1 Introduction T 2 T 2 [1] ω 1 f in vivo in vivo T 1 [2] [3–5] [3,6] [5] [5] in vivo T 1 T 2 T 1 [6] in vivo [7] in vivo 2 Theory 2.1 Coupled Bloch equations A B ( M z A , M z B ) ( M x A , M y A , M x B , M y B ) T 2 [2,6] (1) d M z A d t = R A M 0 A - M Z A - RM 0 B M Z A + RM 0 A M Z B + ω 1 ( t ) M y A (2)  d M z B d t = R B M 0 B - M Z B - RM 0 A M Z B    + RM 0 B M Z A - ( R RFB ( Δ f , ω 1 ( t ) ) ) M Z B (3) d M x A d t = - M x A T 2 A - 2 π Δ fM y A (4) d M y A d t = - M y A T 2 A + 2 π Δ fM x A - ω 1 ( t ) M z A . (1)–(4) T 2 A M 0 A M 0 B R A R B R f ω 1 t −1 R RFB f ω 1 t B f ω 1 t T 2 B [8] ω 1 2 ¯ (5) R RFB ( Δ f , ω 1 ) = ω 1 2 ¯ 2 π T 2 B ∫ 0 1 1 | 3 u 2 - 1 | exp - 2 2 π Δ fT 2 B 3 u 2 - 1 2 d u A B F R [6] f m [9,10] W R RFB [6] RM 0 B ( A → B ) RM 0 A ( B → A ) k f k k r 2.2 Sled and Pike’s RP signal equation [6] (1)–(4) τ RP S A (1), (3) and (4) R R A [6] (6) M z ( t ) = M z A ( t ) M z B ( t ) . M z S θ (7) S = S 1 A cos θ 0 0 1 . t 1 t 0 (1) and (2) (8) M z ( t 0 + t 1 ) = exp { A CW t 1 } M z ( t 0 ) + [ I - exp { A CW t 1 } ] A CW - 1 BM 0 (9) M z ( t 0 + t 1 ) = exp { A FP t 1 } M z ( t 0 ) + [ I - exp { A FP t 1 } ] A FP - 1 BM 0 ,  A CW = - R A - RM 0 B RM 0 A RM 0 B - R B - RM 0 A - R RFB  A FP = - R A - RM 0 B RM 0 A RM 0 B - R B - RM 0 A  B = - R A 0 0 - R B . T k M z τ RP T τ RP τ RP (10) M z ( T ) = M z ( 0 ) , M z (11) SI ( ω 1 , Δ f ) = M z A ( TR ) S 1 A sin θ , T (10) 2.3 Sled and Pike’s CW signal equation [6] T T M z T (12) SI ( ω 1 , Δ f ) = ( E 1 - 1 ) ( E 2 - 1 ) ( λ 2 - λ 1 ) S 1 A M z , CW A sin θ ( E 1 - 1 ) ( S 1 A E 2 cos θ - 1 ) ( λ 2 - λ 1 ) + ( S 1 A cos θ - 1 ) ( E 2 - E 1 ) ( λ 2 - R A - RM 0 B ) . M z , CW A [1] T (13) M z , CW A = M 0 R A RM 0 A + R A R B + R B RM 0 B + R RFB R A R A RM 0 A + R A R B + R B RM 0 B + R RFB R A + R RFB RM 0 B , M z , CW = A CW - 1 BM 0 (12) (14)  λ 1 , 2 = 1 2 R A + RM 0 B + R B + RM 0 A + R RFB ±    1 2 R A + RM 0 B + R B + RM 0 A + R RFB 2 - 4 R A R B + RM 0 B R B + R A R RFB + R A RM 0 A + RM 0 B R RFB ,  E 1 , 2 = e - λ 1 , 2 TR . 3 T [6] F (15) F = M 0 B M 0 A . F RM 0 A = RM 0 B F (13) [3,6,11] 2.4 Ramani’s signal equation [1] (1)–(4) [5] (16) ω 1 CWPE = γ P SAT , P SAT [5] f f F F F RM 0 B (17) SI ( ω 1 , Δ f ) = M 0 R B RM 0 B R A + R RFB ( ω 1 CWPE , Δ f ) + R B + RM 0 B F RM 0 B R A ( R B + R RFB ( ω 1 CWPE , Δ f ) ) + 1 + ω 1 CWPE 2 π Δ f 2 1 R A T 2 A R RFB ( ω 1 CWPE , Δ f ) + R B + RM 0 B F , M 0 c M 0 (17) 2.5 Fitting M 0 , R A , R B , RM 0 B , F , T 2 A T 2 B [1,12] R Aobs T obs R A [1] (18) R A = R Aobs - RM 0 B ( R B - R Aobs ) R B - R Aobs + RM 0 B F . S ω 1 f R B R B −1 [1,5,6] ω 1 t f 3 Materials and methods 3.1 Numerical simulations In order to compare the three signal equations, and to highlight their shortcomings, we need to test their performance against data corresponding to a known set of parameters. The easiest way to obtain such data is to synthetically produce them, using numerical simulations. τ SAT (1)–(4) M z A ( τ SAT ) T 2 ∗ (19) SI ∝ M xy ( readout ) = M z ( τ SAT ) sin θ . B SAT MAX τ SAT σ B EXC MAX τ EXC BW ω 1 t B (19) M z (1)–(4) M z n τ SAT M z n τ SAT M z n τ SAT [13] ( R A , R B , T 2 A , T 2 B , F , RM 0 B ) [14,15] Table 1 τ EXC BW Experiment 1: τ SAT σ σ ω CWPE −1 τ SAT [6,15] f [13] Table 1 R Aobs (18) S A (7), (11) and (12) Experiment 2: τ SAT ω CWPE −1 f Experiment 3: RM B τ SAT θ τ SAT θ τ SAT θ Experiment 4: τ SAT θ M 0 3.2 S A 3.2 In vivo [15] 3 −1 f f θ R Aobs The study was approved by the Joint Research Ethics Committee of The National Hospital for Neurology and Neurosurgery and the Institute of Neurology, UCL, and the subject gave written informed consent before taking part. 3.3 Image analysis [15] [16] http://air.bmap.ucla.edu:16080/AIR [17] R Aobs [18] T 1 RM 0 B , F , T 2 B T 2 A T T p 4 Results Fig. 1 (18) Table 1 Fig. 1 4.1 Duty cycle effect R A , RM 0 B , F T 2 A Fig. 2 T 2 B T 2 A RM 0 B T 2 A RM 0 B T 2 A = 76.2  ms RM 0 B = 3.31  s - 1 T 2 A = 75.0  ms RM 0 B = 3.93  s - 1 4.2 Saturation effect of the excitation Fig. 3 T 2 B T 2 A Table 1 Table 1 RM 0 B F T 2 A τ SAT Table 1 T 2 A T 2 A RM 0 B 4.3 Sensitivity to noise Fig. 4 RM 0 B T 2 A T 2 A T 2 B RM 0 B T 2 A 4.4 In vivo in vivo F Fig. 5 RM 0 B T 2 A T 1 (17) Table 2 4.4.1 5° data (between equations) F , RM 0 B T 2 A p T 2 A F RM 0 B Table 2 p RM 0 B F T 2 B T 2 A 4.4.2 15° data (between equations) p T 2 A p F T 2 A 4.4.3 15° data vs. 5° data T 2 B p F T 2 A p 5 Discussion RM 0 B , F T 2 B in vivo [7] T 2 A T 2 [11] T 2 T 2 T 2 A [1,11] in vivo underestimated [11] in vivo Table 2 in vivo [19] T 2 A [20] T 2 A [6,11] RM 0 B in vivo T 1 T 2 A T 2 A [7] T 2 A T 2 Fig. 1 [5] [7] [7] underestimate [7] Figs. 2–4 F R A F −1 (18) RM 0 B F R A F [9] in vivo [7] T 2 A T 1 (17) T 1 in vivo F T 1 Table 2 T 2 A F in vivo T 2 B Fig. 3 in vivo in vivo [20] [21] [22] [20] [6,11] in vivo in vivo RM 0 B , F , T 2 B , T 2 A M 0 [11] M 0 T 1 in vivo in vivo B 1 B 1 R Aobs [6,18] B 1 [2,6] [23] Furthermore, it would be interesting to explore possible modifications of Ramani’s equation to account for the imaging parameters of the pulse sequence (for example by incorporating an additional contribution of [1-cos(alpha)]/TR to the CW saturation rate acting on the free pool). Finally, providing reliable information about the optimal number of sampling points and their distribution would yield an additional element towards the choice of the most appropriate equation for a given application. All these areas deserve further investigation, which we hope to pursue in the future. 6 Conclusion T 1 F RM 0 B T 1 T 1 in vivo RM 0 B , F T 2 B T 1