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Package maths_fns :: Module correlation_time
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Module correlation_time

source code

Functions [hide private]
 
calc_sphere_ti(data, diff_data)
Diffusional correlation times.		 source code
 
calc_sphere_dti(data, diff_data)
Partial derivatives of the diffusional correlation times.		 source code
 
calc_spheroid_ti(data, diff_data)
Diffusional correlation times.		 source code
 
calc_spheroid_dti(data, diff_data)
Diffusional correlation time gradients.		 source code
 
calc_spheroid_d2ti(data, diff_data)
Diffusional correlation time Hessians.		 source code
 
calc_ellipsoid_ti(data, diff_data)
Diffusional correlation times.		 source code
 
calc_ellipsoid_dti(data, diff_data)
Diffusional correlation time gradients.		 source code
 
calc_ellipsoid_d2ti(data, diff_data)
Diffusional correlation time Hessians.		 source code

Imports: sqrt


Function Details [hide private]

calc_sphere_ti(data, diff_data)

source code  
Diffusional correlation times.

The correlation time is

    t0 = tm.

calc_sphere_dti(data, diff_data)

source code  
Partial derivatives of the diffusional correlation times.

The tm partial derivatives are

    dt0
    ---  =  1.
    dtm

calc_spheroid_ti(data, diff_data)

source code  
Diffusional correlation times.

The equations for the parameters {Diso, Da} are

    t-1  =  (6Diso - 2Da)**-1,

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

    t1   =  (6Diso + 2Da)**-1,

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

calc_spheroid_dti(data, diff_data)

source code  
Diffusional correlation time gradients.

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

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

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

    dt1         dDiso
    ---   =  -6 ----- (6Diso + 2Da)**-2.
    dtm          dtm


As

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

the equations simplify to

    dt-1
    ----  =  tm**-2 (6Diso - 2Da)**-2,
    dtm

    dt0
    ---   =  tm**-2 (6Diso - Da)**-2,
    dtm

    dt1
    ---   =  tm**-2 (6Diso + 2Da)**-2.
    dtm


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

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

    dt0
    ---   =  (6Diso - Da)**-2,
    dDa

    dt1
    ---   =  -2(6Diso + 2Da)**-2.
    dDa


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

calc_spheroid_d2ti(data, diff_data)

source code  
Diffusional correlation time Hessians.

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

    d2t-1        / dDiso \ 2                         d2Diso
    -----  =  72 | ----- |   (6Diso - 2Da)**-3  -  6 ------ (6Diso - 2Da)**-2,
    dtm2         \  dtm  /                            dtm2

    d2t0         / dDiso \ 2                        d2Diso
    ----   =  72 | ----- |   (6Diso - Da)**-3  -  6 ------ (6Diso - Da)**-2,
    dtm2         \  dtm  /                           dtm2

    d2t1         / dDiso \ 2                         d2Diso
    ----   =  72 | ----- |   (6Diso + 2Da)**-3  -  6 ------ (6Diso + 2Da)**-2.
    dtm2         \  dtm  /                            dtm2


As

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

and

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

the equations simplify to

    d2t-1
    -----  =  2tm**-4 (6Diso - 2Da)**-3  -  2tm**-3 (6Diso - 2Da)**-2,
    dtm2

    d2t0
    ----   =  2tm**-4 (6Diso - Da)**-3  -  2tm**-3 (6Diso - Da)**-2,
    dtm2

    d2t1
    ----   =  2tm**-4 (6Diso + 2Da)**-3  -  2tm**-3 (6Diso + 2Da)**-2.
    dtm2


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

     d2t-1          dDiso
    -------  =  -24 ----- (6Diso - 2Da)**-3,
    dtm.dDa          dtm

     d2t0           dDiso
    -------  =  -12 ----- (6Diso - Da)**-3,
    dtm.dDa          dtm

     d2t1          dDiso
    -------  =  24 ----- (6Diso + 2Da)**-3.
    dtm.dDa         dtm

As

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

the equations simplify to

     d2t-1
    -------  =  4tm**-2 (6Diso - 2Da)**-3,
    dtm.dDa

     d2t0
    -------  =  2tm**-2 (6Diso - Da)**-3,
    dtm.dDa

     d2t1
    -------  =  -4tm**-2 (6Diso + 2Da)**-3.
    dtm.dDa



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

    d2t-1
    -----  =  8 (6Diso - 2Da)**-3,
    dDa2

    d2t0
    ----  =  2 (6Diso - Da)**-3,
    dDa2

    d2t1
    ----  =  8 (6Diso + 2Da)**-3. 
    dDa2


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

calc_ellipsoid_ti(data, diff_data)

source code  
Diffusional correlation times.

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

    t-2  =  (6Diso - 2DaR)**-1,

    t-1  =  (6Diso - Da(1 + 3Dr))**-1,

    t0   =  (6Diso - Da(1 - 3Dr))**-1,

    t1   =  (6Diso + 2Da)**-1,

    t2   =  (6Diso + 2DaR)**-1,

where:
          __________
    R = \/1 + 3Dr**2.

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

calc_ellipsoid_dti(data, diff_data)

source code  
Diffusional correlation time gradients.

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

    dt-2         dDiso
    ----  =  - 6 ----- (6Diso - 2DaR)**-2,
    dtm           dtm

    dt-1         dDiso
    ----  =  - 6 ----- (6Diso - Da(1 + 3Dr))**-2,
    dtm           dtm

    dt0          dDiso
    ---   =  - 6 ----- (6Diso - Da(1 - 3Dr))**-2,
    dtm           dtm

    dt1          dDiso
    ---   =  - 6 ----- (6Diso + 2Da)**-2,
    dtm           dtm

    dt2          dDiso
    ---   =  - 6 ----- (6Diso + 2DaR)**-2.
    dtm           dtm


As

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

the equations simplify to

    dt-2
    ----  =  tm**-2 (6Diso - 2DaR)**-2,
    dtm

    dt-1
    ----  =  tm**-2 (6Diso - Da(1 + 3Dr))**-2,
    dtm

    dt0
    ---   =  tm**-2 (6Diso - Da(1 - 3Dr))**-2,
    dtm

    dt1
    ---   =  tm**-2 (6Diso + 2Da)**-2,
    dtm

    dt2
    ---   =  tm**-2 (6Diso + 2DaR)**-2.
    dtm


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

    dt-2
    ----  =  2R (6Diso - 2DaR)**-2,
    dDa

    dt-1
    ----  =  (1 + 3Dr) (6Diso - Da(1 + 3Dr))**-2,
    dDa

    dt0
    ---   =  (1 - 3Dr) (6Diso - Da(1 - 3Dr))**-2,
    dDa

    dt1
    ---   =  -2 (6Diso + 2Da)**-2,
    dDa

    dt2
    ---   =  -2R (6Diso + 2DaR)**-2.
    dDa


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

    dt-2
    ----  =  6 Da.Dr/R (6Diso - 2DaR)**-2,
    dDr

    dt-1
    ----  =  3Da (6Diso - Da(1 + 3Dr))**-2,
    dDr

    dt0
    ---   =  -3Da (6Diso - Da(1 - 3Dr))**-2,
    dDr

    dt1
    ---   =  0,
    dDr

    dt2
    ---   =  -6 Da.Dr/R (6Diso + 2DaR)**-2.
    dDr

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

calc_ellipsoid_d2ti(data, diff_data)

source code  
Diffusional correlation time Hessians.

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

    d2t-2        / dDiso \ 2                          d2Diso
    -----  =  72 | ----- |   (6Diso - 2DaR)**-3  -  6 ------ (6Diso - 2DaR)**-2,
    dtm2         \  dtm  /                             dtm2

    d2t-1        / dDiso \ 2                                 d2Diso
    -----  =  72 | ----- |   (6Diso - Da(1 + 3Dr))**-3  -  6 ------ (6Diso - Da(1 + 3Dr))**-2,
    dtm2         \  dtm  /                                    dtm2

    d2t0         / dDiso \ 2                                 d2Diso
    ----   =  72 | ----- |   (6Diso - Da(1 - 3Dr))**-3  -  6 ------ (6Diso - Da(1 - 3Dr))**-2,
    dtm2         \  dtm  /                                    dtm2

    d2t1         / dDiso \ 2                         d2Diso
    ----   =  72 | ----- |   (6Diso + 2Da)**-3  -  6 ------ (6Diso + 2Da)**-2,
    dtm2         \  dtm  /                            dtm2

    d2t2         / dDiso \ 2                          d2Diso
    ----   =  72 | ----- |   (6Diso + 2DaR)**-3  -  6 ------ (6Diso + 2DaR)**-2.
    dtm2         \  dtm  /                             dtm2

As

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

and

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

the equations simplify to

    d2t-2
    -----  =  2tm**-4 (6Diso - 2DaR)**-3  -  2tm**-3 (6Diso - 2DaR)**-2,
    dtm2

    d2t-1
    -----  =  2tm**-4 (6Diso - Da(1 + 3Dr))**-3  -  2tm**-3 (6Diso - Da(1 + 3Dr))**-2,
    dtm2

    d2t0
    ----   =  2tm**-4 (6Diso - Da(1 - 3Dr))**-3  -  2tm**-3 (6Diso - Da(1 - 3Dr))**-2,
    dtm2

    d2t1
    ----   =  2tm**-4 (6Diso + 2Da)**-3  -  2tm**-3 (6Diso + 2Da)**-2,
    dtm2

    d2t2
    ----   =  2tm**-4 (6Diso + 2DaR)**-3  -  2tm**-3 (6Diso + 2DaR)**-2.
    dtm2



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

     d2t-2           dDiso
    -------  =  -24R ----- (6Diso - 2DaR)**-3,
    dtm.dDa           dtm

     d2t-1                    dDiso
    -------  =  -12(1 + 3Dr) ----- (6Diso - Da(1 + 3Dr))**-3,
    dtm.dDa                    dtm

     d2t0                     dDiso
    -------  =  -12(1 - 3Dr) ----- (6Diso - Da(1 - 3Dr))**-3,
    dtm.dDa                    dtm

     d2t1          dDiso
    -------  =  24 ----- (6Diso + 2Da)**-3,
    dtm.dDa         dtm

     d2t2           dDiso
    -------  =  24R ----- (6Diso + 2DaR)**-3.
    dtm.dDa          dtm

As

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

the equations simplify to

     d2t-2
    -------  =  4R tm**-2 (6Diso - 2DaR)**-3,
    dtm.dDa

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

     d2t0
    -------  =  2(1 - 3Dr) tm**-2 (6Diso - Da(1 - 3Dr))**-3,
    dtm.dDa

     d2t1
    -------  =  -4 tm**-2 (6Diso + 2Da)**-3,
    dtm.dDa

     d2t2
    -------  =  -4R tm**-2 (6Diso + 2DaR)**-3.
    dtm.dDa


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

     d2t-2                  dDiso
    -------  =  -72 Da.Dr/R ----- (6Diso - 2DaR)**-3,
    dtm.dDr                  dtm

     d2t-1             dDiso
    -------  =  -36 Da ----- (6Diso - Da(1 + 3Dr))**-3,
    dtm.dDr             dtm

     d2t0             dDiso
    -------  =  36 Da ----- (6Diso - Da(1 - 3Dr))**-3,
    dtm.dDr            dtm

     d2t1
    -------  =  0,
    dtm.dDr

     d2t2                  dDiso
    -------  =  72 Da.Dr/R ----- (6Diso + 2DaR)**-3.
    dtm.dDr                 dtm

As

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

the equations simplify to

     d2t-2
    -------  =  12 Da.Dr/R tm**-2 (6Diso - 2DaR)**-3,
    dtm.dDr

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

     d2t0
    -------  =  -6 Da tm**-2 (6Diso - Da(1 - 3Dr))**-3,
    dtm.dDr

     d2t1
    -------  =  0,
    dtm.dDr

     d2t2
    -------  =  -12 Da.Dr/R tm**-2 (6Diso + 2DaR)**-3.
    dtm.dDr


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

    d2t-2
    -----  =  8R**2 (6Diso - 2DaR)**-3,
    dDa2

    d2t-1
    -----  =  2(1 + 3Dr)**2 (6Diso - Da(1 + 3Dr))**-3,
    dDa2

    d2t0
    ----   =  2(1 - 3Dr)**2 (6Diso - Da(1 - 3Dr))**-3,
    dDa2

    d2t1
    ----   =  8(6Diso + 2Da)**-3,
    dDa2

    d2t2
    ----   =  8R**2 (6Diso + 2DaR)**-3.
    dDa2


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

     d2t-2
    -------  =  24Da.Dr (6Diso - 2DaR)**-3  +  6Dr/R (6Diso - 2DaR)**-2,
    dDa.dDr

     d2t-1
    -------  =  6Da (1 + 3Dr)(6Diso - Da(1 + 3Dr))**-3  +  3(6Diso - Da(1 + 3Dr))**-2,
    dDa.dDr

     d2t0
    -------   =  -6Da (1 - 3Dr)(6Diso - Da(1 - 3Dr))**-3  -  3(6Diso - Da(1 - 3Dr))**-2,
    dDa.dDr

     d2t1
    -------   =  0,
    dDa.dDr

     d2t2
    -------   =  24Da.Dr (6Diso + 2DaR)**-3  -  6Dr/R (6Diso + 2DaR)**-2.
    dDa.dDr


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

    d2t-2
    -----  =  72(Da.Dr/R)**2 (6Diso - 2DaR)**-3  +  6Da/R**3 (6Diso - 2DaR)**-2,
    dDr2

    d2t-1
    -----  =  18Da**2 (6Diso - Da(1 + 3Dr))**-3,
    dDr2

    d2t0
    ----   =  18Da**2 (6Diso - Da(1 - 3Dr))**-3,
    dDr2

    d2t1
    ----   =  0,
    dDr2

    d2t2
    ----   =  72(Da.Dr/R)**2 (6Diso + 2DaR)**-3  -  6Da/R**3 (6Diso + 2DaR)**-2.
    dDr2


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

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