Functional MRI (second edition) -- 4. Basic Principles of MR Image Formation

Bloch Equation

dM⃗ dt=γM⃗ ×B⃗ +1T1(M0→−Mz→)−1T2(Mx→+My→)\frac{d\vec{M}}{dt}=\gamma \vec{M} \times \vec{B}+\frac{1}{T_{1}}(\vec{M_{0}}-\vec{M_{z}})-\frac{1}{T_{2}}(\vec{M_{x}}+\vec{M_{y}})————————–1

可以分解为三个标量方程：

dMxdt=MyγB−MxT2\frac{dM_{x}}{dt}=M_{y}\gamma B-\frac{M_{x}}{T_{2}}————–2

dMydt=−MxγB−MyT2\frac{dM_{y}}{dt}=-M_{x}\gamma B-\frac{M_{y}}{T_{2}}————–3

dMzdt=−(Mz−M0)T1\frac{dM_{z}}{dt}=-\frac{(M_{z}-M_{0})}{T_{1}}—————-4

求解4得到

Mz=M0(1−e−t/T1)M_{z}=M_{0}(1-e^{-t/T_{1}})———————5

求解2、3的解为

Mx=(Mx0cosωt+My0sinωt)e−t/T2M_{x}=(M_{x0}cos\omega t+M_{y0}sin\omega t)e^{-t/T_{2}}—————6

My=(−Mx0sinωt+My0cosωt)e−t/T2M_{y}=(-M_{x0}sin\omega t+M_{y0}cos\omega t)e^{-t/T_{2}}——————7

写成

Mxy=Mx+iMy=(Mx0+iMy0)e−t/T2(cosωt−isinωt)=Mxy0e−t/T2e−iωtM_{xy}=M_{x}+iM_{y}=(M_{x0}+iM_{y0})e^{-t/T_{2}}(cos\omega t-isin\omega t)=M_{xy0}e^{-t/T_{2}}e^{-i\omega t}——————8

磁场不是均匀稳定磁场而是

B(τ)=B0+Gx(τ)x+Gy(τ)y+Gz(τ)zB(\tau)=B_{0}+G_{x}(\tau)x+G_{y}(\tau)y+G_{z}(\tau)z————–9

由于ω=γB\omega=\gamma B————-10

得到

Mxy(x,y,z,t)=Mxy0(x,y,z)e−t/T2e−iγB0te−iγ∫t0(Gx(τ)x+Gy(τ)y+Gz(τ)z)dτM_{xy}(x,y,z,t)=M_{xy0}(x,y,z)e^{-t/T_{2}}e^{-i\gamma B_{0}t}e^{-i\gamma \int_{0}^{t}(G_{x}(\tau)x+G_{y}(\tau)y+G_{z}(\tau)z)d\tau}

接收到的MR信号为

S(t)=∫x∫y∫zMxy(x,y,z,t)dxdydzS(t)=\int_{x}\int_{y}\int_{z}M_{xy}(x,y,z,t)dxdydz

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Slice Selection

ω+−Δω/2=γGz(z+−Δz/2)\omega +- \Delta \omega/2=\gamma G_{z}(z+-\Delta z/2)

一层中的

M(x,y)=∫z0+Δz2z0−Δz2Mxy0(x,y,z)dzM(x,y)=\int_{z_{0}-\frac{\Delta z}{2}}^{z_{0}+\frac{\Delta z}{2}}M_{xy0}(x,y,z)dz

由于已经选取了一层，所以在2D下的信号

S(t)=∫x∫yM(x,y)e−iγ∫t0(Gx(τ)x+Gy(τ)y)dτS(t)=\int_{x}\int_{y}M(x,y)e^{-i\gamma \int_{0}^{t}(G_{x}(\tau)x+G_{y}(\tau )y)d\tau}

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k-space

定义:

kx(t)=γ2π∫t0Gx(τ)dτk_{x}(t)=\frac{\gamma}{2\pi}\int_{0}^{t}G_{x}(\tau)d\tau

ky(t)=γ2π∫t0Gy(τ)dτk_{y}(t)=\frac{\gamma}{2\pi}\int_{0}^{t}G_{y}(\tau)d\tau

所以

S(t)=∫x∫yM(x,y)e−i2πkx(t)xe−i2πky(t)ydxdyS(t)=\int_{x}\int_{y}M(x,y)e^{-i2\pi k_{x}(t)x}e^{-i2\pi k_{y}(t)y}dxdy

因此S(t)与k之间的关系是2-D傅里叶变换。

Field of view:

FOVx=1Δkx=FOV_{x}=\frac{1}{\Delta k_{x}}=sampling rate along kx=1γ2π(GxΔt)k_{x}=\frac{1}{\frac{\gamma}{2\pi}(G_{x}\Delta t)}

FOVy=1Δky=FOV_{y}=\frac{1}{\Delta k_{y}}=sampling rate along ky=1γ2π(GyΔt)k_{y}=\frac{1}{\frac{\gamma}{2\pi}(G_{y}\Delta t)}

voxel size:

FOVxMx=1MxΔkx=12kxmax\frac{FOV_{x}}{M_{x}}=\frac{1}{M_{x}\Delta k_{x}}=\frac{1}{2k_{xmax}}

FOVyMy=1MyΔky=12kymax\frac{FOV_{y}}{M_{y}}=\frac{1}{M_{y}\Delta k_{y}}=\frac{1}{2k_{ymax}}