10. : (NMR) ( 1 H) MRI ρ H (x,y,z) NMR (Nuclear Magnetic Resonance) spectrometry: NMR NMR s( B ) m m = µ 0 IA = γ J (1) γ: :Planck constant J: Ĵ 2 = J(J +1),Ĵz = J J: (J = 1 2 for 1 H) I m A 173/197
10.1 B = 0 B 0 - - - + + + + + B = 0 B 0 E 1/2 E 0 E +1/2 m = 0 m 0 B = B 0 e z 0 m(t) ( ) m(t) t E ± 1 2 = E 0 1 2 γ B Boltzmann n 1 = n 2 + 1e E kt 2 B = 0 n + 1 > n m (i) 2 1 2 ( n/n 5ppm) M = m = 0 M = m = m ez 0 174/197
10.2 ( ( dv v B, v e m) ) dm B = B 0 e z ( B 0 = const) = γm B (2) e z (Eq.(2)) m (Eq.(2)) e z dm = 0 m z = const (3) m dm = 1 dm m = 0 2 m dm (4) m 2 = const (5) (Independent of B ) d(eq.(2)) d 2 m 2 = γ dm B =(γb 0 ) 2 (m e z ) e z =(γb 0 ) 2 ( m+m z e z ) (6) 175/197
e x (Eq.(6)) and e y (Eq.(6)) t = 0 m = (m,0,m z ) Eq.(7) at t = 0 d 2 m n 2 +(γb 0 ) 2 m n = 0 (n {x,y}) m n = C n cosω 0 t+s n sinω 0 t (7) ω 0 = γb 0 (8) ω 0 : C x = S y = m C y = S x = 0 e x (Eq.(2)) : ( dm x = γb 0 m y ) Eq.(7) ω 0 ( C x sinω 0 t+s x cosω 0 t) = γb 0 }{{} =ω 0 (C y cosω 0 t+s y sinω 0 t) C x = S y, S x = C y m x = +m cosω 0 t (9) m y = m sinω 0 t (10) m 2 = m 2 +m2 z = const m z = ± m 2 m 2 (11) 176/197
dm = γm (B 0 e z ) m x = +m cosω 0 t m y = m sinω 0 t m z = ± m 2 m 2 (12) y Y x ω 0 t X { ex (t) = cos(ω 0 t)e x sin(ω 0 t)e y m z z e Y (t) = sin(ω 0 t)e x +cos(ω 0 t)e y (13) x y x y m(t) = m e X (t)+m z e z (14) m z 177/197
(NMR) 10.3 (NMR) (ω = ω 0 ) B(t) = B 0 +B 1 (t) (15) B 0 B 1 B 0 = B 0 e z Y Y B 1 (t) = B 1Y e Y (t) (16) ( ) X X B1 (t) B 0 B 1Y = const, B 1X = 0 z m 0 m(t) m(t) =m X (t)e X (t)+m Y (t)e Y (t)+m z (t)e z (17) (m(0) = m e X +m 0z e z ) 178/197
(NMR) m X m Y m z de X de γm B = 0 B 1Y B 0 Y = ω 0 e Y, = +ω 0 e X e X e Y e z (18) m Y B 0 m z B 1Y = γ m X B 0 +γ 0 0 +m X B 1Y dm = dm ω 0 m Y ω 0 m X 0 = γm B + dm X dm Y dm z dm X dm Y dm z = γb 1Ym z 0 +γb 1Y m X Y : m Y(t) = 0 ( m Y(0) = 0) X,z :? 179/197
(NMR) B = B 0 e z B = B 0 e z +B 1Y e Y (t) dm x dm X +γb 0 m y γb 1Y m z dm y = γb 0 m x dm Y = 0 dm z 0 dm z +γb 1Y m X z Y : ω 0 = γb 0 :ω 1Y = γb 1Y B = B 0 e z +B 1Y e Y (t) m B 0 z B 1Y ( ) Y 180/197
(NMR) m B 1 B 0 B = B0 m X e X m (i) = m (i) X e (i) X +m(i) z e z e (i) X ) e(i X, m(i) z B 1 0 B (i) 1X 0 B 1 (t) = B (i) 1X e(i) X (t)+b(i) 1Y e(i) Y (t) (i) B 1X : m (i) B (i) 1X e(i) X dm(i) Y = m (i) z B(i) 1X e(i) Y m(i) Y B(i) 1X e z = γm (i) z B (i) 1X 0 m (i) B 1 ( m = const) M = m = m z e z B = B0 +B 1 X B 1 m (i ) = m (i) X e X +m z (i) e z X M = m = m X e X +m z e z B 1 = 0 B 1 0 181/197
(NMR) B = B 0 B 1 B 1 B 0 B 1 M B 1 M B 1 B 1 Observed 182/197
10.4. ( - ) T1, T 2 dm z = M z M 0 T 1 M z (t) = M 0 (1 e t T 1)+M z (0)e t T 1 z X m t ( - ) Mz dm = M T 2 M (t) = M (0)e t T 2 Y X m M t 183/197
10.5 (1) z G z e z (B e z ) B z (z) = (B 0 +G z (z z 0 ))e z B ω(z) = ω 0 +γg z (z z 0 ) 0 z B 1 (t) cos(ω 0 t) ( ) ω 0 t z = z 0 B z z B 0 +G z z B 0 Resonance with B 1 B 0 G z z 184/197
(2) z (x,y) G ξ e ξ (B z e z ) B z (ξ) = (B 0 +G ξ ξ)e z, (e ξ = cosθe x +sinθe y ) ξ ω 0 (ξ) = γ(b 0 +G ξ ξ) = ω 0 +γg ξ ξ η B z B 0 : ρ(x,y) s(t,θ) = ρ(x,y)e jω 0 (ξ(x,y))t dxdy F t ( ) S(ω,θ) = ρ(ξ(ω),θ)dη ( ξ(ω) = ω ω 0 γg ξ ) y θ ξ x ω 0 (ξ) = ω 0 +γg ξ ξ CT 185/197
s(t,θ) ξ ξ(x,y;θ), η η(x,y;θ) ω 0 (ξ) = ω 0 +γg ξ ξ ω 0 (x,y,θ) s(t,θ) = y x ρ(x,y)ejω 0 t dxdy = η S(ω,θ) = s(t,θ)e jωt = t = η = η ξ ρ(ξ,η) t ej(ω 0+γG ξ ξ ω)t dξdη η ξ ρ(ξ,η)ejω 0 t dξdη ξ ρ(ξ,η)ej(ω 0+γG ξ ξ ω)t dξdη ρ(ξ,η)δ(ω 0 +γg ξ ξ ω)dξdη ξ = ( { ρ(ξ,η)dη L η r(ξ,η;θ) ξ = ω ω }) 0 L η γg ξ S(ω,θ) = ρ(ξ(ω),θ)dη, ξ(ω) = ω ω 0 γg ξ 186/197