Our leaf sequencing algorithm employed Xia’s method (
10) to initiate the input beam intensity map based on the model of Langer et al. ( 11). The total leaf travel distance (LTD) is used as a key constraint from our model in intensity modulation. 3.1. Intensity Modulation Model
The photon beam intensity modulation can be modelled as a mathematical problem by considering to decompose a given m × n integer matrix into a positive linear combination of (0, 1) matrices with the strict consecutive 1’s property in rows. The integer matrix can be expressed as (Equation 1):
Where A is the intensity matrix of m row and n column. S
i is the corresponding (0, 1) matrices with the strict consecutive 1’s. u i is the LINAC beam monitor unit in each MLC aperture. The mathematical problem is to reduce an intensity matrix A(i, j) into a group of MLC beam segments which fields can be generated by a MLC. Matrix elements in A(i, j) are assumed to be nonnegative integers representing the photon beam intensity level.
For an intensity matrix of irregular subfield, the leaf movement direction is decided with the MLC set to an angle by which the leaves move (1) along the shorter field size and (2) along the least intensity gradient direction. However, in some complicated cases, the least intensity gradient direction is ambiguous. The MLC therefore has to set at an angle such that the leaves move along the shorter field size direction. When the beam intensity matrices have regular patterns, the least intensity gradient direction becomes obvious. However, the shorter field size direction is perpendicular to the least intensity gradient. In that case, we suggest the MLC should be set at an angle such that leaves move along the least intensity gradient direction. It is because such arrangement results in the least number of beam segments.
During the matrix reduction process, a residual beam intensity matrix A
k(i, j) defined as the original intensity matrix, A(i, j) minus all the previous MLC beam segments, is calculated. The maximum matrix element L max,k in the A k(i, j) is found to determine the intensity level, d k for the next segment (Equation 2):
Where nint is the nearest integer (
10). 3.2. The New Algorithm
Our algorithm used an integer linear program formulation to determine the beam segment with the minimal NS and TNMU (
11). The formulation contains two phases ( 12). The first phase involves in minimizing the beam-on time (Equation 3):
Where T is an upper bound of the photon beam monitor unit, which can be calculated using the maximum number of the summation in each row from the beam intensity matrix A. This upper bound can be written as (Equation 4):
The second phase involves in minimizing the number of beam segments (Equation 5):
t = 1, if an element switches from covered to uncovered, and g t = 0, otherwise.
The value of T is equal to the Z
* produced from stage one in Equation 3. The parameter m in Equation 4 represents the number of rows, and the parameter n is the number of columns in the beam intensity matrix A. In reality of radiation dose delivery, there are many restrictions on the MLC leaf position such as collision within or between rows, contiguity, leaf-moving direction, and tongue-and-groove effects. These MLC constraints can be characterized by the binary variables namely, l mnt and r mnt. If the beam intensity element in m and n of A is covered by a left/right MLC leaf, when the t th monitor unit is delivered, l mnt/r mnt is set to 1. Otherwise, it is equal to zero. X mnt is a binary variable equal to 1 when a monitor unit is delivered through the beam intensity element i and j at period t, and is equal to 0 when the element is covered by either the left or right leaf. In addition, combinations of l mnt and r mnt can be used to reflect different MLC conditions such as leaf collision within a row, between rows, restriction of contiguity, tongue-and-groove effects and leaf motion direction ( 12).
When the MLC leaves change from one beam segment to the next, all leaves start moving at the same time. However, due to the different travel distances, the leaves do not always stop exactly together. For a Topslane Venus M MLC (
11), the leaves are designed to move at the same speed between 10 and 18 mm s −1. The leaf travel time is therefore determined by the leaf having the longest travel distance in the segment. Since leaf travel time can be calculated by finding the maximum leaf travel time from each row, the travel time for the whole leaf sequence is approximately proportional to the LTD (Equations 6 - 8)
Where m is the number rows in A. The two variables M
mt and N mt represent the left/right leaf edge location. For minimizing the LTD as shown (Equation 9):
Since Equation 6 has absolute values, the related constraint will convert into a mixed integer nonlinear programming. As it is difficult to determine the global optimum, we alternatively cancel the absolute symbol to consider the positive and negative value simultaneously. Equation 6 is therefore transferred into a linear form as (Equations 10
Our improved algorithm can therefore be expressed as follows (Equation 12):