Relaxivity of an agent is it's ability to change the natural relaxation (R1 and/or R2) of the tissue. To get the relaxivity of an agent you therefore measure the R1 and/or R2 at various concentrations of the agents. Assume that you have a gadolinium based agent, then R1_observed = R1_background + [Gd] * relaxivity_Gd_agent which is a linear relationship and thus easy to solve via linear fit (relaxivity is the slope).

Keep in mind R=1/T2 and R1=1/T1: R indicates rate and T indicates time. For R2 and R2* measurement, the most common way is to acquire multiple echoes. The data should follow an exponential decay such that S= S0*e(-TE/T2). There are various methods to measure R1, with inversion recovery probably being the golden standard and saturation recovery being probably the easiest (spin-echo with a constant TE and variable TR with largest TR being very large). The data should follow an exponential recovery model but the exact equation depends on the method. For the saturation recovery method S=S0(1-e(-TR/T1)). In both equations S is the measured signal and S0 is a constant representing the signal at equilibrium.

T1 and T2-weighted images are different from T1 and T2 measurements. You can optimize contrast once you know the characteristic T1 and T2 times of two different tissues of interests to improve CNR. But, in general, you can achieve T1-weighted spin-echo images with short TE and TR and T2-weighted spin-echo images with long TE and TR.

Carlos J. Perez-Torres gave a very good answer about the measurement of relaxivities above. If you start with a new, unknown compound, you would first have to find out what range of concentrations might be relevant for MRI. With respect to r2 measurements, you could start with a reasonably strong T2-weighting spin-echo sequence (i.e., long TR>3000 ms, intermediate TE of e.g. 50 ms) and add first low and then increasing amounts of your agent to water (as your solvent) and compare the signal of the pure solvent with the one of the dissolved contrast agent. If the signal shows a substantial difference (e.g. about 50% signal drop due to the contrast agent), then the concentration seems appropriate for r2 (and probably also for r1) measurements. I.e., start your T2 measurements with a tenth of this concentration and increase up to 10 times this concentration to cover a reasonable range of T2 and R2 values, from which r2 can be determined.

(For r1, basically the same procedure with a reasonably strong T1-weighting test pulse sequence should be performed.)

thanks for all answers, i want to know do the parameters to be selected depends on the which paramagnetic metal is present?

The parameters do not directly depend on the paramagnetic metal of your contrast agent (e.g., they can be similar for gadolinium-based and manganese-based agents); they depend on the (beforehand unknown) relaxivities r1 and r2. (r1 and r2can (but need not) be different for different metals. E.g., for superparamagnetic iron-oxide contrast agents, r2 is generally much greater than r1, i.e. you should either use shorter TEs or lower concentrations for r2 measurements.)

Obviously, this means that you have to start with some guess or rough estimation of r1 and r2, then perform some preliminary measurements to determine approximate values of r1 and r2, and only then you will be able to set up a more precise measurement with more appropriate measurement parameters (TE, TI, concentrations etc.).

Experimentally, you take a range of concentrations of your contrast agent in water or buffer (say, 0.01 mM to 0.1 mM, or something like that), and use a relaxometric protocol to measure the relaxation times of protons in those samples. For a sample protocol, see below:

The MRI experiments were conducted as describe here:

3 T experiments: The T2 and T1 relaxation measurements and phantom images were obtained using a 3.0 T system (Philips, The Netherlands). Standard 5-mm diameter NMR tubes were used for experiments. A transmit/receive radio frequency (RF) head coil was applied. A single slice multi-echo pulse sequence was used for T2 measurements with the following parameters: repetition time (TR) = 2000 ms, 1 average, matrix size 128 × 128, field of view (FOV) 16 cm × 16 cm, slice thickness 5 mm, 32 echoes 15 ms apart. The T2 relaxation time was calculated using a single exponential fitting of the echo train. For T1 measurements the inversion recovery (IR) method was used with the following parameters: slice thickness 5 mm, FOV 16 × 16 cm, 1 average, matrix size 128 × 128, echo time (TE) = 17 ms, TR= 15 s, inversion time TI = (25, 250, 500, 800, 1200, 1700, 2300, 3000 ms). The T1 relaxation time was calculated using a single exponential fitting of the MR signal at different inversion times.

9.4 T experiments: T1 and T2 relaxation measurements and phantom images were obtained using a 9.4 T / 21 cm magnet (Magnex, UK) and a Bruker console (Bruker, Germany). Standard 5-mm diameter NMR tubes were used for experiments. A transmit/receive radio frequency (RF) volume birdcage coil was applied. A single slice multi-echo pulse sequence was used for T2 measurements with the following parameters: repetition time (TR) = 7.5 s, 1 average, matrix size 128 × 128, field of view (FOV) 3 cm × 3 cm, slice thickness 2 S2 mm, 64 echoes 4 ms apart. The T2 relaxation time was calculated using a single exponential fitting of the echo train (Bruker, Germany). For T1 measurements TRUE FISP method was used with the following parameters: slice thickness 2 mm, FOV 3 × 3 cm, 1 average, matrix size 128 × 128, echo time (TE) = 1.5 ms, TR = 1s.

You then do a relaxometric plot as follows:

Relaxivities (r1, r2) were determined from relaxation times measured at different concentrations using the following equation:

1/Ti= 1/T(i,0) + r(i).C

where Ti are the observed relaxation times in the presence of the contrast agent, T(i,0) is the relaxation time of pure water protons, C is the concentration of the MRI CA. The subscripts indicate longitudinal (i= 1) or transverse (i = 2) relaxivities and relaxation times, respectively.

Please see: RSC Adv., 2017,7, 38125-38134 for more details.