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Package lib :: Package diffusion :: 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
Variables [hide private]
  __package__ = 'lib.diffusion'

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

The equations are:

   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

The equations are:

   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

The equations are:

   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

The equations are:

    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

The equations are:

   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

The equations are:

   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

The equations are:

   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

The equations are:

   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

The equations are:

   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

The equations are:

    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

The equations are:

    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

The equations are:

   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

The equations are:

    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

The equations are:

   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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