Evaluating and understanding the performances of magnetic colloids as contrast agents for MRI requires a theory describing their magnetic interactions with water protons. The field dependence of the proton longitudinal relaxation rate (nuclear magnetic relaxation dispersion profiles) in aqueous colloidal suspensions of superparamagnetic particles is based on the so-called Curie relaxation, which essentially accounts for the high field part of the NMRD profiles (B0>0.02 T). The low-field part of the NMRD profiles can only be explained by the crystal’s internal anisotropy energy, a concept which clarifies the important difference between superpara- and paramagnetic compounds: the anisotropy energy modifies both the electronic precession frequencies and the thermodynamic probability of occupation of the crystal magnetic states. Our theory clearly explains why a low-field dispersion exists for suspensions of small size crystals, and why it does not for large crystals’ suspensions. This important effect is due to the Boltzmann factors depending on the anisotropy energy, which is itself proportional to the particle volume.

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Computer simulations of water transverse relaxation induced by superparamagnetic particles are shown to disagree with the available theories, covering the slow diffusion domain. Understanding these new simulations, not in the slow diffusion domain, thus requires a new theoretical approach. A "partial refocusing model" is introduced for this purpose; it is based on a spatial division between an inner region where the gradients are too strong for the refocusing pulses to be efficient and an outer region where they are efficient. This model agrees with published simulations of relaxation induced by magnetic dipoles approximated as points. Susceptibility-induced T 2 -shortening refers to the dephasing of magnetic moments due to field gradients created by small magnetized particles. We are concerned here with strongly magnetized particles, defined by the condition that the dephasing during an echo interval is large, viz., CP ⌬ r Ͼ 1, where CP is half the interval between successive 180°pulses in a CPMG sequence, or half the echo time for a single (Hahn) spin-echo sequence ( CP ϭ TE/2), and where ⌬ r is the rms angular frequency shift at the particle surface (compared with a point infinitely far away). The present discussion is limited to spherical particles.No general theory is able to describe the relaxation typical of this process over the entire range of variation of the parameters, but some theories address the problem over parts of the whole range.First, the quantum-mechanical outer-sphere theory, which applies to relaxation induced by weakly magnetized particles (1,2), remains valid for strongly magnetized particles, provided they are small enough to satisfy the motional averaging condition, i.e., D Ͻ 1/⌬ r , where D ϭ r 2 /D is the diffusion time, where r is the particle radius and D the water diffusion coefficient ( D is thus the time required for a water molecule to diffuse a distance ͱ2r in any specified direction).Second, there are two limits in which the relaxation behavior for larger particles is well known: one is the short echo limit. The boundary between the weak and the strong magnetization domains is defined by the equality CP ⌬ r ϭ 1. The theories valid below the limit, especially mean gradient diffusion theory (MGDT (3)), appear thus as a useful theoretical starting point for the strong magnetization domain, even if this limit is unrealistic from a practical point of view (too-short echo times for magnetizations typical of superparamagnetic (SPM) particles, for instance).The other is the long echo limit, bringing us to the FID andT* 2 , where relaxation is governed by the static dephasing regime (SDR) (4,5). Intermediate echo times are less well explored, although recently theories have been introduced for the case of "slow diffusion" (TE Ͻ D ) around strongly magnetized particles (6,7). The actual condition is stated differently in the two publications and will be defined more precisely below.We present new simulations of relaxation induced by particles with a strong magnetization typical of supe...

Since 1/T2 of protons of tissue water is generally much greater than 1/T1 at typical imaging fields, small single-ion contrast agents--such as Gd(DTPA), which make comparable incremental contributions and therefore smaller fractional contributions to 1/T2 compared to 1/T1--are not as desirable for contrast-enhancement as agents that could enhance 1/T2 preferentially. In principle, such specialized agents will only be effective at higher fields because the field dependence (dispersion) of 1/T1 is such that it approaches zero at high fields whereas 1/T2 approaches a constant value. The residual 1/T2 is called the "secular" contribution and arises from fluctuations in time--as sensed by the protons of diffusing solvent or tissue water molecules--of the component of the magnetic field parallel to the static applied field. For solutions or suspensions of sufficiently large paramagnetic or ferromagnetic particles (greater than or equal to 250 A diameter), the paramagnetic contributions to the relaxation rates satisfy 1/T2 much greater than 1/T1 at typical imaging fields. We examine the theory of secular relaxation in some detail, particularly as it applies to systems relevant to magnetic resonance imaging, and then analyze the data for solutions, suspensions, or tissue containing ferritin, erythrocytes, agar-bound magnetite particles, and liver with low-density composite polymer-coated magnetite. In most cases we can explain the relaxation data, often quantitatively, in terms of the theory of relaxation of protons (water molecules) diffusing in the outer sphere environments of magnetized particles. The dipolar field produced by these particles has a strong spatial dependence, and its apparent fluctuations in time as seen by the diffusing protons produce spin transitions that contribute to both 1/T1 and /T2 comparably at low fields; for the larger particles, because of dispersion, the secular term dominates at fields of interest. On the basis of the agreement of theory with data for solutions of small paramagnetic complexes, large magnetite particles, and liver containing low-density polymer-coated magnetite agglomerates, it is argued that the theory is sufficiently reliable so that, e.g., for ferritin--for which 1/T2 is unexpectedly large--the source of its large relaxivity must reside in nonideal chemistry of the ferritin core. For blood, it appears that diffusion through intracellular gradients determines 1/T2.

Chemical exchange (CE) theory is compared with two theories of T 2 -shortening caused by microscopic magnetic centers: inner-and outer-sphere relaxation theory (long-echo limit) and mean gradient diffusion theory (short-echo limit). The CE equation is shown to be identical to these theories in the respective limits and appropriate parameter relationships are derived for spherical particles. The theories are then compared with computer simulations of spherical particles and with a recent general theory, with good agreement in the asymptotic regions. T 2 -shortening by paramagnetic (PM) particles is the basis for many MRI contrast agents and also for functional MRI (in which case the magnetic particles are deoxygenated red blood cells or capillaries). There are also endogenous magnetic materials such as ferritin and hemosiderin that affect MRI image contrast through the same mechanism. In an applied magnetic field these particles become partially aligned with the field; this induced magnetization, in turn, sets up microscopic field gradients that dephase nearby water protons, and hence shorten T 2 and T* 2 .While the complexity of the systems (irregular distributions, unequal magnetizations, varying orientation of capillaries, etc.) makes a general theory of relaxation difficult, there are theories that apply in certain limits, depending on the relative magnitude of three time parameters. In addition, a general theory was recently presented for weakly magnetized particles and applied to randomly distributed spheres (1). One of the time parameters is CP , defined as half the interval between successive 180°pulses in a CPMG sequence, or half the echo time for a single (Hahn) spin-echo sequence ( CP ϭ TE/2). The other two time parameters are inherent in the object or tissue being studied; they are the time to diffuse past a magnetized particle ( D ), and the time for a significant amount of dephasing to occur (i.e., the inverse of the spread in Larmor frequency, 1/⌬ ). These times will be defined more rigorously as specific scenarios are presented.In the "long-echo" limit ( CP ӷ D ), the standard quantum-mechanical theory of relaxation applies, as extended by Gueron and others (2-4) to cover induced magnetization (called the "Curie effect" by Gueron). In the "shortecho" limit ( CP Ӷ D ), the classical theory of diffusion in magnetic gradients, referred to herein as mean gradient diffusion theory (MGDT), may be used under certain conditions (5-7). In addition, a model based on chemical exchange (CE) theory (8 -9) has sometimes been used (10 -13), although this use has not been validated.We will attempt to provide such a validation by comparing CE theory with the two diffusion theories in the appropriate limits and with the more recent theory (1). In the process, we will determine the theoretical relationship between the CE parameters and the diffusion parameters. In some cases this will require new versions of the theoretical equations that are better suited for the comparison. We will also present new computer simula...