Reconstruction Algorithms in Prostrate Brachytherapy

1
Reconstruction Algorithms in Prostrate
Brachytherapy

• By
• Sreeram Narayanan

2
Approach

• Calculate the 3-D co-ordinates of seed end-points
  from a given set of images
• Match corresponding seed images in a minimum of
  three radiographs.

3
Problems

• Patient movement
• Improper film placement
• Digitization error
• Overlapping seeds

4
Algorithms(1/2)

• Siddon
• Minimize the Sum of Square of Distances(SSD) to
  calculate the seed endpoints.
• Construct a cost function matrix consisting of
  SSD of all possible combinations of seed images
• Do an exhaustive search to find out the most
  probable seed triplets.

5
Algorithms(2/2)

• Altschuler
• Create two different cost function matrices
• Apply a computationally efficient technique to
  calculate the minimum cost triplet.
• See after various trials, if the cost converges
  to a unique minimum value
• This minimum corresponds to the best possible
  triplet set.

6
Siddon Method

• Seed Reconstruction

S1
S3
S2
d2
d3
d1
R
I3
I2
I1
7

• Source Image
• Sk 0 Ik Xk
• 0
  Yk
• SID FID
• Lk Sk ak Ik - Sk is the line connecting the
  image to the source
• dk Lk - R Ik is the distance from the
  assumed seed position to the line connecting the
  Source Image
• Let nk be a unit vector along the direction of
  the line
• dk nk 0

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• DIk2 where
• dmin,k Sk - RIk - (Sk - RIk)nk nk
• The minimization condition is
• Substituting dmin,k in the above condition we get
  the values of RIk corresponding to the 3-D
  co-ordinates

9
Image Matching

• Consider only two images
• A Matrix is formed consisting of M(j1,j2)
• M minimum of
• ( DI1(1,j1),I2(1,j2)2 DI1(2,j1), I2(2,j2)2
  ) or
• ( DI1(1,j1),I2(2,j2)2 DI1(2,j1), I2(1,j2)2
  )

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Matrix of Cost Functions

• Choose one element from each row column such
  that their sum is minimum

11

• For three films choose one element from each row,
  column slice of the matrix.
• If there are N seeds in each image the exhaustive
  search requires N! combinations for two films
  (N!)2 for three films.
• This is an NP-complete problem

12
Altschuler Cost Functions(1/2)

• The area of the triangle formed by Ray midpoints
  plus the square of the maximum of the minimum
  distances between any pair of the triplet rays.

.
Ray2
Ray3
Ray1
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Altschuler Cost Functions(2/2)

• The distance between the actual seed image(Si)
  the back-projected image (Ii)of the inferred 3D
  position

P
I3
S3
S2
S1
I1
I2
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To Find the Lowest Cost Triplet Set (1/4)

• For Three radiographs, the cost functions can be
  represented in the form of a 3-D cube.
• If there are N1 seeds in the first radiograph, N2
  in the second N3 in the third, The dimension of
  the cube will be N1? N2 ? N3

N3
N2
N1
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To Find the Lowest Cost Triplet Set (2/4)

• Let there be equal number of seeds in all
  radiographs (N1 N2 N3 N).
• Generate N1 random numbers
• The first random number will determine which way
  to look at the cube.(It will select one of the
  three radiographs).
• The rest N random number will determine the order
  of the slices of the cube (It will determine the
  order in which the seed images are taken from
  that radiograph)

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To Find the Lowest Cost Triplet Set (3/4)

• For each seed image in the first radiograph,
  select an unmatched seed image in the other two
  radiographs such that its cost is minimum.
• For the pth slice we will have (N-p1)2 costs to
  choose from.
• For the last slice we will have only one value
  remaining.
• Add each of the selected cost to get a total cost.

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To Find the Lowest Cost Triplet Set(4/4)

• Repeat this process with different random
  numbers.
• For a good data set, the total cost will be the
  same for all the trials.
• The triplet set which gives the lowest total cost
  is taken as the best possible matching.

18
Experiments (1/3)

• Simulations were done using software.
• We choose to do these trials with three
  radiographs at 0, 15 -15 degrees.
• The number of seeds were kept the same for all
  the radiographs.
• The size of the seed was 0.3 cm.
• Two Cost functions - Siddon Altschuler-II -
  were used.
• No noise was added in the beginning.

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Experiments (2/3)

• Digitization error was introduced as a Gaussian
  with variance 0.01 cm.
• Film placement error was simulated by shifting
  the imaging plane by up to 0.5 cm in both X Y
  direction inducing a small rotation.
• The results were checked changing N from 10 up to
  60 seeds for different sets of Digitization
  film placement error.

20
Experiments (3/3)

• Patient movement was simulated as a Gaussian with
  variance ranging from 0.01cm to 0.5cm around the
  seed end-points
• The results were compared with the original
  configuration for both the 3D end co-ordinates
  triplet matching.
• One seed was made to overlap so that the number
  of seeds were 60, 59 60 in the 15, 0 -15
  degree films respectively.

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Results - Siddon

• Errors(cm) Maximum Mismatches
• E1 E2 N 10 20 30 40 50 60
• 0.01 0.2 0 0
  0 0 0 0
• 0.01 0.3 0 0
  0 0 0 0
• 0.01 0.4 0 0
  0 0 0 0
• 0.01 0.5 0 0
  0 0 2 2
• E3
• 0.2 0 0 0
  0 0 0
• 0.3 0 0
  0 0 0 0
• 0.4 0 0 0
  0 0 2
• 0.5 0 0
  0 0 2 2

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Results - Altschuler

• Errors(cm) Maximum Mismatches
• E1 E2 N 10 20 30 40 50 60
• 0.01 0.2 0 0
  0 0 0 0
• 0.01 0.3 0 0
  0 0 0 0
• 0.01 0.4 0 0
  0 0 2 4
• 0.01 0.5 0 0
  0 2 6 6
• E3
• 0.2 0 0 0 0 0
  0
• 0.3 0
  0 0 0 0 0
• 0.4 0 0 0 2 4
  4
• 0.5 0
  0 0 2 6 6

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Results with Overlapping Seed

• Two seed pairs match to the same seed.
• One that has a cost lower than that of other is
  the one that is the correct match.
• The other seed can be partially reconstructed
  assuming same end co-ordinates as the matched
  seed.

24
Another Imaging Method

• The best way to resolve ambiguities in matching
  is to place the radiographic sources such that
  they are NOT co-planar with implant volume.
• This involves moving the patient in between the
  shots, in the inferior-superior direction.
• In such a case Altschuler Cost Function-I will
  probably give the best matching.