-
Notifications
You must be signed in to change notification settings - Fork 1
/
mdpMain.m
393 lines (342 loc) · 14.3 KB
/
mdpMain.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
classdef mdpMain < handle
% mdpMain Compute the optimal multi-modality cancer treatment policy using
% backward induction for stationary transition probabilities.
properties
n % number of state variables
m % number of states per state variable
numStates % total number of states
numActions % number of actions
T % number of epochs
r % reward functions
Pn % side-effect and tumor transition probabilities
P % transition probability matrix
A % optimal policy
tol % tie threshold
end
methods
% Initialize mdp variables according to pars and/or default values.
function mdp = mdpMain(pars)
flag = exist('pars','var');
mdp.n = 3;
if flag && isfield(pars,'m')
mdp.m = pars.m;
else
mdp.m = [2,11,11];
end
mdp.numStates = prod(mdp.m);
mdp.numActions = 3;
if flag && isfield(pars,'T')
mdp.T = pars.T;
else
mdp.T = 3;
end
if flag && isfield(pars,'r')
mdp.r = pars.r;
else
mdp.r{1} = @(S)0;
mdp.r{2} = @(S)100*((1/2)*(mdp.m(2)^2-S(2)^2)/mdp.m(2)^2+...
(1/2)*(mdp.m(3)^2-S(3)^2)/mdp.m(3)^2);
end
if flag && isfield(pars,'Pn')
mdp.Pn = pars.Pn;
else
mdp.Pn{1,1} = [0 40 60]; % M1 side-effect
mdp.Pn{1,2} = [0 60 40]; % M2 side-effect
mdp.Pn{1,3} = [60 40 0]; % M3 side-effect
mdp.Pn{2,1} = [70 30 0]; % M1 tumor
mdp.Pn{2,2} = [60 40 0]; % M2 tumor
mdp.Pn{2,3} = [0 30 70]; % M3 tumor
end
mdp.tol = 1e-6;
% check transition probability assumptions
mdp.checkAssumptions();
end
% Check transition probability assumptions.
function flag = checkAssumptions(mdp)
flag = 0;
% no negative probability values
for i = 1:mdp.numActions
sPn = mdp.Pn{1,i};
tPn = mdp.Pn{2,i};
for j = 1:mdp.n
if sPn(j) < 0 || tPn(j) < 0
flag = 1;
disp('Assumption violation: Negative transition probability.')
end
end
end
% surveillance transition probabilities
sPn_surveillance = mdp.Pn{1,mdp.numActions};
tPn_surveillance = mdp.Pn{2,mdp.numActions};
% side-effect does not increase with surveillance
if sPn_surveillance(3) > 0
flag = 1;
disp('Assumption violation: Increase in side-effect due to surveillance.')
end
% tumor does not decrease with surveillance
if tPn_surveillance(1) > 0
flag = 1;
disp('Assumption violation: Decrease in tumor due to surveillance.')
end
for i = 1:mdp.numActions-1
sPn_treatment = mdp.Pn{1,i};
tPn_treatment = mdp.Pn{2,i};
% tumor does not increase with treatment
if tPn_treatment(3) > 0
flag = 1;
disp('Assumption violation: Increase in tumor due to treatment.')
end
% side-effect does not decrease with treatment
if sPn_treatment(1) > 0
flag = 1;
disp('Assumption violation: Decrease in side-effect due to treatment.')
end
end
sPn_M1 = mdp.Pn{1,1};
tPn_M1 = mdp.Pn{2,1};
for i = 2:mdp.numActions-1
sPn_treatment = mdp.Pn{1,i};
tPn_treatment = mdp.Pn{2,i};
% M1 has highest risk (side-effect increase)
if sPn_M1(3) < sPn_treatment(3)
flag = 1;
disp('Assumption violation: M1 does not have highest side-effect increase.')
end
% M1 has highest reward (tumor decrease)
if tPn_M1(1) < tPn_treatment(1)
flag = 1;
disp('Assumption violation: M1 does not have highest tumor decrease.')
end
end
end
% Create the matrix of all possible states.
function S = calcStates(mdp)
S = zeros(mdp.numStates,mdp.n);
for i = 1:mdp.numStates
temp1 = i-1;
for j = mdp.n:-1:1
temp2 = mod(temp1,prod(mdp.m(j:end)));
S(i,j) = temp2/prod(mdp.m(j+1:end))+1;
temp1 = temp1 - temp2;
end
end
S = S - 1;
end
% Calculate the immediate or terminal reward vector.
function R = calcReward(mdp,t,S)
rt = mdp.r{t};
R = zeros(mdp.numStates,1);
for i = 1:mdp.numStates
R(i) = rt(S(i,:));
end
end
% Calcuate the transition probability matrix.
function calcProb(mdp)
mdp.P = zeros(mdp.numStates,mdp.numStates,mdp.numActions);
for i = 1:mdp.numActions
% extract and normalize
sProbVec = mdp.Pn{1,i}/sum(mdp.Pn{1,i});
tProbVec = mdp.Pn{2,i}/sum(mdp.Pn{2,i});
% create OAR and tumor transition probability matrices
oarProbMat = mdp.calcProbMat(mdp.m(2),sProbVec,'o');
tProbMat = mdp.calcProbMat(mdp.m(3),tProbVec,'t');
% combine OAR and tumor matrices with absorbing states
P0 = kron(oarProbMat,tProbMat);
idx = (mdp.m(2) - 1)*mdp.m(3) + 1;
P0(idx:end,idx:end) = eye(mdp.m(3)); % OAR death
for j = mdp.m(3):mdp.m(3):prod(mdp.m(2:3))
P0(j,j) = 1; % tumor death
end
% combine OAR/tumor matrix with M1 history
if i == 1 % choose M1
P1 = zeros(mdp.m(2)*mdp.m(3)); P1(:,end) = 1;
mdp.P(:,:,i) = kron([0,1],[P0;P1]);
else % don't choose M1
mdp.P(:,:,i) = kron(eye(2),P0);
end
end
end
% Calculate the probability matrix for the given state variable.
function probMat = calcProbMat(~,m,probVec,var)
onesVec = cell(1,2);
onesVec{1} = ones(1,m-1);
onesVec{2} = ones(1,m);
% diagonal elements
probMat = zeros(m);
for i = 1:3
probMat = probMat + probVec(i)*diag(onesVec{2-abs(i-2)},i-2);
end
% boundary conditions
if strcmp(var,'t')
probMat(1,:) = [1 zeros(1,m-1)]; % tumor remission
probMat(end,:) = zeros(1,m); % tumor death
else
probMat(1,1) = probVec(1) + probVec(2); % healthy OAR
probMat(end,:) = zeros(1,m); % OAR death
end
end
% Calculate the optimal multi-modality policy.
function calcPolicy(mdp)
% compute immediate and terminal rewards
S = mdp.calcStates();
rt = mdp.calcReward(1,S);
rN = mdp.calcReward(2,S);
% compute transition probability matrix
mdp.calcProb();
% initialize optimal policy and patient utility
mdp.A = zeros(mdp.numStates,mdp.T);
V = rN;
% compute optimal policy for each epoch with backward induction
Vtemp = zeros(mdp.numStates,mdp.numActions);
for t = mdp.T:-1:1
% calculate maximum utility
for a = 1:mdp.numActions
Vtemp(:,a) = mdp.P(:,:,a)*(rt+V);
end
V = max(Vtemp,[],2);
% assign optimal modalities
Atemp = zeros(mdp.numStates,1);
for i = 1:mdp.numStates
for j = 1:mdp.numActions
if abs(Vtemp(i,j)-V(i)) < mdp.tol
Atemp(i) = 10*Atemp(i)+j;
end
end
end
mdp.A(:,t) = Atemp;
end
end
% Print the optimal policy.
function printPolicy(mdp)
for t = 1:mdp.T
% surgery history = 0
fprintf('Policy computed for h = 0 and t=%d\n',t)
disp(flipud(reshape(mdp.A(1:mdp.m(2)*mdp.m(3),t),mdp.m(3),mdp.m(2))'))
% surgery history = 1
fprintf('Policy computed for h = 1 and t=%d\n',t)
disp(flipud(reshape(mdp.A(mdp.m(2)*mdp.m(3)+1:end,t),mdp.m(3),mdp.m(2))'))
end
end
% Plot the optimal multi-modality policy for three actions.
function p = plotPolicy(mdp)
% assign actions to colormap
map = mdp.defColormap();
[B,labels,X] = mdp.action2color();
% plot the policy for each epoch
figure();
res = 100;
for t = 1:mdp.T
% M1 history = 0
h1 = subplot(2,mdp.T+1,t);
p1 = get(h1,'pos');
p1(1) = p1(1) - 0.03*(t-1);
set(h1,'pos',p1);
H0 = reshape(B(1:mdp.m(2)*mdp.m(3),t),mdp.m(3),mdp.m(2))';
H0 = imresize(H0,res,'nearest');
imagesc(H0);
colormap(map)
caxis([1 8])
options = {'Interpreter','LaTeX','FontSize',13};
temp = title(sprintf('t = %d',t));
set(temp,options{:});
set(gca,'YDir','normal'), axis square
set(gca,'xtick',[]);
if t == 1
set(gca,'ytick',res/2:2*res:10*res+res/2);
set(gca,'yticklabel',0:2:10);
else
set(gca,'ytick',[]);
if t == 3
set(gca,'YAxisLocation','Right')
ylabel('h = 0',options{:})
end
end
% M1 history = 1
h2 = subplot(2,mdp.T+1,mdp.T+t+1);
p2 = get(h2,'pos');
p2(1) = p2(1) - 0.03*(t-1);
p2(2) = p1(2) - p1(4) + 0.1;
set(h2,'pos',p2);
H1 = reshape(B(mdp.m(2)*mdp.m(3)+1:end,t),mdp.m(3),mdp.m(2))';
H1 = imresize(H1,res,'nearest');
imagesc(H1);
colormap(map)
caxis([1 8])
set(gca,'YDir','normal'), axis square
set(gca,'xtick',res/2:2*res:10*res+res/2);
set(gca,'xticklabel',0:2:10);
if t == 1
tx = text(8*res,-3*res,'Tumor Progression State ($$\tau$$)',options{:});
ty = text(-3.25*res,4.5*res,{'Side Effect State ($$\phi$$)'},options{:},'Rotation',90);
p = p2;
set(gca,'ytick',res/2:2*res:10*res+res/2);
set(gca,'yticklabel',0:2:10);
else
set(gca,'ytick',[]);
if t == 3
set(gca,'YAxisLocation','Right')
ylabel('h = 1',options{:})
end
end
end
% plot the legend
h3 = subplot(2,mdp.T+1,[mdp.T+1 2*(mdp.T+1)]);
p3 = get(h3,'pos');
p3(1) = p1(1) + p1(3) + 0.06;
p3(2) = p2(2) + 0.065;
p3(4) = p2(4) + 0.11;
set(h3,'pos',p3);
X = imresize(X,res,'nearest');
imagesc(X), axis off
colormap(map)
caxis([1 8])
% annotate the legend
for i = 1:length(labels)
text(1.1*res,(i-1/2)*res,labels(i),options{:})
end
text(p3(2)+0.15,p3(1)-0.25*res,'Modalities',options{:})
end
% Define the policy colormap for three actions.
function map = defColormap(~)
map = [0.6806 0.0836 0.1972 % raspberry (M1)
0.9957 0.7438 0.1340 % yellow (M2)
0 0.4791 0.7942 % steel blue (M3)
0.9110 0.3483 0.1050 % orange (M1/M2)
0.3226 0.7985 1.0000 % sky blue (M1/M3)
0.4995 0.7224 0.2015 % apple green (M2/M3)
0 0 0 % black (M1/M2/M3)
0.9400 0.9400 0.9400]; % gray
end
% Assign actions to colormap colors for three actions.
function [B,labels,X] = action2color(mdp)
temp1 = [1 2 3 12 13 23 123];
temp2 = {'$$M_1$$','$$M_2$$','$$M_3$$','$$M_1$$ or $$M_2$$',...
'$$M_1$$ or $$M_3$$','$$M_2$$ or $$M_3$$',...
'$$M_1$$, $$M_2$$, or $$M_3$$'};
B = zeros(size(mdp.A)); labels = cell(1,length(temp1)); X = [];
count = 1;
for i = 1:length(temp1)
if sum(sum(mdp.A == temp1(i))) > 0
B(mdp.A == temp1(i)) = i;
labels(count) = temp2(i);
X = [X; i length(temp1)+1];
count = count + 1;
end
end
labels = labels(1:count-1);
end
% Display the modality legend.
function displayLegend(mdp,map,labels,X)
% plot the colormap
subplot(2,mdp.T+1,[mdp.T+1 2*(mdp.T+1)])
imagesc(X), axis off
colormap(map)
caxis([1 8])
% annotate the legend
for i = 1:length(labels)
text(1.55,i,labels(i),'Interpreter','LaTeX','FontSize',14)
end
title('Modalities','Interpreter','LaTeX','FontSize',14)
end
end
end