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Perfusion imaging is crucial in acute ischemic stroke for quantifying the salvageable penumbra and irreversibly damaged core lesions. As such, it helps clinicians to decide on the optimal reperfusion treatment. In perfusion CT imaging, deconvolution methods are used to obtain clinically interpretable perfusion parameters that allow identifying brain tissue abnormalities. Deconvolution methods require the selection of two reference vascular functions as inputs to the model: the arterial input function (AIF) and the venous output function, with the AIF as the most critical model input. When manually performed, the vascular function selection is time demanding, suffers from poor reproducibility and is subject to the professionals' experience. This leads to potentially unreliable quantification of the penumbra and core lesions and, hence, might harm the treatment decision process. In this work we automatize the perfusion analysis with AIFNet, a fully automatic and end-to-end trainable deep learning approach for estimating the vascular functions. Unlike previous methods using clustering or segmentation techniques to select vascular voxels, AIFNet is directly optimized at the vascular function estimation, which allows to better recognise the time-curve profiles. Validation on the public ISLES18 stroke database shows that AIFNet reaches inter-rater performance for the vascular function estimation and, subsequently, for the parameter maps and core lesion quantification obtained through deconvolution. We conclude that AIFNet has potential for clinical transfer and could be incorporated in perfusion deconvolution software.