An MRI machine creates images by acquiring the magnetic resonance signal generated by the water molecules in human body. Water molecules generate the signal in response to a constant high magnetic field, and an electromagnetic field oscillating in RF frequency. To describe the signal generation process briefly:

The magnetization vector of the water molecules is disturbed from the equilibrium state by using an RF pulse of finite length, which is also called the excitation pulse. Following the RF pulse the magnetization vector returns back to their equilibrium state by means of a process called relaxation. The relaxation process is exponential and governed by two relaxation factors, T₁ and T₂. Typically, while magnetization recovers exponentially, the generated MR signal decreases.

The magnetization curve is given by , where is a constant representing equilibrium magnetization, and  represents the magnetization immediately after excitation. For a system that is excited from its equilibrium state for the first time, can be assumed to be 0.

The signal curve is then given by , where is the magnetization right before the excitation pulse. For a system that is excited from its equilibrium state for the first time, can be assumed to be equal to .

The MR signal is acquired during the relaxation process, and during each acquisition one line of a 2D slice is acquired. Hence, in order to acquire a full 2D image the above process has to be repeated multiple times, with a period that is called the repetition time (TR).

As seen in above figure, the starting magnetization state at the beginning of each repetition depends on the end state the magnetization reaches after the relaxation process. If the TR were long enough, the magnetization would regain its original state. However, since an MR exam cannot last very long, and an image requires many TRs, the TR has to be kept to a minimum. This limitation prevents the magnetization from a full recovery, and causes the magnetization at the end of TR to be less than the beginning. The same also happens in the subsequent TR, and also in the TR after that, and so on. This introduces a non-uniformity over TRs. If signal is acquired and image is reconstructed without taking this non-uniformity into account, the image will contain artifacts, and potentially lead to faulty diagnostics.

Luckily, however, the signal non-uniformity is known to decrease significantly over a certain number of repetitions, where the magnetization reaches a situation called the steady-state. In order to prevent artifacts, the image is reconstructed by data acquired in steady-state only. For a simple MR sequence as such, steady-state can be assumed to be reached at the end of the first TR already. As a result, the signal becomes a function of T₁, T₂ and TR. 

The performance of an MR sequence is typically measured by its signal-to-noise ratio (SNR), which is calculated as the ratio of signal and noise variation:

Noise variation is known to be reduced if the measurement is repeated, and the average is of repeated acquisitions is used. In case of n averages, the noise reduces by . Hence SNR scales with :

In order to improve signal-to-noise, one can maximize signal, by increasing TR, and decrease the noise by increasing number of averages, n. 

• With the above information in mind, please write a program that calculates the optimal TR and number of averages n that yields the best SNR, for an MR experiment that needs to acquire m lines, thus execute m TRs, within a maximum examination time of T. What is the maximum SNR that the program yields? Which performs better, higher number of averages or longer TR?
• Test Data: Use , , m = 256 and
• Sometimes it is desired to acquire MR data with at least two averages to reduce random artifacts. Using the above program calculate the optimal TR for an experiment that averages the signal at least 2 times.

Hint1: Use the area under the signal curve to calculate SNR.

Hint2: Do not forget the first TR required for steady-state.