The trivalent lanthanide cations also differ from the more common transition metal ions in having usually large coordination numbers, typically 8-10. Gd³⁺ exists in aqueous solution below pH ~6 as a hydrated “aqua” ion with ~8-9 inner-sphere water molecules (see Figure 1) each contributing a lone pair of electrons to its coordination sphere. Water molecules are quite polar so these “inner-sphere” water molecules each contribute a small amount of electron density to the highly charged Gd³⁺ ion thereby reducing the effective charge on the central ion. Although the preferred geometry of Gd(H₂O)₈³⁺ in solution likely approximates a square antiprism with the gadolinium ion situated between two planes each containing four water molecules on the corners of a square with the two planes staggered 45° with respect to each other. The entire system is quite dynamic with any one water molecule residing in the inner coordination sphere for only about 1 ns (4). Water exchange between an inner-sphere coordination site and bulk water occurs via an associative pathway with an additional water molecule “squeezing” into the coordination sphere to form a 9-coordinate intermediate followed by rapid dissociation of one of the existing water molecules to reform the Gd(H₂O)₈³⁺ structure. Upon chelation by an organic ligand such as those used in clinically approved MRI contrast agents, most of the water molecules are displaced from the inner-sphere of the Gd³⁺ ion by the more basic donor atoms of the ligand, typically amines (N) or carboxylates (O). A typical ligand used to create a MRI contrast agent has eight donor atoms so, upon chelation, only a single water molecule remains in the inner coordination sphere of the Gd³⁺. This single water binding site is important for MRI contrast because it allows, through chemical exchange, a large number of water molecules to experience the Gd³⁺ each second. Interestingly, it wasn't until in early 90's that we learned that this single Gd³⁺-bound water molecule in agents such as GdDTPA^2- and GdDOTA^- exchanges about 200-fold more slowly (5) than do the water molecules in the aqua ion, Gd(H₂O)₈³⁺. This moderately slow water exchange has consequences in limiting the relaxation efficiency or relaxivity of Gd³⁺-based MRI contrast agents (6, 7) in certain circumstances.

One can see by inspection of equation 1 that the Gibbs free energy of this equilibrium process will have a large favorable entropy term due to release of seven of the eight inner-sphere water molecules from the Gd³⁺. This entropy contribution is referred to as the “chelate effect”. Also, we have seen that the metal ion-ligand interaction must have a large electrostatic component that contributes an additional favorable enthalpy term so the overall free energy change becomes quite favorable. As a general rule of thumb, the Gd³⁺ aqua ion forms the most stable complexes with ligands having the most basic donor atoms. Some thermodynamic data (8-11) for the four ligands shown in Figure 2 are summarized in Table 1.

These data illustrate some common themes. First, the two most basic atoms in ligands such as these are always amines (represented by the first two protonation constants, log K₁ and log K₂) and, second, linear amines are typically less basic than macrocyclic amines. This is clearly evident in that the macrocyclic-derived ligand, DOTA, has the most basic nitrogen (highest log K₁), followed by DTPA, followed by the bis-amide of DTPA (DTPA-BMA), and the tetra-amide of DOTA (DOTA-(gly)₄). This trend also shows that amine groups with amide-containing side-chains (DTPA-BMA and DOTA-(gly)₄) are considerably less basic than amine groups with acetate side-chains (DTPA and DOTA). These apparently small differences in log K₁ and log K₂ values, however, have a significant impact on the thermodynamic stabilities of the resulting GdL complexes. Unlike the relatively small variations in the log K₁ and log K₂ values for the ligands, the log K_st values for the complexes vary by over 10 orders of magnitude (Table 1). Linear correlations between the sum of ligand protonation constants (Σ log K_1-n) and thermodynamic stabilities (log K_st) have been noted many times over the years (12) and the four ligands described here follow that trend nicely. Of course, certain assumptions must be made about how many protonations constants to include in the term Σ log K_1-n for each ligand (see footnote to Table 1) but the general theorem that ligands having more basic donor atoms form the most stable complexes certainly holds true. Given the low value of log K_st for GdDOTA-(gly)₄, one would then be tempted to conclude that this system would not be stable enough for in vivo applications. However, as we shall see from data presented below, this is not the case. In fact, the later system is extraordinarily kinetically inert toward dissociation and it is this chemical feature that plays a critical role in reducing in vivo toxicity.

The stability constant is widely used to compare contrast agents because it reduces comparisons to a single convenient number. From Table 1, it is clear that these different ligands have different basicity and thus different affinities for the proton. The equilibrium described by equation 1 is valid under conditions when the ligand is entirely deprotonated but, at physiological pH values, the ligand will be partially protonated so one can argue that a better way to compare ML stabilities is to use conditional stability constants, defined by equations 3 and 4.

The calculated conditional constants (log K_eff) at pH 7.4 are also given in Table 1. Because DOTA is such a basic ligand, there is strong competition for protons at pH 7.4 and log K_eff is 7 orders of magnitude lower for GdDOTA at pH 7.4 than the thermodynamic stability constant. Consequently, GdDTPA has a more favorable binding constant than does GdDOTA at pH 7.4. For less basic ligands like GdDTPA-BMA and DOTA-(gly)₄, the conditional constant is closer to the thermodynamic stability constant because there is less competition by available protons at pH 7.4. Another consequence of equation 3 is that as the pH is lowered, the concentration of protons is increased and the equilibrium in equation 3 is pushed to the left. Thus for GdDTPA, log K_eff is 18.4 at pH 7.4 while at pH 4, log K_eff = 11.2. Stronger acid conditions clearly results in lower complex stability.

In biological media, there are additional competitors besides the proton. For instance endogenous ions like zinc, copper, and iron form very stable complexes with these ligands. At the same time, gadolinium has a high affinity for phosphate, citrate, and carbonate ions and will bind to proteins like serum albumin. In a widely cited paper, Cacheris et al. (9) built a model taking into account a range of equilibrium constants and argued that GdDTPA-BMA had a high LD₅₀ (low acute toxicity) because the DTPA-BMA ligand had a high selectivity for Gd³⁺ compared to other endogenous ions like Zn²⁺. Cacheris et al. (9) proposed a “selectivity factor” that takes into account the stability constants of the Ca²⁺, Zn²⁺, and Cu²⁺ complexes as well as the pH. On the basis of their calculations, GdDTPA-BMA is expected to release half as much of its Gd³⁺ in vivo as compared with GdDTPA (9). However animal experiments have consistently demonstrated that more Gd³⁺ is retained in bone and other organs when the animals are given GdDTPA-BMA than when GdDTPA is administered (13-15). Clearly there are additional factors to consider besides thermodynamic stability. Models like the “selectivity factor” are based on the assumption that the system is either at thermodynamic equilibrium or will eventually reach thermodynamic equilibrium – likely a poor assumption in most cases where agents are cleared from the body relatively quickly. Clearly, the kinetics of metal dissociation or metal exchange is much more important in determining long term toxicity. Animal studies and increasingly the experience with NSF patients support the notion that kinetic inertness is the most critical factor.

An illustration of the relative importance of kinetic stability of each contrast agent is to consider a system where the contrast agent is dissolved in a solution containing phosphate anions at physiological pH. Even chelates with very high thermodynamic stability constants will equilibrate to form some free chelate and unchelated gadolinium (so called “free gadolinium”). Since gadolinium phosphate is insoluble, any unchelated gadolinium will precipitate as insoluble gadolinium phosphate. The system will react according to Le Chatelier's Principle to establish a new equilibrium to replenish the “free gadolinium” until the phosphate anions are almost all precipitated from the solution. The rate at which this happens is determined by the rate of dissociation of Gd³⁺ from each ligand. A contrast agent with a relatively low thermodynamic stability constant and relatively slow rate of dissociation will produce less solid gadolinium phosphate per unit time than an agent with a high thermodynamic stability constant and a relatively fast rate of dissociation. These kinds of considerations are important if the precipitation of solid gadolinium phosphate initiates a sequence of events that leads to toxicity.

An analysis of residual ¹⁷⁷LuDOTA-(gly)₄^- in tissue showed that the compound was mainly in blood, kidney and urine at 30 minutes (Figure 5). At 2 hours, the amount remaining in these organs had dropped significantly consistent with rapid excretion from the body via renal filtration with little apparent accumulation in any of the organs tested. These data are quite similar to results obtained for other macrocyclic-based ligand systems such as GdDOTA^-.