Essential Concepts in MRI. Yang Xia

Essential Concepts in MRI

matrix product Aρ, via Eq. (A2.33). Since Ix=12[0110] in the formalism of Pauli’s spin matrices (cf. Appendix A2.5), we have, from Eq. (3.12),

(3.16)

where Tr represents the trace of the matrix (the sum of the diagonal elements) – see Eqs. (A2.32) and (A2.33), and the two examples at the end of Appendix A2.6. The term in brackets describes the degree of “single quantum coherence” of the ensemble, while the average (the bars) reflects the phase coherence between the +1/2 and –1/2 states. At thermal equilibrium, both a1/2*a−1/2¯ and a1/2a−1/2*¯ are zero (no transverse component), while the term (|a1/2|2¯−|a−1/2|2¯) has a stable value (the longitudinal magnetization along the z axis).

3.5 MACROSCOPIC MAGNETIZATION FOR SPIN 1/2

In the current context, the observable quantity is just the (macroscopic) magnetization M, given by

upper M equals ModifyingAbove less-than upper N gamma italic h over two pi upper I greater-than With bar (3.17a)

or upper M equals upper N gamma italic h over two pi left-parenthesis ModifyingAbove less-than upper I Subscript x Baseline greater-than With bar i plus ModifyingAbove less-than upper I Subscript y Baseline greater-than With bar j plus ModifyingAbove less-than upper I Subscript z Baseline greater-than With bar k right-parenthesis comma (3.17b)

where N is the number of spins, and i, j, and k are the unit vectors in the Cartesian coordinates. Equation (3.17) is important because it may be shown that any state of the density matrix (defined in Appendix A2.6) for an ensemble of non-interacting spin-1/2 particles can be described using the macroscopic magnetization defined in this manner, thus permitting a classical description of simple spin systems.

In the absence of an external magnetic field, the ensemble average of the magnetization vector should be zero due to the random directions of the magnetic dipoles of the nuclei.

If a sample is immersed in an external field and in thermal equilibrium, the density operator associated with this magnetization vector is given by

rho equals StartFraction exp left-parenthesis negative script upper H slash k Subscript upper B Baseline upper T right-parenthesis Over upper T r left-bracket exp left-parenthesis script upper H slash zero width space k Subscript upper B Baseline upper T right-parenthesis right-bracket EndFraction period (3.18)

The transverse component of M is zero due to the even distribution of the azimuthal phase angles of the precessing nuclei in the transverse plane. This corresponds to phase incoherence leading to the zero value of the off-diagonal elements of ρ,

ModifyingAbove a Subscript 1 slash 2 Baseline zero width space asterisk a Subscript negative 1 slash 2 Baseline With bar equals ModifyingAbove a Subscript 1 slash 2 Baseline a Subscript negative 1 slash 2 Baseline zero width space asterisk With bar equals 0 period (3.19)

The z component of the magnetization M arises from the difference in populations between the upper and lower energy states. At room temperature, the magnitude of this magnetization in the equilibrium state, M₀, can be derived as

(3.20)

For a spin-1/2 system at room temperature, the population difference between the spin-up (m=+1/2) state and the spin-down (m=−1/2) state can be calculated from the diagonal elements of ρ, as

ModifyingAbove Math bar pipe bar symblom a Subscript 1 slash 2 Baseline Math bar pipe bar symblom squared With bar minus ModifyingAbove Math bar pipe bar symblom a Subscript negative 1 slash 2 Baseline Math bar pipe bar symblom squared With bar equals StartFraction italic h over two pi gamma upper B 0 Over 2 k Subscript upper B Baseline upper T EndFraction period (3.21)

For protons at B₀ = 1.4 T (60 MHz), it is equal to about 5 × 10^-6, a small value resulting from the small value of γħB₀ (Zeeman splitting) compared to k_BT (Boltzmann energy). It is this small magnetization of nuclei at room temperature that limits NMR detection sensitivity and leads to resolution limitations in MRI experiments [7, 8].

3.6 RESONANT EXCITATION

When both B₀ and B₁(t) are present and perpendicular to each other (B₁ in the transverse plane), we can write down the Hamiltonian in the laboratory frame as

script upper H Subscript lab Baseline equals minus italic h over two pi gamma upper B 0 upper I Subscript z Baseline minus 2 italic h over two pi gamma upper B 1 cosine left-parenthesis omega t right-parenthesis upper I Subscript x Baseline period (3.22)

In the rotating frame, the Hamiltonian becomes

script upper H Subscript rotating Baseline equals en-dash italic h over two pi gamma left-parenthesis upper B 0 en-dash omega slash gamma right-parenthesis upper I Subscript z Baseline en-dash italic h over two pi gamma upper B 1 upper I Subscript x Baseline period (3.23)

At ω = ω₀, we have

script upper H Subscript rotating Baseline equals en-dash italic h over two pi gamma upper B 1 upper I Subscript x Baseline period (3.24)

Since Ix=12(I++I−), where I₊ and I_- are the raising and lowering operators defined in Appendix A2.4, the time evolution of the spin system corresponds to an inter-conversion of each spin between |1/2> and |–1/2> at a rate of γB₁ (an oscillation).

3.7 MECHANISMS OF SPIN RELAXATION

Spin relaxation is truly fundamental and central to the theory of NMR and MRI; the influence of spin relaxation on both NMR and MRI measurements is wide, deep, and quite often subtle. It is therefore worth taking some time to learn to appreciate the subtleties of spin relaxation.

A nucleus in a liquid experiences a fluctuating field, due to the magnetic moments of nuclei in other molecules as they undergo

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