university , electroscience laboratory , technical report 2902 - 12 , august 1973 ; and , j . h . richmond , &# 34 ; radiation and scattering by thin - wire structures in a homogeneous conducting medium ( computer program description )&# 34 ;, ieee trans . antennas propagation , vol . ap - 22 , no . 2 , p . 365 , march 1974 . wires was modified to analyzing the near - field and far - field adaptive nulling performance of thin - wire phased arrays in free space . a new version of the thin - wire code that can analyze adaptive hyperthermia arrays in an infinite homogeneous conducting medium was written to conduct the adaptive hyperthermia simulation discussed below . the new version of the thin - wire code is attached as appendix a . wires is a moment - method diode that uses the electric field integral equation ( efie ) to enforce the boundary condition of the tangential field being zero at the surface of the antenna of interest . the moment - method basis and testing functions used in this code are piecewise sinusoidal . appendix b lists sample input and output files for the adaptive hyperthermia simulation . the first data file was used to generate the e - field results for a four auxiliary probe system , and the second data file was used to generate the e - field results for a two auxiliary probe system . the corresponding output files give the values for the array mutual coupling , quiescent and adaptive transmit weights , channel correlation matrix , eigenvalues , and cancellation . referring to fig1 , there is shown the hyperthermia phased - array antenna system 200 of fig7 redrawn to simplify the following method of moments analysis . the rf source 208 , power divider 209 and weights 210 n of fig7 are modeled as a plurality of rf signal generators 250 1 through 250 n , feeding its corresponding transmit antenna element 204 1 through 204 n . each generator 250 1 through 250 n has a corresponding amplitude and phase weight denoted by w 1 through w n , and a known output impedance z l . the jth probe 226 ( i . e ., the same as the ith probe 226 of fig7 with different notation ) is modeled as a dipole antenna having an overall length l p and an open - circuit voltage v j o . c . induced by the rf energy transmitted from the antenna array 200 . the open - circuit voltage at the jth probe antenna 226 is computed from the array terminal currents and from z n , j the open - circuit mutual impedance between the nth array element and the jth probe antenna . let v nj o . c . represent the open - circuit voltage at the jth probe due to the nth transmit - array element . here , the jth probe can denote either the focal point calibration probe ( calibration probe 212 of fig7 ) or one of the auxiliary probes used to null a sidelobe . the number of auxiliary probes is denoted by n aux . referring also to fig1 , the jth probe 226 is modeled as a voltage source 260 , having an output voltage v j o . c . , driving a first impedance 262 , representing the input impedance z in of the jth probe , in series with a second impedance 264 , representing the termination impedance z r of the jth probe . the jth probe receive current i j rec flows through these two impedances . the output voltage of the jth probe v j rec appears across the termination impedance z r . referring again to fig1 , next , let z denote the open - circuit mutual impedance matrix ( with dimensions n x n for the n - element array ). the open - circuit mutual impedance between array elements 204 m and 204 n is denoted z m , n . it is assumed that multiple interaction between the hyperthermia array and the auxiliary probe can be neglected . thus , the hyperthermia array terminal current vector i can be computed in terms of the transmit weights w as next , let z n , j be the open - circuit mutual impedance between the jth probe and the nth array element . the induced open - circuit voltage v nj o . c . at the jth receive probe , due to the nth array element transmit current i n , can then be expressed as in matrix form , the induced open - circuit probe - voltage matrix v probe o . c . is where z probe , array is a rectangular matrix of order n aux x n for the open - circuit mutual impedance between the probe array and the hyperthermia array . note that the jth row of the matrix z probe , array is written as ( z 1 j , z 2 j . . . , z n j ), where j = 1 , 2 , . . . , n aux . the receive voltage matrix is then computed by the receiving circuit equivalence theorem for an antenna . the receive - antenna equivalent circuit is depicted in fig1 , where it is readily determined that ## equ17 ## where z in is the input impedance of the probe . it should be noted that the v probe rec matrix is a column vector of length n aux and v j rec is the jth element of the matrix . the probe - receive current matrix is given by ## equ18 ## the jth element of the column vector i probe rec is denoted i j rec , j = 1 , 2 , . . . , n aux . finally , the power received by the jth probe is ## equ19 ## where re means real part . substituting equations ( 40 ) and into equation ( 42 ) yields ## equ20 ## the total interference power received by the auxiliary probe array is given by ## equ21 ## the incident electric field e is related to the open - circuit voltage v o . c . by the effective height h of the probe antenna as if the length l p of the probe antenna 226 is approximately 0 . 1λ or less , the current distribution is triangular and the effective height is h = 0 . 5l p . thus , for a short - dipole probe the open - circuit voltage can be expressed as ## equ22 ## it then follows from equation ( 46 ) that the e field for a short - dipole probe at position ( x , y , z ) is given by ## equ23 ## finally , the quiescent and adapted e - field radiation patterns are computed using the quiescent and adapted weight vectors w q and w a , respectively , in equations ( 39 ) and ( 47 ). the moment - method expansion and testing functions are assumed to be sinusoidal . the open - circuit mutual impedances in equation ( 39 ) between thin - wire dipoles in a homogeneous conducting medium are computed based on subroutines from the moment - method computer code developed by j . h . richmond . in evaluating z n j for the jth auxiliary probe , double precision computations are used . as mentioned previously , the array is calibrated ( phased focused ) initially using a short dipole at the focal point . to accomplish this numerically , having computed v focus rec , the transmit array weight vector w will have its phase commands set equal to the conjugate of the corresponding phases in v focus rec . transmit antenna radiation patterns are obtained by scanning ( moving ) a dipole probe with half - length 1 in the near - field and computing the receive probe - voltage response . the received voltage matrix for the jth probe ( denoted v j rec ) is computed at k frequencies across the nulling bandwidth . thus , v j rec ( f 1 ), v j rec ( f 2 ), . . . v j rec ( f k ) are needed . for the purposes of this computer simulation , the impedance matrix is computed at k frequencies and is inverted k times . the probe channel correlation matrix elements are computed by evaluating equation ( 2 ) numerically , using simpson &# 39 ; s rule numerical integration . for multiple auxiliary probes , the channel correlation matrix is evaluated using equation ( 4 ). adaptive array radiation patterns are computed by superimposing the quiescent radiation pattern with the weighted sum of auxiliary - channel - received voltages . to gain insight into the effect of a lossy medium , e . g .. the target body , on the propagation of an electromagnetic wave , it is useful to review certain fundamental equations which govern the field characteristics . in a conducting medium , maxwell &# 39 ; s curl equations in time - harmonic form are where e and h are the electric and magnetic fields , respectively , j is the conduction current density , ω = 2πf is the radian frequency , ε is the permittivity of the medium , and μ is the permeability of the medium . the permittivity is expressed as ε = ε r ε o , where ε r is the dielectric constant ( relative permittivity ) and ε o is the permittivity of free space . similarly , μ = μ r μ o , where μ r is the relative permeability and μ o is the permeability of free space . for a medium with electrical conductivity σ , j and e are related as from equations ( 48 ) and ( 49 ), the vector wave equation in terms of e is derived as it is readily shown that ## equ24 ## the quantity σ / ωε is referred to as the loss tangent . it is common to express the complex propagation constant as where α is the attenuation constant and β is the phase constant . the constants α and β are found by setting equation ( 53 ) equal to equation ( 54 ) and then squaring both sides , equating the real and imaginary parts , and solving the pair of simultaneous equations , with the result ## equ25 ## the wavelength λ in the lossy dielectric is then computed from ## equ26 ## the intrinsic wave impedance η is given by ## equ27 ## the instantaneous power density of the electromagnetic field is given by poynting &# 39 ; s vector , denoted p , ## equ28 ## which had units of ( w / m 2 ). the time - average power flow density is equal to the real part of the complex poynting &# 39 ; s vector . the time - average power dissipation per unit volume p d ( w / m 3 ) is derived from maxwell &# 39 ; s equations , with the result ## equ29 ## the specific absorption rate ( sar ) is the power dissipated or absorbed per unit mass ( w / kg ) of the medium ( tissue ), or ## equ30 ## where ρ is the density of the medium in kg / m 3 . it is convenient to have a simple equation for computing the propagation loss between any two points in the near field of an isolated transmitting antenna . thus , mutual coupling effects are ignored for the time being . consider a time - harmonic source radiating a spherical wave into an infinite homogeneous conducting medium . for an isotropic radiator , and suppressing the e j ωt time dependence , the electric field as a function of range r can be expressed as ## equ31 ## where e o is a constant . for a source at the origin , the amplitude of the electric field at range r 1 is given by ## equ32 ## and at range r 2 by ## equ33 ## the total propagation loss between ranges r 1 and r 2 is found by taking the ratio of equations ( 64 ) and ( 63 ), or ## equ34 ## the field attenuation a . sub . α in db from range r 1 to range r 2 due to the lossy dielectric is simply similarly , the 1 / r attenuation loss a r in db is ## equ35 ## a thermal analysis computer program called the transient thermal analyzer ( tta ), developed by arthur d . little , inc ., has been used to accomplish the thermal modeling of homogeneous muscle tissue surrounded by a constant - temperature water bolus . the tta program uses the finite - difference technique to solve a set of nonlinear energy balance equations . consider a system of interconnected nodes that model an inhomogeneous volume for which the temperature t i of the ith node is to be determined . the heat - balance equation , which is solved by tta , is expressed as ## equ36 ## where q ij is the net outward heat flow from node i in the direction of node j , p i ( t ) is the power into node i at time t , and m i is the thermal mass ( mass times specific heat ) of node i . fig1 shows an electric circuit analog 400 which is used to model the two - dimensional thermal characteristics of the material volume 402 which simulates the target body as a plurality of uniformly distributed nodes 406 spaced δ1 apart . with reference to the ith node 406 i , but applying generally to the other nodes , power p i in watts is delivered 404 i to the ith node . capacitor 408 i , having thermal capacitance denoted c i ( with units joules /° c . ), is used to model the thermal capacitance at the ith node . resistor 410 ij , having heat resistance denoted r ij ( with units ° c ./ w ), is used to model the heat resistance between ith node 406 i and the jth node 406 j . with a spacing of δ1 between nodes ( assuming cubic cells ), the values of r ij , c i and p i are computed as ## equ37 ## where k ij is the thermal conductivity ( with units w / m ° c .) between nodes i and j ; where c pi is the specific heat at the ith node and ρ i is the density ( kg / m 3 ) at the ith node ; and where ( sar ) i is the sar for the ith node , which is given by ## equ38 ## where σ i is the electrical conductivity of the ith node and | e i | is the magnitude of the electric field delivered by the hyperthermia array to the ith node . it should be noted that in substituting equation ( 72 ) into equation ( 71 ), the density ρ i cancels . thus , an equivalent approach to computing the power delivered to the ith node is written in terms of the time - average power dissipated per unit volume of the ith node ( denoted p di ) as fig1 is a block diagram showing how tta is used in the hyperthermia simulation described herein . first , the method of moments 500 , controlled by the smi nulling algorithm 502 , is used to compute the electric field radiation pattern throughout a homogeneous region , simulating muscle tissue , inside an annular phased array 501 . these e - field simulations assume that the signal received by a short - dipole probe within the region is due to a transmitting phased array embedded in an infinite homogeneous lossy dielectric ( muscle tissue ). the resulting e - field power distribution is then read 504 into the tta program 506 , which computes the temperature distribution inside an elliptical muscle - tissue target surrounded with a constant - temperature water bolus 507 . because the rf wavelengths in the target and water bolus are similar , the e - field simulations are believed to give a reasonable approximation to the field distribution inside the elliptical target . the computed temperature distribution is output 508 from the tta for further analysis or display . the e - field calculation in the assumed infinite homogeneous medium introduces additional field attenuation not present in a clinical hyperthermia system with an annular array transmitting through a water bolus into a patient . as mentioned earlier , the water bolus has very little rf propagation loss . in addition , the transmit array weights are normalized according to equation ( 6 ). thus , no attempt is made to compute the absolute e - field strength in volts / meter in the elliptical target . instead , the peak power in the elliptical target is adjusted ( by a scale factor ) to produce a desired maximum focal - region temperature ( t max ) after t minutes . it should be noted that an approximate absolute scale factor could be computed by making an initial computer simulation with an infinite homogeneous water bolus and then matching the target boundary field to the infinite homogeneous muscle tissue simulation . the computer simulation model is related , in part , to the hyperthermia annular phased - array antenna system shown in fig1 . the simulated array is assumed to have a 60 - cm array diameter with eight uniformly spaced dipole elements which operate over the frequency band 60 - 120 mhz . the eight elements of the array are assumed to be fully adaptive , whereby seven independent nulls can be formed while simultaneously focusing on a tumor . it is further assumed for the purpose of this simulation that the adaptive radiation pattern null - width characteristics in a homogeneous target are similar to the characteristics observed in an inhomogeneous target . the null - width characteristics are directly related to the rf wavelength , and , only a 5 percent change in wavelength occurs between the assumed muscle tissue and water bolus . with this assumption , the transmit array may be simulated as embedded in homogeneous tissue , which allows direct use of the thin - wire moment - method formulation discussed above . after computing the two - dimensional e - field distribution in the homogeneous medium , we then consider only an elliptical portion of the homogeneous region and use the ellipse as the homogeneous target . in the thermal analysis , the elliptical target is surrounded with a constant 10 ° c . water bolus . the e - field amplitude is scaled to produce a 46 ° c . peak temperature , at time t = 20 minutes , at the center of the elliptical phantom . the initial temperature of the phantom is assumed to be 25 ° c . ( room temperature ). all computer simulations assume a 120 mhz operating frequency with initially four auxiliary nulling probes , i . e ., n aux = 4 : the parameters used in the electrical and thermal analyses are summarized in table 1 . these parameters are for a frequency of 100 mhz , but is assumed that similar values of the parameters will exist at 120 mhz . it should be noted that although the relative dielectric constants of phantom muscle tissue and distilled water are very similar , the electrical conductivities are vastly different . the relevant thermal characteristics -- density , specific heat , and thermal conductivity -- are very similar for phantom muscle tissue and distilled water . substituting the values f = 120 mhz , σ = 0 . 5 s / m , and ε r = 73 . 5 into equation ( 53 ) yields γ m = 10 . 0 + j23 . 8 for the muscle tissue . with β m = 23 . 8 radians / m , the wavelength in the phantom muscle tissue is λ m = 26 . 5 cm . the attenuation constant for the muscle tissue is α m = 10 . 0 radians / m . similarly , for distilled water γ w = 0 . 0021 + j22 . 5 , so the wavelength is λ w = 27 . 9 cm . the attenuation constant for the distilled water medium is α w = 0 . 0021 radians / m . the propagation loss in the phantom muscle tissue is 20log 10 e - 10 . 0 , or - 0 . 87 db / cm . similarly , the propagation loss in the distilled water is found to be - 0 . 0002 db / cm . thus , the total loss due to propagation through 15 cm of distilled water is 0 . 003 db . for 15 cm of muscle tissue the corresponding loss is 13 . 1 db . the wave impedance in the muscle tissue is computed from equation ( 58 ) as η m = 33 . 9 + j14 . 2 ω , and similarly in the distilled water η w = 42 . 1 + j0 . 004 ω . fig1 shows the geometry used in the simulations , which parallels the array shown in fig3 . a 60 - cm - diameter ring array 102 of eight perfectly conducting center - fed dipoles , 104 1 through 104 8 , uniformly surrounds a fictitious elliptical target zone 106 with major axis 30 cm and minor axis 20 cm . the length of each dipole array element 104 n at 120 mhz in the infinite homogeneous muscle tissue is λ / 2 , or 13 . 25 cm . the array focus 107 is assumed at the origin ( x = 0 , y = 0 , z = 0 ) and four auxiliary short - dipole probes , 112 1 through 112 4 , with length 1 . 27 cm ( 0 . 05λ ) are positioned at ( x , y , z ) coordinates at ( 15 cm , 0 , 0 ), (- 15 cm , 0 , 0 ), ( 0 , 0 , 10 cm ), and ( 0 , 0 , - 10 cm ), respectively , i . e ., the auxiliary e - field probes are located every 90 ° in azimuth on the perimeter of the target . in rectangular coordinates , each dipole is oriented along the y direction and the feed terminals of each dipole are located at y = 0 . the moment - method computer simulations were run on a sun 3 / 260 workstation . the total cpu time for a complete moment - method run is 19 . 2 minutes . this cpu time includes computing the quiescent and adaptive radiation patterns on a 41 by 41 grid of points . the cpu time without radiation pattern calculations is 33 seconds . fig1 shows the two - dimensional radiation pattern in the plane y = 0 , before nulling , at 120 mhz with uniform amplitude and phase illumination . the calculated data are collected on a 41 by 41 grid of points over a square region , with side length 76 . 2 cm , centered at the focus 107 . the spacing between data points is 1 . 905 cm , or 0 . 072λ , and the contour levels are displayed in 10 - db steps . the e - field data are computed for the case of a 1 . 27 - cm short - dipole observation probe . the positions of the eight dipole radiators 104 1 through 104 8 are clearly evident by the - 20 db contours surrounding each element . the radiation pattern is symmetric because of the symmetry of the array and the assumed homogeneous medium . fig1 shows finer contour levels ( 1 - db steps ) for the quiescent radiation pattern of fig1 . here , it is evident that the focused main beam of the ring array is increasing in amplitude as the observation point moves closer to the focus . away from the main beam region , the pattern amplitude is seen to increase as the observation position moves toward the array perimeter . fig2 shows the quiescent radiation pattern of fig1 cut at z = 0 . the large amplitude that occurs at ± 30 cm , i . e ., at the position of the array 102 , is due to the e - field probe &# 39 ; s close proximity to the transmitting elements 104 1 and 104 5 . the large attenuation that occurs from the array diameter to the focus is due to the 1 / r attenuation loss and the loss in the uniform homogeneous muscle tissue . fig2 shows the radiation pattern of fig1 cut at x = 0 . here , the pattern is identical to the pattern of fig2 due to the symmetry of the array . in both fig2 and 21 the boundary of the fictitious elliptical target zone 106 is indicated . the target zone of fig2 is larger than that of fig2 since the major axis of elliptical target 106 lies along the x - axis , and the minor axis of target 106 lies along the z - axis . the increasing radiation pattern amplitude near the left and right sides of the elliptical target of fig2 is shown to produce hot spots in the thermal distribution . because the top ( anterior ) and bottom ( posterior ) of the elliptical target of fig2 are not as strongly illuminated as to the left and right sides of the elliptical target of fig2 , no quiescent hot spots occur at the top or bottom . further , fig2 shows that the ring - array half - power beamwidth in the target region is approximately 13 cm , or approximately one - half the wavelength ( 26 . 5 cm ) in the phantom muscle tissue . the adaptive nulling resolution or closest allowed spacing between a deep adaptive null and the main beam has been shown to be equal to the half - power beamwidth of the antenna . thus , the closest allowed null position is 13 cm from the focus . since the target width is 30 cm across the major axis , two nulls can be formed at ( x =± 15 cm , z = 0 ) at the left and right side of the target without disturbing the focus . however , if two deep nulls are formed at the posterior and anterior ( x = 0 , z =± 10 cm ) of the target the focus will be compromised . in practice , the water bolus surrounding the target would restrict the placement of short - dipole probes 112 n to the surface of the target . thus , only weak nulls can be formed at ( x = 0 , z =± 10 cm ) so that the focus will not be affected by the adaptive nulling process . that is , the effect of the two minor axis nulls is to keep the z =± 10 cm e - field from increasing beyond the quiescent values . next , adaptive radiation patterns are computed with four auxiliary dipole probes 112 1 through 112 4 positioned as shown in fig1 . the value of the receiving gain for auxiliary dipole probes 112 1 and 112 2 is adjusted to produce a snr & gt ; 35 db . this amount of snr results in greater than 35 db of nulling in the direction of auxiliary dipole probes 112 1 and 112 2 . in contrast , the gain values for auxiliary dipole probes 112 3 and 112 4 are turned down to produce about a 3 db snr . thus , only about 3 db of nulling will occur at probe positions 112 3 and 112 4 as the adaptive algorithm reduces the interference to the noise level of the receiver . the reason for choosing these null depths will become apparent with the data that follow . fig2 shows the two - dimensional radiation pattern after nulling with four auxiliary probes 112 1 through 112 4 . two deep adaptive nulls at x =± 15 cm occur as expected , and weak nulling occurs at z =± 10 cm , also as expected . the two deep nulls in the z = 0 cut are quantified in fig2 , where greater than 35 db of interference nulling or pattern reduction occurs at x =± 15 cm . the peak level at the focus 107 is adjusted to 0 db for both the quiescent and adaptive patterns . two weak adaptive nulls are seen in the x = 0 radiation pattern cut shown in fig2 . the weak nulls in effect in the adaptive patterns reduce variation from the quiescent radiation pattern . fig2 ( a ) shows the transmit array amplitude weights before ( solid line ) and after ( broken line ) nulling , and fig2 ( b ) shows the transmit array phase weights before ( solid line ) and after ( broken line ) nulling . as shown , the adaptive transmit weights exhibit a 5 - db dynamic range in fig2 ( a ). fig2 shows the channel correlation matrix eigenvalues before ( solid line ) and after ( broken line ) nulling . there are two large eigenvalues , λ 1 and λ 2 , and two weak ( non - zero ) eigenvalues , λ 3 and λ 4 , shown in fig2 . these eigenvalues are directly associated with the two high - snr auxiliary probes 112 1 and 112 2 , and the two weak - snr auxiliary probes 112 3 and 112 4 , respectively . note that the 0 - db level in fig2 is equal to the receiver noise level . the probe - array output power before and after adaptive nulling is 31 . 4 db and 0 . 9 db , respectively , as calculated from equation ( 10 ). this difference in power before and after nulling indicates that the adaptive cancellation is - 30 . 5 db . to simulate the temperature distribution in the target body resulting from the calculated e - fields , the transient thermal analysis ( tta ) software is used to compute the temperature distribution in an elliptical phantom surrounded with a constant - temperature water bolus . the 41 × 41 two - dimensional e - field radiation pattern data of fig1 through 24 are used as the power source for the thermal node network . two node spacings are considered . first , the node spacing δx = δz = δ1 = 1 . 905 cm ( coarse grid ) is used to obtain thermal data . then , the node spacing is decreased by a factor of two to δ1 = 0 . 9525 cm ( fine grid ) to check convergence . the coarser spacing is shown to be adequate . the scale factors used to convert the normalized e - field distributions to a power level that induces a 46 ° c . peak temperature at t = 20 minutes are 94 . 1 db and 96 . 0 db for the quiescent and adaptive patterns , respectively . these scale factors are determined empirically . from equations ( 68 ) through ( 73 ) and the parameter values given in table 1 , all resistors r ij in the phantom muscle tissue had a value of 96 . 5 ° c ./ w and all resistors r ij in the water bolus had a value of 87 . 2 ° c ./ w . the value of the capacitors c i in the phantom muscle tissue is 23 . 6 j /° c . capacitors are not used in the water - bolus region in the input to the transient thermal analysis software . instead , a constant temperature of 10 ° c . is enforced at each water - bolus node . with a 41 × 41 grid , a total of 3280 resistors and 1681 capacitors are used in the thermal simulation . the cpu time required to compute this temperature distribution is under four minutes . fig2 shows the two - dimensional temperature distribution produced at time t = 20 minutes in the elliptical phantom muscle tissue target 106 without adaptive nulling . to generate fig2 , the power source used in the transient thermal analysis is the quiescent radiation pattern given in fig1 . the initial temperature ( at time t = 0 ) is 25 ° c . notice the occurrence of two hot spots 122 and 124 on the left and right sides of the elliptical phantom , respectively . the peak temperature at focus 107 is 46 ° c ., which is achieved by scaling the normalized quiescent e - field as described earlier . the two hot spots 122 and 124 are quantified in the z = 0 temperature pattern cut shown in fig2 , and have a peak temperature at each hot spot of approximately 41 ° c . the temperature profile for x = 0 in fig2 shows no hot spots . as any undesired hot spot is a potential source for compromising the therapy session , adaptive nulling is used to reduce the sidelobes corresponding to the hot spots . fig3 shows the simulated two - dimensional thermal distribution at time t = 20 minutes , with adaptive nulling at four auxiliary probes 112 1 through 112 4 in effect . the focal - spot diameter at focus 107 with adaptive nulling is equivalent to the focal - spot diameter before adaptive nulling , shown in fig2 . hot spots on the left and right sides of the target 106 are eliminated . fig3 shows a comparison of the temperature distribution before ( solid line ) and after ( broken line ) nulling along the major axis ( z = 0 ) of the target ellipse 106 . similarly , fig3 shows the temperature distribution before ( solid line ) and after ( broken line ) nulling along the minor axis ( x = 0 ) of the target ellipse 106 . the convergence of the previous thermal simulations was verified by increasing the density of e - field observation probe positions by a factor of two , with a new spacing between points of 0 . 9525 cm , still with a 41 × 41 grid . the ring array operates as before at 120 mhz , and there are four auxiliary probes 112 1 through 112 4 laid out as shown in fig1 . as the auxiliary positions are the same , the adaptive weights and channel correlation matrix eigenvalues in fig2 and 26 , respectively , remain the same . from the parameter values in table 1 , all resistors r ij in the finer - grid muscle - tissue phantom had a value of 193 . 0 ° c ./ w and all resistors r ij in the water bolus had a value of 174 . 4 ° c ./ w . the value of the capacitors c i in the phantom muscle - tissue is 2 . 95 j /° c . again , a constant temperature of 10 ° c . is enforced at each water - bolus node . the e - field scaling factors to raise the focal - point temperature to 46 ° c . before and after nulling are 76 . 5 db and 78 . 4 db , respectively . the finer - grid two - dimensional thermal distributions before and after nulling are shown in fig3 and 34 , respectively . although the temperature contours are smoother , the general agreement between these patterns and the coarser - grid patterns in fig2 and 30 are evident . similarly , one - dimensional thermal pattern cuts with the finer grid are shown in fig3 ( x axis ) and 36 ( z axis ), and good agreement with the coarse - grid patterns of fig3 and 32 , respectively , is observed . in particular , the finer detail in fig3 shows that the hot spots 122 and 124 are at approximately 42 ° c . compared to 41 ° c . observed for the coarse grid of fig3 . thus , convergence of the coarse - grid thermal patterns is demonstrated . while this invention has been particularly shown and described with references to preferred embodiments thereof , it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims . for instance , the adaptive nulling methods described herein are applicable to any hyperthermia system using waves to induce heating , such an ultrasound hyperthermia system , or a combination ultrasound and rf hyperthermia system . further , although the hyperthermia system is described herein with respect to a particular range of rf frequencies , the invention is applicable to hyperthermia systems operating from low frequencies to microwave frequencies . although the preferred embodiment of the invention describes using four auxiliary probes , it is understood that a larger or smaller number of auxiliary probes may also be used with differing nulled radiation patterns , e . g ., more control over the radiation pattern with more probes and less control with fewer probes . still further , the invention is applicable to non - medical hyperthermia systems , such as those used for industrial materials heating . ## spc1 ##
a hyperthermia array uses adaptive nulling with non - invasive auxiliary probes to reduce the electric field intensity at selected positions in the target body while maintaining a desired focus at a tumor thereby avoiding or reducing the occurrences of &# 34 ; hot spots &# 34 ; during ultrasonic or r . f . hyperthermia treatment . a hyperthermia applicator has an annular phased array of electric field radiators coupled to a source of electric radiation through a controllable transmit weighting network to control the phase and amplitude of the electric field radiation transmitted . the transmit weighting networks respond to feedback signals from a controller coupled to ele government support the invention described herein was supported in whole or in part by contract no . f19628 - 90 - c - 0002 from the united states air force .