93 Hemodynamics
94 J.C. Kim and E.B. Shim
Properties of the Arterial Wall
In reality, an artery is a viscoelastic tube whose diameter varies with the pulsating pressure that is generated by a pulsating flow; in addition, it propagates pressure and flow waves at a certain velocity, which is largely determined by the elastic properties of the arterial wall.
To study the hemodynamics of the arterial system, knowledge of the elastic properties of the arterial wall is of fundamental importance. This section deals with the basic concept of elasticity and the effects of its changes during circulation on the hemodynamic properties of the arterial wall.
When a force F is applied to a specimen with cross-sectional area A and length L , the length increases by D L . To obtain a unique characterization of the material, independent of sample size, we normalize the force by the area, and the stress is obtained as s = F / A . Similarly, we normalize the length change to the starting length L 0 and obtain the strain as e = D L / L 0. The relationship between stress and strain is given in Fig. 5a . The slope of the stress–strain curve in the region where a linear relationship holds is defined as the elastic modulus (Young’s modulus of elasticity, E = s / e ). The units of the elastic modulus are the same as units of pressure, i.e., N/m 2 = Pa or mm Hg.
By contrast, a curved relationship between stress and strain can be observed in main ves- sels (Fig. 5b ). Vascular tissue is composed of elastin, vascular smooth muscle, and collagen.
Elastin fibers are highly extensible, and even at large deformations, they can be characterized by an almost constant Young’s modulus. Conversely, collagen fibers are very stiff [11] . At low strains, collagen fibers are wavy and bear no load, and the elastin and smooth muscle mainly determine the wall elasticity. At larger strains, collagen starts bearing a load, leading to an increasingly stiff wall.
Elasticity plays an important role in circulation. All blood vessels are elastic, and their elastic moduli do not differ greatly. A typical value of the incremental elastic modulus of arteries in the normal human at a pressure of 100 mm Hg is about 5 × 10 6 dynes/cm 2 or 500 kPa (3,750 mm Hg). For diastolic heart tissue, the incremental elastic modulus at a filling pressure of 5 mm Hg is about 4 × 10 5 dynes/cm 2 or 40 kPa, and in systole, these values are 20 or more times larger. With the passage of time, certain histological changes that alter structure and function occur in the walls of the large central arteries. The principal changes occur in the intima and media [12, 13] ; endothelial cells of the intima become more irregular in size and
u
Stress Stress
Strain
a b
Strain
Solid Arterial wall
Fig. 5 Stress–strain relationship for general solid materials ( a ) and the arterial wall ( b )
95 Hemodynamics
shape, and their function is depressed progressively. These changes in arterial wall structure lead to an increased elastic modulus (Fig. 6 ) with age and a decrease in arterial distensibility and compliance [14, 15] . This increase in arterial rigidity increases the pulse wave velocity and reflected wave amplitude and causes an increase in systolic blood pressure; this, in turn, increases the incidence of stroke, cardiac failure, and all-cause mortality [16, 17] . Other macroscopic findings with aging are a progressive increase in artery diameter, wall thickness, intima–medial thickness, and pulse wave velocity [18] . The increase in the mean distal abdominal aortic stiffness and internal diameter with age was found to be gender dependent, with a greater increase occurring in males.
Vascular Impedance
As its name implies, the term “impedance” is a measure of the opposition to flow presented by a system. Etymologically, the term “resistance” conveys the same meaning as does imped- ance, but it is confined to non-oscillatory or steady motions. In another sense, clinicians frequently use the term “impedance” to refer to the impediment or hindrance to blood flow [19, 20] . When the general term is applied to the vascular system, it is usually in reference to the input impedance, this being the ratio between pulsatile pressure and pulsatile flow recorded in an artery feeding a particular vascular bed. The input impedance has to do with the input to the entire vascular tree beyond this site; it depends not only on the local arterial Fig. 6 Age-related increase in the abdominal aortic pressure–strain modulus ( E p ) of females compared with males. Note that the increase in stiffness ( E p ) is almost linear with age in females, whereas in males it is expo- nential in nature and of greater magnitude. * P < 0.05, ** P < 0.01, *** P < 0.001. Copied from [17]
96 J.C. Kim and E.B. Shim
properties but also on the properties of all vessels in the vascular bed beyond, down to the point where all pulsations generated by the heart have been attenuated [10] .
The input impedance modulus of any region of the arterial system is the ratio of harmonic terms of pressure at the input to the corresponding harmonic terms of flow. Therefore, the ratio of pulsatile pressure to pulsatile flow in the common carotid artery determines the input impedance of the cerebral arterial system. The simplest model of the arterial system is the Windkessel, with lumped capacitance and resistance elements [21, 22] . The term “Windkessel”
means “air-chamber” in German. The input impedance can be expressed as the modulus and phase (Fig. 7 ). At the input of a Windkessel, the impedance modulus falls from its high value at zero frequency, which is determined by its resistive value, to low values at high frequency, which are determined by the magnitude of capacitance in relation to resistance, whereas the phase is zero at zero frequency and becomes progressively more negative (flow leading pres- sure) at higher frequencies.
Let us consider the effects of the elasticity of the arterial wall on the hemodynamics using an equivalent electrical circuit for the aortic system represented in Fig. 7 . If we assume that the arterial wall is rigid, the value of the capacitance is zero, and we can analyze blood flow using the Poiseuille equation. The input flow to the arterial system equals the sum of the outflow of blood from the arterial system into the venous system and the rate of storage:
in out stored p t( ) 0.
Q Q Q
= + = R = (10)
If the arterial wall loses its elastic properties ( C = 0), the rate of storage is zero, and the blood from the heart flows into the capillaries directly. In this situation, no phase difference exists between the curves for flow and for pressure. Conversely, when the arterial wall has compliance, a portion of the ejected cardiac output flows into the arterial system, and the rest is stored in the arterial wall (i.e., the rate of storage is not zero). The remaining volume is supplied to the peripheral vessels during diastole. This process can be described mathematically as
in out stored
( ) dVolume ( ) d ( )
d d .
p t p t p t
Q Q Q C
R t R t
= + = + = + (11)
When the elasticity of an artery (capacitance, C ) varies from 1.9 to 0.1 with constant resistance R , the peak blood pressure decreases markedly from 350 to near 100 mm Hg. Conversely, when Fig. 7 Diagrammatic representation of the input impedance concept ( left ) and a representation of modulus (| Z |) and phase ( q ) ( right )
97 Hemodynamics
the resistance varies from 10 to 100 with constant elasticity ( C ), only a minor increase in blood pressure is observed. These facts demonstrate the important role of the elasticity of the arterial wall in regulating blood pressure and the risk of hypertension due to atherosclerosis.
Conclusion
To understand hemodynamics in the cerebral arteries, the basic concepts of fluid mechanics are essential. Moyamoya disease is a cerebrovascular disease that involves, at least in part, hemo- dynamic factors. Recent advances in computer technology and medical imaging equipment have provided quantitative insight into the hemodynamics of complex arterial systems by combining these numerical and experimental techniques [23] . Phase-contrast magnetic resonance angiography and transcranial Doppler are noninvasive techniques for measuring regional cerebral blood flow in real time [24, 25] . The determination of the hemodynamics in cerebral arteries using these ultra-sophisticated technologies will contribute to the diagnosis and treatment of moyamoya disease.
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Introduction
Moyamoya disease (MMD) is an idiopathic entity characterized by progressive occlusion of the distal internal carotid arteries (ICAs), as well as the proximal anterior artery (ACA) and middle cerebral artery (MCA). Although the vast majority of cases are sporadic, familial factors have been implicated in a small subset of cases. The frequency of familial occurrence of MMD is estimated to be approximately 7–10% of all reported cases [1] and the incidence among Asians is high. In addition, studies looking at the pathogenesis of MMD have uncovered both genetic predisposing factors as well as acquired factors. Single nucleotide polymorphism, sickle cell anemia, and neurofibromatosis are among some of the purported genetic factors.
Acquired conditions such as tuberculosis meningitis infection and atherosclerosis have also been identified [2] .
From an anatomical standpoint, the primary structural abnormalities in vessels afflicted with MMD entail smooth muscle cell proliferation and migration, which ultimately manifest as intimal thickening [3] .
A similar phenomenon is observed in vessels of the heart, kidney, and other organs [4] . These observations might suggest that systemic factors could play a role in the pathogenesis of MMD. Such factors include nitric oxide (NO), vascular endothelial growth factor, platelet derived growth factor, and alpha-1 antitrypsin. Nevertheless, the etiology of MMD as well as its initial predilection for the distal ICA, proximal MCA, and proximal ACA remains unclear.
Hemodynamic changes as well as factors within the intracranial vasculature have been hypothesized to play a role. Recent investigations have therefore focused on regional changes with respect to intracranial blood flow. This chapter will introduce computational modeling to study cerebral regional hemodynamic parameters with the goal of ascertaining why specific vascular regions are affected by MMD.
H.J. Seol (!)
Department of Neurosurgery , Kangwon National University Hospital , 17-1, Hyoja 3-dong , Chuncheon , Kangwon-do , 200-722 , Republic of Korea
e-mail: [email protected]