For any pool of NMR spins undergoing continuous RF saturation “on-resonance”, the NMR signal intensity at steady-state is described by (

15),

where

*M*_{z}^{o} is the equilibrium magnetization in the absence of RF saturation,

*ω*_{1} is the

*B*_{1} magnetic field strength of the saturation RF pulse in radians/s, and

*R*_{1} and

*R*_{2} are the longitudinal and transverse relaxation rates of the spin pool, respectively. If this pool is also exchanging spins with another pool during this process, then an exchange term can be added to each relaxation rate as follows,

where k is the rate at which these spins leave this pool and enter the other pool. Assuming that this rate is fast relative to the relaxation rates, i.e.,

*k* R

_{1} and

*k* R

_{2} (experimental evidence to support this assumption is given under Results),

equation 2 simplifies to,

For simplicity, M_{z}^{ss}/M_{z}^{o} is temporarily given the symbol m. This equation holds for any pool of spins undergoing continuous RF saturation. To describe specific pools, we define m^{a} as the bulk water pool and, for the particular case of a Eu^{3+}-based PARACEST agent, m^{b} as the Eu^{3+}-bound water pool.

Now, if one saturates the Eu

^{3+}-bound water pool using a frequency selective RF pulse (ω

_{1}), then these saturated or partially saturated spins will move into the bulk water pool at a rate,

*k*^{b}, resulting in loss of magnetization in the bulk water signal. If the Eu

^{3+}-bound water is fully saturated, m

^{b} would by definition be zero. If partially saturated, 0 < m

^{b} < 1. The rate of loss of magnetization in the bulk water pool due to exchange depends upon both

*k*_{b} and the fractional saturation in the Eu

^{3+}-bound water pool, 1 −

*m*^{b}. So, if one defines the product, (1 −

*m*^{b})

*k*_{b} as R

_{CEST},

equation 4 follows from

equation 3.

After a prolonged period of RF saturation on the Eu^{3+}-bound water resonance and the system has reached steady state, the magnetization remaining in the bulk water pool is determined by R_{CEST} and the rate of recovery of bulk water magnetization due to T_{1}. In the equations that follow, we use the term c/55.5 as the fraction of protons in the bound pool (the agent) relative to the total water. This term strictly applies to aqueous samples only (in vitro samples). For the more general case (i.e., tissue), c/55.5 should be replaced with (n)c_{agent}/c_{water} where n is the number of exchangeable protons on the agent, c_{agent} is the concentration of the agent in the tissue, and c_{water} is the concentration of water protons in the tissue. The value of c_{water} can be estimated independently using a proton density weighted imaging sequence. Hence, for in vitro conditions,

Since CEST spectra are presented as plots as M

_{z}^{ss}/M

_{o}, it is useful to convert

equation 6 into,

Thus, a plot of M

_{z}^{ss}/(M

_{o} − M

_{z}^{ss}) versus 1/

*ω*_{1}^{2} should be linear with a slope of 55.5

*R*_{1}^{a}k_{b}/

*c* and a Y-axis intercept of 55.5

*R*_{1}^{a}/(

*k*_{b}_{*} *c*) while the X-axis intercept (when M

_{z}^{ss}/(M

_{o} − M

_{z}^{ss}) = 0) provides a direct readout of the exchange rate, −1/k

_{b}^{2}. The power of this method is that

*k*_{b} can be determined without knowing concentration or relaxation rate. The determination of the agent concentration in tissue requires knowledge of the value of R

_{1}^{a}. Since the concentration of agent in tissue will be low and the relaxivities of these agents are also low, one can use the intrinsic value for R

_{1} of the tissue as an estimate for R

_{1}^{a}. Given this assumption and the

*k*_{b} as determined from the intercept, then the concentration, c, can be derived from the slope.

Equation [7] can also be derived from the Bloch-McConnell equations using identical assumptions.

This analysis assumed that measurements of magnetization were performed at steady state following a very long saturating pulse. It is straightforward to show that the same analysis applies to magnetization measurements obtained using the imaging sequence shown in , which uses saturation pulses applied for durations that are much shorter than T

_{1}. In this approach, CEST pulses alternate with observation pulses and acquisitions and these pulse trains repeat at intervals of TR/N, where N is the number of slices imaged. In this case, the longitudinal magnetization varies as a result of relaxation processes, observation pulses, and the application of CEST pulses. In this imaging sequence, the signal-to-noise and contrast are determined primarily by the central k-space lines and thus it is the magnetization at this point in the data collection that determines the CEST effect. If linear k-space sampling is employed for a 128×128 acquisition, the steady state condition will occur after 64 TR intervals. If we employ a 100 ms TR with a 50 ms Fermi pulse for CEST saturation (50% duty cycle), steady state is reached after 6.4 sec with 3.2 sec of CEST saturation. These parameters are similar to those used when the CW saturation employed. Note that if a large number of slices are chosen, TR can become longer than T

_{1} and steady state is then virtually identical to that following a long saturation pulse. Taking all of these considerations into account, we can derive the same expression as is shown in

equation 7 for CEST imaging sequence shown in .

An implicit assumption of this straight line analysis is that the bound to free pool chemical shift difference, which was not considered at all, is much larger than ω

_{1}. This condition keeps the free water magnetization pointing along the z axis. If ω

_{1} is increased to violate this condition, the plot curves upward. Going the other way, decreasing ω

_{1}, the plot becomes too noisy; m approaches 1 and 1-m can be positive, negative, or zero. An anonymous reviewer notes that when

Eq. [7] is multiplied, term by term, by ω

_{1}^{2}, the result plots as a straight line against ω

_{1}^{2}, again with the x intercept determining k

_{b}. For two point analyses, the two methods give the same k

_{b}. With more than two points, the plot against ω

_{1}^{2} has more favorable noise propagation than the plot against 1/ω

_{1}^{2}.