Module cr72
source code
The Carver and Richards (1972) 2-site all time scale exchange CR72 and CR72 full
models.
Description
This module is for the function, gradient and Hessian of the CR72 and CR72 full
models.
References
The model is named after the reference:
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Carver, J. P. and Richards, R. E. (1972). General 2-site solution
for chemical exchange produced dependence of T2 upon Carr-Purcell
pulse separation. J. Magn. Reson., 6, 89-105. (DOI: 10.1016/0022-2364(72)90090-X).
Equations
The equation used is:
R2eff = 1/2 [ R2A0 + R2B0 + kex - 2.nu_cpmg.cosh^-1 (D+.cosh(eta+) - D-.cos(eta-)) ] ,
where:
1 / Psi + 2delta_omega^2 \
D+/- = - | +/-1 + -------------------- | ,
2 \ sqrt(Psi^2 + zeta^2) /
1
eta+/- = 2^(-3/2) . -------- sqrt(+/-Psi + sqrt(Psi^2 + zeta^2)) ,
nu_cpmg
Psi = (R2A0 - R2B0 - pA.kex + pB.kex)^2 - delta_omega^2 + 4pA.pB.kex^2 ,
zeta = 2delta_omega (R2A0 - R2B0 - pA.kex + pB.kex).
kex is the chemical exchange rate constant, pA and pB are the
populations of states A and B, and delta_omega is the chemical shift
difference between the two states in ppm.
CR72 model
Importantly for the implementation of this model, it is assumed
that R2A0 and R2B0 are identical. This simplifies some of the
equations to:
R2eff = R20 + kex/2 - nu_cpmg.cosh^-1 (D+.cosh(eta+) - D-.cos(eta-) ,
where:
Psi = kex^2 - delta_omega^2 ,
zeta = -2delta_omega (pA.kex - pB.kex).
Links
More information on the CR72 model can be found in the:
More information on the CR72 full model can be found in the:
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r2eff_CR72(r20a=None,
r20b=None,
pA=None,
dw=None,
kex=None,
cpmg_frqs=None,
back_calc=None,
num_points=None)
Calculate the R2eff values for the CR72 model. |
source code
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eta_scale = 0.353553390593
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__package__ = ' lib.dispersion '
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Imports:
arccosh,
array,
cos,
cosh,
isfinite,
min,
max,
sqrt,
sum
r2eff_CR72(r20a=None,
r20b=None,
pA=None,
dw=None,
kex=None,
cpmg_frqs=None,
back_calc=None,
num_points=None)
| source code
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Calculate the R2eff values for the CR72 model.
See the module docstring for details.
- Parameters:
r20a (float) - The R20 parameter value of state A (R2 with no exchange).
r20b (float) - The R20 parameter value of state B (R2 with no exchange).
pA (float) - The population of state A.
dw (float) - The chemical exchange difference between states A and B in rad/s.
kex (float) - The kex parameter value (the exchange rate in rad/s).
cpmg_frqs (numpy rank-1 float array) - The CPMG nu1 frequencies.
back_calc (numpy rank-1 float array) - The array for holding the back calculated R2eff values. Each
element corresponds to one of the CPMG nu1 frequencies.
num_points (int) - The number of points on the dispersion curve, equal to the length
of the cpmg_frqs and back_calc arguments.
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