Abstract:
Methods for performing dose determination and cost function gradients in a radiation therapy are disclosed, which include: discretizing a volume-of-interest (VOI) into a set of voxels; identifying a set of beamlets which deposit dose contributions of radiation to the VOI, and each beamlet has a weight factor; transforming the dose contributions into a first domain, and transforming the weight factors into a second domain orthogonal to the first domain; calculate the local derivatives of a cost function of dose and cost function gradients with respect to the weights of the beamlets.

Description:
RELATED APPLICATION 
       [0001]    This is a non-provisional application based upon U.S. provisional patent application Ser. No. 61/790,670, filed Mar. 15, 2013, which is incorporated herein by reference. 
     
    
     TECHNICAL FIELD 
       [0002]    The disclosure relates planning and delivery of radiation in radiation therapy. In particular, methods and systems are provided for determining dose distributions in a radiation therapy. 
       BACKGROUND 
       [0003]    Radiation therapy involves the transmission of radiation energy to a tumor site (or volume-of-interest) within the patient to control cell growth. Radiation therapy may be curative in a number of types of cancer if they are localized to one area of the body. 
         [0004]    Radiation therapy treatment planning may be carried out according to a forward planning technique or an inverse planning technique. Forward planning involves delivering an initial planned radiation dose and then delivering subsequent doses by observation or inference of the efficacy of the preceding dose in a trial-and-error manner. The determination of dose delivery by forward planning is therefore performed according to human observation and experience. Inverse planning instead seeks to calculate an optimized dose delivery and then work backwards to determine the appropriate radiation beam characteristics to deliver that optimized dose. Inverse planning of radiation therapy for tumors may be performed for Tomotherapy, or other Intensity Modulated Radiation Therapy (IMRT) radiation delivery techniques employing ionizing photon radiation or any other ionizing radiation like e.g. protons, heavy ions, or electrons. These techniques involve transmission of radiation beams, usually collimated by an appropriate collimation device like a multi-leaf collimator (MLC), toward the volume-of-interest (VOI) from various angular orientations. 
         [0005]    Radiation dose can damage or kill both cancerous and healthy tissue cells. A radiation beam originating from a radiation source and projection through a subject will deposit radiation dose along its path. It is typical that some healthy tissue will receive radiation dose during a radiation treatment. In order to ensure that the patient is optimally treated, it is necessary to predict and shape a dose distribution in treatment planning of radiation therapy. Dose is deposited energy by some treatment devices applying ionizing radiation to a patient or measurement device (or in other words, VOI), and being computed in a point or a number of points. The dose at any given position inside a VOI is composed of a weighted superposition of elementary doses. Elementary dose is dose deposited during a certain time interval and/or a certain configuration of the treatment device. 
         [0006]    Two quantities are of interest in treatment planning and form the basis of all planning algorithms: 
         [0007]    1. The dose to any point in a VOI; 
         [0008]    2. The derivative of a cost function of dose at any point with respect to its elementary constituents. 
         [0009]    Current techniques for evaluating these two quantities are cumbersome and time-consuming. There is a desire for improvement of systems and methods for determining the above two quantities quickly and accurately, which is important not only for designing good radiation treatment plans, but also for the successful implementation of further interactive adaptive treatment techniques. 
       BRIEF SUMMARY 
       [0010]    According to an aspect of the present disclosure, a method for performing dose determination in a radiation therapy, including: creating a set of voxels from a volume of interest in a physical object or system, or virtual model thereof; identifying a plurality of beamlets, each of said beamlets depositing a dose contribution of radiation to at least one subset of said voxels; transforming the dose contributions of all beamlets for each of said voxels into a first domain, yielding a set of transformed dose contributions for each of said voxels; assigning a weight to each of said beamlets to create a set of weights; transforming the set of weights into a second domain orthogonal to the first domain, yielding a set of transformed weights; and determining a dose for each of said voxels through summing up the elementwise product of the set of transformed dose contributions and the set of transformed weights. The first domain and second domain are the same, and the first domain is a sparse domain. 
         [0011]    According to a further aspect of the present disclosure, the method further includes setting non-significant elements of the set of transformed dose contributions to zero. The non-significant elements may be identified by a threshold. 
         [0012]    According to a further aspect of the present disclosure, the method further includes dropping non-significant elements that are defined for each of said voxels. 
         [0013]    According to another aspect of the present disclosure, a method for determining a gradient of a cost function of dose is disclosed, having: creating a set of voxels from a volume of interest in a physical object or system, or virtual model thereof; identifying a plurality of beamlets, each of said beamlets depositing a dose contribution of radiation to at least one subset of said voxels; transforming the dose contributions of all beamlets for each of said voxels into a first domain, yielding a set of transformed dose contributions from each of said beamlets; associating a weight with each of said beamlets; identifying a dose for each of said voxels; identifying a cost function of said dose; computing the gradient of the cost function with respect to the dose in each voxel; and determining the transform of the gradient of the cost function with respect to each of said weights in the first domain through summing up the elementwise products of the set of transformed dose contributions and the cost function gradient with respect to the dose in each voxel. Having determined said quantity, applying a transform from the first domain yielding the gradient of the cost function with respect to each of said weights. 
         [0014]    According to a further aspect of the present disclosure, the method further includes setting non-significant elements of the set of transformed dose contributions to zero. Said non-significant elements may be identified by a threshold. 
         [0015]    According to a further aspect of the present disclosure, the method further includes dropping non-significant elements that are defined for each of said voxels. 
         [0016]    According to another aspect of the present disclosure, a non-transitory computer readable medium is disclosed, having stored thereon instructions that when executed cause one or more processors to perform the steps of: creating a set of voxels from a volume of interest in a physical object or system; identifying a plurality of beamlets, each of said beamlets depositing a dose contribution of radiation to at least one subset of said voxels; transforming the dose contributions of all beamlets for each of said voxels into a first domain, yielding a set of transformed dose contributions for each of said voxels; assigning a weight to each of said beamlets to create a set of weights; transforming the set of weights into a second domain orthogonal to the first domain, yielding a set of transformed weights; and determining a dose for each of said voxels through summing up the elementwise product of the set of transformed dose contributions and the set of transformed weights. 
         [0017]    According to another aspect of the present disclosure, a non-transitory computer readable medium is disclosed, having stored thereon instructions that when executed cause one or more processors to perform the steps of: creating a set of voxels from a volume of interest in a physical object or system, or virtual model thereof; identifying a plurality of beamlets, each of said beamlets depositing a dose contribution of radiation to at least one subset of said voxels; transforming the dose contributions of all beamlets for each of said voxels into a first domain, yielding a set of transformed dose contributions from each of said beamlets; associating a weight with each of said beamlets; identifying a dose for each of said voxels; identifying a cost function of said dose; computing the gradient of the cost function with respect to the dose in each voxel; and determining the transform of the gradient of the cost function with respect to each of said weights in the first domain through summing up the elementwise products of the set of transformed dose contributions and the cost function gradient with respect to the dose in each voxel. Having determined said quantity, applying a transform from the first domain yielding the gradient of the cost function with respect to each of said weights. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0018]      FIG. 1  is an illustrative example of the grid system which is compiled from a 3 3  arrangement of voxels having bent rectangular regions. 
           [0019]      FIG. 2  is a flowchart of an overall process used in dose determination according to described embodiments. 
           [0020]      FIG. 3  is a flowchart of a VOI discretization subprocess of the process shown in  FIG. 2 . 
           [0021]      FIG. 4  is a flowchart of a beam and beamlet setup subprocess of the process shown in  FIG. 2 . 
           [0022]      FIG. 5A  is a flowchart of a dose calculation subprocess. 
           [0023]      FIG. 5B  is a flowchart of a cost function gradient calculation subprocess. 
           [0024]      FIG. 5C  is a flowchart of dose determination subprocess. 
           [0025]      FIG. 6  is a block diagram of a system for dose determination according to described embodiments. 
           [0026]      FIG. 7  is a schematically depicts an example radiation delivery apparatus that may be used in a radiation therapy. 
       
    
    
     DETAILED DESCRIPTION 
       [0027]    The disclosed embodiments relate to methods and systems for dose determination to tumor sites (or VOI) in a patient in radiation therapy. During radiation dose delivery, radiation scattering commonly occurs due to the passage of the radiation through the body volume. These scatter effects are taken into account during the computation of the distribution of dose deposited by the radiation. Dose in a point or a plurality of points in a VOI is computed by a superposition of a plurality of elementary doses, each with its individual weight factor: 
         [0000]        D=t·φ,   (1)
 
         [0000]    where D denotes a dose in a point of VOI, t is a vector of elementary doses, and φ is a vector of the respective weight factors. 
         [0028]    Referring now to  FIG. 1 , VOI  102  is irradiated by a radiation beam  107  originating from a radiation source  106 . For the reason of simplicity, current embodiment only discloses one radiation beam irradiating a VOI. It is to be understood that, however, in some other embodiments, more than one radiation beam may be projected on a VOI. VOI  102  can be organs, tissues, or any body parts of a patient that need to be irradiated. However, it is understood that VOI  102  can also be any experimental object that is used such as for researching or testing purposes. 
         [0029]    Typically, VOI  102  has an irregular shape. It is broken down into a plurality of subvolumes, so-called voxels  104  as illustrated in  FIG. 1 . Voxels  104  may be regular or irregular in shape, as depicted. VOI  102 , or in other words, the plurality of voxels  104 , are projected by a single beam  107 . 
         [0030]    The single beam  107  can be divided into a multiple of elementary beams, so-called beamlets  108 . Each of these beamlets  108  is configured to deliver a dose to a respective subset of voxels  104 , which may include one or more voxels  104 . However, due to the scatter effects, any beamlet  108  may not only deliver a dose to the subset of voxels it is directly impinging on, but also deposit a dose within other voxels within the VOI  102  (in fact, other body volume of a patient besides the VOI, may also get irradiated by any beamlet  108 ). In other words, each of these beamlets  108  may deposit a dose, so called elementary dose, to each voxel  104 . Consequently, the dose in a voxel  104  consists of each elementary dose deposited on the voxel  104  by all beamlets  108 . 
         [0031]    Assuming the number of voxels in a VOI is N v , the number of beamlets in a beam is N b , and further assuming that each beamlet is assigned with a weight factor, e.g. a beamlet j is assigned a weight factor φ j , the dose in a voxel i is then expressed as a linear combination of weighted dose contribution of each beamlet as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       D 
                       i 
                     
                     = 
                     
                       
                         ∑ 
                         j 
                       
                        
                       
                           
                       
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                          
                         
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                           j 
                         
                       
                     
                   
                   , 
                   
                       
                   
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                     = 
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                   , 
                   
                     N 
                     b 
                   
                   , 
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where D i  denotes the dose deposited in voxel i from a beam. T ij  denotes the dose contribution of beamlet j to voxel i, or in other words, T ij  denotes the dose contribution received by voxel i deposited from beamlet j. The dose contribution of each beamlet j, where j=1, . . . , N b , to each voxel i, where i=1, . . . , N v , forms a set of dose contributions, or more precisely, forms a N v ×N b  matrix T ij . φ j  is the weight factor of beamlet j. In some embodiments, each beamlet inside a beam can be assigned with equal weight. In some embodiments, each beamlet inside a beam can be assigned with various weight values. In other embodiments, each beamlet inside a beam can also be assigned with non-constant weights. The weight factor of each beamlet j, where j=1, . . . , N b , forms a set of weight factors, or more precisely, forms a vector Φ with a length of N b . 
         [0032]    For dose determination in a radiation therapy, not only the dose to each voxel D i , but also the derivative of a function of dose at any voxel with respect to its elementary constituents is required. This function is also sometimes termed as a “cost function” or an “objective function”. The elementary constituents can be e.g. the weight factors of beamlets. The derivative of a cost function with respect to each weight factor is computed for optimization. The latter yields an optimal weight factor of each beamlet. 
         [0033]    By the virtue of chain rule, the gradient of a cost function ƒ(D) of dose D with respect to a weight factor φ j  can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           f 
                            
                           
                             ( 
                             D 
                             ) 
                           
                         
                       
                       
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                               f 
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                                 ( 
                                 D 
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                               D 
                               i 
                             
                           
                         
                         · 
                         
                           
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                               D 
                               i 
                             
                           
                           
                             ∂ 
                             
                               ϕ 
                               j 
                             
                           
                         
                       
                     
                   
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                     v 
                   
                 
               
               
                 
                   ( 
                   3 
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         [0000]    where D denotes the dose in a VOI consisting of a plurality of voxels, D i  denotes the dose deposited in voxel i, ƒ(D) is a cost function, and φ j  denotes the weight factor of beamlet j. From equation (2), equation (3) can be further expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           f 
                            
                           
                             ( 
                             D 
                             ) 
                           
                         
                       
                       
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                   4 
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         [0000]    where T ij  denotes the dose contribution of beamlet j to voxel i. 
         [0034]    However, in operation, it is typical that a VOI is divided into an enormous number of voxels, and a beam is also split into a vast number of beamlets. In other words, both N v  and N b  can be numbers having very large order of magnitude. This leads to computation of formulae (2) and (4) for all the voxels inside a VOI and all beamlets becoming very time-consuming due to large numbers of arithmetic operations. Therefore, improved methods for accelerating of computing the dose and the derivative of a cost function are required. 
         [0035]    One approach is performing a transformation of the set of dose contributions and the set of weight factors into orthogonal domains. 
         [0036]    Said transformation of the set of dose contributions is denoted as Z and the transformation of the set of weight factors is denoted as Y. They need to fullfill one basic requirement: 
         [0000]    the scalar product remains unchanged, i.e. Z and Y are orthogonal: 
         [0000]        t·φ=Z ( t )· Y (φ)
 
         [0000]    To achieve said acceleration, the domain Z transforms into is sparse, i.e. the information present in vector t is statistically concentrated in only a few coefficients of the transformed vector Z(t). 
         [0037]    Equation (1) for calculating the dose in a point can be expressed as: 
         [0000]        D=Z ( t )· Y (φ)
 
         [0038]    By applying the Z-transform to all N v  row vectors t in matrix T, i.e. equation (2) for calculating the dose in voxel i, can be expressed as 
         [0000]        D   i   =Z ( T ) ij   ·Y (φ) j  
 
         [0039]    Non-significant elements can be identified and dropped, in effect, rendering the matrix Z(T) sparse. The i-th row of Z(T) now contains k i  significant elements, where k i  is a much smaller number compared to N b . This may lead to a significant reduction of the arithmetic operations of dose calculation, since k i &lt;&lt;N b . In some embodiments, a single method can be applied to identify non-significant elements. In other embodiments, the method may vary across Z(T). 
         [0040]    Further, since the majority of members of Z(T) are 0, only the non-zero members of Z(T) are saved in a storage, which leads to a reduction of the storage space. Choosing the relative amount of zero-entries of Z(T) allows to adjust the trade-off between computation speed and accuracy. Or more precisely, dropping less elements of Z(T) keeps the dose D=Z(T)Y(φ) more accurate. 
         [0041]    A further approach is performing a transformation of the set of dose contributions to compute the derivative of a cost function of dose with respect to the weights of its constituents. 
         [0042]    Said transformation of the set of dose contributions is denoted as Z. It needs to fullfill one basic requirement, namely that there exists an transform U such that 
         [0000]        x=U ( Z ( x )) 
         [0000]    for any given N v -element vector x. 
         [0043]    Then equation (4) for calculation the derivative of a cost function with respect to a weight factor φ can then be expressed as: 
         [0000]    
       
         
           
             
               
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         [0044]    Non-significant elements can be identified and dropped, in effect, rendering the matrix Z(T) ij  sparse. The i-th row of Z(T) now contains k i  significant elements, where k i  is a much smaller number compared to N b . This may lead to a significant reduction of the arithmetic operations of the cost function gradient calculation, since k i &lt;&lt;N b . In some embodiments, a single method can be applied to identify non-significant elements. In other embodiments, the method may vary across Z(T). 
         [0045]    Further, since the majority members of Z(T) are 0, only the non-zero members of Z(T) are saved in a storage, which leads to a reduction of the storage space. Choosing the relative amount of zero-entries of Z(T) allows to adjust the trade-off between computation speed and accuracy. Or more precisely, dropping less elements of Z(T) keeps the derivative of ƒ(D) with respect to the weight factors φ j  more accurate. 
         [0046]    Turning now to the drawings,  FIG. 2  is a block diagram illustrating a process  200  for dose determination in a radiation therapy according to embodiments of the disclosure. Process  200  executes a number of sequential steps which include VOI discretization  300 , beam and beamlet setup  400 , and dose determination  500 , wherein dose determination  500  includes dose calculation  500   a  and cost function derivative calculation  500   b . These subprocesses are described in further detail below, with reference to  FIGS. 3 to 5 . 
         [0047]    Referring now to  FIG. 3 , VOI discretization subprocess  300  is described in further detail. In some embodiments, a VOI can be a tumor site in a patient. In other embodiments, a VOI can also be an object detected in a measurement device. VOI discretization subprocess  300  executes two sequential steps which include inputting data  310  and choosing discretization parameters  320 . 
         [0048]    The VOI discretization subprocess  300  handles input data from e.g. CT scans, as shown in step  310 . Once the input data is received at step  310 , the discretization parameters (may include e.g. voxel size or resolution) to be used for radiation therapy, or more precisely for radiation dose determination, can be chosen. This information is then used to generate a discretized representation of the input data. After the VOI has been discretized, subprocess  300  feeds into beam and beamlet setup subprocess  400 . 
         [0049]    Referring now to  FIG. 4 , beam and beamlet setup subprocess  400  is described. Beam and beamlet setup subprocess  400  determines the boundaries of each beam and divides the beams into beamlets, calculates the beamlet boundaries and determines which voxels receive a dose contribution from each beamlet. 
         [0050]    Subprocess  400  begins with data input from the user (i.e. medical personnel planning the dose delivery) as to the desired radiation beam characteristics, at step  410 . At this step, the user also inputs beam setup information, such as the distance between the radiation source and the VOI isocenter. 
         [0051]    For radiation therapy, the radiation beams are typically delivered to a patient lying on a bed while a gantry carrying a radiation beam emitter moves around the patient as depicted in  FIG. 7 . A radiation therapy apparatus  700  may include gantry  710  and bed  712 . Radiation beam  704  emitted from radiation source  702  toward subject  720  is permitted to pass through multi-leaf collimator  718  for beam shaping. The gantry  710  can be positioned at numerous different angles around trace  740 , depending on the dose delivery plan developed by the radiation oncologist and the limitations of the radiation delivery apparatus. 
         [0052]    Turing back to  FIG. 4  now, after the beam characteristics are chosen at step  410 , the beam boundaries are then calculated at step  420 . At a given gantry angle, the position and width of the beam is calculated in order to fully cover the VOI as seen by the beam source from the radiation beam emitter. In the current embodiment, merely one beam is irradiated from a radiation beam emitter to a VOI, however, it is understandable that multiple beams may be emitted in a radiation therapy. 
         [0053]    Once the planned beam boundaries are determined at step  420 , the beam is divided into beamlets at step  430 . The number of beamlets within each beam can depend on the tumor-shape, gantry angle, equipment limitations, beam boundaries and so on. 
         [0054]    At step  440 , for a given gantry angle, the respective beam, beamlets, and the voxels within the VOI through which each beam and beamlet contribute dose to, are stored. Then the gantry can move to the next predetermined angles, and sub-steps  420  to  440  are repeated till beam and beamlets are stored for all predetermined gantry angles. 
         [0055]    After the last gantry angle has been processed at step  440 , subprocess  400  feeds into dose determination subprocess  500 . 
         [0056]    During the dose determination subprocess  500 , two aspects of information are determined: dose to each voxel in a VOI, and the derivative of a cost function with respect to each beamlet weight factor. The former is determined through subprocess  500   a  in  FIG. 5A , and the latter is determined through subprocess  500   b  in  FIG. 5B . 
         [0057]    Referring now to  FIG. 5A , dose calculation subprocess  500   a  is described. Dose determination subprocess  500   a  calculates and stores the set of dose delivered from each beamlet to each voxel in a VOI. 
         [0058]    Subprocess  500   a  begins with step  510   a , at which a set T of dose contribution is created. The elements of set T, so called dose contributions or elementary doses, represent the doses received by each voxel projected from each beamlet. For example, an element T ij  of T denotes the dose contribution to a voxel i deposited by a beamlet j. 
         [0059]    At step  520   a , a set φ of beamlet weight factors is created. In some embodiments, each beamlet is set default to have equal weighting, which may be e.g. 1. In other embodiments, the weight factors may be varying for different beamlets. An element φ j  of set Φ denotes the weight factor of a beamlet j. 
         [0060]    At step  530   a , set T of dose contribution is transformed into a first domain. Said transformation of the set of dose contributions is denoted as Z. The domain Z transforms into is sparse, i.e. the information present in T is statistically concentrated in only a few coefficients of the transformed matrix Z(T). 
         [0061]    In some embodiments, non-significant elements can be identified and dropped, in effect, rendering the matrix Z(T) sparse. The i-th row of Z(T) now contains k i  significant elements, where k i  is a much smaller number compared to N b . This may lead to a significant reduction of the arithmetic operations of dose calculation, since k i &lt;&lt;N b . In some embodiments, a single method can be applied to identify non-significant elements. In other embodiments, the method may vary across Z(T). 
         [0062]    Further, since the majority members of Z(T) are 0, only the non-zero members of Z(T) are saved in a storage, which leads to a reduction of the storage space. Choosing the relative amount of zero-entries of Z(T) allows to adjust the trade-off between computation speed and accuracy. Or more precisely, dropping less elements of Z(T) keeps the dose D more accurate. 
         [0063]    At step  540   a , the set co of beamlet weight factors is transformed into a second domain orthogonal to the first domain. The transformation of the set of weight factors is denoted as Y. Said transformations fullfill one basic requirement: the scalar product remains unchanged, i.e. t·φ=Z(t)·Y(φ). 
         [0064]    At step  550   a , dose for each voxel is calculated. For example, dose D i  for a voxel i can be calculated by the following expression: 
         [0000]        D   i   =Z ( T ) ij   ·Y (φ) j   (9)
 
         [0000]    where the calculation runs through all the voxels in a VOI. 
         [0065]    According to the current embodiments, dose D i  is calculated by the multiplication of Z(T) ij ·Y(φ) j , instead of T ij ·φ j . The i-th row of Z(T) now contains k i  significant elements, where k i  is a much smaller number compared to N b . This may lead to a significant reduction of the arithmetic operations of dose calculation, since k i &lt;&lt;N b . Further, since the majority members of Z(T) are 0, only the non-zero members of Z(T) are saved in a storage, which leads to a reduction of the storage space. 
         [0066]    Referring now to  FIG. 5B , the derivative calculation subprocess  500   b  is described. Derivative calculation subprocess  500   b  calculates and stores the set of derivative of a cost function with respect to each beamlet weight factor. 
         [0067]    Subprocess  500   b  begins with step  510   b , the same as step  510   a , at which a set T of dose contributions is created. The elements of set T, so called dose contributions or elementary doses, represent the doses received by each voxel deposited by each beamlet. For example, an element T ij  of T denotes the dose contribution to a voxel i deposited by a beamlet j. 
         [0068]    At step  520   b , set T of dose contribution is transformed into a first domain. Said transformation of the set of dose contributions is denoted as Z. The domain Z transforms into is sparse, i.e. the information present in T is statistically concentrated in only a few coefficients of the transformed matrix Z(T). 
         [0069]    In some embodiments, non-significant elements can be identified and dropped, in effect, rendering the matrix Z(T) sparse. The i-th row of Z(T) now contains k i  significant elements, where k i  is a much smaller number compared to N b . This may lead to a significant reduction of the arithmetic operations of dose calculation, since k i &lt;&lt;N b . In some embodiments, a single method can be applied to identify non-significant elements. In other embodiments, the method may vary across Z(T). 
         [0070]    Further, since the majority members of Z(T) are 0, only the non-zero members of Z(T) are saved in a storage, which leads to a reduction of the storage space. Choosing the relative amount of zero-entries of Z(T) allows to adjust the trade-off between computation speed and accuracy. Or more precisely, dropping less elements of Z(T) keeps the dose D=Z(T)Y(φ) more accurate. 
         [0071]    At step  530   b , dose for each voxel is identified. 
         [0072]    At step  540   b , a cost function of dose is defined as ƒ(D). 
         [0073]    The transformation in  520   b  needs to fullfill one basic requirement, namely that there exists an transform U such that 
         [0000]        x=U ( Z ( x )) 
         [0000]    for any given N v -element vector x. 
         [0074]    At step  550   b , the gradient of the cost function with respect to each beamlet weight factor is calculated. For example, a gradient with respect to the weight factor of beamlet j can be calculated as: 
         [0000]    
       
         
           
             
               
                 ∂ 
                 
                   f 
                    
                   
                     ( 
                     D 
                     ) 
                   
                 
               
               
                 ∂ 
                 
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                       f 
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                   T 
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         [0000]    and the calculation of this equation will run through all the beamlets. 
         [0075]    According to the current embodiments, gradient 
         [0000]    
       
         
           
             
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         [0000]    is calculated by the multiplication and transformation 
         [0000]    
       
         
           
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         [0000]    instead of 
         [0000]    
       
         
           
             
               
                 ∂ 
                 
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                 ∂ 
                 
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         [0000]    Non-significant elements can be identified and dropped, in effect, rendering the matrix Z(T) ij  sparse. The i-th row of Z(T) now contains k i  significant elements, where k i  is a much smaller number compared to N b . This may lead to a significant reduction of the arithmetic operations of the cost function gradient calculation, since k i &lt;&lt;N b . 
         [0076]    Further, since the majority members of Z(T) are 0, only the non-zero members of Z(T) are saved in a storage, which leads to a reduction of the storage space. Choosing the relative amount of zero-entries of Z(T) allows to adjust the trade-off between computation speed and accuracy. Or more precisely, dropping less elements of Z(T) keeps the derivative of ƒ(D) with respect to the weight factors φ j  more accurate. 
         [0077]    In some embodiments, the dose for each voxel and the gradient of a cost function with respect to each weight factor are calculated separately as described in subprocesses  500   a  and  500   b . In other embodiments, aforementioned two sets of value can also be determined in a single process  500   c , as will be described in detail. 
         [0078]    Subprocess  500   c  begins with step  510   c , the same as step  510   a , at which a set T of dose contribution is created. The elements of set T, so called dose contributions or elementary doses, represent the doses received by each voxel projected from each beamlet. For example, an element T ij  of T denotes the dose contribution to a voxel i deposited by a beamlet j. 
         [0079]    At step  520   c , the same as step  520   a , a set φ of beamlet weight factors is created. In some embodiments, each beamlet is set default to have equal weighting, which may be e.g. 1. In other embodiments, the weight factors may be varying for different beamlets. An element φ j  of set φ denotes the weight factor of a beamlet j. 
         [0080]    At step  530   c , the same as step  530   a , set T of dose contribution is transformed into a first domain. Said transformation of the set of dose contributions is denoted as Z. The domain Z transforms into is sparse, i.e. the information present in T is statistically concentrated in only a few coefficients of the transformed matrix Z(T). 
         [0081]    In some embodiments, non-significant elements can be identified and dropped, in effect, rendering the matrix Z(T) sparse. The i-th row of Z(T) now contains k i  significant elements, where k i  is a much smaller number compared to N b . This may lead to a significant reduction of the arithmetic operations of dose calculation, since k i &lt;&lt;N b . In some embodiments, a single method can be applied to identify non-significant elements. In other embodiments, the method may vary across Z(T). 
         [0082]    Further, since the majority members of Z(T) are 0, only the non-zero members of Z(T) are saved in a storage, which leads to a reduction of the storage space. Choosing the relative amount of zero-entries of Z(T) allows to adjust the trade-off between computation speed and accuracy. Or more precisely, dropping less elements of Z(T) keeps the dose D=Z(T)Y(φ) and the derivative of ƒ(D) with respect to the weight factor φ j  more accurate. 
         [0083]    At step  540   c , the same as in step  540   a , the set φ of beamlet weight factors is transformed into a second domain orthogonal to the first domain. The transformation of the set of weight factors is denoted as Y. Said transformations fullfill one basic requirement: the scalar product remains unchanged, i.e. t·φ=Z(t)·Y(φ). At step  550   c , the same as step  550   a , dose for each voxel is calculated. For example, dose D i  for a voxel i can be calculated by the following expression: 
         [0000]        D   i   =Z ( T ) ij   ·Y (φ) j  
 
         [0000]    where the calculation runs through all the voxels in a VOI. 
         [0084]    At step  560   c , the same as step  540   b , a cost function of dose is defined as ƒ(D). 
         [0085]    The transformation in  530   c  needs to fullfill one basic requirement, namely that there exists an transform U such that 
         [0000]        x=U ( Z ( x )) 
         [0000]    for any given N v -element vector x. 
         [0086]    At step  570   c , the same as step  550   b , the gradient of the cost function with respect to each beamlet weight factor is calculated. For example, a gradient with respect to the weight factor of beamlet j can be calculated as: 
         [0000]    
       
         
           
             
               
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         [0000]    and the calculation of this equation will run through all the beamlets. 
         [0087]    The subprocess  500  is merely depicted for a given gantry angle. However, the described method is not limited to a single gantry angle. The extension of the described method to more than one gantry angle is straightforward. 
         [0088]    While the methods and subprocesses for dose determination have been described above in relation to various embodiments, the invention may be embodied also in a system  600 , running dose determination module  604  and configured to perform the described methods and subprocesses, as is shown in  FIG. 6 . 
         [0089]    In  FIG. 6 , system  600  includes a computer system  603  having memory  606  and dose determination module  604  running as executable computer program instructions thereon. The computer program instructions are executed by one or more processors (not shown) within computer system  603 . Memory  606  comprises fast memory, such as fast-access RAM, for storing data sets and calculation terms used during the dose determination process  200 . 
         [0090]    Computer system  603  further includes normal computer peripherals (not shown), including graphics displays, keyboard, secondary memory, graphical processing unit (GPU), and serial and network interfaces, as would normally be used for a computer system which receives input data  602  and generates corresponding output data  608 . 
         [0091]    While embodiments of the disclosure have been described in relation to dose determination for radiation therapy treatment, it is to be understood that the dose determination process  200  and determination system  600  may be equally useful for planning dose delivery to body volumes other than those of human patient under treatment for cancerous tumors. For example, the described systems and methods may be employed for animals other than humans and may be employed for irradiation non-living tissue or material or organic matter where selective dose delivery of radiation is desired. Further, the described systems or methods can be employed for virtual models of physical objects or systems in a similar fashion. 
         [0092]    In this description, certain terms have been used interchangeably. For example, dose contribution and elementary dose have been used interchangeably and are intended to have the same meaning. Similarly, gradient and derivative have been used interchangeably and are intended to have the same meaning. Moreover, Volume element and voxel have been used interchangeably and are intended to have the same meaning. For the sake of clarity, the Einstein notation convention is used interchangeably with the customary notation to describe summations over a range of indices.