text: high density ultrasound transducers have been developed in which the transducer is produced in the form of a two - dimensional grid of uniformly shaped piezoelectric ( pzt ) “ rods ” glued to a conductive matching layer substrate . for both manufacturing and performance reasons , the pzt rods have rectangular ( or square ) profiles , with an aspect ratio ( i . e ., ratio of height / width ) of greater than or equal to one , and are preferably uniform in size to produce the same frequency response . spacing between the rods also influences the acoustic performance of the transducer and is preferably minimized , i . e ., much smaller than the size of the individual piezoelectric rods , for therapeutic transducers . a high density phased array transducer may have hundreds , even thousands of densely packed piezoelectric rods , each having a relatively small energy transmitting surface , e . g ., 1 mm square . an example of one such high density transducer array 130 of uniformly shaped piezoelectric rods is shown in fig3 . the transducer array 130 comprises a two - dimensional arrangement of individual piezoelectric rods 132 glued to a planar substrate . the piezoelectric rods 132 are substantially identical in size and shape , including having substantially uniform ( square ) distal facing energy transmitting surfaces 134 ( also square ). the rods 132 are arranged in uniformly aligned columns 135 and rows 136 , with minimal spacing provided between adjacent rods . it will be appreciated that the relatively small transducer rod size allows for greater electronic steering capability of the overall array . however , as the “ steering angle ” increases , hot spots start to appear outside of the intended focal zone . more particularly , the independent piezoelectric rods 132 in such known arrays 130 are typically produced using a dicing machine that can dice along straight lines only . each rod is connected to its own electronic drive signal input , such that each rod forms a distinct transducer element . from a physical point of view , the acoustic performance ( e . g ., frequency response , efficiency , etc ) of the array 130 is influenced by the three dimensional structure of the individual rods 132 , and preferably each rod &# 39 ; s height should be equal or higher than its width . however , the steering / focusing ability of the transducer array 130 is fully defined by the geometrical surface ( i . e ., the area of a transducer element that emits a respective acoustic wave at a same phase ) of the respective rods 132 . all internal transducer structure aspects , such as piezoelectric rod height , aspect ratio , etc , are irrelevant to steering / focusing ability . as used herein , the term “ hot spot ” refers to a tissue region having an energy level ( which may be measured , for example , in terms of temperature or acoustic pressure ) that is above a prescribed ( safe ) level at which the tissue in the hot spot will be temporarily or permanently injured . because such hot spot ( s ) start to appear as the electronic steering angle increases , electronic steering to each possible “ steered - to ” focal zone must be carefully analyzed for safety purposes before undertaken . further , the energy absorbed at the hot spot ( s ) decreases the remaining energy available for contributing to the intended “ steered - to ” focal zone . fig4 illustrates the principle of electronic steering of a two dimensional planar array 150 comprised of uniformly shaped and arranged elements ( such as array 130 of fig3 ). in particular , the “ steering angle ” of any one transducer element 154 of the array 150 is the angle α formed between a first focal axis 152 extending generally orthogonally from the element to a “ non - steered ” focal zone 158 at which the element 154 contributes a maximum possible power , and a second focal axis 156 extending from the transducer element 154 to a “ steered - to ” focal zone 160 . the “ steering ability ” of the transducer array 150 is defined as a steering angle α at which energy delivered to the steered - to focal zone 160 from a given one - dimensional element row falls to half of the maximum power delivered to the non - steered focal zone 158 . notably , the steering angle if each transducer element of a phased array will be different . however , as the distance to a steered - to focal zone increases , the respective steering angles for the array elements approach the same value . for ease in illustration , the distance of the steered - to focal zone in the simulations in fig5 - 7 is assumed to be infinity , so that the steering angles of each array element is the same . from a physical point of view , a single transducer element emits a wave in the form of a spreading beam . the angular distribution of this spreading beam is called “ directivity .” while a single element of an array ( if it is the only element that is activated ), cannot produce a focused beam , an array of activated elements can produce focused beam , where the size of the “ focus ” is smaller when transducer elements have larger emitting surface areas . each transducer element contributes to the focus proportionally to the value of its directivity at the “ focus .” thus , the steering region of a phased array transducer is dependent on each element &# 39 ; s directivity patterns . by way of further illustration , fig5 depicts the relationship between a transducer element surface size and its steering ability , represented in terms of its half - energy angle . for purposes of illustration , there are two curves shown in fig5 . there are two curves shown in fig5 ; one ( labeled “ 1 el direct ”) is a simplified analytical result of the element &# 39 ; s directivity ; the other ( labeled “ real calc ”) is a numerical simulation for a phased array transducer . as can be seen , for a transducer element having a size d / λ , where λ is the wavelength of the wave emitted by the element , the half energy steering angle , or “ steering ability ,” of the transducer array with d / λ = 1 is 30 °, which is the angle at which a steered - to focal zone has an energy level equal to half the maximum energy that the transducer would contribute to a non - steered focal zone . in order to better illustrate the relationship between the electronic steering angle and formation of hot spot ( s ), consider a one - dimensional array ( i . e ., row ) of transducer elements having a cross sectional dimension ( i . e ., element surface size ) of d / λ = 1 . if δφ is a phase difference between neighboring elements of the array , maximum energy emission occurs at angles satisfying the relationship : sin ( α )=( n + δφ / 2π ) λ / d , where λ is an ultrasound wavelength , integer n = 0 for the main focus and n ≠ 0 for hot spots . thus , where d ≦ λ / 2 , no hot spots will be formed . as such , the advantages of the embodiments described below particularly apply where the element size is equal to or greater than one - half of the drive signal wavelength . the electronic steering ability of a transducer device can be defined by : i s ≡ energy ⁢ ⁢ at ⁢ ⁢ main ⁢ ⁢ focus all ⁢ ⁢ emmited ⁢ ⁢ energy . i d = ( sin ⁡ ( π ⁢ ⁢ d ⁢ ⁢ sin ⁡ ( α ) / λ ) π ⁢ ⁢ d ⁢ ⁢ sin ⁡ ( α ) / λ ) 2 . fig6 shows a comparison of this directivity formula to energy flux calculation in fourier space for two different ultrasound wavelengths , 1 λ and 3 λ . as a result of the hot spot generation , large steering angles cannot be practically used , since nearly all of the energy that does not go to the steered - to focal zone is concentrated at hot spots . as can be seen in fig7 , for d = λ , while attempting to steer to 30 °, hot spots are produced at − 30 ° of equal intensity as the main focus , reducing the steering ability that can be safely used to about half of the main focus steering ability . it will be appreciated by those skilled in the art , that as the steering angle amplitude ( absolute value ) increases , hot spots begin to appear at numerous different points , and are both uncontrollable and undesirable . in accordance with a general aspect of the invention , a high density , two - dimensional transducer array is formed using transducer elements having irregular shaped energy transmitting surfaces . in various embodiments , the transducer element surface shapes may have rectilinear or curve - linear profiles , or a combination of both , and may include many different types of “ irregular ” shapes . by way of example , a multi - element transducer array 200 constructed according to one embodiment of the invention is shown in fig8 . in particular , the multi - element array 200 comprises irregular shaped transducer elements 202 having at least five different element shapes , including an l - shape 202 a , a rectangular ( or “ i ”) shape 202 b , a square shape 202 c , a t - shape 202 d , and an s - shape 202 f , respectively , which are mounted to a substrate 204 in an interlocking ( or mating ) configuration , resembling a “ tetris ” game formation . the array 200 may be constructed , by way of example and not limitation , using a conventional dicing machine , but making much smaller cuts to create a uniform array of piezoelectric rods in the same formation as shown in fig3 . however , the individual rods are then coupled to a same electronic drive signal in order to form the irregular shaped elements 202 of the array 200 . for example , as shown in fig8 a , an l - shape element 202 a may comprise three adjacent and aligned square rods 206 a - c , along with a forth rod 206 d located adjacent to the third rod 206 c and orthogonal to the alignment of rods 206 a - c . similarly , as shown in fig8 b , an s - shape element 202 e may be formed by electrically coupling four square rods 208 a - d in an s - shape formation . it should be appreciated , however , that the transducer elements 202 may also be formed by one - piece piezoelectric elements , instead of a mosaic arrangement of smaller component elements . while the array 200 of fig8 has a generally planer configuration , it is possible for such irregular shaped transducer arrays to employ other configurations . for example , a spherical cap transducer array 220 having irregular (“ tetris ”) shaped elements 222 is shown in fig9 . moreover , the “ tetris ” shapes of fig8 and 9 are but two examples of transducer arrays having non - uniform transducer elements . multi - element arrays 240 and 230 having more complex , pseudo random shaped elements 242 and 232 , respectively , are shown in fig1 a and 10b . notably , the elements 242 and 232 have curve - linear profiles instead of the rectilinear profiles of elements 202 and 222 of arrays 200 and 220 , respectively . in the various embodiments contemplated by the invention , the irregular shaped elements of a transducer array preferably have the same or similar total surface areas , e . g ., with each element being formed by connecting a same number of smaller piezoelectric elements in differing patterns . however , it is acceptable that some elements of the transducer array have differing surface areas , ( such as elements 210 and 212 in array 200 , which may be formed out of five , instead of four , square rods ). although the surface areas of the respective elements of an array may differ , they will still respond to the same frequency , while possibly producing slightly different amplitudes ( depending on whether any compensation is made to the drive signal current , which is spread over a lesser or greater surface area ). further , it will be appreciated that the elimination of hot spots in embodiments of the present invention is due to the non - uniform locations of the geometric centers of the respective transducer elements 202 . in particular , the geometrical “ centers ” of neighboring elements 202 of the array are randomly shifted by the length of the element comparable with its size ( e . g ., anywhere from a ratio of 1 / 4 to a ratio of 1 / 1 ). in contrast , in a conventional “ uniform ” transducer arrays the geometric centers of the elements are “ ordered ” along straight lines or circles . in a therapeutic focused ultrasound system , it is desirable to deliver maximal energy delivery to the focal zone and minimal energy to any other locations . this is normally achieved by maximal transducer area coverage by emitting elements . thus , in embodiments of the present invention , “ random order ” may be obtained while retaining full area coverage by emitting elements by using “ irregular shaped ” and / or “ irregular oriented ” transducer elements . while this results is some “ smearing ” of the acoustic waves and , thus , some ( relatively small ) corresponding loss of energy intensity at the focal zone , the steering capability of the respective array greatly exceeds that of a conventional uniform - shaped transducer array of otherwise similar element size . in particular , the appearance of hot spots is greatly decreased by the element disorder of an irregular shaped array , while the main focal zone has only a minor power degradation . further , the element size may be much larger than the acoustic wavelengths that are used . many different irregular shape patterns are possible within the ambit of the invention , with the particular element pattern realized on the substrate specifically chosen depending on the particular steering angles to be achieved without the formation of prohibitive hot spots . because of the disorder in the waves caused by the irregular element shapes and positions , only the main focus survives . the use of an irregular shaped array for improved electronic steering when delivering therapeutic levels of ultrasound energy is fundamentally different from a ( known ) “ parsing ” technique used for ultrasound imaging , in which gaps ( uniform or varied ) between elements are used to increase steering performance . while a similar effect may be achieved by irregular spacing of the elements , this would result in large gaps and unacceptable losses in output power in the case of a therapeutic transducer application . further , the gaps between elements causes more spreading in space of the acoustic energy for the same focal zone power delivery , which in turn may cause excessive heating of the surrounding tissue . the effect of the reduction in hot spots and the benefit of using the irregular shaped transducer array 200 of fig8 versus the uniform - shaped transducer array 130 of fig3 may be demonstrated with reference to fig1 a - b , 12 a - b , 13 a - b , 14 a - b and 15 a - b . in particular , images generated by 3 - d simulations of respective converging acoustic energy beams transmitted from a multi - element transducer having uniform - shaped elements is shown in fig1 a , 12 a , 13 a , 14 a and 15 a , and images generated by 3 - d simulations of respective converging acoustic energy beams transmitted from a multi - element transducer having irregular shaped elements are shown in corresponding fig1 b , 12 b , 13 b , 14 b and 15 b ). each of the pictures represents an acoustic pressure field obtained by the respective transducer array at the same conditions . each field was obtained by phasing the transducer to obtain a maximum power at the desired focus position measure in x , y , x distances from the transducer array . fig1 a and 11b are the respective acoustic pressure fields at the x , y , z coordinates of 0 , 0 , 30 mm ( i . e ., taken along the z - axis with zero electronic steering ). as can be observed , the pressure fields are very similar , with the field in 11 b ( from the irregular shaped elements ) being a slightly “ reduced amplitude ” version of the field generated by the uniform - shaped elements shown in fig1 a . fig1 a and 12b show the respective x - y cross - section of the fields shown in fig1 a and 11b , for z = 25 mm . fig1 a and 13b are the respective acoustic pressure fields at the x , y , z coordinates of 11 mm , 0 , 30 mm , i . e ., with an electronic steering angle out of the x - pane of 20 . 1 °. as can be seen , the field generated by the uniform - shaped elements ( fig1 a ) has a significant hot spot reduction formed at 20 . 1 ° (− 11 mm , 0 , 30 mm ) in the x - plane . fig1 a and 14b show the respective x - y cross - sections of the fields shown in fig1 a and 13b , for z = 23 mm . fig1 a and 15b are taken at respective x - y cross - sections ( for z = 21 mm ) of acoustic pressure energy fields having x , y , z coordinates of 11 mm , 11 mm , 30 mm , i . e ., with an out - of - plane steering angle of 27 . 4 ° in both the x and y directions . as can be seen in fig1 a , electronic steering to this focal zone using a conventional , uniform - shaped transducer array results in hot spots at each of the “ mirror - image ” locations , i . e ., at coordinates (− 11 mm , 11 mm , 30 mm ), ( 11 mm , − 11 mm , 30 mm ), and (− 11 mm , − 11 mm , 30 mm ). on the other hand , as can be seen in fig1 b , by using a transducer array having irregular shaped elements , there are no distinct hot spots such as seen in fig1 a , even though there is an energy increase throughout the cross - sectional plane . in some embodiments , before the respective transducer array is activated to deliver treatment - level ultrasound energy , an acoustic wave simulation can be performed to determine if any hot spots will be generated . for example , a computer model of the transducer may be created to model the configuration ( e . g ., shape , size , and relative position ) of the transducer elements . various operational parameters ( such as operation frequencies , amplitudes , and operation phases for the respective transducer elements ) can then be applied to the computer model to determine if a hot spot will result from a certain operational condition . as will be appreciated by those skilled in the art , while all transducer elements of an array may be activated in some instances , e . g ., in order to maximize an amount of energy delivered to a steered - to focal zone , in other instances , sufficient therapeutic energy may be delivered without activating all elements of the array . also , while the above embodiments have been described with reference to creating a single focal zone , in other embodiments , the same or similar methods can be used to create a plurality of simultaneous focal zones , thereby allowing simultaneous treatment of multiple target tissue regions . thus , although particular embodiments of the invention have been shown and described , it should be understood that the above discussion is not intended to limit the invention to these illustrated and described embodiments , which are provided for purposes of example only . instead , the invention is defined and limited only in accordance with the following claims .