expconv

expconv

Exponential convolution functions for pharmacokinetic modeling.

This module implements efficient recursive formulas for convolving signals with exponential and multi-exponential functions. These are the core building blocks for compartmental pharmacokinetic models.

GPU/CPU agnostic using the xp array module pattern. NO scipy dependency - see XP Compatibility Requirements in plan.md.

References

Flouri D, Lesnic D, Sourbron SP (2016). Fitting the two-compartment model in DCE-MRI by linear inversion. Magn Reson Med. 76(3):998-1006. doi:10.1002/mrm.25991

Attribution

expconv adapts the recursive exponential-convolution algorithm from dcmri v0.6.20 (https://github.com/dcmri/dcmri, module dcmri/utils.py), licensed under the Apache License, Version 2.0. Modified: (f, T, t) signature, batched/GPU array-module path, and non-normalized output. See the project NOTICE file.

expconv

expconv(f, T, t)

Convolve a signal with exponential decay function(s).

Computes the convolution: (f * exp(-t/T))(t) = integral_0^t f(u) * exp(-(t-u)/T) du

using an efficient recursive formula that avoids explicit numerical integration. This is the fundamental operation for compartmental pharmacokinetic models.

Handles both single-voxel (scalar T) and batch (array T) cases. When T is an array, every voxel is processed in a single pass with the loop running over time points — efficient on CPU and GPU.

PARAMETER	DESCRIPTION
f	Input signal (e.g., arterial input function). Shape (n_times,) or (n_times, 1) or (n_times, n_voxels). TYPE: ndarray
T	Time constant(s) of the exponential decay in seconds. Scalar for a single convolution, or array of shape (n_voxels,) for batch convolution. TYPE: float or ndarray
t	Time points in seconds. Shape (n_times,) or (n_times, 1). Must be monotonically increasing. TYPE: ndarray

RETURNS	DESCRIPTION
ndarray	Convolution result. Shape (n_times,) for scalar T, (n_times, n_voxels) for array T.

Notes

The recursive formula from Flouri et al. (2016) is used:

E[i] = E[i-1] * exp(-dt/T) + integral_{t[i-1]}^{t[i]} f(u) * exp(-(t[i]-u)/T) du

where the integral within each interval is evaluated analytically assuming piecewise-linear interpolation of f.

This is O(n) in computation time, compared to O(n^2) for naive numerical integration.

Examples:

>>> import numpy as np
>>> from osipy.common.convolution import expconv
>>> t = np.linspace(0, 10, 101)
>>> aif = np.exp(-t / 2)  # Input function
>>> T = 5.0  # 5 second time constant
>>> result = expconv(aif, T, t)

References

.. [1] Flouri D, Lesnic D, Sourbron SP (2016). Fitting the two-compartment model in DCE-MRI by linear inversion. Magn Reson Med. 76(3):998-1006. doi:10.1002/mrm.25991