All supported fit models and the corresponding settings are listed here.
Mono-Exponential:
$$ S(b) = \exp (-b_{values} \cdot D_1) \cdot S_0 $$
The mono-exponential model uses $S_0$ instead of $f_1$.
Bi-Exponential:
$$ S(b) = f_1 \exp (-b_{values} \cdot D_1) + f_2 \exp(-b_{values} \cdot D_2)$$
Tri-Exponential:
$$ S(b) = f_1 \exp (-b_{values} \cdot D_1) + f_2 \exp(-b_{values} \cdot D_2) + f_3 \exp(-b_{values} \cdot D_3)$$
To use the model containg $S_0$ set fit_s0 to true and add boundaries accordingly Example File
.
Bi-Exponential:
$$ S(b) = (f_1 \exp (-b_{values} \cdot D_1) + (1 - f_1) \exp(-b_{values} \cdot D_2)) * S_0$$
Tri-Exponential:
$$ S(b) = (f_1 \exp (-b_{values} \cdot D_1) + f_2 \exp(-b_{values} \cdot D_2) + (1 - f_1 - f_2) \exp(-b_{values} \cdot D_3)) * S_0$$
To use the reduced fitting model set fit_reduced to true and remove the related fraction or $S_0$ boundaries. Example File
Mono-Exponential:
$$ S(b) = \exp (-b_{values} \cdot D_1)$$
Bi-Exponential:
$$ S(b) = f_1 \exp (-b_{values} \cdot D_1) + (1 - f_1) \exp(-b_{values} \cdot D_2)$$
Tri-Exponential:
$$ S(b) = f_1 \exp (-b_{values} \cdot D_1) + f_2 \exp(-b_{values} \cdot D_2) + (1 - f_1 - f_2) \exp(-b_{values} \cdot D_3)$$
All models get an extra $T_1$ Fitting Term at the end of the equation.
For fitting $T_1$ aswell set fit_t1 to true and add related boundaries. Example File
Mono-Exponential:
$$ S(b) = \exp (-b_{values} \cdot D_1) \cdot S_0 \cdot \exp (\frac{-T_1}{t_{mix}})$$
The NNLS algorithm performs a fitting without knowing the exact number of components present in the data.
Data points $y_i$ are modeled using fractions $s_i$ and the matrix containing the exponentials $A_{ij}$.
$$ y_i = \sum^{M}_{j=1} A_{ij} s_{j} \qquad i = 1,2,\ldots, N$$
The algorithm tries to minimize the error for model and data.
$$ \chi^{2} = min \left[ \sum^{N}_{i=1} \left\vert \sum^{M}_{j=1} A_{ij}s_j - y_i \right\vert^{2} \right] $$
It's also prossible to use regularization term to get a more physilogical representation of the signal. For the second order this results in:
$$ \chi^{2} = min \left[ \sum^{N}_{i=1} \left\vert \sum^{M}_{j=1} A_{ij}s_j - y_i \right\vert^{2} + \mu \sum^{M}_{j=1} \vert s_{j+2} - 2 s_j+1 + s_j \vert^{2} \right] $$
For more information see: Whittall (1989), Lawson (1995), Stabinska (2021), Periquito (2021), Bjarnason (2010).