B14

The Baldwin 2014 2-site exact solution relaxation dispersion model for SQ CPMG-type data. This model is labelled as B14 in relax.

This model is not implemented yet

Equation

The paper main equation 50, needs following definitions

[math] v_{1c} = F_0\cosh{\left(\tau_{\textrm{CP}}E_0\right)}-F_2\cosh{\left(\tau_{\textrm{CP}}E_2\right)} \\ v_{1s} = F_0\sinh{\left(\tau_{\textrm{CP}}E_0\right)}-F_2\sinh{\left(\tau_{\textrm{CP}}E_2\right)} \\ v_{2}N = v_{1s}\left(O_B-O_A\right)+4O_B F_1^a \sinh{\left(\tau_{\textrm{CP}}E_1\right)} \\ p_D N = v_{1s} + \left(F_1^a+F_1^b\right)\sinh{\left(\tau_{\textrm{CP}}E_1\right)}\\ v_3 = \left( v_2^2 + 4 k_{\textrm{BA}} k_{\textrm{AB}} p_D^2 \right)^{1/2} \\ y = \left( \frac{v_{1c}-v_3}{v_{1c}+v_3} \right)^{N_{\textrm{CYC}}} [/math]

Note, [math] E_2 [/math] is complex. If |x| denotes the complex modulus:
[math] \cosh{\left(\tau_{\textrm{CP}}E_2\right)} = \cos{\left(\tau_{\textrm{CP}}|E_2|\right)} \\ \sinh{\left(\tau_{\textrm{CP}}E_2\right)} = \sin{\left(\tau_{\textrm{CP}}|E_2|\right)} [/math]

Equation compared to Carver Richards 72

Please see the relax summary of the model parameters here.

These definitions comes from the papers "Supplementary Section 4. Relation to Carver Richards equation".

[math] \tau_{\textrm{CP}} = \frac{1}{4\nu_\textrm{CPMG}} \\ \alpha_- = \Delta R_2 + k_{\textrm{AB}} - k_{\textrm{BA}} \\ \zeta = 2 \Delta \omega \, \alpha_- = h_1\\ \Psi = \alpha_-^2 + 4 k_{\textrm{BA}} k_{\textrm{AB}} - \Delta \omega^2 = h_2\\ \xi = \frac{2\tau_{\textrm{CP}}}{\sqrt{2}}\sqrt{\Psi + \sqrt{\Psi^2 + z^2}} = 2h_3 \tau_{\textrm{CP}} = \tau_{\textrm{CP}}E_0\\ \eta = \frac{2\tau_{\textrm{CP}}}{\sqrt{2}}\sqrt{-\Psi + \sqrt{\Psi^2 + z^2}} = 2h_4 \tau_{\textrm{CP}} = \tau_{\textrm{CP}}E_2\\ D_+=\frac{1}{2}\left(1+\frac{\Psi+2\Delta \omega^2}{\sqrt{\Psi^2+z^2}} \right) = F_0 \\ D_-=\frac{1}{2}\left(-1+\frac{\Psi+2\Delta \omega^2}{\sqrt{\Psi^2+z^2}} \right) = F_2 [/math]

Physical meanings:
[math]\xi[/math] and [math]\eta[/math] are differences of the real and imaginary components of the free precession frequencies [math]f_{00}[/math] and [math]f_{11}[/math] (equation (41)).
[math]D_+[/math] and [math]D_-[/math] are the stay/stay ([math]F_{0}[/math]), and swap/swap ([math]F_{2}[/math]) coefficients (equation (36)).
[math]\zeta[/math] and [math]\Psi[/math] are parameters that enable the free precession frequencies to be written in a more concise form (equation (12)).
Note that in reference 2 (10.1016/S0076-6879(01)39315-1), in equation 25, the definition used for [math]\tau_{\textrm{CP}}[/math] is twice that used in this work but is otherwise identical.

Parameters

The B14 model has the parameters

Reference

The reference for the B14 model is:

• A.J. Baldwin (2014). An exact solution for R2,eff in CPMG experiments in the case of two site chemical exchange. J. Magn. Reson., 2014. (10.1016/j.jmr.2014.02.023).

Related models

The B14 model is a linear correction to the CR72 model, and algorithms based on this have significant advantages in both precision and speed over existing formulaic approaches.

Links

The implementation of the CR72 model in relax can be seen in the: