# The Approach to Steady State Using Homogeneous and Cartesian Coordinates

Authors: D. F. Gochberg; Z. Ding
Journal: Computational and Mathematical Methods in Medicine
 (2013)

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## Abstract

Repeating an arbitrary sequence of RF pulses and magnetic field gradients will eventually lead to a steady-state condition in any magnetic resonance system. While numerical methods can quantify this trajectory, analytic analysis provides significantly more insight and a means for faster calculation. Recently, an analytic analysis using homogeneous coordinates was published. The current work further develops this line of thought and compares the relative merits of using a homogeneous or a Cartesian coordinate system.


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## Body

## 1. Introduction

A recent paper by Nazarova and Hemminga [1] analyzed the signal arising from repeated identical RF pulses using a formalism based on homogeneous coordinates. Unlike the conventional Cartesian coordinates, homogeneous coordinates allow T1 relaxation to be treated in a manner similar to RF field-induced rotations, that is, by matrix multiplication. Signals from repeated identical RF pulses are an issue, most notably, in steady-state free precession (SSFP) fast imaging [2] techniques, where the lack of gradient or RF spoiling leads to complicated dynamics when the repetition time <T2. The formalism can also be applied to any repeated sequence of pulses, such as those, for example, that arise in pulsed saturation magnetization transfer methods [3].In this paper, we will expand on the work of Nazarova and Hemminga to show a simple way to calculate the steady-state magnetization, and we will relate the properties of the homogeneous matrices and calculations to those that arise in a more traditional Cartesian representation of magnetization. Specifically, we explicitly establish algebraic relations between the two systems, thus facilitating understanding of the pros and cons of each representation. We will employ the same notation as Nazarova and Hemminga, except as noted.


## 2. Background

Consider a pulse sequence made up of RF pulses and magnetic field gradients repeated everyτ seconds (see Figure 1). The conventional representation of the magnetization is a 3×1 vector M with equilibrium value Meq and components Mx, My, and Mz in the rotating frame. RF pulses with field strengths = B1≫1/(γT2) and angular frequency offset Δω0 can be represented by a 3×3 rotation matrix Rx,y(α), where T2 is the transverse relaxation rate and γ is the gyromagnetic ratio. Likewise, off-resonance precession and the scaling effect of relaxation can also be represented by matrices. For example, precession about the z direction and the decay of magnetization can be represented as follows:
(1)Rz(Δω0τ)=[cosΔω0τsinΔω0τ0-sinΔω0τcosΔω0τ0001],S(τ,T1,T2)=[e-τ/T2000e-τ/T2000e-τ/T1].
Similarly, a gradient pulse can be modeled by making the precession position dependent. In distinction, the effect of  T1 relaxation to a nonzero thermal equilibrium value cannot be represented by a 3×3 matrix. It is instead equivalent to an additive translation.Figure 1
An arbitrary sequence of pulses repeated everyτ seconds.Nazarova and Hemminga use this approach to give the relation for magnetization aftern repeated α pulses about the x-axis:
(2)Mn=Rz(Δω0τ)S(τ,T1,T2)Rx(α)Mn-1+(1-e-τ/T1)Meq,
where Mn is the magnetization after the nth repetition (initial condition = M0). The general form for the repetition of an arbitrary pulse sequence is
(3)Mn=CMn-1+DMeq,forn≥1,Mn=M0,forn=0.
Note that D is scalar only when a single RF pulse is repeated.An alternative approach is to use homogeneous coordinates whereT1 relaxation enters in the same way as do rotations:
(4)Mn=AMn-1,forn≥1,Mn=M0,forn=0.M is now a 4×1 vector with components Mx, My, Mz, and 1, and A is a 4×4 matrix whose upper left portion matches the 3×3 rotations and scalings discussed above and whose far right column represents the T1 relaxation towards Meq. For example,
(5)Rz(Δω0τ)=[cosΔω0τsinΔω0τ00-sinΔω0τcosΔω0τ0000100001].
And T1 relaxation for a period τ is represented by
(6)T=[10000100001Meq(1-e-τ/T1)0001].Meq in this notation is represented by
(7)[00Meq1].
(Note the corrections to Nazarova and Hemminga in (6) and (7)).


## 3. Homogeneous and Cartesian Representations

Both homogeneous and conventional representations have identical underlying math and therefore will give equivalent results.  However, their differences in formalism have small consequences in computation time, and they provide different notational approaches.The solution in the homogeneous case is(8)Mn=AnM0Mn=pA1λA1nvA1+pA2λA2nvA2+pA3λA3nvA3+pA4λA4nvA4,
where vAi and λAi are the ith eigenvector and eigenvalue, respectively, of A and pAi is the corresponding projection of M0 onto vAi. The second form of the solution can be derived in two ways: (1) by rewriting An as (VAΛAVA-1)n=VAΛAnVA-1, where the columns of VA are vAi, ΛA is diagonal with elements λAi, and pA (with elements pAi) equals VA-1M0 and (2) by rewriting M0 as pA1vA1+pA2vA2+pA3vA3+pA4vA4 and applying (4) n times. (Note that in Nazarova and Hemminga VA is called B).The conventional Cartesian case has the less intuitive solution to (3):
(9)Mn=CnM0+(∑i=0n-1Ci)DMeq,forn≥1Mn=CnM0+(-∑i=n∞CiD+∑i=0∞CiD)MeqMn=CnM0+(-Cn+I)∑i=0∞CiDMeqMn=Cn(M0-(I-C)-1DMeq)+(I-C)-1DMeq(10)Mn=VCΛnVC-1(M0-(I-C)-1DMeq)+(I-C)-1DMeqMn=pC1λC1nvC1+pC2λC2nvC2+pC3λC3nvC3+(I-C)-1DMeq,
where the columns of VC are the eigenvectors vCi of C, the elements of the diagonal matrix ΛC are the eigenvalues λCi, and pC (with elements pCi) equals VC-1(M0-(I-C)-1DMeq).Often, one only cares about the steady-state solution, which is the solution that is independent ofn and is approached as n→∞. We can solve this condition by solving Mn when Mn=Mn-1. For the homogeneous notation (4),
(11)Mn=AMn-1=AMn.
That is, the steady-state solution is the eigenvector of A with eigenvalue = 1.In the conventional formalism, we again solve for a steady-state solution(≡Mss) by setting Mn=Mn-1, this time in (3):
(12)Mss=(I-C)-1DMeq,
(another option is to take n→∞ in (9) with Cn→0 due to the relaxation terms on the diagonal). Since taking an inverse takes ~dimension3 operations, while finding an eigenvector via row reduction takes ~1/3 dimension3 operations [4], finding the steady-state via the homogeneous equation may provide a slightly more efficient method, though no rigorous evaluation of the computation time has been made. The computation time is normally not essential but may be so in certain problems, such as least squares fitting of magnetization transfer parameters [3].


## 4. Relations between Homogeneous and Cartesian Representations

In general,A has the form
(13)A=[C(3×3)DMeq(3×1)0001].From this form it follows that(14)λAi=λCi,fori=1,2,3,λA4=1,vAi=[vCi0],fori=1,2,3,vA4=[Mss1].


## 5. An Example

Consider a simple example: a single short (relative toT2) 90° pulse repeated every τ seconds. To keep the illustration as analytically simple as possible, we assume T1=T2 and a system starting at thermal equilibrium. This gives (from Nazarova and Hemminga with corrections)
(15)A=[Ecosθ0Esinθ0-Esinθ0Ecosθ00-E0Meq(1-E)0000],
where E=exp(-τ/T1)=exp(-τ/T2) and θ=Δω0τ. We can solve Mn using (8) and then convert back to Cartesian coordinates, or we can use (13) to get C and DM0 and then use (10). With help from Mathematica, we get
(16)Mn=Meq1+E2+2Eγ[2Eγ(1-γ)E(1-E-2γ)1+E(-1+2γ)]+(-E)n(γ+i1-γ2)n×Ei1-γ2(i1-γ2+E+γ)Meq(1+γ)(1+E2+2Eγ)(-1+γ+i1-γ2)×[2γ(1-γ)-1+γ-i1-γ2γ+i1-γ21]+En(-γ+i1-γ2)n×Ei1-γ2(-i1-γ2+E+γ)Meq(1+γ)(1+E2+2Eγ)(1-γ+i1-γ2)×[2γ(1-γ)-1+γ+i1-γ2γ-i1-γ21],
where γ=(1-cosθ)/2 (not to be confused with the gyromagnetic ratio). Note that even though it is calculated from complex eigenvectors and eigenvalues, for every n, Mn has real components. (We have chosen to keep the solution in a form where its connection to (8) is clear.) One of the eigenvalues = 1, corresponding to the steady-state solution. Since one of the p components equals zero, there are only two eigenvectors that contribute, thereby ensuring that Mn travels in a plane as n increases. If there is no relaxation (E=1), then this plane is perpendicular to the steady-state vector, as can be seen by taking the dot product of the steady-state vector with the other two eigenvectors. (In this case the “steady-state” vector is never actually reached, as discussed below.)If insteadM0≠Meq, then none of the p components are equal to zero.  Therefore, an additional eigenvector contributes, and the path is no longer planar. The additional eigenvector is
(17)[γ-1γ-11]
with corresponding eigenvalue E. Since this eigenvalue is real, it cannot contribute to a rotational trajectory, as will be illustrated below.


## 6. Numerical Methods

Figure2 plots the case for a repeated 90° pulse with E=0.98, θ=60°, and with two different initial conditions M0. Figure 2(a) illustrates that when M0=Meq, the path is a planar spiral, as expected. When M0≠Meq (reached by magnetization preparations), the spiral instead wraps around a cone. The head-on view in Figure 2(b) indicates that the spirals rotate at the same angular rate, which follows from the additional eigenvalue being real. The conditions in Figure 3 are the same as in Figure 2 except that E=1. With no relaxation, there is no decay and the “steady-state” solution is never reached; it is perpendicular to the plane containing the trajectory.M
n for n=0,1,…,200. α=90°, θ=60°, and E=0.98 with two different initial conditions: M0=[001]=Meq (solid circles starting at *) and [-0.4472-0.22360] (open circles starting at ×). The solid line is along the +z-axis and the dashed line is the steady-state magnetization Mss. (a) and (b) are two different views, with (b) looking down along the direction of Mss.
(a)
(b)Identical to Figure2, but with E=1. Note that the “steady-state” vector is never actually reached and that it is perpendicular to the plane containing the circular trajectory.
(a)
(b)


## 7. Discussion

The homogeneous and Cartesian coordinates provide two equivalent ways of representing magnetization. In pulsed repetition experiments, homogeneous coordinates lead to a simpler equation for the dynamics and steady state, but with a less intuitive connection to the measured magnetization. The choice of which system to use is in the end one of personal preference.



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*Source: 729236-2013-07-28.xml*