The Purkinje network is responsible for the fast and coordinated distribution of the electrical impulse in the ventricle that triggers its contraction. Therefore, it is necessary to model its presence to obtain an accurate patient-specific model of the ventricular electrical activation. In this paper, we present an efficient algorithm for the generation of a patientspecific Purkinje network, driven by measures of the electrical activation acquired on the endocardium. The proposed method provides a correction of an initial network, generated by means of a fractal law, and it is based on the solution of Eikonal problems both in the muscle and in the Purkinje network. We present several numerical results both in an ideal geometry with synthetic data and in a real geometry with patient-specific clinical measures. These results highlight an improvement of the accuracy provided by the patientspecific Purkinje network with respect to the initial one. In particular, a cross-validation test shows an accuracy increase of 19% when only the 3% of the total points are used to generate the network, whereas an increment of 44% is observed when a random noise equal to 20% of the maximum value of the clinical data is added to the measures.

The propagation of the electrical signal in the Purkinje network is the starting point for the activation of the ventricular muscular cells leading to the contraction of the ventricle. In the computational models, describing the electrical activity of the ventricle is therefore important to account for the Purkinje fibers. Until now, the inclusion of such fibers has been obtained either by using surrogates such as space-dependent conduction properties or by generating a network based on an a priori anatomical knowledge. The aim of this work was to propose a new method for the generation of the Purkinje network using clinical measures of the activation times on the endocardium related to a normal electrical propagation, allowing to generate a patient-specific network. The measures were acquired by means of the EnSite NavX system. This system allows to measure for each point of the ventricular endocardium the time at which the activation front, that spreads through the ventricle, has reached the subjacent muscle. We compared the accuracy of the proposed method with the one of other strategies proposed so far in the literature for three subjects with a normal electrical propagation. The results showed that with our method we were able to reduce the absolute errors, intended as the difference between the measured and the computed data, by a factor in the range 9-25 %, with respect to the best of the other strategies. This highlighted the reliability of the proposed method and the importance of including a patient-specific Purkinje network in computational models.

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To properly describe the electrical activity of the left ventricle, it is necessary to model the Purkinje fibers, responsible for the fast and coordinate ventricular activation, and their interaction with the muscular propagation. The aim of this work is to propose a methodology for the generation of a patient-specific Purkinje network driven by clinical measurements of the activation times related to pathological propagations. In this case, one needs to consider a strongly coupled problem between the network and the muscle, where the feedback from the latter to the former cannot be neglected as in a normal propagation. We apply the proposed strategy to data acquired on three subjects, one of them suffering from muscular conduction problems owing to a scar and the other two with a muscular pre-excitation syndrome (Wolff-Parkinson-White). To assess the accuracy of the proposed method, we compare the results obtained by using the patient-specific Purkinje network generated by our strategy with the ones obtained by using a non-patient-specific network. The results show that the mean absolute errors in the activation time is reduced for all the cases, highlighting the importance of including a patient-specific Purkinje network in computational models.

Abstract. We present a numerical solver for the fast conduction system in the heart using both a CPU and a hybrid CPU/GPU implementation. To verify both implementations, we construct analytical solutions and show that the L 2 -error is similar in both implementations and decreases linearly with the spatial step size. Finally, we test the performance of the implementations with networks of varying complexity, where the hybrid implementation is, on average, 5.8 times faster.

SUMMARYCardiac Purkinje fibres provide an important pathway to the coordinated contraction of the heart. We present a numerical algorithm for the solution of electrophysiology problems across the Purkinje network that is efficient enough to be used in in-silico studies on realistic Purkinje networks with physiologically detailed models of ion exchange at the cell membrane. The algorithm is based on operator splitting and is provided with three different implementations: pure CPU, hybrid CPU/GPU, and pure GPU. Compared to our previous work, we modify the explicit gap junction term at network bifurcations in order to improve its mathematical consistency. Due to this improved consistency of the model, we are able to perform an empirical convergence study against analytical solutions. The study verified that all three implementations produce equivalent convergence rates, which shows that the algorithm produces equivalent result across different hardware platforms. Finally, we compare the efficiency of all three implementations on Purkinje networks of increasing spatial resolution using membrane models of increasing complexity. Both hybrid and pure-GPU implementations outperform the pure-CPU implementation, but their relative performance difference depends on the size of the Purkinje network and the complexity of the membrane model used.