Noise correlation matrix for multiple RF coils derived from first principles of statistical physics

Y. Wang, R. Brown, P. Spincemaille, R. Lee Radiology, Weill Medical College of Cornell University, New York, NY, United States, Radiology, New York University, New York, NY, United States Introduction Noise correlation has been a controversial and confusing topic in MRI literature (1,2), due to the lack of a clear definition of thermal noise and the complexity of circuitry in MR scanners. The formulation outlined in the original Nyquist paper does not provide a straightforward account for noise correlation among two circuits (3). With the widespread use of multiple rf channel technology on MR scanners, the noise correlation matrix among coil elements is becoming a very important issue for coil design (4,5), and it is desired to have a fundamental understanding of the noise correlation. Here we attempt to elucidate the noise correlation from the first principle of thermal statistical physics. Theory The energy stored in a system of circuits is (6), H(q1...qn, i1...in) = ∑abLabiaib/2 + ∑aqa /2Ca, where ia is the current in coil a, qa the charge, Ca the capacitance, and Lab the mutual inductance between coils a and b. The thermal probability for various current values is determined by the Boltzman distribution. The noise correlation matrix is derived from the thermal average of current fluctuation (7): = ∫iaibexp(-∑cdLcdicid/2kT)di1...din/ ∫exp(-∑cdLcdicid/2kT)di1...din = kT(L )ab. Hence, Ψab ≡ = kT(L)ab. [1] Eq.1 characterizes the total noise correlation. Noise correlation at a given frequency is derived from the fluctuation-dissipation theorem with E/iω = f the “force” for current and iω/Z = α(ω) the generalized susceptibility (7,8): Ψab ≡ = 4kT Re((Z)ab). [2] Here Z is the coil impedance matrix. The relation between the noise correlation over all frequencies and that at one frequency is Ψab = ∫0 dω/2π Ψab, as derived from the Kramers-Kronig relation (7), α’(0)− α’(∞)=2/π ∫0dω α”(ω)/ω, where α’ and α” are the real and imaginary part of α. From the definition of impedance matrix, α’(0) = 0, α’(∞) = -L, α”(ω)/ω = Re((Z)). Therefore, (L)ab = 2/π ∫0dω Re((Z)ab). [3] The above theory is examplified in the case of a single coil made of R, L and C in series. <(i)> = 4kT R/(R+(ωL-1/ωC)), = 4kT ∫0 dω/2π R/(R+(ωL-1/ωC)) = kT/L (9). Materials and methods Two circular coils with 13.2 cm diameter were constructed. Experiments were performed on this coil system over a large tank of water. The spacing between the two coils was variable to allow a range of mutual inductance. The impedance matrix at various frequencies was measured using a HP network analyzer. Measurements of noise (voltage converted to current using the impedance matrix) were performed on a 1.5T GE SIGNA CV/i MR system. The RF excitation was disabled during imaging for noise collection. The mutual inductance matrix was estimated according to method by Grover (10). Results The network analyzer measured total noise (right side of Eq.3, Area(Re((Z)12)))was linearly correlated to the estimated inverse inductance (left side of Eq.3, (L)12) for the two coil system (Fig.1a), r = .95. The MR scanner measured noise matrix was linearly correlated to real part of the inverse of the coil impedance matrix (Fig.1b), r = .94. Discussion These preliminary experiments confirm the theoretical predictions on noise correlation (Eqs.2&3). Small errors in the experimental measurements may be attributed to weighting by the coil resonance spectrum (Q values) and effects from amplifiers. The theory presented here clarifies confusions in MRI literature. Eq.1 indicates that the total noise correlation is zero when there is no mutual inductance, consistent with the thermodynamic argument in Ref. 1. However, the noise correlation at a given resonance frequency is determined by the impedance matrix (Eq.2), as suggested in Ref. 2. Eqs.1&2 are consistent, as indicated by the KramersKronig relation Eq.3. REFERENCES 1. Jesmanowicz A et al, MRM 20:36-47 (1991). 2. Redpath TW, MRM 24: 85-9 (1992); British J Radiology 71:704-7 (1998). 3. Nyquist H, Phys Rev 32:110-3 (1928). 4. Weiger M, et al, MRM 45:495-504 (2001). 5. Ohliger MA, et al, MRM 52:628-39 (2004). 6. Landau and Lifshitz, Electrodynamics of Continuous Media, 2002. 7. Landau and Lifshitz, Statistical Physics, Part I&II;, 2003. 8. Callen HB, et al, Phys Rev, 83:34-40 (1951). 9. Feynman’s Lecture on Physics, v.1, 1977. 10. Grover FW, Inductance calculations, working formulas and tables, 1946.