A numerical prediction of spin-lattice relaxation enhancement of water protons by paramagnetic metal ions and free radicals. A contribution to the basic principles of relaxometry in MRI

The Solomon-Bloembergen and several simplified equations from literature are critically evaluated with the purpose of selecting an equation for rapid numerical predictions of spin-lattice relaxation data of water protons induced by paramagnetic transition metal and lanthanide ions and organic free radicals. The following equation is proposed for practical applications: $$frac{1}{{T_1 }} approx Ffrac{{pi ^2 gamma _{rm I} ^2 gamma _S ^2 hbar ^2 N}}{k} frac{{meta }}{T} S(S + 1) approx F 1.06^7 frac{{meta }}{T} S(S + 1),$$ , whereT ₁ is the spin-lattice relaxation time, 1/T ₁ is the spin-lattice relaxation rate, γ₁ is the gyromagnetic ratio of the proton of water, γ_S is the gyromagnetic ratio of the electron,k is the Boltzman constant,h is the Plank constant, ? =h/2π,N is the number of paramagnetic species per cubic centimeter, i.e. the Avogadro number/1000,m is the concentration of the paramagnetic species in mM, η is the viscosity in centipoise at appropriate temperature,T is the appropriate absolute temperature,S is the spin quantum number, andF is a methodical factor which can range between 1–2.7, depending on the magnetic properties of the instrumentation, i.e. ～1.07 MHz (0.025 T)–90 MHz (2.1 T), and the relaxation enhancing agent. It is proposed to useF=1 for rapid numerical predictions. This equation is based on models involving translational motions and is suitable for rapid estimatesin vitro of spin-lattice relaxation times and rates of protons in water, and water protons in plasma and other biological fluids in the presence of paramagnetic metal ions, their low molecular weight complexes, and free radicals, as exemplified by nitroxyl (aminoxyl) radicals. Although this equation will be used for measurementsin vitro, the results could also be applied to project the effectiveness of new contrast agentsin vivo. It is believed that this equation will be useful in research involving the development of new contrast agents for MRI.