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Package maths_fns :: Module correlation_time
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Module correlation_time

source code

Functions [hide private]
 
calc_iso_ti(data, diff_data)
Diffusional correlation times for isotropic diffusion.		 source code
 
calc_iso_dti(data, diff_data)
Partial derivatives of the diffusional correlation times.		 source code
 
calc_axial_ti(data, diff_data)
Diffusional correlation times.		 source code
 
calc_axial_dti(data, diff_data)
Diffusional correlation time gradients.		 source code
 
calc_axial_d2ti(data, diff_data)
Diffusional correlation time Hessians.		 source code
 
calc_aniso_ti(data, diff_data)
Diffusional correlation times.		 source code
 
calc_aniso_dti(data, diff_data)
Diffusional correlation time gradients.		 source code
 
calc_aniso_d2ti(data, diff_data)
Diffusional correlation time Hessians.		 source code

Imports: sqrt


Function Details [hide private]

calc_iso_ti(data, diff_data)

source code 

Diffusional correlation times for isotropic diffusion.

t0 = tm

calc_iso_dti(data, diff_data)

source code  
Partial derivatives of the diffusional correlation times.

The tm partial derivatives are:

    dt0
    ---  =  1
    dtm

calc_axial_ti(data, diff_data)

source code  
Diffusional correlation times.

The equations for the parameters {Dper, Dpar} are:

             1
    t0  =  -----
           6Dper

                1
    t1  =  ------------
           5Dper + Dpar

                 1
    t2  =  -------------
           2Dper + 4Dpar


The equations for the parameters {Diso, Da} are:

    t0   =  1/6 (Diso - Da)**-1

    t1   =  1/6 (Diso - Da/2)**-1

    t2   =  1/6 (Diso + Da)**-1

The diffusion parameter set in data.diff_params is {tm, Da, theta, phi}.

calc_axial_dti(data, diff_data)

source code  
Diffusional correlation time gradients.

tm partial derivatives
~~~~~~~~~~~~~~~~~~~~~~

    dt0        1 dDiso
    ---   =  - - ----- (Diso - Da)**-2
    dtm        6  dtm

    dt1        1 dDiso
    ---   =  - - ----- (Diso - Da/2)**-2
    dtm        6  dtm

    dt2        1 dDiso
    ---   =  - - ----- (Diso + Da)**-2
    dtm        6  dtm


    dDiso
    -----  =  -1/6 * tm**-2
     dtm


Da partial derivatives
~~~~~~~~~~~~~~~~~~~~~~

    dt0
    ---   =  1/6 (Diso - Da)**-2
    dDa

    dt1
    ---   =  1/12 (Diso - Da/2)**-2
    dDa

    dt2
    ---   =  -1/6 (Diso + Da)**-2
    dDa


The diffusion parameter set in data.diff_params is {tm, Da, theta, phi}.

calc_axial_d2ti(data, diff_data)

source code  
Diffusional correlation time Hessians.

tm-tm partial derivatives
~~~~~~~~~~~~~~~~~~~~~~~~~

    d2t0      1 / dDiso \ 2                     1 d2Diso
    ----   =  - | ----- |   (Diso - Da)**-3  -  - ------ (Diso - Da)**-2
    dtm2      3 \  dtm  /                       6  dtm2

    d2t1      1 / dDiso \ 2                       1 d2Diso
    ----   =  - | ----- |   (Diso - Da/2)**-3  -  - ------ (Diso - Da/2)**-2
    dtm2      3 \  dtm  /                         6  dtm2

    d2t2      1 / dDiso \ 2                     1 d2Diso
    ----   =  - | ----- |   (Diso + Da)**-3  -  - ------ (Diso + Da)**-2
    dtm2      3 \  dtm  /                       6  dtm2


    d2Diso
    ------  =  1/3 * tm**-3
     dtm2


tm-Da partial derivatives
~~~~~~~~~~~~~~~~~~~~~~~~~

     d2t0         1 dDiso
    -------  =  - - ----- (Diso - Da)**-3
    dtm.dDa       3  dtm

     d2t1         1 dDiso
    -------  =  - - ----- (Diso - Da/2)**-3
    dtm.dDa       6  dtm

     d2t2       1 dDiso
    -------  =  - ----- (Diso + Da)**-3
    dtm.dDa     3  dtm


Da-Da partial derivatives
~~~~~~~~~~~~~~~~~~~~~~~~~

    d2t0
    ----  =  1/3 (Diso - Da)**-3
    dDa2

    d2t1
    ----  =  1/12 (Diso - Da/2)**-3
    dDa2

    d2t2
    ----  =  1/3 (Diso + Da)**-3 
    dDa2


The diffusion parameter set in data.diff_params is {tm, Da, theta, phi}.

calc_aniso_ti(data, diff_data)

source code  
Diffusional correlation times.

The equations for the parameters {Diso, Da, Dr} are:

    t-2  =  1/6 (Diso + Da)**-1

    t-1  =  1/6 (Diso - (Da + Dr)/2)**-1

    t0   =  1/6 (Diso - mu)**-1

    t1   =  1/6 (Diso - (Da - Dr)/2)**-1

    t2   =  1/6 (Diso + mu)**-1

where:
           __________________
    mu = \/ Da**2 + Dr**2 / 3

The diffusion parameter set in data.diff_params is {tm, Da, Dr, alpha, beta, gamma}.

calc_aniso_dti(data, diff_data)

source code  
Diffusional correlation time gradients.

tm partial derivatives
~~~~~~~~~~~~~~~~~~~~~~

    dt-2       1 dDiso
    ----  =  - - ----- (Diso + Da)**-2
    dtm        6  dtm

    dt-1       1 dDiso
    ----  =  - - ----- (Diso - (Da + Dr)/2)**-2
    dtm        6  dtm

    dt0        1 dDiso
    ---   =  - - ----- (Diso - mu)**-2
    dtm        6  dtm

    dt1        1 dDiso
    ---   =  - - ----- (Diso - (Da - Dr)/2)**-2
    dtm        6  dtm

    dt2        1 dDiso
    ---   =  - - ----- (Diso + mu)**-2
    dtm        6  dtm


    dDiso
    -----  =  -1/6 * tm**-2
     dtm


Da partial derivatives
~~~~~~~~~~~~~~~~~~~~~~

    dt-2
    ----  =  -1/6 (Diso + Da)**-2
    dDa

    dt-1
    ----  =  1/12 (Diso - (Da + Dr)/2)**-2
    dDa

    dt0
    ---   =  1/6 Da/mu (Diso - mu)**-2
    dDa

    dt1
    ---   =  1/12 (Diso - (Da - Dr)/2)**-2
    dDa

    dt2
    ---   =  -1/6 Da/mu (Diso + mu)**-2
    dDa


Dr partial derivatives
~~~~~~~~~~~~~~~~~~~~~~

    dt-2
    ----  =  0
    dDr

    dt-1
    ----  =  1/12 (Diso - (Da + Dr)/2)**-2
    dDr

    dt0
    ---   =  1/18 Dr/mu (Diso - mu)**-2
    dDr

    dt1
    ---   =  -1/12 (Diso - (Da - Dr)/2)**-2
    dDr

    dt2
    ---   =  -1/18 Dr/mu (Diso + mu)**-2
    dDr

The diffusion parameter set in data.diff_params is {tm, Da, Dr, alpha, beta, gamma}.

calc_aniso_d2ti(data, diff_data)

source code  
Diffusional correlation time Hessians.

tm-tm partial derivatives
~~~~~~~~~~~~~~~~~~~~~~~~~

    d2t-2     1 / dDiso \ 2                     1 d2Diso
    -----  =  - | ----- |   (Diso + Da)**-3  -  - ------ (Diso + Da)**-2
    dtm2      3 \  dtm  /                       6  dtm2

    d2t-1     1 / dDiso \ 2                              1 d2Diso
    -----  =  - | ----- |   (Diso - (Da + Dr)/2)**-3  -  - ------ (Diso - (Da + Dr)/2)**-2
    dtm2      3 \  dtm  /                                6  dtm2

    d2t0      1 / dDiso \ 2                     1 d2Diso
    ----   =  - | ----- |   (Diso - mu)**-3  -  - ------ (Diso - mu)**-2
    dtm2      3 \  dtm  /                       6  dtm2

    d2t1      1 / dDiso \ 2                              1 d2Diso
    ----   =  - | ----- |   (Diso - (Da - Dr)/2)**-3  -  - ------ (Diso - (Da - Dr)/2)**-2
    dtm2      3 \  dtm  /                                6  dtm2

    d2t2      1 / dDiso \ 2                     1 d2Diso
    ----   =  - | ----- |   (Diso + mu)**-3  -  - ------ (Diso + mu)**-2
    dtm2      3 \  dtm  /                       6  dtm2


    d2Diso
    ------  =  1/3 * tm**-3
     dtm2


tm-Da partial derivatives
~~~~~~~~~~~~~~~~~~~~~~~~~

     d2t-2      1 dDiso
    -------  =  - ----- (Diso + Da)**-3
    dtm.dDa     3  dtm

     d2t-1        1 dDiso
    -------  =  - - ----- (Diso - (Da + Dr)/2)**-3
    dtm.dDa       6  dtm

     d2t0         1 dDiso Da
    -------  =  - - ----- -- (Diso - mu)**-3
    dtm.dDa       3  dtm  mu

     d2t1         1 dDiso
    -------  =  - - ----- (Diso - (Da - Dr)/2)**-3
    dtm.dDa       6  dtm

     d2t2       1 dDiso Da
    -------  =  - ----- -- (Diso + mu)**-3
    dtm.dDa     3  dtm  mu


tm-Dr partial derivatives
~~~~~~~~~~~~~~~~~~~~~~~~~

     d2t-2
    -------  =  0
    dtm.dDr

     d2t-1        1 dDiso
    -------  =  - - ----- (Diso - (Da + Dr)/2)**-3
    dtm.dDr       6  dtm

     d2t0         1 dDiso Dr
    -------  =  - - ----- -- (Diso - mu)**-3
    dtm.dDr       9  dtm  mu

     d2t1       1 dDiso
    -------  =  - ----- (Diso - (Da - Dr)/2)**-3
    dtm.dDr     6  dtm

     d2t2       1 dDiso Dr
    -------  =  - ----- -- (Diso + mu)**-3
    dtm.dDr     9  dtm  mu


Da-Da partial derivatives
~~~~~~~~~~~~~~~~~~~~~~~~~

    d2t-2
    -----  =  1/3 (Diso + Da)**-3
    dDa2

    d2t-1
    -----  =  1/12 (Diso - (Da + Dr)/2)**-3
    dDa2

    d2t0      1 Da**2                      1  / Da**2     \ 
    ----   =  - ----- (Diso - mu)**-3  -  --- | ----- - 1 | (Diso - mu)**-2
    dDa2      3 mu**2                     6mu \ mu**2     /

    d2t1
    ----   =  1/12 (Diso - (Da - Dr)/2)**-3
    dDa2

    d2t2      1 Da**2                      1  / Da**2     \ 
    ----   =  - ----- (Diso + mu)**-3  +  --- | ----- - 1 | (Diso + mu)**-2
    dDa2      3 mu**2                     6mu \ mu**2     /


Da-Dr partial derivatives
~~~~~~~~~~~~~~~~~~~~~~~~~

     d2t-2
    -------  =  0
    dDa.dDr

     d2t-1
    -------  =  1/12 (Diso - (Da + Dr)/2)**-3
    dDa.dDr

     d2t0        1 Da.Dr                     1  Da.Dr
    -------   =  - ----- (Diso - mu)**-3  -  -- ----- (Diso - mu)**-2
    dDa.dDr      9 mu**2                     18 mu**3

     d2t1
    -------   =  -1/12 (Diso - (Da - Dr)/2)**-3
    dDa.dDr

     d2t2        1 Da.Dr                     1  Da.Dr
    -------   =  - ----- (Diso + mu)**-3  +  -- ----- (Diso + mu)**-2
    dDa.dDr      9 mu**2                     18 mu**3


Dr-Dr partial derivatives
~~~~~~~~~~~~~~~~~~~~~~~~~

    d2t-2
    -----  =  0
    dDr2

    d2t-1
    -----  =  1/12 (Diso - (Da + Dr)/2)**-3
    dDr2

    d2t0      1  Dr**2                      1   / Dr**2      \ 
    ----   =  -- ----- (Diso - mu)**-3  -  ---- | ------ - 1 | (Diso - mu)**-2
    dDr2      27 mu**2                     18mu \ 3mu**2     /

    d2t1
    ----   =  1/12 (Diso - (Da - Dr)/2)**-3
    dDr2

    d2t2      1  Dr**2                      1   / Dr**2      \ 
    ----   =  -- ----- (Diso + mu)**-3  +  ---- | ------ - 1 | (Diso + mu)**-2
    dDr2      27 mu**2                     18mu \ 3mu**2     /


The diffusion parameter set in data.diff_params is {tm, Da, Dr, alpha, beta, gamma}.

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