M can be calculated from P by diagonalisation to obtain PD, and t

M can be calculated from P by diagonalisation to obtain PD, and then transforming it with its matrix of Eigenvectors A, according to: equation(5) M=PNcyc=(APDA-1)Ncyc=APDNcycA-1 The CPMG element P consists of two concatenated Hahn echoes, H, each of which consists of two equal delays of duration τcp, separated by a 180° pulse (Eq. (30)): equation(6) H=O*OH=O*O The effect of a single CPMG unit GSI-IX price is then given by equation(7) P=H*H=OO*O*OP=H*H=OO*O*Oas

derived in Eq. (42), from which M can be calculated using Eq. (5) (Eq. (46)). As implicitly assumed by Carver and Richards, the effects of chemical exchange during signal detection will be neglected (though this assumption can be removed– see Supplementary Section 7), and IG(Trel) calculated from: equation(8) IG(Trel)=M(0,0)PG+M(0,1)PEIG(Trel)=M(0,0)PG+M(0,1)PEwhere 0, 0 and 0, 1 specify the required matrix elements of M. Insertion of this result into Eq. (1) gives the final result for R2,eff (Eq. (50)), summarised selleck products in Appendix A. Combining the matrix Eq. (46) with the results in Supplementary Section 7 to give R2,eff including the effects of chemical exchange during detection will further improve the theoretical description of the experiment [41]. The free precession matrix R+ can be related to its diagonalised form RD via the transformation R = JRDJ−1

such that: equation(9) O=eR+t=eJRDJ-1=JeRD+tJ-1 From which it follows that the matrix exponential is given in terms of two characteristic frequencies, the Eigenvalues f00 and f11, corresponding to the ground and excited state ensembles respectively: equation(10) eRD+t=e-tR2Ge-tf0000e-tf11 A factor of R2G has been factored from both f00 and f11, which allows us to express them conveniently in terms of the difference in relaxation,

ΔR2 = R2E − R2G in what follows and so: equation(11) f00=12(ΔR2+kEX+iΔω)-12h2+ih1f11=12(ΔR2+kEX+iΔω)+12h2+ih1where h1=2Δω(ΔR2+kEG-kGE)h1=2Δω(ΔR2+kEG-kGE) equation(12) h2=(ΔR2+kEG-kGE)2+4kEGkGE-Δω2h2=(ΔR2+kEG-kGE)2+4kEGkGE-Δω2 The identity h2+ih1=h3+ih4, enables us to explicitly separate the real and the imaginary components of the Eigenvalues: h3=12h2+h12+h22 equation(13) h4=12-h2+h12+h22 In terms of these substitutions, Liothyronine Sodium f00 and f11 are then succinctly expressed as: equation(14) f00=12(ΔR2+kEX-h3)+i2(Δω-h4)f11=12(ΔR2+kEX+h3)+i2(Δω+h4) The real part of the two Eigenvalues, f  00R   and f  11R   describe the effective relaxation rates of the two ensembles, and the imaginary parts f  00I   and f  11I   define the frequencies where the resonance will ultimately be observed. The imaginary component, f  00I   denotes the exchange-induced shift in the observed position of the ground state resonance [24]. The following useful sum and difference relations: equation(15) f11R+f00R=ΔR2+kEXf11I+f00I=Δωf11R-f00R=h3f11I-f00I=h4play an important role in the CPMG experiment and emerge explicitly as arguments of trigonometric terms in the final expression for R  2,eff   (Eq. (41)).

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