equations.txt

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|)}