Lecture 3: The MR Signal Equation

• We have solved the Bloch equation and examined– Precession

– T2 relaxation

– T1 relaxation

• MR signal equation– Understand how magnetization changes phase due to

gradients

– Understand MR image formation or MR spatial encoding

Complex transverse magnetization: mm is complex.

m =Mx+iMy

Re{m} =Mx Im{m}=My

This notation is convenient:It allows us to represent a two

element vector as a scalar.

Re

Im

m

Mx

My

Inhomogeneous Object - Nonuniform field.Now we get much more general.

Before, M(t) was constant in space and so no spatial variable was needed to describe it.

Now, we let it vary spatially.

k

j

i

z

y

x

r

We also let the z component of the B field vary in space.

),B(γ),ω(Let

γBω Recall,

k)B(B),(B

oo

o

trtr

,trtr

This variation gives us a new phase term, or perhaps a more general one than we have seen before.

),,,(),,,(),(

ion,magnetizat erseFor transv

),()(

tzyxiMtzyxMtrm

trMtM

yx

Transverse Magnetization Equation

We can number the terms above 1-41. - variation in proton density

2. - spatial variation due to T2

3. - rotating frame frequency- “carrier” frequency of signal

4.

- spins will precess at differential frequency according to the gradient they experience (“see”).

ttrTto τreerMtrm

0iω)(/ )d,ω(iexp )(),( o2

)(rM o

)(/ 2 rTte

te oiω

tτr

0)d,ω(iexp

),ω( r

Transverse Magnetization Equation

Phase is the integral of frequency.

tτr

0)d,ω(iexp

),ω( r

- spins will precess at differential frequency according to the gradient they “see”.

t

τr0

)d,ω(

Thus, gradient fields allow us to change the frequency and thus the phase of the signal as a function of its spatial location.

Let’s study this general situation under several cases, from the specific to the general.

The Static Gradient Field

In a static (non-time-varying) gradient field, for example in x,

xrrGx xxx γG)B(γ)ω(

B

Plugging into solution (from 2 slides ago)

ttrTto τreerMtrm

0iω)(/ )d,ω(iexp )(),( o2

txeerMtrm trTto x

iω)(/ Gγiexp )(),( o2

and simplifying yields

The Static Gradient Field

Now, let the gradient field have an arbitrary direction. But the gradient strength won’t vary with time.

k)GGGB()(B o zyxr zyx

Using dot notation,

k)GB()(B o rr

Takes into account the effects of both the static magnetic and gradient fields on the magnetization gives

treerMtrm trTto r

iω)(/ Gγiexp )(),( o2

Time –Varying Gradient Solution (Non-static)

. We can make the previous expression more general by allowing the gradients to change in time. If we do, the integral of the gradient over time comes back into the above equation.

ttrTto τreerMtrm

0iω)(/ d)(Giexp )(),( o2

Receiving the signalLet’s assume we now use a receiver coil to “read” the transverse

magnetization signal We will use a coil that has a flat sensitivity pattern in space. Is this a good assumption?

dxdydzreerMtSx y z

ttrTtoreceive τ

0iω)(/ d)(Giexp )()( o2

- Yes, it is a good assumption, particularly for the head and body coils.

Then the signal we receive is an integration of

over the three spatial dimensions.

),( trm

),( trm

Receiving the signal (2)

3) Demodulate to baseband.i.e.

dxdydzreerMtSx y z

ttrTtoreceive τ

0iω)(/ d)(Giexp )()( o2

zoslice thickness z

2/

2/),,(),(

zz

zzo

odzzyxmyxm

Let’s simplify this:

1) Ignore T2 for now; it will be easy to add later.

2) Assume 2D imaging. (Again, it will be easy to expand to 3 later.)

tett oωireceive )(S)S( S(t) is complex.

Receiving the signal (3)

3) Demodulate to baseband.i.e. tett oωi

receive )(S)S( S(t) is complex.

Then,

dxdyrγx,ymtSx y

tτ

0d)(Giexp )()(

end goal

Gx(t) and Gy(t)

Now consider Gx(t) and Gy(t) only.

dxdyyγ

xγ

x,ymtS

t

x y

t

τ

τ

0 Y

0 x

)d(G2

2iexp

)d(G2

2iexp )()(

Deja vu strikes, but we push on!

Let t

xxτγ

tk 0)d(G

2)(

t

yyτγ

tk 0)d(G

2)(

dxdyykxkx,ymtS yx y

x 2iexp2iexp )()(

kxand ky related to t

Let t

xxτγ

tk 0)d(G

2)(

t

yyτγ

tk 0)d(G

2)(

dxdyykxkx,ymtS yx y

x 2iexp2iexp )()(

As we move along in time (t) in the signal, we simply change where we are in kx, ky

dxdyex,ymkkMx y

ykxkyx

yx ][2i

)(),(

(dramatic pause as students inch to edge of seats...)

kxand ky related to t (2)

x,y are in cm, kx,ky are in cycles/cm or cycles/mm

kx, ky are reciprocal variables.x y

))(),(()( tktkMtS yx

Remarks:1) S(t), our signal, are values of M along a trajectory in F.T. space.

- (called k-space in MR literature)

2) Integral of Gx(t), Gy(t), controls the k-space trajectory.

3) To image m(x,y), acquire set of S(t) to cover k-space and apply inverse Fourier Transform.

Spatial Encoding Summary Consider phase.

)(i),( x,y,teyxm

Receiver detects signal

planeyx

x,y,tR dxdyeyxmtS

,

)(i),()(

What is ? )(x,y,t

dt

tdt

)()(

Frequency

t

dttyxx,y,t0

')',,()(

is time rate of change of phase.

dt

tyxdtyx

),,(),,(

So

Spatial Encoding Summary

So t

dttyxx,y,t0

')',,()(

But ),,γB()( tyxx,y,t

Expanding, ])(G)(Gγ[B)( o ytxtx,y,t yx

We control frequency with gradients.

ydxdt

ydxx,y,t

ty

txo

tyx

00

0 o

)(Gγ)(Gγ

)(G)(GBγ)(

So,

dxdydyGxGieyxmtSx y

t

yxti

r ]))()(([exp(),()(0

0

Complex Demodulation

• The signal is complex. How do we realize this?

• In ultrasound, like A.M. radio, we use envelope detection, which is phase-insensitive.

• In MR, phase is crucial to detecting position. Thus the signal is complex. We must receive a complex signal. How do we do this when voltage is a real signal?

Complex Demodulationsr(t) is a complex signal, but our receiver voltage can only read a singlevoltage at a time. sr(t) is a useful concept, but in the real world, wecan only read a real signal. How do we read off the real and imaginary components of the signal, sr(t)?

titir

tir

eetts

etsts0

0

)()()(

)()(

Complex receiver signal ( mathematical concept)

The quantity s(t) or another representation, )()( tiet

is what we are after. We can write s(t) as a real and imaginary component, I(t) and Q(t)

))(sin()()(

))(cos()()(

)()()( )(

tttQ

tttI

tiQtIet ti

Complex Demodulation

An A/D converter reading the coil signal would read the real portion of sr(t), the physical signal sp(t)

))(cos()()(

)}(Re{)(

0 tttts

tsts

p

rp

Complex Demodulation

sp(t)

X

X

cos(t)

Inphase: I(t)

Quadrature: Q(t)

sin(t)

Low Pass Filter

Low Pass Filter

))(cos()(

)}(Re{)(

0 ttt

tsts rp

Let’s next look at the signal right after the multipliers. We canconsider the modulation theorem or trig identities.

Complex Demodulation

sp(t)

X

X

cos(t)

Inphase: I(t)

Quadrature: Q(t)

sin(t)

Low Pass Filter

Low Pass Filter

))](2cos())()[cos((2

1)(

)cos())(cos()()(

0int

00int

ttttts

ttttts

ermediatei

ermediatei

))](sin())(2)[sin((2

1)(

)sin())(cos()()(

0int

00int

ttttts

ttttts

ermediateq

ermediateq

))(cos()(

)}(Re{)(

0 ttt

tsts rp

Complex Demodulation

sp(t)

X

X

cos(t)

Inphase: I(t)

Quadrature: Q(t)

sin(t)

Low Pass Filter

Low Pass Filter

))(cos()(2

1)( tttI

))(sin()(2

1)( tttQ

Complex Demodulation

The low pass filters have an impulse response that removes thefrequency component at 2We can also consider them to have gain to compensate for the factor ½.