The time-dependence of exchange-induced relaxation during modulated radio frequency pulses

Quantitative Assessment of Water Pools by T1p and T2p MRI in Acute Cerebral Ischemia of the Rat

“…The time-dependent magnitude of the effective frequency vector is given by, $${\omega }_{\mathit{\text{eff}}}\left(t\right)=\sqrt{{\omega }_1^2\left(t\right)+{\left({\omega }_0-{\omega }_{RF}\left(t\right)\right)}^2}.$$ In accordance with the 2SX model, with two water populations ( P A and P B , where P A + P B = 1) coupled by equilibrium exchange, the rate constant describing longitudinal rotating frame relaxation ( R 1ρ,obs =( T 1ρ,obs ) -1 ) during an adiabatic pulse is given by (Michaeli et al 2004; Sorce et al 2006) $${\overline{R}}_{1\rho ,\text{italicobs}}=\frac{P_A}{T_P}\underset{0}{\overset{T_P}{\int }}{R}_{1\rho A,dd}\left(t\right)dt+\frac{P_B}{T_P}\underset{0}{\overset{T_P}{\int }}{R}_{1\rho B,dd}\left(t\right)dt+\frac{1}{T_P}\underset{0}{\overset{T_P}{\int }}{R}_{1\rho ,\text{italicex}}\left(t\right)dt,$$ where T P is the length of individual pulses. Contributions from dipolar fluctuations (first and second terms of Eq.…”

Rotating frame relaxation during adiabatic pulses vs. conventional spin lock: simulations and experimental results at 4 T

“…The formalism was initially formulated to describe relaxation during CW SL experiments, where ω eff and α are constant during the RF irradiation [1,2,16,17]. Recently, the theoretical expressions of R 1ρ ( t ) and R 2ρ ( t ) have been generalized to the case of time-dependent ω eff ( t ) and α( t ) [6,18,19]. It has been shown that the relaxations during adiabatic rotation in the weak field approximation could be represented as an average of instantaneous time-dependent contributions due to the different relaxation channels:$${\stackrel{‒}{R}}_{1,2\rho }=\frac{1}{T_{\mathrm{normalp}}}{\int }_0^{T_{\mathrm{normalp}}}{R}_{1,2\rho ,\mathrm{ex}}\left(t\right)dt+\frac{1}{T_{\mathrm{normalp}}}{\int }_0^{T_{\mathrm{normalp}}}{R}_{1,2\rho ,\mathrm{dd}}\left(t\right)dt,$$where R 1,2ρ,dd and R 1,2ρ,ex are the rotating frame relaxations due to dipolar interactions and anisochronous exchange, respectively.…”

“…The difference in chemical shifts between sites m and q is defined by δω in rad/s. Here k 1 and k -1 represent the forward and backward exchange rate constants obeying the McConnell relationship [21]:$${\tau }_{\mathrm{ex}}=\frac{P_{\mathrm{normalm}}}{k_{-1}}=\frac{P_{\mathrm{normalq}}}{k_1},$$and for anisochronous exchange in the fast regime$${\tau }_{\mathrm{ex}}=\frac{1}{k_{\mathrm{ex}}}<<\delta {\omega }^{-1}.$$Exchange-induced R 1ρ,ex and R 2ρ,ex under CW SL irradiation were derived in [1,2]:$$R_{1\rho ,\mathrm{ex}}=P_{\mathrm{normalm}}{P}_{\mathrm{normalq}}\delta {\omega }^2{\sin }^2\alpha \frac{{\tau }_{\mathrm{ex}}}{1+{\left({\omega }_{\mathrm{eff}}{\tau }_{\mathrm{ex}}\right)}^2},$$$$R_{2\rho ,\mathrm{ex}}=P_{\mathrm{normalm}}{P}_{\mathrm{normalq}}{\left(\delta \omega \right)}^{2}true[{\cos }^2\alpha {\tau }_{\mathrm{ex}}+\frac{1}{2}{\sin }^2\alpha \frac{{\tau }_{\mathrm{ex}}}{1+{\left({\omega }_{\mathrm{eff}}{\tau }_{\mathrm{ex}}\right)}^2}true].$$The corresponding time-dependent relaxation functions during adiabatic rotation were detailed in [19]. Theory shows R 1ρ,ex and R 2ρ,ex to be dependent on the choice of the amplitude- and frequencymodulation functions of adiabatic pulses via their α( t ) and ω eff ( t ) dependencies.…”

Exchange-induced relaxation in the presence of a fictitious field

“…Previous work from our laboratory focused on the derivation of exchange-induced relaxations during adiabatic pulses of the hyperbolic secant (HS n ) family [3], where the theoretical treatment of exchange-induced relaxation was given in the effective field frame, H 1 (t) [12, 13]. Following our nomenclature, H 1 is the effective field in first rotating frame (FRF), and thus upon convention the H 1 (here we adapted terminology used in [4]) is the vector sum of ω 1 (t) x⃗ ' and Δω ( t ) z⃗ '.…”

“…We note that the symbol ± indicates the taking of the hermitian adjoint, and in particular we adopt the convention: $${T_{1q}}^{\pm }={false(-1false)}^q{T}_{1-q}$$ Using Eqs. (15–19) and using the secular approximation [27], we find: $$\frac{d\widetilde{\tilde{\tilde{\sigma }}}false(tfalse)}{dt}=-\underset{0}{\overset{\infty }{\int }}true\{\delta false(tfalse)\delta false(t-\tau false)true\}\sum _{q_2=-1}^1\sum _{q_1=-1}^1\text{italicexp}false(i\tau q_2{H}_{2}false(tfalse)false){false({d_{q_1{q}_2}}^{false(1false)}false({\alpha }_{2}false(tfalse)false)false)}^2{false({d_{0q_1}}^{false(1false)}false({\alpha }_{1}false(tfalse)false)false)}^2{false(-1false)}^{q_2}false[T_{1q_2},false[T_{1-q_2},\widetilde{\tilde{\tilde{\sigma }}}false(tfalse)false]d\tau .$$ We assume an exponential dependence for the correlation function: $$false\{\delta false(tfalse)\delta false(t-\tau false)false\}\propto \text{italicexp}false(-\tau /{\tau }_{ex}false),$$ where τ ex is the lifetime of the spin in either of the two exchanging sites [13, 14]. Then, by carrying out the integration, we derive the spectral density function: $$Jfalse(q_2{\mathrm{H}}_{2}false(tfalse)false)=k_{ex}\frac{{\tau }_{ex}}{false(1+{false(false(q_2{H}_{}}^{}}$$…”