In standard segmentation-based MRI-guided attenuation correction (MRAC) of PET data on hybrid PET/MRI systems, the inter/intra-patient variability of linear attenuation coefficients (LACs) is ignored owing to the assignment of a constant LAC to each tissue class. This can lead to PET quantification errors, especially in the lung regions. In this work, we aim to derive continuous and patient-specific lung LACs from time-of-flight (TOF) PET emission data using the maximum likelihood reconstruction of activity and attenuation (MLAA) algorithm. The MLAA algorithm was constrained for estimation of lung LACs only in the standard 4-class MR attenuation map using Gaussian lung tissue preference and Markov random field smoothness priors. MRAC maps were derived from segmentation of CT images of 19 TOF-PET/CT clinical studies into background air, lung, soft tissue and fat tissue classes, followed by assignment of predefined LACs of 0, 0.0224, 0.0864 and 0.0975 cm(-1), respectively. The lung LACs of the resulting attenuation maps were then estimated from emission data using the proposed MLAA algorithm. PET quantification accuracy of MRAC and MLAA methods was evaluated against the reference CT-based AC method in the lungs, lesions located in/near the lungs and neighbouring tissues. The results show that the proposed MLAA algorithm is capable of retrieving lung density gradients and compensate fairly for respiratory-phase mismatch between PET and corresponding attenuation maps. It was found that the mean of the estimated lung LACs generally follow the trend of the reference CT-based attenuation correction (CTAC) method. Quantitative analysis revealed that the MRAC method resulted in average relative errors of -5.2 ± 7.1% and -6.1 ± 6.7% in the lungs and lesions, respectively. These were reduced by the MLAA algorithm to -0.8 ± 6.3% and -3.3 ± 4.7%, respectively. In conclusion, we demonstrated the potential and capability of emission-based methods in deriving patient-specific lung LACs to improve the accuracy of attenuation correction in TOF PET/MR imaging, thus paving the way for their adaptation in the clinic.