Signal Modelling - Quantitative MRI mOOC

The steady-state longitudinal magnetization of an inversion recovery experiment can be derived from the Bloch equations for the pulse sequence { $\theta_{180}$ – TI – $\theta_{180}$ – (TR-TI)}, and is given by:

M_{z}(TI) = M_0 \frac{1-\text{cos}(\theta_{180})e^{- \frac{TR}{T_1}} -[1-\text{cos}(\theta_{180})]e^{- \frac{TI}{T_1}}}{1 - \text{cos}(\theta_{180}) \text{cos}(\theta_{90}) e^{- \frac{TR}{T_1}}}

(2.1)

where $M_{z}$ is the longitudinal magnetization prior to the $\theta_{90}$ pulse. If the in-phase real signal is desired, it can be calculated by multiplying Eq. 2.1 by $k \text{sin}\left( \theta_{90} \right ) e^{-TE/T_{2}}$ , where $k$ is a constant. This general equation can be simplified by grouping together the constants for each measurements regardless of their values (i.e. at each TI, same TE and $\theta_{90}$ are used) and assuming an ideal inversion pulse:

M_z(TI) = C(1-2e^{- \frac{TI}{T_1}} + e^{- \frac{TR}{T_1}})

(2.2)

where the first three terms and the denominator of Eq. 2.1 have been grouped together into the constant $C$ . If the experiment is designed such that TR is long enough to allow for full relaxation of the magnetization (TR > 5_T_₁), we can do an additional approximation by dropping the last term in Eq. 2.2:

M_z(TI) = C(1-2e^{- \frac{TI}{T_1}})

(2.3)

The simplicity of the signal model described by Eq. 2.3, both in its equation and experimental implementation, has made it the most widely used equation to describe the signal evolution in an inversion recovery T₁ mapping experiment. The magnetization curves are plotted in Figure 2.2 for approximate T₁ values of three different tissues in the brain. Note that in many practical implementations, magnitude-only images are acquired, so the signal measured would be proportional to the absolute value of Eq. 2.3.

Source:Jupyter Notebook

Figure 2.2:Inversion recovery curves (Eq. 2.2) for three different T₁ values, approximating the main types of tissue in the brain.

Practically, Eq. 2.1 is the better choice for simulating the signal of an inversion recovery experiment, as the TRs are often chosen to be greater than 5_T_₁ of the tissue-of-interest, which rarely coincides with the longest T₁ present (e.g. TR may be sufficiently long for white matter, but not for CSF which could also be present in the volume). Eq. 2.3 also assumes ideal inversion pulses, which is rarely the case due to slice profile effects. Figure 2.3 displays the inversion recovery signal magnitude (complete relaxation normalized to 1) of an experiment with TR = 5 s and T₁ values ranging between 250 ms to 5 s, calculated using both equations.

Source:Jupyter Notebook

Figure 2.3:Signal recovery curves simulated using Eq. 2.3 (solid) and Eq. 2.1 (dotted) with a TR = 5 s for T₁ values ranging between 0.25 to 5 s.

Click here to view the qMRLab (MATLAB/Octave) code that generated Figure 2.2.


% Verbosity level 0 overrides the disp function and supresses warnings.
% Once executed, they cannot be restored in this session
% (kernel needs to be restarted or a new notebook opened.)
VERBOSITY_LEVEL = 0;

if VERBOSITY_LEVEL==0
    % This hack was used to supress outputs from external tools
    % in the Jupyter Book.
    function disp(x)
    end
    warning('off','all')
end

try
    cd qMRLab
catch
    try
        cd ../../../qMRLab
    catch
        cd ../qMRLab
    end
end

startup
clear all

%% Setup parameters
% All times are in seconds
% All flip angles are in degrees

params.TR = 5.0;
params.TI = linspace(0.001, params.TR, 1000);
            
params.TE = 0.004;
params.T2 = 0.040;
            
params.EXC_FA = 90;  % Excitation flip angle
params.INV_FA = 180; % Inversion flip angle

params.signalConstant = 1;

%% Calculate signals
%
% The option 'GRE-IR' selects the analytical equations for the
% gradient echo readout inversion recovery experiment The option
% '4' is a flag that selects the long TR approximation of the 
% analytical solution (TR>>T1), Eq. 3 of the blog post.
%
% To see all the options available, run:
% `help inversion_recovery.analytical_solution`

% White matter
params.T1 = 0.900; % in seconds

signal_WM = inversion_recovery.analytical_solution(params, 'GRE-IR', 4);

% Grey matter
params.T1 = 1.500;  % in seconds
signal_GM = inversion_recovery.analytical_solution(params, 'GRE-IR', 4);

% CSF
params.T1 = 4.000;  % in seconds
signal_CSF = inversion_recovery.analytical_solution(params, 'GRE-IR', 4);

Click here to view the qMRLab (MATLAB/Octave) code that generated Figure 2.3.

% Verbosity level 0 overrides the disp function and supresses warnings.
% Once executed, they cannot be restored in this session
% (kernel needs to be restarted or a new notebook opened.)
VERBOSITY_LEVEL = 0;

if VERBOSITY_LEVEL==0
    % This hack was used to supress outputs from external tools
    % in the Jupyter Book.
    function disp(x)
    end
    warning('off','all')
end

try
    cd qMRLab
catch
    try
        cd ../../../qMRLab
    catch
        cd ../qMRLab
    end
end

startup
clear all

%% Setup parameters
% All times are in seconds
% All flip angles are in degrees

params.TR = 5.0;
params.TI = linspace(0.001, params.TR, 1000);
            
params.TE = 0.004;
params.T2 = 0.040;
            
params.EXC_FA = 90;  % Excitation flip angle
params.INV_FA = 180; % Inversion flip angle

params.signalConstant = 1;

T1_range = 0.25:0.25:5; % in seconds

%% Calculate signals
%
% The option 'GRE-IR' selects the analytical equations for the
% gradient echo readout inversion recovery experiment. The option
% '1' is a flag that selects full analytical solution equation 
% (no approximation), Eq. 1 of the blog post. The option '4' is a
% flag that selects the long TR approximation of the analytical 
% solution (TR>>T1), Eq. 3 of the blog post.
%
% To see all the options available, run:
% `help inversion_recovery.analytical_solution`

for ii = 1:length(T1_range)
    params.T1 = T1_range(ii);
    
    signal_T1_Eq1{ii} = inversion_recovery.analytical_solution(params, 'GRE-IR', 1);

    signal_T1_Eq3{ii} = inversion_recovery.analytical_solution(params, 'GRE-IR', 4);
end