This is a proof of concept for the inclusion of positional uncertainty within the dwell time optimisation algorthim. The focus is on the optimisation method, and as such it is assumed that each seed radially deposits its energy purely based upon the inverse square law.
Two regions are defined, a central target region and a off centre avoid region.
In [1]:
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.colors import BoundaryNorm
from matplotlib.colorbar import ColorbarBase
%matplotlib inline
from utilities import BasinhoppingWrapper, create_green_cm
In [2]:
green_cm = create_green_cm()
Create the calculation grid
In [3]:
grid_spacing = 0.1 # Was 0.1, using 0.2 to speed up for testing
x_ = np.arange(-0.6, 0.6 + grid_spacing, grid_spacing)
y_ = np.arange(-0.6, 0.6 + grid_spacing, grid_spacing)
z_ = np.arange(-0.6, 0.6 + grid_spacing, grid_spacing)
x, y, z = np.meshgrid(x_, y_, z_)
x = np.ravel(x)
y = np.ravel(y)
z = np.ravel(z)
Define the target and avoid cubes
In [4]:
def target(x, y, z):
return (
(x < 0.45) & (x > -0.45) &
(y < 0.45) & (y > -0.45) &
(z < 0.45) & (z > -0.45))
def avoid(x, y, z):
return (
(x < 0.25) & (x > 0.05) &
(y < 0.15) & (y > -0.15) &
(z < 2) & (z > -2))
target_cube = target(x, y, z)
avoid_cube = avoid(x, y, z)
target_cube = target_cube & ~avoid_cube
Display the target and avoid cubes
In [5]:
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(
x[target_cube], y[target_cube], z[target_cube],
alpha=0.2, s=2, color='blue')
ax.scatter(
x[avoid_cube], y[avoid_cube], z[avoid_cube],
alpha=0.2, s=2, color='red')
ax.set_xlim([np.min(x_), np.max(x_)])
ax.set_ylim([np.min(y_), np.max(y_)])
ax.set_zlim([np.min(z_), np.max(z_)])
Out[5]:
In [6]:
ax.view_init(azim=-90, elev=90)
fig
Out[6]:
Create initial equidistant parrallel lines to represent catheters. Inbuild a slight random skew to emulate what occurs physically
In [7]:
lines_in_row = 4
spacing = np.linspace(-0.45, 0.45, lines_in_row)
number_of_lines = lines_in_row**2
wonkiness = 0.01
line_start = np.meshgrid(spacing, spacing, 1)
line_finish = np.array([
line_start[0] + np.random.normal(scale=wonkiness, size=[lines_in_row, lines_in_row, 1]),
line_start[1] + np.random.normal(scale=wonkiness, size=[lines_in_row, lines_in_row, 1]),
-line_start[2]])
In [8]:
line_start = np.array([np.ravel(mesh) for mesh in line_start])
line_finish = np.array([np.ravel(mesh) for mesh in line_finish])
Display the lines overlayed
In [9]:
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(
x[target_cube], y[target_cube], z[target_cube],
alpha=0.2, s=2, color='blue')
ax.scatter(
x[avoid_cube], y[avoid_cube], z[avoid_cube],
alpha=0.2, s=2, color='red')
ax.scatter(*line_start)
ax.scatter(*line_finish)
for i in range(len(line_start[0])):
plt_coords = [
[line_start[j][i], line_finish[j][i]]
for j in range(len(line_start))]
ax.plot(*plt_coords, color='black', alpha=0.5)
ax.set_xlim([-1, 1])
ax.set_ylim([-1, 1])
ax.set_zlim([-1, 1])
Out[9]:
Create a function to return x, y, z coords when a distance along a line is requested
In [10]:
diff = (line_finish - line_start)
line_length = np.sqrt(diff[0]**2 + diff[1]**2 + diff[2]**2)
def find_distance_coords(line_num=None, distance=None):
relative_dist = distance / line_length[line_num]
if (relative_dist > 1) | (relative_dist < 0):
return np.array([np.nan]*3)
x = (
line_start[0][line_num] * (1 - relative_dist) +
line_finish[0][line_num] * relative_dist)
y = (
line_start[1][line_num] * (1 - relative_dist) +
line_finish[1][line_num] * relative_dist)
z = (
line_start[2][line_num] * (1 - relative_dist) +
line_finish[2][line_num] * relative_dist)
coords = np.array([x, y, z])
return coords
Pick dwell positons starting at a random position along the line
In [11]:
dwell_spacing = 0.1 # Was 0.1, increased to 0.3 for testing
dwell_distances_from_initial = np.arange(0, 2, dwell_spacing)
number_of_dwells = len(dwell_distances_from_initial)
In [12]:
inital_dwell_position = np.random.uniform(
low=0, high=dwell_spacing, size=number_of_lines)
inital_dwell_position
Out[12]:
In [13]:
dwell_distances = np.reshape(inital_dwell_position, (-1, 1)) + np.reshape(dwell_distances_from_initial, (1, -1))
In [14]:
def find_dwell_coords(line_num=None, dwell_num=None):
distance = dwell_distances[line_num, dwell_num]
coords = find_distance_coords(
line_num=line_num, distance=distance)
return coords
Find all the dwell positions that are on the grid
In [15]:
dwell_positions = np.array([
[
find_dwell_coords(
line_num=line_num, dwell_num=dwell_num)
for dwell_num in range(number_of_dwells)]
for line_num in range(number_of_lines)])
Plot the dwell positions
In [16]:
line_colours = np.random.uniform(size=(number_of_lines,3))
In [17]:
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(
x[target_cube], y[target_cube], z[target_cube],
alpha=0.2, s=2, color='blue')
ax.scatter(
x[avoid_cube], y[avoid_cube], z[avoid_cube],
alpha=0.2, s=2, color='red')
for line_num in range(number_of_lines):
ax.scatter(*np.transpose(dwell_positions[line_num]),
c=line_colours[line_num], alpha=0.7)
ax.set_xlim([-1, 1])
ax.set_ylim([-1, 1])
ax.set_zlim([-1, 1])
Out[17]:
Select the dwell positions that fall withing the target region. Only use these dwell positions.
In [18]:
dwell_positions_to_be_filtered = np.reshape(dwell_positions, (-1, 3))
distance_to_dwell_pos_initial = np.array([
np.sqrt(
(x[i] - dwell_positions_to_be_filtered[:,0])**2 +
(y[i] - dwell_positions_to_be_filtered[:,1])**2 +
(z[i] - dwell_positions_to_be_filtered[:,2])**2
)
for i in range(len(x))
])
In [19]:
# closest_voxel_to_dwell = np.reshape(np.argmin(distance_to_dwell_pos_initial, axis=0), [1, -1])
# target_cube_voxels = np.reshape(np.where(target_cube)[0], [-1, 1])
# is_dwell_closest_to_target = np.any(closest_voxel_to_dwell == target_cube_voxels, axis=0)
shift_uncertainty = 0.05
target_cube_voxels = np.where(target_cube)[0]
relevant_dwell_test = (
np.min(distance_to_dwell_pos_initial[target_cube_voxels], axis=0) <
3 * shift_uncertainty)
relevant_dwell_positions = dwell_positions_to_be_filtered[relevant_dwell_test]
num_relevant_dwells = np.sum(relevant_dwell_test)
In [20]:
line_number_index = np.reshape(np.arange(number_of_lines), (number_of_lines, 1))
line_number_index = np.ones([number_of_lines, number_of_dwells]) * line_number_index
line_number_index = np.ravel(line_number_index)[relevant_dwell_test]
line_number_index = line_number_index.astype(int)
In [21]:
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(
x[target_cube], y[target_cube], z[target_cube],
alpha=0.2, s=2, color='blue')
ax.scatter(
x[avoid_cube], y[avoid_cube], z[avoid_cube],
alpha=0.2, s=2, color='red')
for line_num in range(number_of_lines):
ref = line_number_index == line_num
ax.scatter(*np.transpose(relevant_dwell_positions[ref]),
c=line_colours[line_num], alpha=0.7)
ax.set_xlim([np.min(x_), np.max(x_)])
ax.set_ylim([np.min(y_), np.max(y_)])
ax.set_zlim([np.min(z_), np.max(z_)])
Out[21]:
Create an array containing the distance to each dwell position for each voxel and translate this to exposure at that voxel per unit dwell time at each dwell position. This is defined in this way so that a reasonable portion of the calculation of exposure can be done outside of the optimisation.
In [22]:
distance_to_dwell_pos = np.array([
np.sqrt(
(x[i] - relevant_dwell_positions[:,0])**2 +
(y[i] - relevant_dwell_positions[:,1])**2 +
(z[i] - relevant_dwell_positions[:,2])**2
)
for i in range(len(x))
])
exposure_per_unit_time = 1 / distance_to_dwell_pos**2
Create exposure calculation function
In [23]:
def calculate_exposure(dwell_times):
exposure = np.sum(dwell_times * exposure_per_unit_time, axis=1)
return exposure
Run a test of arbitrary dwell times
In [24]:
num_relevant_dwells = len(relevant_dwell_positions)
random_pick = np.random.uniform(
size=2, high=num_relevant_dwells, low=0).astype(int)
dwell_times = np.zeros([1, num_relevant_dwells])
dwell_times[0, random_pick] = 10
exposure = calculate_exposure(dwell_times)
In [25]:
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
reference = exposure > 80
colour = exposure[reference]
colour[colour > 200] = 200
ax.scatter(
x[reference], y[reference], z[reference],
alpha=0.2, s=20, c=colour, cmap=green_cm)
ax.set_xlim([np.min(x_), np.max(x_)])
ax.set_ylim([np.min(y_), np.max(y_)])
ax.set_zlim([np.min(z_), np.max(z_)])
Out[25]:
Create function to display the results of the optimisation as it is being calculated.
In [26]:
def display_results(dwell_times):
dwell_times = np.reshape(dwell_times, (1, num_relevant_dwells))
exposure = calculate_exposure(dwell_times)
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
reference = exposure > 25
colour = exposure[reference]
colour[colour > 100] = 100
small = exposure[reference] < 50
large = ~small
ax.scatter(
x[reference][small], y[reference][small], z[reference][small],
alpha=0.2, s=3, c=colour[small], cmap=green_cm)
ax.scatter(
x[reference][large], y[reference][large], z[reference][large],
alpha=0.4, s=20, c=colour[large], cmap=green_cm)
ax.set_xlim([np.min(x_), np.max(x_)])
ax.set_ylim([np.min(y_), np.max(y_)])
ax.set_zlim([np.min(z_), np.max(z_)])
cost_function(dwell_times, debug=True)
plt.show()
The maximum exposure to a grid position not within the target is aimed to be less than 100. This cost function aims to achieve this.
In [27]:
def hot_exterior_cost_function(max_target_exterior):
return ((max_target_exterior)/50)**4
testx = np.linspace(0, 120)
testy = hot_exterior_cost_function(testx)
plt.plot(testx, testy)
plt.ylim([0, 20])
Out[27]:
No grid point within the target is to be less than about 50. This cost function aims to achieve this.
In [28]:
def cold_target_cost_function(min_target):
return ((min_target-80)/15)**4
testx = np.linspace(0, 120)
testy = cold_target_cost_function(testx)
plt.plot(testx, testy)
plt.ylim([0, 20])
Out[28]:
The avoid cost function aims to make no point within the avoid structure more than 45.
In [29]:
def hot_avoid_cost_function(max_avoid):
return ((max_avoid)/25)**4
testx = np.linspace(0, 120)
testy = hot_avoid_cost_function(testx)
plt.plot(testx, testy)
plt.ylim([0, 20])
Out[29]:
Create the cost function to be used by the optimiser
In [30]:
def cost_function(dwell_times, debug=False):
dwell_times = np.reshape(dwell_times, (1, num_relevant_dwells))
exposure = calculate_exposure(dwell_times)
min_target = np.min(exposure[target_cube])
max_avoid = np.max(exposure[avoid_cube])
max_target_exterior = np.max(exposure[~target_cube])
cold_target_cost = cold_target_cost_function(min_target)
hot_exterior_cost = hot_exterior_cost_function(max_target_exterior)
hot_avoid_cost = hot_avoid_cost_function(max_avoid)
total_cost = hot_exterior_cost + cold_target_cost + hot_avoid_cost
if debug:
print("Minimum target = %0.4f, resulting cost = %0.4f" %
(min_target, cold_target_cost))
print("Maximum exterior = %0.4f, resulting cost = %0.4f" %
(max_target_exterior, hot_exterior_cost))
print("Maximum avoid = %0.4f, resulting cost = %0.4f" %
(max_avoid, hot_avoid_cost))
print("Total cost = %0.4f" % (total_cost))
return total_cost
Create initial conditions
In [31]:
num_relevant_dwells
Out[31]:
In [32]:
initial_conditions = np.ones(num_relevant_dwells)*0.1
Step noise
In [33]:
step_noise = np.ones(num_relevant_dwells) * 0.3
Bounds
In [34]:
bounds = ((0, None),)*num_relevant_dwells
Run the optimiser
In [35]:
optimisation = BasinhoppingWrapper(
to_minimise=cost_function,
initial=initial_conditions,
step_noise=step_noise,
basinhopping_confidence=5,
optimiser_confidence=0.0001,
n=5,
debug=display_results,
bounds=bounds
)
In [36]:
display_results(optimisation.result)
Display histogram of resulting dwell times
In [37]:
plt.hist(optimisation.result)
Out[37]:
Give overview of dwell times segmented by catheter
In [38]:
for line_num in range(number_of_lines):
print("Line number %d" % (line_num))
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(
x[target_cube], y[target_cube], z[target_cube],
alpha=0.2, s=2, color='blue')
ax.scatter(
x[avoid_cube], y[avoid_cube], z[avoid_cube],
alpha=0.2, s=2, color='red')
ref = line_number_index == line_num
ax.scatter(*np.transpose(relevant_dwell_positions[ref]),
c=line_colours[line_num], alpha=0.7)
ax.set_xlim([np.min(x_), np.max(x_)])
ax.set_ylim([np.min(y_), np.max(y_)])
ax.set_zlim([np.min(z_), np.max(z_)])
plt.show()
plt.scatter(np.arange(np.sum(ref)), optimisation.result[ref],
c=line_colours[line_num])
plt.xlabel("Dwell number")
plt.ylabel("Units of dwell time")
plt.ylim(bottom=0)
plt.show()
Convert result into column vector for post analysis
In [39]:
dwell_times = np.reshape(optimisation.result, (1, num_relevant_dwells))
Create a custom resolution exposure calculation function for post analysis
In [40]:
def create_calculate_exposure(dwell_positions,
x_new=None, y_new=None, z_new=None):
x, y, z = np.meshgrid(
x_new, y_new, z_new)
x = np.ravel(x)
y = np.ravel(y)
z = np.ravel(z)
distance_to_dwell_pos = np.array([
np.sqrt(
(x[i] - dwell_positions[:,0])**2 +
(y[i] - dwell_positions[:,1])**2 +
(z[i] - dwell_positions[:,2])**2
)
for i in range(len(x))
])
exposure_per_unit_time = 1 / distance_to_dwell_pos**2
def calculate_exposure(dwell_times, reshape=False):
exposure = np.sum(dwell_times * exposure_per_unit_time, axis=1)
if reshape:
exposure = np.reshape(exposure, (
len(x_new), len(y_new), len(z_new)))
return exposure
return calculate_exposure
In [41]:
dx = 0.01; dy = 0.01; dz = 0.1
contour_x_ = np.arange(-1, 1 + dx, dx)
contour_y_ = np.arange(-1, 1 + dy, dy)
contour_z_ = z_ #np.arange(-0.7, 0.7 + dz, dz)
contour_calculate_exposure = create_calculate_exposure(
relevant_dwell_positions, x_new=contour_x_, y_new=contour_y_, z_new=contour_z_)
contour_exposure_init = contour_calculate_exposure(dwell_times, reshape=True)
In [42]:
vmin = 0
vmax = 200
num_contours = 41
fig = plt.figure(figsize=(10,0.3))
cbar_ax = fig.add_subplot(1,1,1)
bounds = np.linspace(vmin, vmax, num_contours)
norm = BoundaryNorm(bounds, green_cm.N)
cb = ColorbarBase(
cbar_ax, norm=norm, cmap=green_cm,
orientation='horizontal')
cb.ax.xaxis.set_ticks_position('top')
cb.ax.xaxis.set_label_position('top')
cb.set_clim([vmin, vmax])
contour_exposure = contour_exposure_init.copy()
contour_exposure[contour_exposure > vmax] = vmax
for i, z_value in enumerate(contour_z_):
plt.figure(figsize=(6,6))
c = plt.contourf(contour_x_, contour_y_,
contour_exposure[:, :, i], num_contours,
vmin=vmin, vmax=vmax, cmap=green_cm)
reference = z[target_cube] == z_value
x_target = x[target_cube][reference]
y_target = y[target_cube][reference]
plt.scatter(x_target, y_target, alpha=0.3, color='blue')
reference = z[avoid_cube] == z_value
x_avoid = x[avoid_cube][reference]
y_avoid = y[avoid_cube][reference]
plt.scatter(x_avoid, y_avoid, alpha=0.3, color='red')
plt.title("Slice z = %0.1f" % (z_value))
plt.xlim([np.min(contour_x_), np.max(contour_x_)])
plt.ylim([np.min(contour_y_), np.max(contour_y_)])
plt.show()
Create analysis grid
In [43]:
dx = 0.02
dy = 0.02
dz = 0.02
post_x_ = np.arange(-1, 1 + dx, dx)
post_y_ = np.arange(-1, 1 + dy, dy)
post_z_ = np.arange(-1, 1 + dz, dz)
post_x, post_y, post_z = np.meshgrid(
post_x_, post_y_, post_z_)
post_x = np.ravel(post_x)
post_y = np.ravel(post_y)
post_z = np.ravel(post_z)
post_calculate_exposure = create_calculate_exposure(
relevant_dwell_positions, x_new=post_x_, y_new=post_y_, z_new=post_z_)
post_exposure = post_calculate_exposure(dwell_times)
Define structures on new finer grid
In [44]:
post_target_cube = target(post_x, post_y, post_z)
post_avoid_cube = avoid(post_x, post_y, post_z)
Make a function to plot DVHs given a reference
In [45]:
def create_dvh(reference, exposure, ax=None):
if ax is None:
fig = plt.figure()
ax = fig.add_subplot(1,1,1)
results = exposure[reference]
results[results>150] = 150
hist = np.histogram(results, 100)
freq = hist[0]
bin_edge = hist[1]
bin_mid = (bin_edge[1::] + bin_edge[:-1:])/2
cumulative = np.cumsum(freq[::-1])
cumulative = cumulative[::-1]
bin_mid = np.append([0], bin_mid)
cumulative = np.append(cumulative[0], cumulative)
percent_cumulative = cumulative / cumulative[0] * 100
ax.plot(bin_mid, percent_cumulative)
Plotting of relevant DVHs
In [46]:
fig = plt.figure()
ax = fig.add_subplot(1,1,1)
create_dvh(post_target_cube, post_exposure, ax=ax)
create_dvh(post_avoid_cube, post_exposure, ax=ax)
create_dvh(~post_target_cube, post_exposure, ax=ax)
plt.legend(["Target", "Avoid", "Patient"])
plt.xlim([0, 150])
Out[46]:
Add a random shift to each catheter, see the effect it has on target DVH.
In [47]:
relevant_dwell_distances = np.ravel(dwell_distances)[relevant_dwell_test]
shift_uncertainty = 0.05
In [48]:
line_displacement = np.random.normal(scale=shift_uncertainty, size=number_of_lines)
In [49]:
shifted_dwell_positions = np.array([
find_distance_coords(
distance=relevant_dwell_distances[i] + line_displacement[line_number_index[i]],
line_num=line_number_index[i])
for i in range(len(relevant_dwell_distances))
])
In [50]:
shifted_calculate_exposure = create_calculate_exposure(
shifted_dwell_positions, x_new=post_x_, y_new=post_y_, z_new=post_z_)
shifted_exposure = shifted_calculate_exposure(dwell_times)
With the shift the change in the DVH is the following. For the optimisation given here there is very little change in DVH.
In [51]:
fig = plt.figure()
ax = fig.add_subplot(1,1,1)
create_dvh(post_target_cube, post_exposure, ax=ax)
create_dvh(post_target_cube, shifted_exposure, ax=ax)
plt.legend(["Original", "With position error"], loc='lower left')
plt.xlim([0, 110])
# plt.ylim([95, 100])
Out[51]:
In [52]:
np.min(shifted_exposure[post_target_cube])
Out[52]:
In [53]:
np.min(post_exposure[post_target_cube])
Out[53]:
In [54]:
fig = plt.figure()
ax = fig.add_subplot(1,1,1)
create_dvh(post_avoid_cube, post_exposure, ax=ax)
create_dvh(post_avoid_cube, shifted_exposure, ax=ax)
plt.legend(["Original", "With position error"], loc='upper right')
plt.xlim([0, 110])
Out[54]:
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In [55]:
exposure = calculate_exposure(dwell_times)
np.min(exposure[target_cube])
Out[55]:
In [56]:
def return_cold_spot_in_random_displace():
line_displacement = np.random.normal(scale=shift_uncertainty, size=number_of_lines)
shifted_dwell_positions = np.array([
find_distance_coords(
distance=relevant_dwell_distances[i] + line_displacement[line_number_index[i]],
line_num=line_number_index[i])
for i in range(len(relevant_dwell_distances))
])
shifted_calculate_exposure = create_calculate_exposure(
shifted_dwell_positions, x_new=x_, y_new=y_, z_new=z_)
shifted_exposure = shifted_calculate_exposure(dwell_times)
return np.min(shifted_exposure[target_cube])
In [57]:
n = 10000
cold_spots = np.zeros(n)
for i in range(n):
cold_spots[i] = return_cold_spot_in_random_displace()
In [58]:
plt.hist(cold_spots,20)
plt.xlim([35,60])
Out[58]:
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