Mathematical modelling of responses of cerebral blood vessels to changing intraluminal pressure

The contribution of vascular smooth muscle, elastin and collagen on the passive mechanics of porcine carotid arteries

The main components responsible for the mechanical behavior of the arterial wall are collagen, elastin, and smooth muscle cells (SMCs) in the medial layer. We determined the structural and mechanical changes in porcine carotid arteries after administration of Triton® X-100, elastase, and collagenase using the inflation-deflation test. The arteries were intraluminarly pressurized from 0 to 200 mmHg, and the outer diameter of the artery was measured. The pressure-strain elastic modulus was determined based on the pressure/diameter ratio. The intima-media thickness, wall thickness, thickness of the tunica adventitia layer, and the area fractions of SMCs, elastin, and collagen within the arterial wall (A(A)(SMC/elastin/collagen, wall)) were measured using stereological methods. The relative changes in the relevant components of the treated samples were as follows: the decrease in A(A)(SMC, wall) after administration of Triton® X-100 was 11% ± 7%, the decrease in A(A)(elastin, wall) after administration of elastase was 40% ± 22%, and the decrease in A(A)(collagen, wall) after the application of collagenase was 51% ± 22%. The Triton® X-100 treatment led to a decrease in the SMC content that was associated with enlargement of the arterial wall (outer diameter) for pressures up to 120 mmHg, and with mechanical stiffening of the arterial wall at higher pressures. Elastase led to a decrease in the elastin content that was associated with enlargement of the arterial wall, but not with stiffening or softening. Collagenase led to a decrease in collagen content that was associated with a change in the stiffness of the arterial wall, although the exact contribution of mechanical loading and the duration of treatment (enlargement) could not be quantified.

Cerebral autoregulation: from models to clinical applications

Short-term regulation of cerebral blood flow (CBF) is controlled by myogenic, metabolic and neurogenic mechanisms, which maintain flow within narrow limits, despite large changes in arterial blood pressure (ABP). Static cerebral autoregulation (CA) represents the steady-state relationship between CBF and ABP, characterized by a plateau of nearly constant CBF for ABP changes in the interval 60-150 mmHg. The transient response of the CBF-ABP relationship is usually referred to as dynamic CA and can be observed during spontaneous fluctuations in ABP or from sudden changes in ABP induced by thigh cuff deflation, changes in posture and other manoeuvres. Modelling the dynamic ABP-CBFV relationship is an essential step to gain better insight into the physiology of CA and to obtain clinically relevant information from model parameters. This paper reviews the literature on the application of CA models to different clinical conditions. Although mathematical models have been proposed and should be pursued, most studies have adopted linear input-output ('black-box') models, despite the inherently non-linear nature of CA. The most common of these have been transfer function analysis (TFA) and a second-order differential equation model, which have been the main focus of the review. An index of CA (ARI), and frequency-domain parameters derived from TFA, have been shown to be sensitive to pathophysiological changes in patients with carotid artery disease, stroke, severe head injury, subarachnoid haemorrhage and other conditions. Non-linear dynamic models have also been proposed, but more work is required to establish their superiority and applicability in the clinical environment. Of particular importance is the development of multivariate models that can cope with time-varying parameters, and protocols to validate the reproducibility and ranges of normality of dynamic CA parameters extracted from these models.

The effect of surgery and severe injury on blood vessels

This paper evolves from our previous work that developed a blood vessel model based on the way in which the total tension within the blood vessel wall varies with pressure due to the interaction of specific ions. We use the previous outcomes to examine the diameter of rat middle cerebral arteries (MCAs) as a function of pressure. The MCAs were isolated at 2 h, 1 and 5 days following sham injury and severe traumatic brain injury (TBI). First, we are able to quantify the diameter versus pressure curve in a way that yields three parameters. One of these parameters is the diameter at zero pressure that incorporates properties of the blood vessel walls and the vascular smooth muscle cells. The other two parameters are associated with the myogenic response and the myogenic tone. These parameters enable us to characterize, from the MCAs as a function of pressure in a calcium buffer, each blood vessel into one of three distinct distributions. One of these distributions reflects the optimum conditions. From our data with our blood vessel model, we demonstrate the effect of sham injury on the way in which blood vessels respond to pressure change that appears to recover over time with a half-life of about 40 h. In contrast, severe TBI greatly affects the blood vessel with no indication of recovery over the five-day monitoring period.