-
Notifications
You must be signed in to change notification settings - Fork 13
/
equations.txt
225 lines (126 loc) · 6.97 KB
/
equations.txt
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
Equations related to diffusion and dwi:
The basic signal equation:
S(\theta, b) = S_0 e^{-b A(\theta)}
Where:
A(\theta) = \theta^t Q \theta
And:
Q = \begin{pmatrix} D_{xx} & D_{xy} & D_{xz} \\
D_{yx} & D_{yy} & D_{yz} \\
D_{zx} & D_{zy} & D_{zz} \\
\end{pmatrix}
Such that:
S = S_0 e^{-b \vec{b}^t Q \vec{b}}
DTI:
\begin{pmatrix} log(S_1) \\ log(S_2) \\ \vdots \\ log(S_n)\end{pmatrix} = -b \begin{pmatrix} g_{1x}^2 & g_{1y}^2 g_{1z}^2 & 2g_{1x1y} & 2g_{1x1z} & 2g_{1y1z} & log(S_n) & 1 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
g_{nx}^2 & g_{ny}^2 g_{nz}^2 & 2g_{nxny} & 2g_{nxnz} & 2g_{nynz} & log(S_n) & 1
\end{pmatrix} \begin{pmatrix} D_{xx} \\ D_{yy} \\ D_{xy}\\ D_{xz} \\ D_{yz} \\ log(S_0) \end{pmatrix}
And:
\hat{\beta}_\mathrm{WLS} = (X^T W X)^{-1} X^T W y \\
Where:
W = \mathrm{diag}((X \hat{\beta}_\mathrm{OLS})^2)
And:
\hat{\beta}_\mathrm{OLS} = (X^T X)^{-1} X^T y
We can rewrite that as:
\frac{S}{S_0} = e^{-b \vec{b} Q \vec{b}^t}
Behrens (2003) "ball and stick" model:
S(g, b) = S_0 (f \exp[-b d (-g^T v)^2] + [1-f] \exp[-b d])
Where:
0<= f <=1
Is a "mixing parameter"
\frac{S}{ S_0}= (1 - \sum_{i=1}^{N}{\beta_i}) e^{-b d} +
\sum_{i=1}^{N}{\beta_i e^{-b \theta R_i d Q \theta^t}}
With:
\forall i: 0 \leq \beta_i \leq 1
and
Q = \begin{pmatrix} 1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0\\
\end{pmatrix}
General "ball and stick":
{S (\theta, b)}= \beta_0 e^{-bD} + \sum_{i=1}^{m}{\beta_i e^{-b \theta^t Q_i \theta}}
T_i (\theta, b) = e^{-b \theta^t Q_i \theta}
{S (\theta, b)}= w_0 + \sum_{i=1}^{m}{w_i (T_i (\theta, b) - \bar{T}_b ) }
S (\theta, b) - \bar{S}_b = \sum_{i=1}^{m}{w_i (T_i (\theta,b) - \bar{T}_b ) }
O_i(\theta,b) = T_i(\theta,b) - \bar{T}_b
S (\theta, b) = w_0 + \sum_{i=1}^{m}{w_i O_i(\theta,b)}
S (\theta, b) - \bar{S}_b = \sum_{i=1}^{m}{w_i O_i(\theta,b)}
\bar{S}_b = \frac{1}{N} \sum_{j=1}^{n} S(\theta_j, b)
\bar{T}_b = \frac{1}{N}\sum_{j=1}^{n} T(\theta_j, b)
CanonicalTensorModel:
\begin{pmatrix} \frac{S_1}{S_0} \\ \frac{S_2}{S_0} \\ \vdots \\ \frac{S_n}{S_0}
\end{pmatrix} = \begin{pmatrix} \frac{T_1}{T_{max}} & 1 \\ \frac{T_2}{T_{max}} & 1 \\ \vdots & \vdots \\ \frac{T_n}{T_{max}} & 1 \end{pmatrix} \begin{pmatrix} w_1 \\ w_2 \end{pmatrix}
And rewritten in terms of the diffusion equation:
\begin{pmatrix} \frac{S_1}{S_0} \\ \frac{S_2}{S_0} \\ \vdots \\ \frac{S_n}{S_0}
\end{pmatrix} = \begin{pmatrix} e^{-bD} & e^{-b\theta_1RQ\theta_1^t} \\ e^{-bD} & e^{-b\theta_2RQ\theta_2^t}\\ \vdots & \vdots \\e^{-bD} & e^{-b\theta_nRQ\theta_n^t} \end{pmatrix} \begin{pmatrix} \beta_0 \\ \beta_1 \end{pmatrix}
MultiCanonicalTensor model:
\begin{pmatrix} \frac{S_1}{S_0} \\ \frac{S_2}{S_0} \\ \vdots \\ \frac{S_n}{S_0}
\end{pmatrix} = \begin{pmatrix} \frac{T^1_1}{T^1_{max}} &
\frac{T^2_1}{T^2_{max}} & 1 \\ \frac{T^1_2}{T^1_{max}} &
\frac{T^1_2}{T^1_{max}} & 1 \\ \vdots & \vdots & \vdots \\
\frac{T^1_n}{T^1_{max}} & \frac{T^2_n}{T^2_{max}} & 1 \end{pmatrix}
\begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix}
SparseDeconvolutionModel (non-normed):
\begin{pmatrix} \frac{S_1}{S_0} - \langle\frac{S}{S_0}\rangle \\
\frac{S_2}{S_0} - \langle\frac{S}{S_0}\rangle \\ \vdots \\ \frac{S_n}{S_0} -
\langle\frac{S}{S_0}\rangle \end{pmatrix} = \begin{pmatrix} T^1_1 - \langle T^1
\rangle & T^2_1 - \langle T^2 \rangle & \cdots & T^m_1 - \langle T^m \rangle
\\ T^1_2 - \langle T^1 \rangle & T^2_2 - \langle T^2 \rangle & \cdots & T^m_2 -
\langle T^m \rangle \\ \vdots & \vdots & \cdots & \vdots \\T^1_n - \langle
T^1 \rangle & T^2_n - \langle T^2 \rangle & \cdots & T^m_n - \langle T^m
\rangle \end{pmatrix} \begin{pmatrix} w_1 \\ w_2 \\ \vdots \\ w_m \end{pmatrix}
\begin{pmatrix} S(\theta_1) - \bar{S} \\
S(\theta_2) - \bar{S} \\ \vdots \\ S(\theta_n) - \bar{S}
\end{pmatrix} = \begin{pmatrix} T_1(\theta_1) - \bar{T} & T_2(\theta_1) - \bar{T}& \cdots & T_m(\theta_1) - \bar{T}
\\ T_1(\theta_2) - \bar{T} & T_2(\theta_2) - \bar{T} & \cdots & T_m(\theta_2) -
\bar{T} \\ \vdots & \vdots & \cdots & \vdots \\T_1(\theta_n) - \bar{T} & T_2(\theta_n) - \bar{T} & \cdots & T_m(\theta_n) - \bar{T} \end{pmatrix} \begin{pmatrix} w_1 \\ w_2 \\ \vdots \\ w_m \end{pmatrix}
\begin{pmatrix} S(\theta_1, b) - \bar{S}_b \\
S(\theta_2, b) - \bar{S}_b \\ \vdots \\ S(\theta_n, b) - \bar{S}_b
\end{pmatrix} = \begin{pmatrix} O_1(\theta_1, b) & O_2(\theta_1,b) & \cdots &
O_m(\theta_1,b) \\ O_1(\theta_2,b) & O_2(\theta_2,b) & \cdots & O_m(\theta_2,b) \\ \vdots & \vdots & \cdots & \vdots \\O_1(\theta_n,b) & O_2(\theta_n,b) & \cdots & O_m(\theta_n,b) \end{pmatrix} \begin{pmatrix} w_1 \\ w_2 \\ \vdots \\ w_m \end{pmatrix}
\begin{pmatrix} S(\theta_1) - \bar{S} \\
S(\theta_2) - \bar{S} \\ \vdots \\ S(\theta_n) - \bar{S}
\end{pmatrix} = \begin{pmatrix} O_1(\theta_1) & O_2(\theta_1) & \cdots &
O_m(\theta_1) \\ O_1(\theta_2) & O_2(\theta_2) & \cdots & O_m(\theta_2) \\ \vdots & \vdots & \cdots & \vdots \\O_1(\theta_n) & O_2(\theta_n) & \cdots & O_m(\theta_n) \end{pmatrix} \begin{pmatrix} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_m \end{pmatrix}
Lasso:
Minimize:
|| y - X\hat{\beta} ||^2
subject to:
\sum_i{|\hat{\beta_i}|} \leq t
Elastic net:
Minimize:
subject to:
w_j \geq 0
\sum_{i=1}^{n}{(S_i - \hat{S}_i)^2} + \Lambda \sum_{j=1}^{m}(\alpha w^2_j + (1-\alpha)|w_j| )
TissueFractionModel:
1 \frac{S}{S_0} = w_1 e^{-bADC} + C_{iso}
2 w_1 \lambda_1 + w_2 \lambda_2 = TF
3 w_1 + w_2 + w_3 = 1
Where w_i are the fractions of different tissue types (fiber, intracellular
water, free water) and \lambda_{i} are the "g ratio" for gray and white
matter (\lambda_1 \approx 0.15 and \lambda_2 \approx=0.6)
And the following should be approximately true:
C_{iso} = w_2 D_g + w_3 D_{csf}
Where D_g\approx1 and D_{csf}\approx 3 are the diffusivities of gray and white
matter, respectively.
\begin{pmatrix} \frac{S_1}{S_0} \\ \frac{S_2}{S_0} \\ \vdots \\ \frac{S_n}{S_0}
\\ TF \end{pmatrix} = \begin{pmatrix} \frac{T_1}{T_{max}} & S_g & S_{iso} \\ \frac{T_2}{T_{max}} & S_g & S_{iso} \\
\frac{T_n}{T_{max}} & S_g & S_{iso} \\ \vdots & \vdots & \vdots \\ \alpha_1 & \alpha_2 & 0
\end{pmatrix} \begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix}
TF = \alpha_1 w_1 + \alpha_2 w_2 \Rightarrow w_2 = \frac{TF - \alpha_1 w_1}{\alpha_2}
TF = w_1 + w_2 \Rightarrow w_2 = \frac{TF - w_1}{
g_f = \sqrt{1-\frac{MVF}{FVF}}\\
\Rightarrow MVF = (1-g_f^2) FVF
Geometrical diffusion measures from MRI for Tensor Basis Analysis (Westin et
al. ISMRM:
Linearity:
\frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2 + \lambda_3}
Planarity:
\frac{2 (\lambda_2-\lambda_3)}{\lambda_1 + \lambda_2 + \lambda_3}
Sphericity:
\frac{3 (\lambda_3)}{\lambda_1 + \lambda_2 + \lambda_3}
RMSE:
RMSE_{x \rightarrow y} = \sqrt{\frac{\sum_i^N(x_i-y_i)^2}{N}}
rRMSE = \frac{RMSE_{model \rightarrow signal}}{RMSE_{signal \rightarrow signal}}
Coefficient of determination:
R^2 = 1 - \frac{\sum_i{x_i-\hat{x}}}{\sum_i{x_i-\bar{x}}}
\sum_{i}^{n}{(s_i - \hat{s_i})^2} + \lambda \sum_{j}^{m}{(\alpha w_{j}^2 + (1-\alpha) |w_j|)}