calc_frtm_tac
=============================================================

.. py:function:: petpal.kinetic_modeling.reference_tissue_models.calc_frtm_tac(tac_times_in_minutes: numpy.ndarray, ref_tac_vals: numpy.ndarray, r1: float, k2: float, k3: float, k4: float) -> numpy.ndarray

   Calculate the Time Activity Curve (TAC) using the Full Reference Tissue Model (SRTM) with the
   given reference TAC and kinetic parameters.

   .. important::
       This function assumes that the reference TAC is uniformly sampled with respect to time
       since we perform convolutions.

   The FRTM TAC can be calculated as:

   .. math::

       C(t) = R_{1}C_\mathrm{R}(t) + \left[ A_{1}e^{-\alpha_{1}t} + A_{2}e^{-\alpha_{2}t} \right]
       \otimes C_\mathrm{R}(t),

   where additionally we have:

   .. math::

       \alpha_{1} &= \frac{k_{2} + k_{3} + k_{4} - \sqrt{\left( k_{2} + k_{3} + k_{4} \right)^2 -
       4k_{2}k_{4}}}{2}\\
       \alpha_{2} &= \frac{k_{2} + k_{3} + k_{4} + \sqrt{\left( k_{2} + k_{3} + k_{4} \right)^2 -
       4k_{2}k_{4}}}{2}\\
       A_{1} &= \left( \frac{k_{3} + k_{4} -\alpha_{2}}{\alpha_{1} - \alpha_{2}} \right)\left(
       \frac{k_{2}}{R_{1}}
       - \alpha_{2} \right)\\
       A_{2} &= \left(  \frac{\alpha_{1}-k_{3} - k_{4} }{\alpha_{1} - \alpha_{2}} \right)\left(
       \frac{k_{2}}{R_{1}}
       - \alpha_{1} \right)


   :param tac_times_in_minutes: The array representing the time-points for both TACs.
   :type tac_times_in_minutes: np.ndarray
   :param r1: The ratio of the clearance rate of tracer from plasma to the reference to the transfer rate of the
              tracer from plasma to the tissue; :math:`R_{1}\equiv\frac{k_1^\prime}{k_1}`.
   :type r1: float
   :param k2: The rate of tracer transfer from the first tissue compartment to plasma.
   :type k2: float
   :param k3: The rate of tracer transfer from the first tissue compartment to the second
              tissue compartment.
   :type k3: float
   :param k4: The rate of tracer transfer from the second tissue compartment to the first
              tissue compartment.
   :type k4: float
   :param ref_tac_vals: The values of the reference TAC.
   :type ref_tac_vals: np.ndarray

   :returns: *np.ndarray* -- TAC values calculated using FRTM.

   :raises AssertionError: If the reference TAC and times are different dimensions.

   .. seealso::

      * :func:`_calc_simplified_frtm`
      * :func:`_calc_frtm_params_from_kinetic_params`