A hybrid numeric-analytic solution for pulsed CEST

Abstract

Chemical exchange saturation transfer (CEST) methods measure the effect of magnetization exchange between solutes and water. While CEST methods are often implemented using a train of off-resonant shaped rf pulses, they are typically analyzed as if the irradiation is continuous. This approximation does not account for exchange of rotated magnetization, unique to pulsed irradiation and exploited by chemical exchange rotation transfer (CERT) methods. In this work, we derive and test an analytic solution for the steady-state water signal under pulsed irradiation by extending a previous work to include the effects of pulse shape. The solution is largely accurate at all offsets, but this accuracy diminishes at higher exchange rates and when applying pulse shapes with large root-mean-squared to mean ratios (such as multi-lobe sinc pulses).

1. Introduction

An analytic solution clarifies the functional dependencies and provides greater insight than does numerical integration of coupled differential equations. This increased clarity may facilitate the design of pulse sequences and metrics with improved specificity and sensitivity for solute characteristics, such as concentration and exchange rate.

Recently, an analytic solution was derived (1) for the water z-magnetization when in exchange with a low-concentration solute and after irradiation by a long-duration train of rf pulses. This solution built upon previous analytic work for continuous wave (2; 3) and pulsed (4; 5; 6; 7; 8) irradiation, but included additional phenomenon, most importantly solute rotation. Consequently, it was applicable to both chemical exchange saturation transfer (CEST) (9) and, more specifically, chemical exchange rotation transfer (CERT) (10) effects.

While this previous solution was largely (but not completely) accurate characterizing the signal after irradiation at the solute resonant frequency, there were large deviations from numerical predictions near the water resonance and small deviations near the solute resonance. These deviations stemmed primarily from approximations related to the rf pulse shape. Pulse effects on water were approximated as purely adiabatic, producing saturation but no net rotation, as produced, for example, by a Gaussian pulse applied far off-resonance (11). Pulse effects on solutes were approximated as those generated by a hard pulse, producing both saturation and rotation effects, but with the characteristic sinc frequency response. In this follow-up paper, we will incorporate the rotation and saturation effects of shaped pulses on both solutes and water. The pulse shape will be characterized numerically, but all dependencies on the sample parameters (such as relaxation and exchange rates) will be derived analytically, producing a hybrid numeric-analytic solution.

While this paper is presented conventionally (with theory, methods, results, and discussion sections), a reader solely interested in using the key results can jump to Equations 22-23, 25-26 (which use the numerically calculated pulse characteristics in Equations 3-7). The key validation is in Figure 3.

Figure 3:

Numeric (i.e., ground-truth) and analytic CEST spectra after a train of 180° pulses. The present solution (∘) closely matches the numeric solution (—) everywhere in the non-exchanging and amide-like exchange cases. For fast-exchanging solutes such as guanidiniums (k_ba ≈ 1000 s⁻¹), the present and previous (•) signal equations both overestimate the CEST effect, but direct water rotation is still captured accurately.

2. Theory

Water and solute magnetizations during a CEST experiment progress through time as described by the Bloch-McConnell equation:

$$\begin{gathered} \frac{d{\overrightarrow{M}}_{\mathrm{a}}}{dt}={\overrightarrow{M}}_{\mathrm{a}}\times ({\omega }_1(t)\widehat{x}+\Delta {\omega }_{\mathrm{a}}\widehat{z})-R_{2\mathrm{a}}(X_{\mathrm{a}}\widehat{x}+Y_{\mathrm{a}}\widehat{y})+R_{1\mathrm{a}}(1-Z_{\mathrm{a}})\widehat{z}+f_{\mathrm{b}}{k}_{\mathrm{b}}\left({\overrightarrow{M}}_{\mathrm{b}}-{\overrightarrow{M}}_{\mathrm{a}}\right) \\ \frac{d{\overrightarrow{M}}_{\mathrm{b}}}{dt}={\overrightarrow{M}}_{\mathrm{b}}\times ({\omega }_1(t)\widehat{x}+\Delta {\omega }_{\mathrm{b}}\widehat{z})-R_{2\mathrm{b}}(X_{\mathrm{b}}\widehat{x}+Y_{\mathrm{b}}\widehat{y})+R_{1\mathrm{b}}(1-Z_{\mathrm{b}})\widehat{z}+k_{\mathrm{b}}\left({\overrightarrow{M}}_{\mathrm{a}}-{\overrightarrow{M}}_{\mathrm{b}}\right). \end{gathered}$$ (1)

Here X, Y, and Z are the scalar components of the normalized vector magnetization, $\overrightarrow{M}$ along the unit vectors in the rotating frame $\widehat{x}$, $\widehat{y}$, and $\widehat{z}$; the "a" and "b" subscripts represent the water and solute pools, respectively; R₁ and R₂ are the longitudinal and transverse relaxation rates, respectively; Δω is the irradiation frequency offset relative to the Larmor frequency of the pool; k_b is the exchange rate from pool b to pool a; f_b is the size of pool b relative to pool a; and ω₁ (t) is the irradiation waveform. In the case of continuous wave irradiation, ω₁ (t) is constant, greatly simplifying the solution to the differential equation (2; 3), and allowing for simplified approximations within limited regimes, e.g. (8). In the case of pulsed irradiation using shaped pulses, ω₁ (t) is a periodic function of time, and solving for the periodic steady-state becomes substantially more complicated.

There is no general analytic solution for the steady-state magnetization after repeated pulsed saturation. For nearly all pulse shapes, there is not even an analytic solution for the magnetization after a single pulse, even when ignoring exchange. Hence, the magnetization must typically be determined by numerical integration and depends on the details of the pulse shape. Our heuristic approach to finding an approximate analytic solution is to reduce this complexity by describing the pulse train by a small number of carefully chosen metrics that (hopefully) largely dictate the resulting magnetization.

More precisely, the primary hypothesis of this work is that the effect of a pulsed irradiation train (with spoilers after each pulse), such as that used in CEST, can be described accurately using tissue parameters (e.g., relaxation and exchange rates) and four metrics derived from the pulse train. Further, while these four pulse train metrics may be calculated numerically, they can be incorporated into an analytic solution. The first of these metrics is the root of the mean squared RF amplitude over the pulse train, which has long been identified as a key factor in the CEST signal (12), and is defined:

$$B_{\text{avg}\text{power}}\equiv \sqrt{\frac{1}{t_{\mathrm{p}}+t_{\mathrm{d}}}{\int }_0^{t_{\mathrm{p}}+t_{\mathrm{d}}}{B}_1^2(t)\mathrm{d}t},$$ (2)

where t_p is the pulse duration, t_d is the delay between pulses, and B₁ is the RF magnetic field strength. The second metric is the duty cycle of the pulse train, which has also been employed before (1), and is defined:

$$\text{DC}\equiv t_{\mathrm{p}}∕(t_{\mathrm{p}}+t_{\mathrm{d}})$$ (3)

The third metric, S, gives the effect of the pulse on the z-magnetization when all relaxation and exchange rates equal zero:

$$S(\Delta \omega )\equiv \frac{Z_{\text{simple}}(t_{\mathrm{p}},\Delta \omega )}{Z_{\text{simple}}(0,\Delta \omega )}.$$ (4)

Here Z_simple is the longitudinal component of ${\overrightarrow{M}}_{\text{simple}}$, the normalized magnetization following a simplified Bloch model accounting only for precession—i.e., ignoring relaxation and exchange effects and assuming that the magnetization is initially along the z-axis. S could also be written as the cosine of the pulse’s flip angle profile. Figure 1 shows example S spectra for hard and Gaussshaped pulses of equal average squared-amplitude and nominal flip angle ($\alpha \equiv {\cos }^{-1}(S(0))={\int }_0^{t_{\mathrm{p}}}{\omega }_1(t)dt$). The final pulse metric, ⟨cos² β⟩, is the average (over the pulse duration) of the projection of ${\overrightarrow{M}}_{\text{simple}}$ onto the effective field (${\overrightarrow{\omega }}_{\text{eff}}={\omega }_1(t)\widehat{x}+\Delta \omega \widehat{z}$) and back:

$$\langle {\cos }^2\beta \rangle \equiv \frac{1}{t_{\mathrm{p}}}{\int }_0^{t_{\mathrm{p}}}\frac{{\left({\overrightarrow{M}}_{\text{simple}}(t)\cdot {\overrightarrow{\omega }}_{\text{eff}}(t)\right)}^2}{\mid {\overrightarrow{\omega }}_{\text{eff}}(t){\mid }^2}dt.$$ (5)

⟨cos² β⟩ characterizes the degree of adiabaticity of the rf pulse. Like S, this characterization is independent of the tissue parameters, since ⟨cos² β⟩ is calculated with no relaxation or exchange effects. ⟨cos² β⟩ is close to 1 when the pulse acts adiabatically, i.e., the magnetization closely follows the effective field. In the extreme opposite case of on-resonant irradiation, the magnetization is always perpendicular to the effective field, and ⟨cos² β⟩ is identically zero. For the special case of a hard pulse (ω₁ (t) = ω_1,hard), ⟨cos² β⟩ = cos² θ, where θ is the angle of the effective field relative to the z-axis. Figure 2 shows ⟨cos² β⟩ for hard and gauss-shaped pulses of equal average squared-amplitude and flip angle. Both S and ⟨cos² β⟩ can be calculated numerically for any pulse through Bloch simulation, and as stated above, are not dependent on any tissue parameters such as R₁, R₂, or exchange rates. It is the goal of this work to incorporate the pulse shape-specific functions S and ⟨cos² β⟩ into a generalized pulsed CEST signal equation.

Figure 1:

S as a function of irradiation frequency offset for a hard and gauss-shaped inversion pulse. The hard inversion pulse was 3μT for 3.9ms, with the gauss-shaped pulse amplitude and duration matched to provide the same integrated squared-amplitude and flip angle. A main field strength of 9.4T was assumed.

Figure 2:

⟨cos² β⟩ as a function of irradiation frequency offset for a hard and gauss-shaped inversion pulse. The hard inversion pulse was 3μT for 3.9ms, with the gauss-shaped pulse amplitude and duration matched to provide the same integrated squared-amplitude and flip angle. A main field strength of 9.4T was assumed. Intuitively, the gauss-shaped pulse is more adiabatic in the sense that the magnetization follows the effective field more closely, leading to a higher ⟨cos² β⟩.

The rest of the theory section is broken into three subsections. The first subsection (2.1) restates the signal equation as derived in (1), which ignored water pool precession effects entirely and approximated shaped pulses as square pulses. The second subsection (2.2) presents an extended solution that a) accounts for water pool precession and decay perpendicular to the applied effective field (i.e., R_2ρ decay), and b) accounts for pulse shape effects on water. This new solution is derived by extending a simplified one-pool solution to include terms related to water pool precession and R_2ρ decay. This extended solution is then restated using the generalized numerically-calculated pulse metrics S and ⟨cos² β⟩, rather than their hard pulse-specific equivalents. By using generalized S and ⟨cos² β⟩ for the effects from shaped pulses, and including exchange of rotated solute magnetization from the previous work (1), one can derive a hybrid numeric-analytic signal equation that employs the easily calculated functions defined in Equations 4 and 5. Finally, the third subsection (2.3) extends these pulse shape effects to the solute rotation and exchange. After developing the theory, the remainder of the present work is a comparison of the resulting predicted solution to our previous analytic solution (1) and to numeric simulation of the Bloch-McConnell equations. This comparison to numeric results justifies the admittedly heuristic derivation of the analytic solution.

2.1. A previous solution for the two-pool CEST signal equation

Our previous derivation of the pulsed CEST signal equation (1) assumed that the train of shaped pulses could be approximated by a train of hard (i.e., rectangular) pulses with the same a) flip angle per pulse α, b) B_{avg power}, and c) pulse repetition time, t_p + t_d. These three constraints defined the hard pulse amplitude, duration, and repetition period as follows:

$$\begin{gathered} {\omega }_{1,\text{hard}}=\frac{p_2}{p_1^2}\frac{1}{t_{\mathrm{p}}}{\int }_0^{t_{\mathrm{p}}}\gamma B_1(t)\mathrm{d}t \\ {t}_{\mathrm{p},\text{hard}}=\frac{p_1^2}{p_2}{t}_{\mathrm{p}} \\ {t}_{\mathrm{p},\text{hard}}+t_{\mathrm{d},\text{hard}}=t_{\mathrm{p}}+t_{\mathrm{d}}, \end{gathered}$$ (6)

where the ratio p₂/p₁² is defined as

$$\frac{p_2}{p_1^2}=\frac{t_{\mathrm{p}}{\int }_0^{t_{\mathrm{p}}}{B}_1^2(t)\mathrm{d}t}{{\left({\int }_0^{t_{\mathrm{p}}}{B}_1(t)\mathrm{d}t\right)}^2}.$$ (7)

(p₁ and p₂ occur in the calculations only in the ratio p₂/p₁², but we choose not to use a single variable in order to be consistent with our previous work (1).) Note that p₂/p₁² = 1 for a hard pulse.

The longitudinal steady-state solution, $Z_{SS}^{\text{pul}}$, to the two-pool coupled Bloch-McConnell equations (13) driven by this simplified irradiation train was approximated as

$$Z_{SS}^{\text{pul}}\approx \frac{P^2{R}_{1\mathrm{a}}}{R_{1\rho }^{\text{pul}}}.$$ (8)

The projection factor, P², is a function of pulse duty cycle (DC_hard ≡ t_p,hard/ (t_p + t_d)) and the effective field angle during the hard pulse (θ_a ≡ arctan (ω_1,hard/Δω_a)),

$$P^2={\cos }^2{\theta }_{\mathrm{a}}{\text{DC}}_{\text{hard}}+(1-{\text{DC}}_{\text{hard}}).$$ (9)

The final term, $R_{1\rho }^{\text{pul}}$, is a sum of R_1ρ during the pulse (as solved in the continuous wave irradiation case (2)), R_1a relaxation between pulses, and a novel rotation exchange term:

$$R_{1\rho }^{\text{pul}}=R_{1\rho }{\text{DC}}_{\text{hard}}+R_{1\mathrm{a}}(1-{\text{DC}}_{\text{hard}})+R_{\text{rot ex}}$$ (10)

$$\begin{gathered} R_{\text{rot ex}}\equiv \frac{1}{t_{\mathrm{p}}+t_{\mathrm{d}}}{f}_{\mathrm{b}}\left(1-{\mathrm{e}}^{-k_{\mathrm{b}}{t}_{\mathrm{d},\text{hard}}}\right)\frac{{{\omega }_{1,\text{hard}}}^2+\Delta {\omega }_{\mathrm{b}}^2}{{{\omega }_{1,\text{hard}}}^2+\Delta {\omega }_{\mathrm{b}}^2+k_{\mathrm{b}}^2} \\ \times \frac{1-{\mathrm{e}}^{-R_{2\rho ,b}{t}_{\mathrm{p},\text{hard}}}\cos ({\omega }_{\text{eff},b}{t}_{\mathrm{p},\text{hard}})-{\mathrm{e}}^{-R_{2\rho ,b}{t}_{\mathrm{p},\text{hard}}}\left(\frac{k_{\mathrm{b}}}{{\omega }_{\text{eff},b}}\right)\sin ({\omega }_{\text{eff},b}{t}_{\mathrm{p},\text{hard}})}{1-{\mathrm{e}}^{-R_{2\rho ,b}{t}_{\mathrm{p},\text{hard}}-k_{\mathrm{b}}{t}_{\mathrm{d},\text{hard}}}\cos ({\omega }_{\text{eff},b}{t}_{\mathrm{p},\text{hard}})+\frac{\Delta {\omega }_{\mathrm{b}}^2}{{{\omega }_{1,\text{hard}}}^2}(1-{\mathrm{e}}^{-R_{1\rho ,\text{fast}}{t}_{\mathrm{p},\text{hard}}-k_{\mathrm{b}}{t}_{\mathrm{d},\text{hard}}})}. \end{gathered}$$ (11)

Here ${\omega }_{\text{eff},b}\equiv \sqrt{{{\omega }_{1,\text{hard}}}^2+\Delta {\omega }_{\mathrm{b}}^2}$ is the effective field strength in the solute’s rotating frame. R_1ρ, R_2ρ,b, and R_1ρ,_fast are system eigenvalues defined in (1) and are not adjusted in this work.

2.2. Including a-pool rotation and R_2ρ decay

The solution in Equations 8-11 does not account for direct water rotation and pulse shape effects and, therefore, grossly fails to capture signal behavior near the water resonance. The derivation of a new analytic solution that includes these effects is divided into five parts:

1. Derive the analytic solution for S and ⟨cos² β⟩ for a single hard pulse applied to a single pool, with no exchange and no relaxation.

   The result:

   $$S_{\text{hard}}(\Delta \omega )={\cos }^2\theta +{\sin }^2\theta \cos ({\omega }_{\text{eff}}{t}_{\mathrm{p},\text{hard}}).$$ (12)

   $$\langle {\cos }^2\beta {\rangle }_{\text{hard}}={\cos }^2(\theta )$$ (13)

   (Since these equations are derived using a single pool, no pool-specific subscripts were applied to Δω, θ, or ω_eff. In section 2.5.2, the "a" subscript will be applied to describe water magnetization. While in section 2.3, the "b" subscript will be applied to describe the effects on the water magnetization that originate in the solute magnetization.)

2. Derive the analytic solution for the z-magnetization after a single pulse repetition period (pause of t_d,hard followed by hard pulse of width t_p,hard), with no exchange but with relaxation.

   After time t_d,hard:

   $$Z(t_{\mathrm{d},\text{hard}})=Z(0){\mathrm{e}}^{-R_{1t_{\mathrm{d},\text{hard}}}}+(1-{\mathrm{e}}^{-R_1{t}_{\mathrm{d},\text{hard}}})$$ (14)
   During the following irradiation period, the magnetization will decay toward a nonzero steady state (Z_SS) at a real rate, $-R_{1\rho }^{1\text{pool}}$, and a complex (i.e., oscillating) rate, $-R_{2\rho }^{1\text{pool}}\pm i{\omega }_{\text{eff}}$(1):

   $$\begin{gathered} Z(t_{\mathrm{d},\text{hard}}+t_{\mathrm{p},\text{hard}})=Z_{SS}\left(1-{\mathrm{e}}^{-R_{1\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}}\right)+Z(t_{\mathrm{d},\text{hard}}){\cos }^2\theta {\mathrm{e}}^{-R_{1\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}} \\ +Z(t_{\mathrm{d},\text{hard}}){\sin }^2\theta \cos ({\omega }_{\text{eff}}{t}_{\mathrm{p},\text{hard}}){\mathrm{e}}^{-R_{2\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}}. \end{gathered}$$ (15)
3. Rewrite the solution from step 2 in terms of S and ⟨cos² β⟩ from step 1. (The motivation for particular substitution choices is addressed in the Discussion section.)

   $$\begin{gathered} Z(t_{\mathrm{d},\text{hard}}+t_{\mathrm{p},\text{hard}})=Z_{SS}\left(1-{\mathrm{e}}^{-R_{1\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}}\right)+Z(t_{\mathrm{d},\text{hard}}){\cos }^2\theta {\mathrm{e}}^{-R_{1\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}} \\ +Z(t_{\mathrm{d},\text{hard}})\left(-{\cos }^2\theta +{\cos }^2\theta +{\sin }^2\theta \cos ({\omega }_{\text{eff}}{t}_{\mathrm{p},\text{hard}})\right){\mathrm{e}}^{-R_{2\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}} \end{gathered}$$ (16)

   $$\begin{gathered} =Z_{SS}\left(1-{\mathrm{e}}^{-R_{1\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}}\right)+Z(t_{\mathrm{d},\text{hard}}){\cos }^2\theta {\mathrm{e}}^{-R_{1\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}} \\ +Z(t_{\mathrm{d},\text{hard}})\left(-1+{\sin }^2\theta +S_{\text{hard}}\right){\mathrm{e}}^{-R_{2\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}} \end{gathered}$$ (17)

   $$\begin{gathered} =Z_{SS}\left(1-{\mathrm{e}}^{-R_{1\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}}\right)+Z(t_{\mathrm{d},\text{hard}}){\cos }^2\theta {\mathrm{e}}^{-R_{1\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}} \\ +Z(t_{\mathrm{d},\text{hard}}){\sin }^2\theta {\mathrm{e}}^{-R_{2\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}}-Z(t_{\mathrm{d},\text{hard}})(1-S_{\text{hard}}){\mathrm{e}}^{-R_{2\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}} \end{gathered}$$ (18)

   $$\begin{gathered} =Z_{SS}\left(1-{\mathrm{e}}^{-R_{1\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}}\right)+Z(t_{\mathrm{d},\text{hard}})\langle {\cos }^2\beta {\rangle }_{\text{hard}}{\mathrm{e}}^{-R_{1\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}} \\ +Z(t_{\mathrm{d},\text{hard}})\langle {\sin }^2\beta {\rangle }_{\text{hard}}{\mathrm{e}}^{-R_{2\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}}-Z(t_{\mathrm{d},\text{hard}})(1-S_{\text{hard}}){\mathrm{e}}^{-R_{2\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}}, \end{gathered}$$ (19)

   where we define ⟨sin² β⟩_hard ≡ 1 − ⟨cos² β⟩_hard for notational convenience.

4. Approximate the dynamic steady-state solution by setting Z (t_d,hard + t_p,hard) = Z (0) and taking a Taylor expansion.

   Combining Equations 14 and 19, substituting in ⟨cos² β⟩_hard, solving for $Z_{SS}^{\text{pul},1\text{pool}}=Z(0)=Z(t_{\mathrm{d},\text{hard}}+t_{\mathrm{p},\text{hard}})$, and applying the Taylor expansion e^x ≈ 1 + x for small x yields

   $$\begin{gathered} Z_{SS}^{\text{pul},1\text{pool}}= \\ \frac{Z_{SS}{R}_{1\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}+S_{\text{hard}}{R}_1{t}_{\mathrm{d},\text{hard}}}{(1-S_{\text{hard}}){\mathrm{e}}^{-R_{2\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}}+R_{1\rho }^{1\text{pool}}\langle {\cos }^2\beta {\rangle }_{\text{hard}}{t}_{\mathrm{p},\text{hard}}+R_{2\rho }^{1\text{pool}}\langle {\sin }^2\beta {\rangle }_{\text{hard}}{t}_{\mathrm{p},\text{hard}}+R_1{t}_{\mathrm{d},\text{hard}}} \end{gathered}$$ (20)

   when higher order cross-terms of the (assumed small) exponential arguments are ignored. Substituting in the continuous wave steady-state solution (2) $Z_{SS}={\cos }^2\theta R_1∕R_{1\rho }^{1\text{pool}}$ and dividing the numerator and denominator by the pulse repetition period provides a familiar form:

   $$\begin{gathered} Z_{SS}^{\text{pul},1\text{pool}}= \\ \frac{\left({\cos }^2\theta {\text{DC}}_{\text{hard}}+S_{\text{hard}}(1-{\text{DC}}_{\text{hard}})\right)R_1}{\frac{1-S_{\text{hard}}}{t_{\mathrm{p}}+t_{\mathrm{d}}}{\mathrm{e}}^{-R_{2\rho }^{1\text{pool}}{t}_{\mathrm{p},\text{hard}}}+R_{1\rho }^{1\text{pool}}\langle {\cos }^2\beta {\rangle }_{\text{hard}}{\text{DC}}_{\text{hard}}+R_{2\rho }^{1\text{pool}}\langle {\sin }^2\beta {\rangle }_{\text{hard}}{\text{DC}}_{\text{hard}}+R_1(1-{\text{DC}}_{\text{hard}})}. \end{gathered}$$ (21)

5. Approximate the effect of shaped pulses and magnetization exchange between pools, and simplify the resulting form.

   Note the similarity of Equation 21 to Equation 8, where the projection factor, P², is slightly altered from that in Equation 9 and the denominator has two additional terms (compared to $R_{1\rho }^{\text{pul}}$ in Equation 10) related to S and R_2ρ. However, this solution lacks pulse shape and spin exchange effects since it was derived using a train of square pulses applied to a single pool.

   To model water magnetization when in exchange with a solute, we first add "a" subscripts to θ and R₁ in equation 21. Next, we propose a corrected projection factor, $P_{\text{corr}}^2$, and corrected $R_{1\rho }^{\text{pul}}$, $R_{1\rho }^{\text{corr}}$, which include shape and exchange effects by approximating the effect of

   • shaped pulses by substituting numerically determined values of S and ⟨cos² β⟩ (using equations 4 and 5 with "a" subscripts on Δω and ω_eff) for S_hard and ⟨cos² β⟩_hard, respectively,
   • saturation exchange by substituting the two-pool R_1ρ and R_2ρ for $R_{1\rho }^{1\text{pool}}$ and $R_{2\rho }^{1\text{pool}}$, respectively, and
   • rotation exchange by adding contributions from R_{rot ex}, as in Equation 10.
     The result is:

     $$P_{\text{corr}}^2\equiv {\cos }^2{\theta }_{\mathrm{a}}{\text{DC}}_{\text{hard}}+S_{\mathrm{a}}(1-{\text{DC}}_{\text{hard}})$$ (22)

     $$\begin{gathered} R_{1\rho }^{\text{corr}}\equiv \frac{1-S_{\mathrm{a}}}{t_{\mathrm{p}}+t_{\mathrm{d}}}{\mathrm{e}}^{-R_{2\rho }{t}_{\mathrm{p},\text{hard}}}+R_{1\rho }\langle {\cos }^2\beta \rangle {\text{DC}}_{\text{hard}}+R_{2\rho }\langle {\sin }^2\beta \rangle {\text{DC}}_{\text{hard}}+R_{1\mathrm{a}}(1-{\text{DC}}_{\text{hard}}) \\ +R_{\text{rot}\text{ex}}^{\text{corr}}. \end{gathered}$$ (23)
     The final term, $R_{\text{rot ex}}^{\text{corr}}$, is adjusted in the following section from the term defined in Equation 11. As this term is related to exchange effects, it is absent from the one-pool solution in Equation 21.

2.3. Including pulse shape effects on the solute through S_b

Substitution of S_b via Equation 12 (using "b" subscripts on Δω, θ, and ω_eff) into Equation 11 can also improve the accuracy of the analytic solution near the solute resonance. As an initial simplification, we identify that the coefficient exp (−R_2ρ,bt_p,hard) × k_b/ω_eff,b is small under most circumstances (This is due to ω_eff,b usually being chosen to be on the order of 1/t_p in order to ensure rotation of the solute magnetization, and $R_{2\rho ,\mathrm{b}}=k_{\mathrm{b}}+\left({\cos }^2{\theta }_{\mathrm{b}}+\frac{1}{2}{\sin }^2{\theta }_{\mathrm{b}}\right)R_{2\mathrm{b}}+\frac{1}{2}{\sin }^2{\theta }_{\mathrm{b}}{R}_{1\mathrm{b}}$ (from equation 7 in (1)) being always greater than k_b. Hence,

$$\text{exp}(-R_{2\rho ,b}{t}_{\mathrm{p},\text{hard}})\times k_{\mathrm{b}}∕{\omega }_{\text{eff},b}\approx \text{exp}(-R_{2\rho ,b}{t}_{\mathrm{p}})\times k_{\mathrm{b}}{t}_{\mathrm{p}}<\text{exp}(-k_{\mathrm{b}}{t}_{\mathrm{p}})\times k_{\mathrm{b}}{t}_{\mathrm{p}}$$ (24)

, which is small compared to 1 for most values of k_bt_p.) We can therefore neglect the sine term in R_{rot. exch}. and, as in the previous section, substitute (S_b − cos² θ_b) / sin² θ_b for both instances of cos (ω_eff,_bt_p,hard). After algebraic manipulation, noting that $\Delta {\omega }_{\mathrm{b}}^2∕{{\omega }_{1,\text{hard}}}^2={\cot }^2{\theta }_{\mathrm{b}}$, we define $R_{\text{rot ex}}^{\text{corr}}$ as:

$$\begin{gathered} R_{\text{rot ex}}^{\text{corr}}\equiv \frac{1}{t_{\mathrm{p}}+t_{\mathrm{d}}}{f}_{\mathrm{b}}\left(1-{\mathrm{e}}^{-k_{\mathrm{b}}{t}_{\mathrm{d},\text{hard}}}\right)\frac{{{\omega }_{1,\text{hard}}}^2+\Delta {\omega }_{\mathrm{b}}^2}{{{\omega }_{1,\text{hard}}}^2+\Delta {\omega }_{\mathrm{b}}^2+k_{\mathrm{b}}^2} \\ \times \frac{1-{\cos }^2{\theta }_{\mathrm{b}}(1-{\mathrm{e}}^{-R_{2\rho ,b}{t}_{\mathrm{p},\text{hard}}})-S_b{\mathrm{e}}^{-R_{2\rho ,b}{t}_{\mathrm{p},\text{hard}}}}{1-{\mathrm{e}}^{-k_{\mathrm{b}}{t}_{\mathrm{d},\text{hard}}}({\cos }^2{\theta }_{\mathrm{b}}({\mathrm{e}}^{-R_{1\rho ,\text{fast}}{t}_{\mathrm{p},\text{hard}}}-{\mathrm{e}}^{-R_{2\rho ,b}{t}_{\mathrm{p},\text{hard}}})+S_b{\mathrm{e}}^{-R_{2\rho ,b}{t}_{\mathrm{p},\text{hard}}})}. \end{gathered}$$ (25)

The proposed solution, validated numerically in the following sections, corrects the definitions of Equations 9-11 to the definitions of Equations 22-25 before finally computing the longitudinal steady-state magnetization as

$$Z_{SS}^{\text{pul},\text{corr}}=\frac{P_{\text{corr}}^2{R}_{1\mathrm{a}}}{R_{1\rho }^{\text{corr}}}.$$ (26)

3. Numerical Validation

3.1. Experimental Methods

The signal equation presented in Equations 22-26 was compared to a previous result (1), reproduced in Equations 8-11, and to numerical Bloch-McConnell simulation. All calculations assumed a main magnetic field strength of 9.4T. The first group of calculations assumed gauss-shaped pulses. Water pool relaxation rates were fixed at R_1a = 1 s⁻¹, R_2a = 10 s⁻¹, with solute pool relaxation rates fixed at R_1b = 1 s⁻¹, R_2b = 100 s⁻¹. Three exchange scenarios were considered. The first was a non-exchanging system (k_b = 0 s⁻¹); the remaining two were designed to be amide-like (δ = 3.5 ppm, k_b = 50 s⁻¹) and guanidinium-like (δ = 2.0 ppm, k_b = 1000 s⁻¹), respectively. The exchanging scenarios were treated as dilute solutions (f_b = 0.4%).

Each of the three simulated systems were driven by irradiation pulse trains of average squared-amplitude equivalent to that of a 0.6μT, 1.8μT, and 3.6μT continuous wave, for a total of nine tested scenarios per designated flip angle. The irradiation duty cycle was 61% for all pulse trains, resulting in timings and amplitudes such that ω_1,hard = γ·1, 3, and 6μT, respectively. The nine scenarios were repeated for pulse flip angles of 180° and 360°.

S_a, S_b, and ⟨cos² β⟩ were calculated for the gauss-shaped pulses using Equations 4 and 5 (and Δω = Δω_a, Δω_b, and Δω_a, respectively), where ${\overrightarrow{M}}_{\text{simple}}$ was calculated using one-pool Bloch simulation with R₁ = R₂ = 0. These functions were in turn used to calculate the proposed analytic solution using Equations 22-26, where eigenvalue rates such as R_1ρ were calculated as in reference (1). A Bloch-McConnell simulation (solving Equation 1) was used to calculate the ground-truth numerical solution for the pulsed steady state, repeating the gauss-shaped pulse train until the normalized signal change from the previous pulse was < 10⁻⁵. The Bloch and Bloch-McConnell simulations were performed in MATLAB 2018a (MathWorks, Natick, MA).

The above procedure used a Gaussian pulse. In order to assess the effect of pulse shape, a similar procedure was used for a cosine pulse, a Hamming-windowed three-lobed sinc pulse ("sinc-3"), and an unwindowed five-lobed sinc pulse ("sinc-5"). An average irradiation power equivalent to that of a 1.2μT continuous wave was assumed for each pulse train, as was a constant pulse repetition time of 24.5 ms and a flip angle of 180°. (As each tested pulse had a different root-mean-squared (rms) to mean amplitude ratio, the duty cycle of each pulse train varied by necessity.) The constraints employed here ensured that each tested pulse train corresponded to the same equivalent hard pulse train, as stated by Equations 6-7. The sample parameters used for this experiment were identical to those of the amide-like exchange case above.

3.2. Validation Results

The calculated spectra resulting from zero-exchange, amide-like, and guanidinium-like systems are presented in Figures 3 (180° pulse train) and 4 (360° pulse train). The solution proposed in Equations 22-26 closely matched the ground-truth numeric solution near the water resonance in all tested cases, which was a key failure of the solution proposed in (1). Furthermore, the inclusion of S_b in Equation 25 eliminated spurious ringing of the previous analytic solution near the solute resonance—visible in Figure 5—presumably caused by the assumption of a hard pulse in that derivation.

Figure 5:

Elimination of ringing near the solute resonance caused by the hard-pulse approximation inherent to Equations 8-11. This figure is a magnification of the spectrum in the central panel of Figure 3; it therefore assumed an amide-like system (δ = 3.5 ppm, k_ba = 50 s⁻¹) and irradiation with average squared-amplitude equivalent to a 1.8 μT continuous wave. Note that the proposed solution captures the numeric solution’s shape more closely than the oscillating spectrum calculated as in reference (1).

However, the present solution retains other key failures of the previous solution. In particular, the proposed analytic solution exaggerates the CEST effect (i.e., the spectral dip at the solute resonance) when exchange is rapid compared to the pulse timings (bottom row of Figures 3 and 4). The exact nature of this error in the previous solution was masked by the failure to capture direct water rotation effects, but is made clearer when pulse shape and water rotation effects are corrected.

Figure 4:

Numeric (i.e., ground-truth) and analytic CEST spectra after a train of 360° pulses. Results are similar to those in Figure 3, with the exception of slight deviation between the proposed and numeric solutions visible near the water resonance frequency. These deviations are present even in the non-exchanging case, but are small in magnitude in all cases.

Figure 6 examines how varying the pulse shape affects the accuracy of the proposed solution. It shows that the proposed solution is accurate even when pulse shape effects dominate the spectrum, as seen most clearly in figure 6a. For the pulse shapes we investigated, the solution is more accurate when the rms-to-mean amplitude ratio of the pulse (i.e., ${\omega }_{1,\text{rms}}∕\langle {\omega }_1\rangle =\sqrt{p_2∕p_1^2}$ is closer to 1, which is the rms-to-mean amplitude ratio of a hard pulse. The detail in Figure 7 shows that in the extreme case of a five-lobed sinc pulse, for which ω_1,rms/ ⟨ω₁⟩ ≈ 2.26, the pulse shape appears to be overrepresented in the approximation—the numerical solution is far more damped by comparison.

Figure 6:

Comparison of the present and the (ground-truth) numeric solutions for different pulse shapes. These figures were generated assuming pulse trains with average irradiation power equivalent to a 1.2μT continuous wave and amide-like exchange characteristics. Because each pulse shape has a different ratio of rms-to-mean amplitude, different duty cycles were employed while pulse repetition time (t_d + t_p) was kept constant at 24.5 ms. All pulse shapes provided well-matched solutions, but pulse shape effects were less damped in the approximation than in the exact solution, especially in more power-intensive pulses such as the five-lobed sinc ("Sinc-5").

Figure 7:

Apparent underdamping of pulse shape effects near the solute resonance following a five-lobed sinc train. This figure contains the same data as (and is a detail of) Figure 6’s bottom right panel. Note that pulse shape effects are largely damped out of the exact solution, but remain in the approximation. This may be caused by the simplifying assumption that rotation exchange does not occur during the pulse.

4. Discussion

The steady-state signal equation presented in Equations 22-23, 25-26 accurately accounts for pulse shape, including direct water rotation effects, and is a significant improvement over the previous analytic signal equation. The previous solution took a bifurcated approach to accounting for the pulse shape: effects on the solute were approximated by using an equivalent hard pulse (that both saturated and rotated the solute), while effects on the water were approximated by using an equivalent adiabatic pulse (that only saturated, with no net rotation). The current work accounts for shape-specific saturation and rotation effects on both the solute and water. This solution is not a phenomenological guess, as the form is dictated by the derivation. However, the derivation is based on simplified modeling and reasonable, but not mathematically rigorous, substitution. Further, we made choices concerning substitution of shape-based metrics, i.e. when we used equation 12, equation 13, and/or left a term as a function of θ, and these choices were guided by our interpretations of individual terms.

The most important correction accounts for direct water rotation by the addition of the term (1 − S_a) / (t_p + t_d) × exp (−R_2ρt_p,hard) to $R_{1\rho }^{\text{corr}}$ in Equation 23, and achieves signal accuracy at small offsets close to that of numerical integration, at least for 180° pulses. As all the summed rates in $R_{1\rho }^{\text{corr}}$ are rates of magnetization elimination, it is possible to interpret this term as a signal loss due to damped rotation of the water magnetization (followed by spoiling that leaves only the z-component) and repeated at a rate 1/ (t_p + t_d) . Intuitively, this rate is defined by the period at which irradiation pulses are applied, t_p + t_d. Somewhat less intuitive is the scaling factor exp (−R_2ρt_p,hard). It can be argued that, when R_2ρ decay is rapid compared to the pulse duration, rotation effects disappear because the magnetization will rapidly align with the effective field.

The addition of ⟨cos² β⟩ and ⟨sin² β⟩ terms in Equation 23 account for (both direct and exchange-induced) water saturation. These projections come from calculating the degree to which the water magnetization adiabatically follows the effective field. If instead a hard pulse approximation was made using projections cos² θ and sin² θ, rather than the average projections over the pulse, ⟨cos² β⟩ and ⟨sin² β⟩, gross signal inaccuracies would occur anywhere these projections differ (see Figure 2).

S_b in $R_{\text{rot ex}}^{\text{corr}}$ in Equation 25 adds solute rotation effects, and the result (though clearly not perfect) eliminates the classic sinc-shaped ringing near the solute resonance, as demonstrated in Figure 5. It is possible that employing ⟨cos² β⟩ in $R_{\text{rot ex}}^{\text{corr}}$ (rather than cos² θ_b, as is presented in Equation 25) might also be more accurate. However, as the basis of each factor in $R_{\text{rot ex}}^{\text{corr}}$ is not fully clear and spectral shape errors near the solute resonance are small, this substitution is difficult to justify by argument or numeric validation.

Similarly, one might wonder if ⟨cos² β⟩ should be substituted into the corrected signal projection factor $P_{\text{corr}}^2$, defined in Equation 22. This would be incorrect, as the cos² θ_a factor in this term originates as the projection between the continuous wave steady-state and $\widehat{z}$. In contrast, ⟨cos² β⟩ samples the projection angle between the effective field and the magnetization during the pulse, in the transient regime. Note also that in $P_{\text{corr}}^2$ the complement to cos² θ_a is S_a. By definition, S_a maps the magnetization before an irradiation pulse to the magnetization afterwards. When S_a is negative—i.e., when each pulse inverts the magnetization—its inclusion in $P_{\text{corr}}^2$ results in an intuitively correct signal decrease. Furthermore, as the irradiation pulses become longer relative to the repetition period and the duty cycle increases, the system approaches the continuous wave scenario, in which the projection factor is simply cos² θ_a.

Note finally that the proposed solution overestimates the CEST effect for fast-exchanging systems. This is most readily seen in the center-bottom panel of Figure 3, but can also be seen in the same and bottom-right panels of Figure 4. As this error is also present in the previous solution (1) upon which the proposed one is based, it is reasonable to conclude that the alterations to the signal equation made here did not introduce this inaccuracy. Given that the error appears only at large exchange rates, it is likely caused by either a) the Taylor approximations that produce the ratio-of-rates form (Equations 8 and 26), or b) the assumption of the previous work which stipulated that intercompartmental exchange occurs at a rate that is substantially slower than the water pool’s precession about the effective field.

Although the proposed solution was able to account for a variety of tested pulse shapes, it appeared to overcompensate for pulse shape effects, generating larger deviations between signal peaks and troughs than the exact solution (see Figure 6-7). Since these effects were more prominent in pulses with higher ω_1,rms/ ⟨ω₁⟩—which by necessity were of a longer duration, in order to match the requisite average power—these erroneously sharp peaks may be underdamped due to the assumption that rotation exchange does not occur during the pulse. For shorter pulses, this assumption is more accurate, and these errors are smaller (though still visible, e.g., in the cosine pulse panel of Figure 6). Like the errors discussed in previous paragraphs, if this hypothesis is correct, faster exchange rates would exacerbate the error: with fast-exchanging pools, even short irradiation pulses might result in an underdamped approximation.

All results in this paper were derived for bulk water exchanging with a single relatively low-concentration solute. In tissue, there will be effectively dozens of molecular pools exchanging magnetization with water, and disentangling their contributions has been a recurring complication in CEST (e.g. (14; 15; 16; 17) amongst many others). To some degree, these magnetization pools can be selectively excited by varying the irradiation frequency. However, resonances are broadened due to relaxation, exchange, and irradiation (18), making multiple exchanging pools unavoidable.

The results of this paper can be extended to multiple exchanging pools by noting that pool contributions add linearly to the water R_1ρ under continuous wave irradiation (19; 18). In pulsed irradiation, these additive rates show up as additional terms in equation 23 inside (the continuous wave) R_1ρ and R_2ρ (using equation 20 of (1)) and $R_{\text{rot ex}}^{\text{corr}}$. Note, however, that the most common source of spectral overlap is the very broad peak originating from the ~ 10μs T₂ macromolecules studied in magnetization transfer (MT) experiments, and that these macromolecules have long-standing modelling issues that make calculating S_b tricky. Specifically, the commonly employed super-Lorentzian lineshape (20) misfits tissue data at small offsets (21), has a zero-offset singularity (20) that necessitates interpolation (e.g., (22)), is not consistent with the orientational dependence of relaxation parameters (23), and does not incorporate exchange with dipolar-order pools (24) as is relevant to inhomogeneous MT (ihMT) results. Hence, the macromolecular lineshape is commonly dealt with heuristically in CEST experiments, making validation of any analytic solution that includes this pool difficult and beyond the scope of this paper.

5. Conclusion

The signal equation presented in Equations 22-23, 25-26, when combined with the Bloch-McConnell system eigenrates (e.g., R_1ρ) derived in reference (1), accurately estimates the exact Bloch-McConnell periodic steady state—i.e., the two-pool CEST signal. This is true even at small frequency offsets, which was a key failure of the previous analytic pulsed CEST signal equation. Thus, the present result can be used to accelerate parameter fitting or experimental optimization problems, even when considering solutes with resonance frequency close to that of water, such as guanidiniums and hydroxyls, though the inaccuracies at higher exchange rates may limit applicability. Also, as an analytic expression provides greater insight into signal dynamics than numeric simulation, it is expected that the proposed solution will enable further investigation into signal processing and experimental design approaches.

Abbreviations:

CERT

chemical exchange rotation transfer

CEST

chemical exchange saturation transfer

DC

duty cycle of shaped pulse train

f _b

b-pool size relative to a-pool size

k _a

exchange rate from a-pool to b-pool

k _b

exchange rate from b-pool to a-pool

R_1ρ, R_2ρ

relaxation rates in the rotating frame

t _p

shaped pulse duration

t _d

duration between shaped pulses

X_a, Y_a, Z_a

normalized magnetization components of a-pool

X_b, Y_b, Z_b

normalized magnetization components of b-pool