Does the principle of minimum work apply at the carotid bifurcation: a retrospective cohort study

Abstract

Background: There is recent interest in the role of carotid bifurcation anatomy, geometry and hemodynamic factors in the pathogenesis of carotid artery atherosclerosis. Certain anatomical and geometric configurations at the carotid bifurcation have been linked to disturbed flow. It has been proposed that vascular dimensions are selected to minimize energy required to maintain blood flow, and that this occurs when an exponent of 3 relates the radii of parent and daughter arteries. We evaluate whether the dimensions of bifurcation of the extracranial carotid artery follow this principle of minimum work.

Methods: This study involved subjects who had computed tomographic angiography (CTA) at our institution between 2006 and 2007. Radii of the common, internal and external carotid arteries were determined. The exponent was determined for individual bifurcations using numerical methods and for the sample using nonlinear regression.

Results: Mean age for 45 participants was 56.9 ± 16.5 years with 26 males. Prevalence of vascular risk factors was: hypertension--48%, smoking--23%, diabetes--16.7%, hyperlipidemia--51%, ischemic heart disease--18.7%.The value of the exponent ranged from 1.3 to 1.6, depending on estimation methodology.

Conclusions: The principle of minimum work (defined by an exponent of 3) may not apply at the carotid bifurcation. Additional factors may play a role in the relationship between the radii of the parent and daughter vessels.

Figure 1

Schematic diagram of a carotid artery bifurcation. The parent artery (CCA) divides into two branches, ICA and ECA. Radius of CCA is measured at a location 2 sphere radiuses from the bifurcation point. Radii of the ICA and ECA are medians of radii measured at points between 6r - 12r and 2r-7r from the bifurcation point respectively. This approach avoided the ICA sinus and reduced influence of irregularities and artifacts due to ECA bifurcation.

Figure 2

Quality of fit for nonlinear regression. r_icaⁿ + r_ecaⁿ vs r_ccaⁿ for all patients using artery size measurement method A and result of nonlinear regression (n = 1.32).

Figure 3

Exponent values, n, for individual bifurcations for 45 patients. Ranges of individual exponent values estimated for all patients (45) and patients with low tortuosity (31) using methods A and B for artery size measurement.

Figure 4

Quality of fit for using mean of individual estimates. r_icaⁿ + r_ecaⁿ vs r_ccaⁿ for all patients using artery size measurement method A and result of mean of individual estimates (n = 1.58).