Project: s41467-025-58356-7 | Function: aath_bfm (python)

system: You are an expert Python programmer implementing algorithms from scientific papers.
Complete the masked function based on the algorithm description.
Follow Python best practices and maintain consistent code style.
Return **only** the complete Python function code.

user: Complete this Python function:

**DESCRIPTION:** Implements the Anderson-Axon-Thie-Homma (AATH) basis function method for PET kinetic analysis of blood-brain barrier permeability. Constructs basis function matrices and performs compartmental modeling to quantify tracer kinetics and permeability parameters from PET imaging data.

**KEY TERMS:** {
  "AATH": "Anderson-Axon-Thie-Homma method",
  "bfm": "basis function method",
  "kinetic_parameters": "tracer kinetic coefficients",
  "compartmental_model": "PET tracer distribution model"
}

**CONTEXT CODE:**         
    nt = len(t)
    
    k2_vals = logspace_with_bounds(*k2_params, prependZero=True)
    nk2 = len(k2_vals)

    td_lut, Tc_lut, k2_lut = np.meshgrid(np.arange(*td_params),
                                        np.arange(*Tc_params),
                                        k2_vals)
    
    td_lut = td_lut.flatten()
    Tc_lut = Tc_lut.flatten()
    k2_lut = k2_lut.flatten()

    ncombos = td_lut.size
    
    Cwb_cdf = cumtrapz(Cwb, x=t, initial=0)

    basis_k2 = np.zeros((nt, nk2))
    for ii in range(nk2):
        basis_k2[:,ii] = convolve_exp(t, Cp, k2_vals[ii])
        
    sysmat = np.zeros((ncombos, nt, 2))
    for ii, (td, Tc, k2) in enumerate(zip(td_lut, Tc_lut, k2_lut)):
        
        idx_k2 = np.where(k2_vals == k2)[0][0]
                    
        Cwb_cdf_t0_shift = np.interp(t-td, t, Cwb_cdf)
        Cwb_cdf_t0w_shift = np.interp(t-td-Tc, t, Cwb_cdf) 
        
        sysmat[ii,:,0] = Cwb_cdf_t0_shift - Cwb_cdf_t0w_shift
        sysmat[ii,:,1] = np.interp(t-td-Tc, t, basis_k2[:,idx_k2])
        
    # Slack constraint such that:
    #   F - K1 - s1 = 0; i.e., F <= K1
    #   using an arbitrary slack value 1e20
    sysmat = np.pad(sysmat, pad_width=((0,0),(0,1),(0,1)), mode='constant', constant_values=0)
    sysmat[:,-1,0] = 1e10
    sysmat[:,-1,1:-1] = -1e10
    sysmat[:,-1,-1] = -1e10
    
    if useVa:
        sysmat = np.append(sysmat, np.zeros_like(sysmat[:,:,0:1]), axis=-1)
        for ii, td in enumerate(td_lut):
            sysmat[ii,:len(t),-1] = np.interp(t-td, t, Cwb, Cwb[0], Cwb[-1])
    
    print(f'Precomputing basis functions took {time.time() - t_str0:0.1f} s')
    
    return sysmat, td_lut, Tc_lut, k2_lut

def aath_bfm(t, Cwb, Q_t, Cp=None, multi=False, **kwargs):
    """
    Implements the Anderson-Axon-Thie-Homma (AATH) basis function method for PET kinetic analysis of blood-brain barrier permeability.

    This function constructs basis function matrices and performs compartmental modeling to quantify tracer kinetics
    and permeability parameters from PET imaging data. The AATH basis function method is a specialized algorithm
    for PET kinetic parameter estimation used in medical imaging research for blood-brain barrier analysis.

    Args:
        t: Time array for analysis
        Cwb: Whole blood time-activity curve
        Q_t: Tissue time-activity curve data
        Cp: Plasma time-activity curve (optional, defaults to Cwb)
        multi: Whether to use multiprocessing for computation
        **kwargs: Additional parameters for basis function construction

    Returns:
        Q_t_fit: Fitted time-activity curves
        kparams: Dictionary of kinetic parameters (td, Tc, F, K1, k2, E, PS, vb, ve, aic)

    Paper reference: "The four parameters of the time delay-corrected S1TC model were estimated in a manner
    similar to the AATH model by a basis function method where v_b and K_1 were estimated by a non-negative
    linear least squares algorithm"

    Formula: AATH basis function method for compartmental modeling
    Implementation: Basis function construction, matrix operations, parameter estimation, and kinetic modeling
    """
    # MASKED: AATH basis function method implementation for PET kinetic analysis
    raise NotImplementedError("This function is masked for testing purposes. The function implements the Anderson-Axon-Thie-Homma basis function method for PET kinetic analysis of blood-brain barrier permeability, constructing basis function matrices and performing compartmental modeling to quantify tracer kinetics and permeability parameters from PET imaging data.")
    
def s1tc_bfm(t, Cwb, Q_t, Cp=None, multi=False, 
             td_params=(0,6.25,0.5),
             k2_params=(0.00001,0.05,100), 
             **kwargs):
    
    t_str0 = time.time()
    
    if Cp is None:
        Cp = Cwb
    
    ncurves, nt = Q_t.shape
    Q_t_fit = np.zeros_like(Q_t)
    
    k2_vals = logspace_with_bounds(*k2_params, prependZero=True)
    nk2 = len(k2_vals)

    td_lut, k2_lut = np.meshgrid(np.arange(*td_params), k2_vals)
    
    td_lut = td_lut.flatten()
    k2_lut = k2_lut.flatten()
    
    # Generate table of basis functions
    nbasis = td_lut.size
    sysmat = np.zeros((nbasis, nt, 2))
    for ii, (td, k2) in enumerate(zip(td_lut, k2_lut)):
        sysmat[ii,:,0] = np.interp(t-td, t, Cwb, Cwb[0], Cwb[-1])
        sysmat[ii,:,1] = np.interp(t-td, t, convolve_exp(t, Cp, k2))

    print('ncurves:', ncurves)
    print('sysmat dimensions:', sysmat.shape)
    
    td = np.zeros(ncurves)
    K1 = np.zeros(ncurves)
    k2 = np.zeros(ncurves)
    vb = np.zeros(ncurves)
    
    if multi:
        f = partial(batch_nnls, sysmat=sysmat)
        batch_size = np.ceil(ncurves / (os.cpu_count())).astype(int)
        batch_size = np.min([batch_size, 2**13])
        
        pool = Pool(processes=os.cpu_count(), maxtasksperchild=1)
        ResultList = pool.map(f, Q_t, chunksize=batch_size)
        pool.close()
        pool.join()
        
        x_list = [res[0] for res in ResultList]
        idx_list = [res[1] for res in ResultList]
        
        td = np.array([td_lut[idx] for idx in idx_list])
        k2 = np.array([k2_lut[idx] for idx in idx_list])
        
        vb = np.array([x[0] for x in x_list])
        K1 = np.array([x[1] for x in x_list])
        
        
        for ii in range(ncurves):
            idx = idx_list[ii]
            Q_t_fit[ii,:] = sysmat[idx,:nt,:].dot(x_list[ii])
    else:
        for ii in range(ncurves):
            # if multi:
            #     f = partial(nnls, b=Q_t[ii,:])
            #     batch_size = np.ceil(nbasis / (os.cpu_count())).astype(int)
                
            #     pool = Pool(maxtasksperchild=1)
            #     ResultList = pool.map(f, sysmat, chunksize=batch_size)
            #     pool.close()
            #     pool.join()
                


Complete the function implementation replacing "raise NotImplementedError".