On deriving Murray's law from constrained minimization of flow resistance

Use of the volume-averaged Murray's deviation method for the characterization of branching geometry in liver fibrosis: a preliminary study on vascular circulation

Background

Vascular changes in liver fibrosis can result in increased intrahepatic vascular resistance and impaired blood circulation. This can hinder the recovery from fibrosis and may eventually lead to portal hypertension, a major cirrhosis complication. This report proposed a volume-averaged Murray's deviation method to characterize intrahepatic circulation in the liver during fibrosis and its subsequent regression via X-ray phase-contrast computed tomography (PCCT).

Methods

Liver fibrosis was induced in 24 Sprague-Dawley rats by exposure to carbon tetrachloride (CCl₄) for up to 10 weeks, after which, spontaneous regression commenced and continued until week 30. High-resolution three-dimensional (3D) imaging of the livers was performed with PCCT. The values of Murray's deviation based on the volume-averaged and the conventional diameter-based methods were compared. After that, the intrahepatic circulation at different stages of fibrosis was evaluated using the volume-averaged method. The increase in collagen during liver fibrosis was assessed by pathological analyses.

Results

A comparison of the 2 methods showed that with an increase in the number of diameter measurements, the value of Murrary's deviation obtained using the diameter-based method gradually approaches those of the volume-averaged method, with minimal variations. The value of Murray's deviation increased with the development of fibrosis. After reversal, the value rapidly decreased and approached that of the normal state in both the main branches (1.05±0.17, 1.17±0.21, 1.34±0.18, and 1.17±0.19 in the normal, moderate, severe, and regressive groups, respectively; P<0.05 between the severe group and other groups) and the small branches (1.05±0.09, 1.42±0.48, 1.79±0.57, and 1.18±0.28 in the normal, moderate, severe, and regressive group, respectively; P<0.05 between adjacent groups). An analysis of Murray's deviation and the pathological results showed that the vascular circulation in this disease model was consistent with the progression and recovery from fibrosis.

Conclusions

This study showed the validity of the volume-averaged method for calculating Murray's deviation and demonstrated that it could accurately evaluate the blood circulation state of the liver during fibrosis and its subsequent regression. Thus, the volume-averaged method of calculating Murray's deviation may be an objective and valuable staging criterion to evaluate intrahepatic circulation during liver fibrosis.