Abstract:
A method for a medical examination is provided. The method includes acquiring at least two datasets that include data acquired at a plurality of points that lie along at least two lines through a center of k-space, reconstructing the at least two datasets to generate an image, and outputting the image.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of prior application Ser. No. 10/613,580, filed Jul. 2, 2003, now U.S. Pat. No. 7,603,156 which is hereby incorporated in its entirety by reference. 
    
    
     BACKGROUND OF THE INVENTION 
     This invention relates generally to medical imaging systems, and more particularly to systems and methods for polar phase encoding for magnetic resonance imaging (MRI). 
     When a substance such as human tissue is subjected to a uniform magnetic field (polarizing field B o ), the individual magnetic moments of the spins in the tissue attempt to align with this polarizing field, but precess about it in random order at their characteristic Larmor frequency. If the substance, or tissue, is subjected to a magnetic field (excitation field B 1 ) which is in the x-y plane and which is near the Larmor frequency, the net aligned moment, M z , may be rotated, or “dipped”, into the x-y plane to produce a net transverse magnetic moment M t . A signal is emitted by the excited spins after the excitation signal B 1  is terminated, this signal may be received and processed to form an image. 
     When utilizing these signals to produce images, magnetic field gradients (G x , G y  and G z ) are employed. Typically, the region to be imaged is scanned by a sequence of measurement cycles in which these gradients vary according to the particular localization method being used. The resulting set of received nuclear magnetic resonance (NMR) signals are digitized and processed to reconstruct the image using one of many well known reconstruction techniques. 
     A variant of the well known Fourier transform (FT) imaging technique is frequently referred to as “spin-warp”. The spin-warp technique is discussed in an article entitled “Spin-Warp NMR Imaging and Applications to Human Whole-Body Imaging” by W. A. Edelstein et al., Physics in Medicine and Biology, Vol. 25, pp. 751-756 (1980). It employs a variable amplitude phase encoding magnetic field gradient pulse prior to the acquisition of NMR spin-echo signals to phase encode spatial information in the direction of this gradient. In a 2-dimensional implementation (2DFT), for example, spatial information is encoded in one direction by applying a phase encoding gradient (G x ) along that direction, and then a spin-echo signal is acquired in the presence of a readout magnetic field gradient (G z ) in a direction orthogonal to the phase encoding direction. The readout gradient present during the spin-echo acquisition encodes spatial information in the orthogonal direction. In a typical 2DFT pulse sequence, the magnitude of the phase encoding gradient pulse G x  is incremented (ΔG x ) in the sequence of views that are acquired during the scan to produce a set of NMR data from which an entire image can be reconstructed. 
     In a 3-dimensional implementation of the spin-warp method phase encoding of the spin-echo signals is performed along two orthogonal axes. As described in U.S. Pat. No. 4,431,968 entitled “Method of 3-dimensional NMR Imaging Using Selective Excitation,” a thick slab of spins is excited by applying a slab-selection gradient (G y ) in the presence of a selective RF excitation pulse and then a first phase encoding gradient (G y ) along the same axis and a second phase encoding gradient (G x ) are applied before the NMR signal acquisition in the presence of a readout gradient (G z ). For each value of the G x  phase encoding gradient, the G y  phase encoding is stepped through all its values to sample a 3-dimensional region of k-space. By selectively exciting a slab, NMR signals are acquired from a controlled 3-dimensional volume. 
     BRIEF DESCRIPTION OF THE INVENTION 
     In one aspect, a method for a medical examination is described. The method includes polar phase encoding to generate a plurality of signals forming datasets representative of an object, where the datasets form a grid in polar coordinates in a k-space. 
     In another aspect, a magnetic resonance (MR) method for medical examinations is described. The MR method includes injecting a patient with a contrast agent that flows into a vasculature of the patient, acquiring MR signals produced by spins in the vasculature from an MR imaging system, and polar phase encoding to generate the MR signals forming datasets representative of the patient, wherein the datasets form a grid in polar coordinates in a k-space. 
     In yet another aspect, a method for a medical examination is described. The method includes sampling datasets on to a grid of polar coordinates in a k-space to generate signals representative of an object of interest that is being medically examined. 
     In still another aspect, a magnetic resonance imaging (MRI) system is described. The MRI system includes a main magnet to generate a uniform magnetic field, a radio frequency pulse generator for exciting the magnetic field, a gradient field generator for generating gradients extending in different directions in the magnetic field, a receiver for receiving magnetic field magnetic resonance (MR) signals representative of an object, and a controller for polar phase encoding to generate the MR signals forming datasets representative of the object, where the datasets form a grid in polar coordinates in a k-space. 
     In another aspect, a controller is described. The controller is programmed to polar phase encode to generate a plurality of MR signals forming datasets representative of an object, where the datasets form a grid in polar coordinates in a k-space. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram of an embodiment of a magnetic resonance imaging (MRI) system in which the herein described systems and methods for polar phase encode placement are implemented. 
         FIG. 2  is a block diagram of a transceiver that forms a part of the MRI system of  FIG. 1 . 
         FIG. 3  is a graphic representation of a pulse sequence employed in the MRI system of  FIG. 1  to practice an embodiment of a method for polar phase encode placement. 
         FIG. 4  illustrates an embodiment of a method for polar phase encode placement with phase encoding in k x  and k y  and frequency encoding in k z . 
         FIG. 5  illustrates an alternative embodiment of a method for polar phase encode placement illustrated as a 2D cross section of  FIG. 4  with phase encodings falling on a polar grid &lt;a,b,n&gt; in polar coordinates of k-space and phase encode locations &lt;m,d&gt; on the grid. 
         FIG. 6  illustrates yet another alternative embodiment of a method for polar phase encode placement in the k-space for reconstruction as 2-dimensional (2D) projection images with a high temporal resolution. 
         FIG. 7  illustrates yet another alternative embodiment of a method for polar phase encode placement in the k-space for reconstruction as 3-dimensional projection reconstruction (3D PR) images with low temporal resolution. 
         FIG. 8  illustrates an embodiment of a method for polar phase encode placement. 
         FIGS. 9 ,  10 ,  11 , and  12  illustrate another alternative embodiment of a method for polar phase encode placement showing flexible k-space sub-sampling for temporal sliding window reconstruction. 
         FIG. 13  illustrates another alternative embodiment of a method for generalized polar phase encode placement illustrated as a 2D cross section with phase encodings falling a generalized polar grid &lt;a,b,r,n&gt; in generalized polar coordinates of k-space and phase encode locations &lt;m,j,d&gt; on the generalized grid. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       FIG. 1  is a block diagram of an embodiment of a magnetic resonance imaging (MRI) system in which the herein described systems and methods for polar phase encode placement are implemented. The operation of the system is controlled from an operator console  100  which includes a keyboard and control panel  102  and a display  104 . Console  100  communicates through a link  116  with a separate computer system  107  that enables an operator to control the production and display of images on the screen  104 . Computer system  107  includes a number of modules which communicate with each other through a backplane. These include an image processor module  106 , a central processing unit (CPU) module  108  and a memory module  113 , known in the art as a frame buffer for storing image data arrays. Computer system  107  is linked to a disk storage  111  and a tape drive  112  for storage of image data and programs, and it communicates with a separate system control  122  through a high speed serial link  115 . 
     System control  122  includes a set of modules connected together by a backplane  118 . These include a CPU module  119  and a pulse generator module  121  which connects to operator console  100  through a serial link  125 . It is through link  125  that system control  122  receives commands from the operator which indicate the scan sequence that is to be performed. Pulse generator module  121  operates the system components to carry out the desired scan sequence. It produces data which indicates the timing, strength and shape of the radio frequency (RF) pulses which are to be produced, and the timing of and length of the data acquisition window. Pulse generator module  121  connects to a set of gradient amplifiers  127 , to indicate the timing and shape of the gradient pulses to be produced during the scan. Pulse generator module  121  also receives patient data from a physiological acquisition controller  129  that receives signals from a number of different sensors connected to a patient  135 , such as electrocardiogram (ECG) signals from electrodes or respiratory signals from a bellows. And finally, pulse generator module  121  connects to a scan room interface circuit  133  which receives signals from various sensors associated with the condition of patient  135  and a magnet assembly  141 . It is also through scan room interface circuit  133  that a patient positioning system  134  receives commands to move patient  135  to the desired position for the scan. It is noted that an object, such as a phantom, can be used instead of patient  135 . 
     The gradient waveforms produced by pulse generator module  121  are applied to a gradient amplifier system  127  comprised of G x , G y  and G z  amplifiers. Each gradient amplifier excites a corresponding gradient coil in an assembly generally designated  139  to produce the magnetic field gradients used for position encoding acquired signals. Gradient coil assembly  139  forms part of magnet assembly  141  which includes a polarizing magnet  140  and a whole-body RF coil  152 . A transceiver module  150  in system control  122  produces pulses which are amplified by an RF amplifier  151  and coupled to RF coil  152  by a transmit/receive switch  154 . The resulting signals radiated by the excited nuclei in patient  135  may be sensed by RF coil  152  and coupled through the transmit/receive switch  154  to a preamplifier  153 . The amplified NMR signals are demodulated, filtered, and digitized in the receiver section of transceiver  150 . Transmit/receive switch  154  is controlled by a signal from pulse generator module  121  to electrically connect RF amplifier  151  to the coil  152  during the transmit mode and to connect preamplifier  153  during the receive mode. Transmit/receive switch  154  also enables a separate RF coil (for example, a head coil or surface coil) to be used in either the transmit or receive mode. 
     The NMR signals picked up by RF coil  152  are digitized by transceiver module  150  and transferred to a memory module  160  in system control  122 . When the scan is completed and an entire array of data has been acquired in memory module  160 , an array processor  161  operates to Fourier transform the data into an array of image data. This image data is conveyed through serial link  115  to computer system  107  where it is stored in disk memory  111 . In response to commands received from operator console  100 , this image data may be archived on tape drive  112 , or it may be further processed by image processor  106  and conveyed to operator console  100  and presented on display  104 . 
     Referring particularly to  FIGS. 1 and 2 , transceiver  150  produces the RF excitation field B 1  through power amplifier  151  at a coil  152 A and receives the resulting signal induced in a coil  152 B. As indicated above, coils  152 A and B may be separate as shown in  FIG. 2 , or they may be a single wholebody coil as shown in  FIG. 1 . The base, or carrier, frequency of the RF excitation field is produced under control of a frequency synthesizer  200  which receives a set of digital signals from CPU module  119  and pulse generator module  121 . These digital signals indicate the frequency and phase of the RF carrier signal produced at an output  201 . The commanded RF carrier is applied to a modulator and up converter  202  where its amplitude is modulated in response to a signal R(t) also received from the pulse generator module  121 . The signal R(t) defines the envelope of the RF excitation pulse to be produced and is produced in module  121  by sequentially reading out a series of stored digital values. These stored digital values may, in turn, be changed from operator console  100  to enable any desired RF pulse envelope to be produced. 
     The magnitude of the RF excitation pulse produced at output  205  is attenuated by an exciter attenuator circuit  206  which receives a digital command from backplane  118 . The attenuated RF excitation pulses are applied to power amplifier  151  that drives RF coil  152 A. 
     The signal produced by patient  135  is picked up by receiver coil  152 B and applied through preamplifier  153  to the input of a receiver attenuator  207 . Receiver attenuator  207  further amplifies the signal by an amount determined by a digital attenuation signal received from backplane  118 . 
     The received signal is at or around the Larmor frequency, and this high frequency signal is down converted in a two step process by a down converter  208  which first mixes the NMR signal with the carrier signal on line  201  and then mixes the resulting difference signal with the 2.5 MHz reference signal on line  204 . The down converted NMR signal is applied via line  212  to the input of an analog-to-digital (A/D) converter  209  which samples and digitizes the analog signal and applies it to a digital detector and signal processor  210  which produces 16-bit in-phase (I) values and 16-bit quadrature (Q) values corresponding to the received signal. The resulting stream of digitized I and Q values of the received signal are output through backplane  118  to the memory module  160  where they are employed to reconstruct an image. 
     The 2.5 MHz reference signal as well as the 250 kHz sampling signal and the 5, 10 and 60 MHz reference signals are produced by a reference frequency generator  203  from a common 20 MHz master clock signal. 
     Although the systems and methods for polar phase encode placement can be used with a number of different pulse sequences, an embodiment of the invention employs a 3-dimensional (3D) gradient recalled echo pulse sequence depicted in  FIG. 3 . 
     Referring particularly to  FIG. 3 , an RF excitation pulse  220  is produced to produce transverse magnetization in a 3D volume. RF excitation pulse  220  is followed by a phase encoding gradient pulse  224  directed along the x axis and a phase encoding gradient pulse  226  directed along the y axis. A readout pulse  228  directed along the z axis follows and an echo NMR signal  230  is acquired and digitized as described above. As used herein, echo refers to any one of a partial echo and a full echo. After the acquisition, rewinder gradient pulses  232  and  234  rephase the magnetization before the pulse sequence is repeated. 
     The pulse sequence is repeated and phase encoding pulses  224  and  226  are stepped through a series of values to sample the 3D k-space. In an embodiment, 256×128 phase encodings are employed by acquiring 256 phase encodes at a projection angle, acquiring 256 phase encodes at another projection angle, and repeating the acquiring for 128 projection angles. As will become apparent from the discussion below, the order in which this k-space sampling is performed is an important aspect of the systems and methods for polar phase encode placement. 
     One aspect of an embodiment of a system and method for polar phase encode placement is that datasets are sampled on to a grid of polar coordinates. Another aspect of an embodiment of a system and method for polar phase encode placement is the location in which k-space is sampled using the pulse sequence in  FIG. 3 . The sampling of k-space is performed by stepping the magnitude of x and y phase encoding pulses  224  and  226  respectively through a sequence of values and the order in which this is done determines how k-space is sampled during the scan. 
       FIGS. 4 and 5  illustrate an embodiment of a method for polar phase encode placement in the k-space, having the k x  axis, the k y  axis, and the k z  axis. The method is executed by computer  12 . Datasets having data  64 ,  66 ,  68 ,  70 , and  72  are collected by phase encoding in x and y directions and by frequency encoding in z direction. For example, information produced by spins of nuclei in a leg of patient  135  is collected by phase encoding in right/left (r/l) and anterior/posterior (a/p) directions, and frequency encoding in sagital/inverse sagital (S/I) direction. In all embodiments described herein, datasets are frequency encoded in z direction. Therefore,  FIGS. 4-12  display only a 2-dimensional (2D) cross section of a 3D method of k-space encoding and sampling. The method includes phase encoding on to a grid  76  of polar coordinates  78  in the k-space to generate MR signals that are representative of patient  135  placed under an examination. 
     Each datum in a dataset that is polar phase encoded is a sample from a location m(a cos(2πd/n)k x +b sin(2πd/n)k y )+ick z , in k-space where 
     a, b, c, and d are real numbers, 
     m, n, and i are an integers, 
     k x , k y , and k z  are unit basis vectors for the k-space, 
     &lt;a, b, c&gt; determines a specific polar grid, and 
     &lt;m,d,i&gt; determines a point on the grid where m is a radial parameter of a phase encode, d is a rotational parameter of a phase encode, and i is a parameter of a frequency encode. 
     Grid  76  is represented as &lt;a,b,c,n&gt;, a datum on the grid is represented as &lt;m,d,i&gt;, and a grid and a datum on the grid together specify a point in the k-space. In an alternative embodiment, each datum in a dataset that is sampled using the method is represented as (m+0.5)(a cos(2πd/n)k x +b sin(2πd/n)k y )+(i+0.5)ck z . Although grid  76  is shown to be elliptical in shape, examples of other shapes of grid include a circular-shaped grid. 
     In an embodiment, the method uses a polar grid &lt;a,b,c,n&gt; and frequency encodes datasets n 1  times along k z  axis by keeping m and d constant, and varying i. An example of n 1  is 256. For every n 1  number of times of frequency encoding, the method includes phase encoding radially once by keeping d constant and varying m. The method includes phase encoding radially for n 2  number of times and for each time of radial phase encoding, the frequency encoding is performed for n 1  times. An example of n 2  is 256. For every n 2  number of times of phase encoding radially, the method includes phase encoding rotationally once by varying d. The method includes phase encoding rotationally for n 3  number of times and for each time of rotational phase encoding, the radial phase encoding is performed for n 2  times. An example of n 3  is 128. In essence, the method forms a nested loop of frequency encoding within radial phase encoding within rotational phase encoding. 
     In an alternative embodiment, the method includes forming a nested loop of frequency encoding within rotational phase encoding within radial phase encoding. It is noted that the methods that are described using  FIGS. 4 and 5  and the following  FIGS. 6-12  are implemented in a pure polar coordinate system and methods that are described using  FIG. 13  are implemented in an extended polar coordinate system. Hence, “polar coordinate system” in general refers to either a pure polar coordinate system or an extended polar coordinate system based on whether the methods illustrated by  FIGS. 4-13  are being described or whether the methods illustrated by  FIG. 13  are being described. 
       FIG. 6  illustrates an alternative embodiment of a method for polar phase encode placement in the k-space. The method includes polar phase encoding to create datasets having data  64 ,  66 ,  68 ,  70 ,  72 ,  84 ,  86 ,  88 , and  90  on a plane  82 . The direction in which the datasets are polar phase encoded is shown by an arrow. The method further includes constructing a 2D image from datasets located on plane  82 . In one embodiment, a 2D image that corresponds to datasets located on plane  82  is constructed by performing a 2D inverse Fourier transformation, such as a 2D fast Fourier transformation (FFT), of the datasets. The 2D inverse Fourier transformation is combined with re-gridding the datasets on to a grid of cartesian coordinates. 
       FIG. 7  illustrates yet another alternative embodiment of a method for polar phase encode placement in the k-space. The method includes polar phase encoding on to planes  82 ,  94 ,  96 , and  98 . For example, plane  82  is formed of 256 phase encodes at a projection angle, plane  94  is formed of 256 phase encodes at another projection angle, and so on until planes  96  and  98  are formed. The direction in which the datasets are polar phase encoded on to planes  82 ,  94 ,  96 , and  98  is shown by arrows. The method further includes constructing either a 3D image or a time course series of 2D images from datasets located on planes  82 ,  94 ,  96 , and  98 . In one embodiment, a time course series of 2D images that corresponds to datasets located on planes  82 ,  94 ,  96 , and  98  is constructed by performing a 2D inverse Fourier transformation of the datasets. For example, a first 2D image that corresponds to datasets located on plane  82  is constructed by performing a 2D inverse Fourier transformation of the datasets. A second 2D image that corresponds to datasets located on plane  94  is constructed by performing a 2D inverse Fourier transformation of the datasets. The 2D inverse Fourier transform is repeated for remaining planes  96  and  98  to form remaining 2D images. 
     A 3D image can be reconstructed from a 3D data set including datasets located on planes  82 ,  94 ,  96 , and  98 . In an embodiment, the 3D image can be reconstructed by performing an inverse Fourier transformation in k z  direction combined with re-gridding, and performing a 2D inverse Fourier transformation in k x  and k y  directions. In an alternative embodiment, the 3D image can be reconstructed by performing an inverse Fourier transformation in k z  direction, and performing backprojection in k x  and k y  directions. 
     It is noted that datasets can be sampled on to planes  82 ,  94 ,  96 , and  98  by a variety of methods including simple phase encoding as described above, Echo-planar imaging (EPI), and spiral imaging to generate the MR signals representative of patient  135 . EPI and spiral imaging are faster sampling methods than simple phase encoding. In addition, different orderings of phase encoding can be used such as centric phase encoding and interleaved phase encoding. It is also noted that datasets can be sampled on to more than four planes  82 ,  94 ,  96 , and  98 . For example, datasets are sampled on to 128 planes, where each plane has a different projection angle. 
     In yet another alternative embodiment, referred to as a “temporal sliding window”, the method includes polar phase encoding on to planes  82 ,  94 ,  96 , and  98 , and constructing a 3D image that corresponds to the datasets. The method then includes polar phase encoding on to plane  82  thereby replacing prior datasets on plane  82 , and reconstructing a 3D image from the datasets located on planes  82 ,  94 ,  96 , and  98 . Thereafter, the method includes polar phase encoding on to plane  94  thereby replacing prior datasets on plane  94 , and reconstructing a 3D image that corresponds to datasets located on planes  82 ,  94 ,  96 , and  98 . The method continues polar phase encoding on to one of planes  82 ,  94 ,  96 , and  98  and to reconstruct a 3D image corresponding to the datasets located on planes  82 ,  94 ,  96 , and  98 . The method provides a medium temporal resolution of a peripheral region, such as a peripheral region  95  of plane  96 . For example, taking 4 seconds, with each second corresponding to each plane  82 ,  94 ,  96 , and  98 , to polar phase encode datasets on to planes  82 ,  94 ,  96 , and  98  and to construct an image corresponding to the datasets is a medium temporal resolution. 
     Central region  97  is usually more interesting to medical personnel than peripheral region, such as peripheral region  95  of plane  96 . An example of a high temporal resolution of central region  97  is when central region  97  is updated every second. The updating includes polar phase encoding on to one of planes  82 ,  94 ,  96 , and  98  and to construct an image corresponding to the datasets located on planes  82 ,  94 ,  96 , and  98 . In an alternative embodiment, instead of updating datasets located on any one of planes  82 ,  94 ,  96 , and  98 , datasets located on different kinds of regions shown in  FIGS. 8 ,  10 , and  11  below are updated. 
       FIG. 8  illustrates an embodiment of a method for polar phase encode placement in the k-space. The method includes polar phase encoding on to a first set of planes  250 ,  252 , and  254 . For example, plane  250  is formed of 256 phase encodes at a projection angle, plane  252  is formed of 256 phase encodes at another projection angle of, for example, 60 degrees from plane  250 , and plane  254  is formed of 256 phase encodes at yet another projection angle of, for example, 60 degrees from plane  252 . Polar phase encoding on to plane  250  creates a dataset having data  260 ,  262 ,  264 ,  266 ,  268 ,  270 ,  272 , and  274 , polar phase encoding on to plane  252  creates a dataset having data  276 ,  278 ,  280 ,  282 ,  284 ,  286 ,  288 , and  290 , and polar phase encoding on to plane  254  creates a dataset having data  292 ,  293 ,  294 ,  295 ,  296 ,  297 ,  298 , and  299 . The direction in which the datasets are polar phase encoded on to planes  250 ,  252 , and  254  is shown by arrows. The method further includes rotating in a counterclockwise direction from planes  250 ,  252 , and  254  by an amount, such as, for example, 5 degrees, and polar phase encoding on to a second set of planes. The method also includes rotating in the counterclockwise direction from the second set of planes by the same amount, such as, for example, 5 degrees, and polar phase encoding on to a third set of planes. The method includes rotating in such as manner to rotate for a total amount of 360 degrees. It is noted that alternatively, the method includes rotating in a clockwise direction instead of the counterclockwise direction. 
       FIG. 9  illustrates still another alternative embodiment of a method for polar phase encode placement in the k-space. The method includes polar phase encoding on to a wedge-shaped region  302 . 
       FIGS. 10 and 11  illustrate another alternative embodiment of a method for polar phase encode placement in the k-space. The method includes polar phase encoding on to a region  312  formed by an intersection of cylinder  304  and wedge-shaped region  302 . In an alternative embodiment, the method includes polar phase encoding on to a region formed by a union of cylinder  304  and wedge-shaped region  302 . It should be noted that the k-space can have a higher or a lower number of concentric cylinders than two cylinders  304  and  308 . 
       FIG. 12  illustrates still another alternative embodiment of a method for polar phase encode placement in the k-space. The method includes polar phase encoding on to a region  314  formed by an intersection of a region  308  between cylinders  304  and  308  and wedge-shaped region  302 . In an alternative embodiment, the method include polar phase encoding on to a region formed by a union of region  314  and wedge-shaped region  302 . 
       FIG. 13  illustrates another embodiment of a method for polar phase encode placement in the k-space. The method extends the method of  FIG. 7  from plane  82  parallel to k z  axis to a set of parallel planes  420 ,  422 ,  82 ,  426 , and  428  parallel to k z  axis. The method includes sampling datasets on to each plane  420 ,  422 ,  82 ,  426 , and  428  in the k-space. Direction in which datasets are sampled on to planes  420 ,  422 ,  82 ,  426 , and  428  is shown by direction of arrows. Planes  420 ,  422 ,  82 ,  426 , and  428  form a group  418  or a slab  418 . Each plane  420 ,  422 ,  82 ,  426 , and  428  encompass a finite region in the k-space. For example, each plane  420 ,  422 ,  82 ,  426 , and  428  is of a shape of a rectangle. As another example, each  420 ,  422 ,  82 ,  426 , and  428  is of a shape of a square. The shape of a plane corresponds to datasets sampled on to the plane. Moreover, the method of  FIG. 13  is implemented in an extended polar co-ordinate system as compared to a pure polar co-ordinate system that is used to implement the methods described in  FIGS. 4-12 . 
     Each datum in a dataset that is generalized polar phase encoded is a sample from a location m(a cos(2πd/n)k x +b sin(2πd/n)k y )+jr(a cos(2πd/n)k x +b sin(2πd/n)k y )+ick z , in k-space where 
     a, b, c, d, and r are real numbers, 
     m, j, n, and i are an integers, 
     k x , k y , and k z  are unit basis vectors for the k-space, 
     &lt;a, b, c, r&gt; determines a specific generalized polar grid, and 
     &lt;m,d,i,j&gt; determines a point on the generalized grid where m is a radial parameter of a phase encode, d is a rotational parameter of a phase encode, j is a translational parameter of a phase encode and i is a parameter of a frequency encode. 
     A generalized grid is represented as &lt;a,b,c,r,n&gt;, a datum on the generalized grid is represented as &lt;m,d,i,j&gt;, and a grid and a datum on the grid together specify a point in the k-space. As an example, datasets having data  429  and  431 , both of which are sampled on to plane  420  are represented as m(a cos(2πd/n)k x +b sin(2πd/n)k y )+jr(a cos(2πd/n)k x +b sin(2πd/n)k y ). 
     In an embodiment, the method includes frequency encoding datasets m 1  times along k z  axis by keeping m, a, d, n, b, j, r, and c constant, and varying i. An example of m 1  is 256. For every m 1  number of times of frequency encoding, the method includes phase encoding radially once by keeping a, d, n, b, j, r, and c constant and varying m. The method includes phase encoding radially for m 2  number of times and for each time of radial phase encoding, frequency encoding is performed for m 1  times. An example of m 2  is 256. For every m 2  number of times of radial phase encoding, the method includes phase encoding translationally once by keeping a, d, n, b, r, and c constant and varying j. The method includes phase encoding translationally for m 3  number of times and for each time of translational phase encoding, radial phase encoding is performed for m 2  times. An example of m 3  is 6. For every m 3  number of times of translational phase encoding, the method includes phase encoding rotationally once by keeping a, n, b, r, and c constant and varying d. The method includes phase encoding rotationally for m 4  number of times and for each time of rotational phase encoding, translational phase encoding is performed for m 3  times. An example of m 4  is 31. In essence, the method forms a nested loop of frequency encoding within radial phase encoding within translational phase encoding within rotational phase encoding. 
     Each group  434 ,  436 , and  438  is similar to group  418  since each group  434 ,  436 , and  438  is a stack of planes similar to planes  420 ,  422 ,  82 ,  426 , and  428 . Directions in which datasets are sampled on to groups  418 ,  434 ,  436 , and  438  is shown by arrows. There is an angle between any two groups. Angle  448  is an angle between group  418  and group  434 . An example of angle  448  is an angle between 1 degree and 359 degrees. Another example of angle  448  is 5 degrees. Another example of angle  448  is 15 degrees. 
     In an alternative embodiment, the method includes sampling datasets on to a first set of regions formed by intersection of planes  420 ,  422 ,  82 ,  426 , and  428  of group  418  with concentric cylinders  304  and  306  in the k-space. In yet another alternative embodiment, the method includes sampling datasets on to a second set of regions formed by intersection of groups  418 ,  434 ,  436 , and  438  with cylinders  304  and  306 . In still another alternative embodiment, the method includes sampling datasets on to a third set of regions formed by intersection of planes  420 ,  422 ,  82 ,  426 , and  428  with group  434 , the first set of regions, and the second set of regions. In another alternative embodiment, the method includes sampling datasets on to a fourth set of regions (not shown) formed by union of planes  420 ,  422 ,  82 ,  426 , and  428  with group  434 , the first set of regions, and the second set of regions. 
     It should be noted that datasets can be sampled on to a higher or a lower number of planes than six planes of group  418 . Similarly, the number of planes in remaining groups  434 ,  436 , and  438  can vary. It should also be noted that datasets can be sampled on to a higher or a lower number of groups of planes than four groups  418 ,  434 ,  436 , and  438 . 
     It should be noted that polar and generalized phase encoding can be combined with continuous and non-continuous moving table imaging and bolus tracking. 
     It should also be noted that number of planes within at least one of groups  418 ,  434 ,  436 , and  438  may change as a scan progresses and that number of groups, i.e., projection angles, may change as the scan progresses. These changes may be made to accommodate changing temporal dynamics or changes in a region of an object being scanned or changes in the object itself. 
     Order of construction of an image from datasets in each group  418 ,  434 ,  436 , and  438  depends on a desired temporal resolution, a desired spatial resolution in k x  direction, and a desired spatial resolution in k y  direction. The desired temporal resolution, desired spatial resolution in k x  direction, and desired spatial resolution in k y  direction depends on various factors including a type of imaging system used to scan patient  135 , whether a contrast agent is administered in patient  135 , and whether a size of a body part of patient  135  to be scanned is large or small. For example, if a contrast agent is administered into patient  135 , a high temporal resolution and a low or a medium spatial resolution are desired. In the example, a high temporal resolution and a low or a medium spatial resolution are desired since images of patient  135  should be obtained at or close to a moment when the contrast agent flows through an internal part, such as, for instance, an artery of patient  135 . As another example, if a body part, such as a leg, of a large size of patient  135  is to be scanned, a low temporal resolution and a high spatial resolution are desired. In the other example, a low temporal resolution and a high spatial resolution are desired since such a combination of resolutions enables a user to visualize a small internal part, such as a vein, within the large body part. In all embodiments of a method for polar phase encode placement, there is a high spatial resolution in k z  direction. As an example, a high spatial resolution in k z  direction is achieved by frequency encoding 256 datapoints in k z  direction. 
     In an embodiment, to provide a high temporal resolution, a high in-plane resolution and no thru-plane resolution, a 2D image that corresponds to datasets located on plane  420  of group  118  is constructed by re-gridding the datasets and performing a 2D inverse Fourier transformation, such as a 2D fast Fourier transformation (FFT), of the datasets. The 2D inverse Fourier transformation is combined with re-gridding the datasets on to a grid of cartesian coordinates. The re-gridding on to the grid of cartesian coordinates and the 2D inverse Fourier transform are performed after sampling datasets on to plane  420 . In an alternative embodiment, to provide a high temporal resolution, a high in-plane resolution, and no thru-plane resolution, a 2D image that corresponds to datasets located on plane  420  of group  118  is constructed by performing a 2D backprojection of the datasets combined with re-gridding the datasets on to grid  76  of polar coordinates  78 . An example of high temporal resolution is taking 1 second to sample datasets on to plane  420  and to construct an image corresponding to the datasets. 
     The length of each group  418 ,  434 ,  436 , and  438  corresponds to in-plane resolution, which is the desired spatial resolution in k y  direction, and the depth of each group  418 ,  434 ,  436 , and  438  corresponds to thru-plane resolution, which is the desired spatial resolution in k x  direction. For example, to obtain a high in-plane resolution, length  425  of group  418  corresponds to a dataset having 256 datapoints in k y  direction. As another example, to obtain a low in-plane resolution, length  425  of group  418  corresponds to a dataset having a datum in k y  direction. Medium in-plane resolution is a resolution between high and low in-plane resolutions. As yet another example, to obtain a low thru-plane resolution, depth  427  of group  418  corresponds to a dataset having a datum in k x  direction. As still another example, to obtain a high thru-plane resolution, depth  427  of group  418  corresponds to a dataset having 256 datapoints in k x  direction. Medium thru-plane resolution is a resolution between high and low thru-plane resolutions. As an example, to obtain a medium thru-plane resolution, depth  427  of group  418  corresponds to a dataset having 6 datapoints in k x  direction. 
     In yet another alternative embodiment, to provide a low temporal resolution, and a full 3D resolution, a 3-dimensional (3D) image that corresponds to datasets located on groups  418 ,  434 ,  436 , and  438  is constructed by performing an inverse Fourier transformation in k z  direction combined with re-gridding, and performing a 2D inverse Fourier transformation in k x  and k y  directions. The full 3D resolution is a high spatial resolution in k x  direction, a high spatial resolution in k y  direction, and a high spatial resolution in k z  direction. In still another alternative embodiment, to provide a low temporal resolution and the full 3D resolution, a 3D image that corresponds to datasets located on groups  418 ,  434 ,  436 , and  438  is reconstructed by performing an inverse Fourier transformation in k z  direction, and performing backprojection in k x  and k y  directions. There is a low temporal resolution since a 3D image is constructed after sampling datasets on to groups  418 ,  434 ,  436 , and  438 , and after reconstructing the datasets. For example, taking 36 seconds, with each second corresponding to each plane of each group  418 ,  434 ,  436 , and  438  to sample datasets on to groups  418 ,  434 ,  436 , and  438 , and to construct an image corresponding to the datasets is a low temporal resolution. 
     In another alternative embodiment, to provide a medium temporal resolution, a medium spatial resolution in k x  direction, and a high spatial resolution in k y  direction, a 2D image that corresponds to datasets located on group  434  is constructed by performing a 3D inverse Fourier transformation of the datasets, and performing a maximum intensity projection (MIP) of the datasets. The 3D inverse Fourier transform is performed after sampling datasets on to group  434 . In yet another alternative embodiment, to provide a medium temporal resolution, a medium spatial resolution in k x  direction, and a high spatial resolution in k y  direction, a 2D image that corresponds to datasets located on group  434  is constructed by performing a 3D backprojection of the datasets and performing an MIP of the datasets. The 3D backprojection is performed after sampling the datasets on to group  434 . The medium temporal resolution is a temporal resolution that is higher than the low temporal resolution and lower than the high temporal resolution. The medium temporal resolution is higher than the low temporal resolution since a 2D image of datasets located on more than a single plane is constructed. The medium temporal resolution is lower than the high temporal resolution since a 2D image is constructed of datasets located on a lower number of groups than four groups  418 ,  434 ,  436 , and  438 . The medium spatial resolution in k x  direction is a spatial resolution that is higher than the low spatial resolution in k x  direction and lower than the high spatial resolution in k x  direction. The medium spatial resolution in k x  direction is higher than the low spatial resolution in k x  direction since the MIP accounts for more than one datum in k x  direction, for instance, by averaging more than one datum in k x  direction or by taking a datum with maximum intensity from multiple datapoints in k x  direction. The medium spatial resolution in k x  direction is lower than the high spatial resolution in k x  direction since a shorter length of datapoints in k x  direction is used to construct a 2D image than a length of datapoints in k x  direction of groups  418 ,  434 ,  436 , and  438 . It is noted that unlike in a cartesian co-ordinate system, in the embodiments of a method for polar phase encode placement described above, spatial resolution is not dependent primarily on the number of planes on to which datasets are polar phase encoded but is dependent primarily on the number of datasets located on a plane. 
     In another alternative embodiment, also termed as a “temporal sliding window”, to provide a medium temporal resolution of a central region  446  formed by intersection of groups  418 ,  434 ,  436 , and  438 , the method includes sampling datasets on to groups  418 ,  434 ,  436 , and  438 , and constructing a 3D image that corresponds to the datasets. The method further includes sampling datasets on to group  418  and reconstructing a 3D image that corresponds to the datasets located on groups  418 ,  434 ,  436 , and  438 . The method also includes sampling datasets on to group  434  and reconstructing a 3D image that corresponds to datasets located on groups  418 ,  434 ,  436 , and  438 , and so on without following any particular order of sampling datasets on to one of groups  418 ,  434 ,  436 , and  438 . The method provides a low temporal resolution of a peripheral region, such as a peripheral region  444 , of any of groups  418 ,  434 ,  436 , and  438 . Although there is a low temporal resolution of a peripheral region, such as peripheral region  444 , there is medium temporal resolution of central region  446  and central region  446  is usually more interesting to medical personnel than a peripheral region. 
     It should be noted that in an alternative embodiment, once datasets are sampled, phase and frequency correction in datasets are obtained by periodically sampling on to a center line of the k-space. In another alternative embodiment, axial plane motion correction is obtained by periodically sampling datasets on to an inner cylinder that is surrounded by one or more cylinders in the k-space. In yet another alternative embodiment, Hermetian symmetry is used to infer full datasets from partially sampled datasets. In still another alternative embodiment, datasets are sampled on to every alternate plane of a group. For example, datasets are sampled on to planes of group before sampling datasets on to planes of group. In an alternative embodiment, multiple phase encodes are sampled in a single TR by EPI or gradient recall and spin echo (GRASE). 
     Thus, the herein described systems and methods for polar phase encode placement provide the desired temporal resolution and the desired spatial resolution in k x  direction, and the desired spatial resolution in k y  direction. The herein described systems and methods for polar encode placement provide the desired temporal resolution, the desired spatial resolution in k x  direction, and the desired spatial resolution in k y  direction by sampling datasets on to a grid of polar coordinates, inverse Fourier transformation, backprojection, and MIP. 
     While the invention has been described in terms of various specific embodiments, those skilled in the art will recognize that the invention can be practiced with modification within the spirit and scope of the claims.