Scientists interested in the Multi-component Driven Equilibrium Single Pulse Observation of T₁ and T₂ (mcDESPOT) approach to tissue analysisCombining spoiled gradient recalled echo (SPGR) and balanced steady state free precession (bSSFP) imaging sequences, as in mcDESPOT (1-2), results in relatively high signal-to-noise ratio (SNR) and rapid image acquisitions. This facilitates mapping of component fractions and corresponding relaxation times. However, the magnetization relaxation that occurs between the pulse preceding signal acquisition and the acquisition itself is neglected in the current treatments of mcDESPOT; that is, nonzero TE effects are neglected in the analysis of both the SPGR and bSSFP signals. Here, we present formulations of the SPGR and bSSFP models that account for nonzero TEs. Numerical simulations are presented to determine the impact of neglecting nonzero TEs on derived parameter estimates from conventional mcDESPOT.

Under conditions of chemical exchange equilibrium between two pools representing respectively a slowly relaxing T₂ species, S, and a fast-relaxing species, F, the steady-state (SS) magnetization, M^SS, immediately preceding each radio-frequency (RF) pulse in a SS sequence is given by

$$\it \mathbf{M}^{SS}=\left(\begin{array}{c}\mathbf{I-\exp\left(\begin{array}{c}\mathbf{A}\cdot\mathit{TR}\end{array}\right)}\mathbf{R}\left(\begin{array}{c}\alpha\end{array}\right)\end{array}\right)^{-1} \left(\begin{array}{c}\mathbf{\exp\left(\begin{array}{c}\mathbf{A}\cdot\mathit{TR}\end{array}\right)-I}\end{array}\right)\mathbf{A}^{-1}\mathbf{C}, [1]$$,

with C = f(M₀, f_F, f_S, T_1,F, T_1,S) and A = f(T_1,F, T_1,S,T_2,F, T_2,S, k_SF, k_FS, θ_RF, Δω, TR), where M₀ is the signal amplitude at TE = 0 ms, TR is the repetition time, θ_RF is the phase increment of the excitation pulse, α is the flip angle (FA), f_F and f_S are the fractions of the fast and slow components, respectively, k_SF and k_FS are the exchange rates between pools, and Δω is the off-resonance frequency. R(α) is a rotation matrix about the RF axis and I is the identity matrix.

In the conventional implementation of mcDESPOT (1-2), the two-component bSSFP signal is taken as

$$S_{Conv}^{SS}=\mid(M_{x,S}^{SS}+M_{x,F}^{SS})+i(M_{y,S}^{SS}+M_{y,F}^{SS})\mid. [2]$$

However, Eqs. 1-2 do not account for the relaxation occurring between the excitation pulse and the subsequent acquisition (3) that occurs at TE_bSSFP = TR_bSSFP/2. The corrected bSSFP magnetization incorporating relaxation time is given by:

$$\it \mathbf{M}_{Corr}^{bSSFP}=\exp\left(\begin{array}{c}\mathbf{A^*}\cdot\mathit{TE_{bSSFP}}\end{array}\right)\mathbf{R}\left(\begin{array}{c}\alpha\end{array}\right)\mathbf{M}^{SS}+\left(\begin{array}{c}\exp\left(\begin{array}{c}\mathbf{A^*\cdot\mathit{TE_{bSSFP}}}\end{array}\right)-\mathbf{I}\end{array}\right)\mathbf{A^*}^{-1}\mathbf{C}, [3]$$

where A* is similar to A but with TR replaced by TE_bSSFP. The corrected bSSFP signal is given by:

$$S_{Corr}^{bSSFP}=\mid(M_{Corr,x,S}^{bSSFP}+M_{Corr,x,F}^{bSSFP})+i(M_{Corr,y,S}^{bSSFP}+M_{Corr,y,F}^{bSSFP})\mid. [4]$$

Similarly, for spoiled SS (SSS) transverse magnetization, M^SSS, as in the case of SPGR, the magnetization immediately following each RF pulse is given by:

$$\it \mathbf{M}^{SSS}=\sin\alpha\left(\begin{array}{c}\mathbf{I-\exp\left(\begin{array}{c}\mathbf{B}\cdot\mathit{TR_{SPGR}}\end{array}\right)\cos\alpha}\end{array}\right)^{-1} \left(\begin{array}{c}\mathbf{I-\exp\left(\begin{array}{c}\mathbf{B}\cdot\mathit{TR_{SPGR}}\end{array}\right)}\end{array}\right)\mathbf{D}, [5]$$

with D = f(M₀, f_F, f_S) and B = f(T_1,F, T_1,S, k_SF, k_FS). In the conventional mcDESPOT, the two-component SPGR signal is taken as

$$S^{SSS}=\mid{M_{z,S}^{SSS}+M_{z,F}^{SSS}}\mid. [6]$$

However, Eqs. 5-6 must be modified to account for relaxation between the excitation and acquisition; the corrected SPGR magnetization is described by:

$$\it \mathbf{M}_{Corr}^{SPGR}=\exp\left(\begin{array}{c}\mathbf{A}\cdot\mathit{TE_{SPGR}}\end{array}\right)\mathbf{R}\left(\begin{array}{c}\alpha\end{array}\right)\mathbf{S}\mathbf{M}_{Corr}^{SSS}+\mathbf{S}\left(\begin{array}{c}\exp\left(\begin{array}{c}\mathbf{A\cdot\mathit{TE_{SPGR}}}\end{array}\right)-\mathbf{I}\end{array}\right)\mathbf{A}^{-1}\mathbf{C}, [7]$$

where S is a spoiler matrix and $$$\it \mathbf{M}_{Corr}^{SSS}= \left(\begin{array}{c}\mathbf{I}-\mathbf{R}\left(\begin{array}{c}\alpha\end{array}\right)\exp\left(\begin{array}{c}\mathbf{A}\cdot\mathit{TR_{SPGR}}\end{array}\right)\mathbf{S}\end{array}\right)^{-1}\mathbf{S}\left(\begin{array}{c}\left(\begin{array}{c}\exp\left(\begin{array}{c}\mathbf{A\cdot\mathit{TR_{SPGR}}}\end{array}\right)-\mathbf{I}\end{array}\right)\mathbf{A}^{-1}\mathbf{C}\end{array}\right)$$$, from which the corrected SPGR signal is given by:

$$S_{Corr}^{SPGR}=\mid(M_{Corr,x,S}^{SPGR}+M_{Corr,x,F}^{SPGR})+i(M_{Corr,y,S}^{SPGR}+M_{Corr,y,F}^{SPGR})\mid. [8]$$

Effects of nonzero TE_SPGR: Corrected SPGR signals were simulated for values of TE_SPGR ranging from 0.5 to 6 ms in 0.25 ms increments (Eq. 8), and corrected bSSFP signals were generated with TE_bSSFP = 3.5 ms and θ_RF = 0 or π (Eq. 4). Signals were generated for values of f_F ranging from 0.025 to 0.5 in increments of 0.025. For each TE_SPGR and f_F combination, corrected SPGR and bSSFP signals were then fitted simultaneously to the corrected models (Eqs. 8 and 4), explicitly accounting for nonzero TEs (TEC-mcDESPOT), or to the conventional models (Eqs. 6 and 2), neglecting nonzero TEs (Conventional mcDESPOT).

Effects of nonzero TE_bSSFP: Corrected SPGR signals were simulated with a fixed TE_SPGR value of 2 ms. Corrected bSSFP signals were generated for values of TE_bSSFP ranging from 1.5 to 6 ms in 0.25 ms increments and for θ_RF = 0 or π. Corrected SPGR and bSSFP signals were generated for values of f_F ranging from 0.025 to 0.5 in increments of 0.025. For each TE_bSSFP and f_F combination, corrected SPGR and bSSFP signals were then fitted simultaneously to the corrected models (TEC-mcDESPOT), or to the conventional models (Conventional mcDESPOT).

Figs.1-2, top rows, show that data simulated with nonzero TEs and fit with TEC-mcDESPOT provide true input values for all parameters; this indicates stability of the fitting procedure. In contrast, when the signals were fitted with Conventional-mcDESPOT, significant errors in all derived parameter estimates were observed. In fact, neglect of nonzero TEs leads to signal models that overestimate SPGR and underestimate bSSFP signal intensities. This results in substantial errors in all mcDESPOT-derived parameter estimates. We have shown that neglect of nonzero relaxation times can result in substantial errors in mcDESPOT-derived parameter estimates and propose the use of the SPGR and bSSFP signal models described herein for quantitative analyses using mcDESPOT.