We advocate the use of an alternative calculus in biomedical image analysis, known as multiplicative (a.k.a. non-Newtonian) calculus. It provides a natural framework in problems in which positive images or positive definite matrix fields and positivity preserving operators are of interest. Indeed, its merit lies in the fact that preservation of positivity under basic but important operations, such as differentiation, is manifest. In the case of positive scalar functions, or in general any set of positive definite functions with a commutative codomain, it is a convenient, albeit arguably redundant framework. However, in the increasingly important non-commutative case, such as encountered in diffusion tensor imaging and strain tensor analysis, multiplicative calculus complements standard calculus in a truly nontrivial way. The purpose of this article is to provide a condensed review of multiplicative calculus and to illustrate its potential use in biomedical image analysis.

Aim This study explores the relationship between in vivo 4D flow cardiovascular magnetic resonance (CMR) derived blood flow energetics in the total cavopulmonary connection (TCPC), exercise capacity and CMR-derived liver fibrosis/congestion. Background The Fontan circulation, in which both caval veins are directly connected with the pulmonary arteries (i.e. the TCPC) is the palliative approach for single ventricle patients. Blood flow efficiency in the TCPC has been associated with exercise capacity and liver fibrosis using computational fluid dynamic modelling. 4D flow CMR allows for assessment of in vivo blood flow energetics, including kinetic energy (KE) and viscous energy loss rate (EL). Methods Fontan patients were prospectively evaluated between 2018 and 2021 using a comprehensive cardiovascular and liver CMR protocol, including 4D flow imaging of the TCPC. Peak oxygen consumption (VO2) was determined using cardiopulmonary exercise testing (CPET). Iron-corrected whole liver T1 (cT1) mapping was performed as a marker of liver fibrosis/congestion. KE and EL in the TCPC were computed from 4D flow CMR and normalized for inflow. Furthermore, blood flow energetics were compared between standardized segments of the TCPC. Results Sixty-two Fontan patients were included (53% male, 17.3 ± 5.1 years). Maximal effort CPET was obtained in 50 patients (peak VO2 27.1 ± 6.2 ml/kg/min, 56 ± 12% of predicted). Both KE and EL in the entire TCPC (n = 28) were significantly correlated with cT1 (r = 0.50, p = 0.006 and r = 0.39, p = 0.04, respectively), peak VO2 (r = − 0.61, p = 0.003 and r = − 0.54, p = 0.009, respectively) and % predicted peak VO2 (r = − 0.44, p = 0.04 and r = − 0.46, p = 0.03, respectively). Segmental analysis indicated that the most adverse flow energetics were found in the Fontan tunnel and left pulmonary artery. Conclusions Adverse 4D flow CMR derived KE and EL in the TCPC correlate with decreased exercise capacity and increased levels of liver fibrosis/congestion. 4D flow CMR is promising as a non-invasive screening tool for identification of patients with adverse TCPC flow efficiency.

Objectives To study flow-related energetics in multiple anatomical segments of the total cavopulmonary connection (TCPC) in Fontan patients from 4D flow MRI, and to study the relationship between adverse flow patterns and segment-specific energetics. Methods Twenty-six extracardiac Fontan patients underwent 4D flow MRI of the TCPC. A segmentation of the TCPC was automatically divided into 5 anatomical segments (conduit, superior vena cava, right/left pulmonary artery (PA) and the Fontan confluence). The presence of vortical flow in the PAs or Fontan confluence was qualitatively scored. Kinetic energy, viscous energy loss and vorticity were calculated from the 4D flow MRI velocity field and normalized for segment length and/or inflow. Energetics were compared between segments and the relationship between vortical flow and segment cross-sectional area (CSA) with segment-specific energetics was determined. Results Vortical flow in the LPA (n = 6) and Fontan confluence (n = 12) were associated with significantly higher vorticity (p = 0.001 and p = 0.015, respectively) and viscous energy loss rate (p = 0.046 and p = 0.04, respectively) compared to patients without vortical flow. The LPA and conduit segments showed the highest kinetic energy and viscous energy loss rate, while most favorable energetics were observed in the superior vena cava. Conduit CSA inversely correlated with kinetic energy (r= -0.614, p = 0.019) and viscous energy loss rate (r= -0.652, p = 0.011). Conclusions Vortical flow in the Fontan confluence and LPA associated with significantly increased viscous energy loss rate. 4D flow MRI derived energetics may be used as a screening tool for direct, MRI-based assessment of flow efficiency in the TCPC.

To assess errors associated with EPI-accelerated intracardiac 4D flow MRI (4DEPI) with EPI factor 5, compared with non-EPI gradient echo (4DGRE).Methods: Three 3T MRI experiments were performed comparing 4DEPI to 4DGRE: steady flow through straight tubes, pulsatile flow in a left-ventricle phantom, and intracardiac flow in 10 healthy volunteers. For each experiment, 4DEPI was repeated with readout and blip phase-encoding gradient in different orientations, parallel or perpendicular to the flow direction. In vitro flow rates were compared with timed volumetric collection. In the left-ventricle phantom and in vivo, voxel-based speed and spatio-temporal median speed were compared between sequences, as well as mitral and aortic transvalvular net forward volume. Results:In steady-flow phantoms, the flow rate error was largest (12%) for high velocity (>2 m/s) with 4DEPI readout gradient parallel to the flow. Voxel-based speed and median speed in the left-ventricle phantom were ≤5.5% different between sequences. In vivo, mean net forward volume inconsistency was largest (6.4 ± 8.5%) for 4DEPI with nonblip phase-encoding gradient parallel to the main flow. The difference in median speed for 4DEPI versus 4DGRE was largest (9%) when the 4DEPI readout gradient was parallel to the flow.Conclusions: Velocity and flow rate are inaccurate for 4DEPI with EPI factor 5 when flow is parallel to the readout or blip phase-encoding gradient. However, mean differences in flow rate, voxel-based speed, and spatio-temporal median

Abstract. In this paper, a new technique (SPASM) based on a 3D-ASM is presented for automatic segmentation of cardiac MRI image data sets consisting of multiple planes with different orientations, and with large undersampled regions. SPASM was applied to sparsely sampled and radially oriented cardiac LV image data.Performance of SPASM has been compared to results from other methods reported in literature. The accuracy of SPASM is comparable to these other methods, but SPASM uses considerably less image data.

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