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README.md

@@ -21,6 +21,7 @@ Created 28 September 2020
 - Signal models: spoiled gradient echo
 - Water exchange models: FXL, NXL, NXL_be
 - T1 fitting using variable flip angle method, IR-SPGR and DESPOT1-HIFI
+- T2(*) fitting for multi-TE acquisitions
 
 ### Not yet implemented/limitations:
 - Additional pharmacokinetic models (add by inheriting from PkModel class)

demo/demo_fit_dce.ipynb

@@ -603,4 +603,4 @@
  },
  "nbformat": 4,
  "nbformat_minor": 5
-}
+}

src/t2star_fit.py

@@ -0,0 +1,167 @@
+"""Functions to fit MRI SPGR signal to obtain T2* or T2.
+
+Created 17 October 2023
+@authors: Michael Thrippleton
+@email: [email protected]
+@institution: University of Edinburgh, UK
+
+Classes:
+    MultiEchoT2sLinear
+    MultiEchoT2sNonLinear
+
+Functions:
+    multiecho_signal: get signal
+"""
+
+import numpy as np
+from scipy.optimize import least_squares
+from fitting import Fitter
+
+
+class MultiEchoT2sLinear(Fitter):
+    """Linear multi-echo T2* estimation.
+
+    Subclass of Fitter.
+    """
+
+    def __init__(self, te, min_signal=0.):
+        """
+
+        Args:
+            te (ndarray): 1D array containing the echo times (s)
+            min_signal (float): exclude echoes where signal is below this
+                value. Defaults to zero.
+
+        """
+        self.te = te
+        self.min_signal = min_signal
+
+    def output_info(self):
+        """Get output info. Overrides superclass method.
+        """
+        return ('s0', False), ('t2s', False)
+
+    def proc(self, s):
+        """Estimate T2*. Overrides superclass method.
+
+        Args:
+            s (ndarray): 1D array containing the signals
+
+        Returns:
+            tuple: (s0, t1)
+                s0 (float): signal without T2* decay
+                t2s (float): T2* (s)
+
+        """
+        if any(np.isnan(s)):
+            raise ArithmeticError(
+                f'Unable to calculate T2s: nan signal received.')
+
+        te_mask = s > self.min_signal
+        if np.sum(te_mask) < 2:
+            raise ArithmeticError(
+                f'Unable to calculate T2s: fewer than 2 included TE values.')
+
+        y = np.log(s[te_mask])
+        x = self.te[te_mask]
+        A = np.stack([x, np.ones(x.shape)], axis=1)
+        slope, intercept = np.linalg.lstsq(A, y, rcond=None)[0]
+
+        if (intercept < 0) or (0. < slope):
+            raise ArithmeticError('T2s estimation failed.')
+
+        t2s, s0 = -1. / slope, np.exp(intercept)
+
+        return s0, t2s
+
+
+class MultiEchoT2sNonLinear(Fitter):
+    """Non-linear multi-echo T2* estimation.
+
+    Subclass of Fitter.
+    """
+
+    def __init__(self, te, min_signal=0.):
+        """
+
+        Args:
+            te (ndarray): 1D array containing the echo times (s)
+            min_signal (float): exclude echoes where signal is below this
+                value. Defaults to zero.
+
+        """
+        self.te = te
+        self.min_signal = min_signal
+        self.linear_fitter = MultiEchoT2sLinear(te, min_signal)
+
+    def output_info(self):
+        """Get output info. Overrides superclass method.
+        """
+        return ('s0', False), ('t2s', False)
+
+    def proc(self, s):
+        """Estimate T2*. Overrides superclass method.
+
+        Args:
+            s (ndarray): 1D array containing the signals
+
+        Returns:
+            tuple: (s0, t1)
+                s0 (float): signal without T2* decay
+                t2s (float): T2* (s)
+
+        """
+        if any(np.isnan(s)):
+            raise ArithmeticError(
+                f'Unable to calculate T2s: nan signal received.')
+
+        te_mask = s > self.min_signal
+        if np.sum(te_mask) < 2:
+            raise ArithmeticError(
+                f'Unable to calculate T2s: fewer than 2 included TE values.')
+
+        # use linear fit to obtain initial guess, otherwise set arbitrary values
+        try:
+            x0 = np.array(self.linear_fitter.proc(s))
+        except ArithmeticError:
+            x0 = np.array([s[0], 10e-3])
+
+        s = s[te_mask]
+        te = self.te[te_mask]
+
+        result = least_squares(self.__residuals, x0, args=(s, te), bounds=(
+                (1e-8, 1e-8), (np.inf, np.inf)), method='trf', x_scale=x0)
+        if result.success is False:
+            raise ArithmeticError(f'Unable to fit multiecho data:'
+                                  f' {result.message}')
+
+        s0, t2s = result.x
+        return s0, t2s
+
+    @staticmethod
+    def __residuals(x, s, te):
+        s0, t2s = x
+        s_est = multiecho_signal(s0, t2s, te)
+        return s - s_est
+
+
+def multiecho_signal(s0, t2s, te):
+    """Return signal for multiple echoes / echo times.
+
+    Parameters
+    ----------
+        s0 : float
+             Signal without T2/T2* decay.
+        t2s : float
+             T2*/T2 value (s).
+        te : float
+             TE value (s).
+
+    Returns
+    -------
+        s : float
+            Signal.
+    """
+    s = s0 * np.exp(-te/t2s)
+
+    return s