Complex 3D blood flow pathways in two cases of aorta to right heart fistulae: a 4D flow MRI study

We present an analysis of 3D blood flow in two cases of Sinus of Valsalva to right heart fistulae based on 4D flow MRI. Despite similar underlying pathology, 3D visualization revealed intricate differences in flow patterns connecting the systemic and pulmonary circulation. The cases illustrates the potential of 4D flow MRI to complement the evaluation of complex structural heart disease by assessing complex flow dynamics and providing quantitative information of flow ratios and flow rates.     

Odd-dimensional counterparts of abelian complex and hypercomplex structures

We introduce the notion of abelian almost contact structures on an odd-dimensional real Lie algebra $\mathfrak{g}$. We investigate correspondences with even-dimensional Lie algebras endowed with an abelian complex structure, and with Kähler Lie algebras when $\mathfrak{g}$ carries a compatible inner product. The classification of 5-dimensional Sasakian Lie algebras with abelian structure is obtained. Later, we introduce abelian almost 3-contact structures on real Lie algebras of dimension $4n+3$, obtaining the classification of these Lie algebras in dimension 7. Finally, we deal with the geometry of a Lie group G endowed with a left invariant abelian almost 3-contact metric structure. We determine conditions for G to admit a canonical metric connection with skew torsion, which plays the role of the Bismut connection for hyperKähler with torsion (HKT) structures arising from abelian hypercomplex structures. We provide examples and discuss the parallelism of the torsion of the canonical connection.