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Background Previous physics-based analyses of arterial morphology in relation to pulsatile pressure and flow, with pulse wave reflection, focused on the large arteries and required assumptions about the relative thicknesses of arterial walls and the velocities of pulse waves in the arteries. A primary objective of this study was to analyze arterial mor- phology and pulse wave reflection, using physics-based wave propagation, which explicitly includes arterial stiffness, with potential autonomic flow regulation, for both large and small arteries. Methods Pulse wave reflections that occur at arterial bifurcations, and their impact on macrocirculation and micro - circulation pulse pressures and flows, are analyzed using the physics of wave propagation and impedance matching. Results The optimum combinations of arterial dimensions and stiffnesses which minimize pulsatile reflections at arterial bifurcations are identified for both macrocirculation and microcirculation. The optimum ratio of arterial bifur - cations’ branch-to-trunk luminal areas is predicted to have a value of 1.26, (with corresponding optimum stiffnesses) based on the principle that autonomic flow regulation minimizes pulsatile reflections. This newly predicted value of area ratio compares favorably with the Murray Scaling Law value of 1.26. For an area ratio of 1.26, the optimum bifurcation stiffness ratio is predicted to have a value of 1.12 for bifurcations in the macrocirculation and a value of 0.89 in the microcirculation. The analysis predicts that minimal pulsatile reflections may occur for area ratios not equal to 1.26, when vasodilation adjusts arterial stiffness to compensate for non-optimal arterial area ratios. The analysis predicts that the capillaries have about one-tenth the stiffness of the aorta, and the capillary bed possesses about one thousand times more total luminal area than the aorta. The analysis predicts there are about thirty generations, aorta to capillaries, of arterial bifurcations in an arterial tree. Conclusions The optimum arterial morphologies predicted by this physics-based analysis correspond to those observed in human vascular physiology. The contributions that arterial stiffnesses and dimensions make to optimal pulsatile flow are relevant to the development of pharmaceuticals related to autonomic vasodilation, to the develop - ment of optimally designed stents and to surgical procedures related to vascular modification. Keywords Arterial bifurcation, Impedance, Stiffness gradient, Area ratio, Pulse wave velocity, Reflection coefficient, Target organ ischemia, Hypertension, Womersley, Murray’s Law 1 Introduction The human arterial system in youth is described  as *Correspondence: being “beautifully designed for its role of receiving spurts Trevor Tucker of blood from the left ventricle and distributing this as firstname.lastname@example.org T Tucker Inc, Ottawa, ON, Canada steady flow through peripheral capillaries”. The arte - rial system develops from the embryonic stage, through © The Author(s) 2023. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Tucker Artery Research youth, into a complex tree-like structure, consisting of the peripheral arteries. To be generally applicable to the arteries of diminishing size and compliance, extend- pressure and flow linkages between macrocirculation ing from the large central arteries into the fine arteri - and microcirculation, however, the physics relationships oles and capillaries. The arterial system’s design includes must explicitly include the stiffnesses of all arteries, large various autonomic regulatory processes for homeostasis and small. maintenance of blood pressure and flow throughout the The physical entity which links macrocirculation and vasculature. These processes mediate flow at both the microcirculation is the arterial junction, which is most large artery (macrocirculation) and small artery (micro- often a bifurcation. The pressure and flow patterns of circulation) levels of the vasculature. Such autonomic waves impinging on bifurcations are fundamental to the processes, include biochemical (i.e., the renin−angio- relationships between pressure and flow in the macrocir - tensin−aldosterone system), cellular (i.e., endothelial culation and in the microcirculation [18–22]. There have dependent vasodilation), neurovascular (i.e., baroreflex been estimated to be [15, 23, 24] twenty to thirty genera- and neuro-glial-vascular) and physical (i.e., the physics of tions, or stages, of cascaded bifurcations in progressing flow) processes. The biochemical, cellular, and neurovas - from the central arteries to the capillaries, representing cular contributions to the linkages between pressure in billions of bifurcations. Hence, the optimal design of the macrocirculation and flow in the microcirculation are bifurcations is of fundamental importance to the homeo- generally qualitatively described [2–8]. In comparison, stasis maintenance of pressure and flow throughout the the physical contributions to such linkages may be quan- vasculature. titively described through the application of the physics At a vascular bifurcation, in general, part of an imping- of fluid dynamics and wave propagation and impedance ing pulse wave is transmitted across the junction in matching to vascular flow. antegrade flow, while a part of the wave is reflected Although specific control algorithms that determine back in retrograde flow. The fraction of the wave which arterial morphology and regulate blood flow in tar - is reflected, as compared to that which impinges on the get organ perfusion are currently unidentified, a recent bifurcation, is termed the reflection coefficient. The paper  has suggested that, in youth at least, an opti- reflection coefficient is defined by Eq. (1): mally designed arterial structure includes the minimiza- Reﬂected Pulse Pressure tion of pulse wave reflections. The minimization of pulse Reﬂection Coeﬃcient = Forward Pulse Pressure wave reflections simultaneously minimizes central pulse (1) pressure and maximizes peripheral pulse wave flow and, A bifurcation’s reflection coefficient is determined by hence, influences perfusion of target organs. the mismatch in the flow impedances on either side of The seminal application of the physics of fluid dynam - the bifurcation [18–22]. If the impedance characterizing ics, by Womersley [10, 11] to the relationships between the trunk artery (into the bifurcation) is equal to the total pulsatile trunk and branch arterial flows, occurred in impedance of the branch arteries (out of the bifurcation), the mid-to-late 1950’s. Womersley’s physics relation- then the reflection coefficient is zero. In this matched ships have been included in most major textbooks related condition, the pulse pressure amplification associated to the dynamics of blood flow [12–17]. The Womersley with the bifurcation is zero and the total pulse flow out of analysis determined relationships between pulse wave the bifurcation is equal to that into the bifurcation. This reflection, pulse wave velocity and arterial luminal areas. condition of matched arterial impedances across bifur- His analysis, as reflected in his plots of reflection coef - cations is, therefore, an optimum condition for pulsatile ficient as a function of area ratio, was limited to arter - blood flow, and related nutrient provision, into target ies whose diameters were greater than about 6.7 mm organs. (Womersley number greater than five). His results were The physics relationship which quantifies a bifurca - based on the asymptotic expansion of Bessel functions tion’s reflection coefficient, as a function of the arterial (solutions to the Bessel equation which describes fluid stiffnesses and luminal cross-sectional areas, may be flow wave propagation in a cylinder), which he included derived from the mismatch in the arterial impedances on in tabulated form. Although the Womersley pulse wave either side of the bifurcation. To focus the body of this analysis described the flow field’s distribution across the analysis on the medical implications of the physics, the diameter of the vessel, it did not explicitly include arterial derivation of the generalized equation for a bifurcation’s stiffness. reflection coefficient is consigned to Appendix A. A fundamental characteristic of the Womersley phys- The quantitative analysis of the role of bifurcations’ ics-based analysis, however, is the existence of a spe- reflections in the linkages between macrocirculation and cific combination of arterial dimensions and pulse wave microcirculation, calls for the definition of these terms. For velocities which optimizes antegrade pulsatile flow into T ucker Artery Research the purpose of this analysis the macrocirculation is defined and reflected pulse pressures) in the trunk artery is also as that portion of the arterial tree for which the arterial transmitted into each of its branch arteries. Hence, the diameters are greater than 6 mm (see Appendix B for the pulse pressure into each of the branch arteries is given by supporting rationale). The microcirculation is defined as [19, 20] Eq. (3): that for which arterial diameters are less than 1 mm. The Pulse Pressure (Branch) region of the arterial tree for which the arterial diameter is = [1 + Reﬂection Coeﬃcient] less than 6 mm but greater than 1 mm is described as the (3) “mesocirculation”. ∗[Forward Pulse Pressure (Trunk)] The above simple Eqs. (1 )–(3) show dependencies of 2 Study Purpose arterial trunk and branch pulse pressures on wave reflec - One purpose of this study was to develop physics-based tion coefficients and apply to all regions of the vasculature. relationships between hemodynamic flow in the larger An increase in wave reflection at a bifurcation produces an central arteries and the smaller peripheral arteries, increase in both trunk and branch pulse pressures. explicitly including the stiffnesses and dimensions of all Although there is substantial evidence that increased arteries, large and small, and also including pulse wave central (aortic) pulse pressure is a predictor of target organ reflections which occur at bifurcations. A second pur - damage, reduced peripheral blood flow, or ischemia, has pose was to quantify the optimum arterial bifurcation also been identified [8 , 25–27] as a contributor to such design which minimizes pulsatile wave reflection and, damage. At bifurcations, in general, the pulsatile flow that is hence, minimizes central pulse pressure and maximizes reflected back on the forward pulse flow in the trunk artery peripheral pulse flow. is in the opposite direction to the forward flow. Hence, the net pulsatile flow in the trunk artery is the forward flow 3 Methodology wave reduced by the reflected flow wave. The predicted net The methodology applied in this analysis is based on pulse flow in the trunk artery is given by [19, 20] Eq. (4): the physics of wave propagation and impedance match- ing in compliant tubes. The physics of wave propagation Pulse Flow (Trunk) includes both the pressure and flow waveforms through = [1 − Reﬂection Coeﬃcient] (4) the ratio of pulse pressure to pulse flow which is the ∗[Forward Pulse Flow (Trunk)] impedance to flow offered by a compliant tube. The anal - ysis focuses on the relationships between the impedances The total antegrade pulse flow into the branches or on either side of arterial bifurcations, in the derivation of periphery of a bifurcation is predicted to be reduced by the the pulse wave reflection inherent to arterial impedance amount of pulse flow that is reflected by the bifurcation. mismatches. Such impedance mismatches are funda- For symmetrical bifurcations, with the two branch arteries mental to the relationships between arterial morphology, of equal luminal cross-sectional areas, the predicted pulse pulse wave reflection and peripheral pulse flow. In order flow in each branch is given by [19, 20] Eq. (5): to focus the body of manuscript on the physical and medical implication of the analysis, the derivation of the Pulse Flow (Branch) physics equations is consigned to the Appendices. = [1 − Reﬂection Coeﬃcient] (5) ∗[Forward Pulse Flow (Trunk)]/2 3.1 F undamental Physical Relationships Between Pulsatile Flow and Bifurcation Reflection Equations (4) and (5) show the dependence of central At arterial bifurcations in general, the portion of the and peripheral pulse wave flows on bifurcations’ reflection pressure wave that is reflected is added to that imping - coefficients. An increase in reflection coefficient produces ing on the bifurcation, so the total pulse pressure in the a decrease in both central and peripheral pulse flow. bifurcation’s trunk artery is given by [19, 20] Eq. (2): The fundamental principle which the above relation - ships identify is that, while increased bifurcation reflec - Pulse Pressure (Trunk) tions increase central pulse pressure, they simultaneously = [1 + Reﬂection Coeﬃcient] (2) decrease peripheral pulse flow. Although the above rela - ∗[Forward Pulse Pressure (Trunk)] tionships are well established in the physics and engineer- ing domains, they appear to be relatively unknown in the Since pressure at any point in a fluid is equal in all medical community, and hence are presented here as rela- directions, the total pulse pressure (the sum of forward tively new fundamental medical principles. Tucker Artery Research 4 Results is used, consistent with standard physics (and Womers- Quantitative results obtained by calculating and plotting ley’s) conventions. Appendix Eqs. (44)–(48) in the macrocirulation, micro- The generalized, physics-based reflection coefficient circulation and mesocirculation regions are provided equations (Appendix Eqs. 44–48) extend Womers- below. ley’s  reflection coefficient analysis, to specifically include the small arteries of the microcirculation, and 4.1 Q uantification of Pulse Wave Reflections to also explicitly include arterial stiffnesses. The bifurca - at the Macrocirculation’s Iliac/Aorta Bifurcation tion reflection coefficient plot of Fig. 1, applicable to the A plot of the reflection coefficient (calculated using macrocirculation case (arteries greater than about 6 mm Eqs. 39 and 44–48 in Appendix A), for the larger arter- diameter), is consistent with Womersley’s  bifurcation ies of the macrocirculation (i.e., diameters greater than reflection plots (Womersley’s plots displayed reflection 6 mm), is shown in Fig. 1 (as a function of the bifurca- coefficient as a function of area ratio for three different tion’s Pulse Wave Velocity Ratio/Area Ratio). The Area assumed values of relative pulse wave velocities). Ratio, using Womersley’s  convention, is the quotient In Fig. 1, the (absolute) values of in vitro measured of the total luminal cross-sectional area of the branch  reflection coefficient data (as measured on aorta/ arteries (out of the bifurcation) divided by the area of the iliac bifurcation cadaveric sections and as superimposed trunk artery (into the bifurcation). Similarly, as a surro- on the predicted reflection coefficient plot) were based gate measure of relative arterial stiffnesses on each side of on Womersley’s analysis approach. In comparison, the the bifurcation, the Pulse Wave Velocity Ratio is the quo- reflection coefficient plot of Fig. 1 is that predicted using tient of the velocity of pulse waves in the branch arter- the generalized equations described in the Appendix ies divided by the velocity of the pulse wave in the trunk (Eqs. 39 and 44–48). The match between the physics- artery. In vitro measured  values of reflection coef - based reflection coefficient plot and the measurement ficient are superimposed on the predicted bifurcation data indicates consistency between the bifurcation reflec - reflection coefficient plot in Fig. 1. An assumption which tion coefficient equations and the measured values of underlies both of the physics-predicted and the meas- pulse wave reflection coefficient for the macrocircula - ured iliac/aortic bifurcation reflection is that the branch tion’s aorta/iliac bifurcation. arteries are well matched to subsequent branch arteries, The reflection coefficient plot of Fig. 1 shows a very dis- and that reflections from such sub-branch arteries are tinct reflection minimum which represents the optimum negligible. The condition in which sub-branch reflections impedance match (with minimal central pulse pressure are not negligible is discussed in Appendix C in relation and maximal peripheral pulse flow). The optimum match to mismatched arterial bifurcations in the mesocircula- occurs under the specific condition that: tion. In Fig. 3, the absolute value of reflection coefficient Pulse Wave Velocity Ratio (Branch/Trunk) (6) = Area Ratio (Branch/Trunk) The plot of Fig. 1 indicates that a variation in the Pulse Wave Velocity Ratio/Area Ratio of 25%, relative to that for minimal reflection, results in an increase in a pulsatile wave’s reflection coefficient from near zero to about 12%. From Eq. (2) this increase in pulsatile reflection (without a change in the forward wave pressure) would therefore increase the central pulse pressure also by about 12%, and would simultaneously decrease peripheral pulse flow by about 12%. At the minimum point in the wave reflection coefficient plot, the central (aortic) pulse pressure is min - imized and represents the optimum bifurcation design. 4.2 Arterial Stiffness Ratio in Relation to Pulse Wave Velocity Ratio Fig. 1 Pulse wave reflection coefficient, in the macrocirculation, as To eliminate the dependence of the measure of stiffness a function of the quotient of the iliac-to-aorta pulse wave velocity on the arterial luminal area (and, hence, include the resis- ratio (PWVR) divided by the areas ratio (experimentally measured data from Greenwald et al.  and solid line is that predicted by this tive influence of viscosity) the arterial stiffness is here analysis) defined by Eq. (7) (see Appendix Eq. 24): T ucker Artery Research Stiﬀness Ratio (Branch/Trunk) = PWV Ratio (Branch/Trunk) (8) ∗ [Area Ratio (Branch/Trunk)] In medical practice the arterial pulse wave velocity (which assumes negligible blood viscosity) is most often used as an indicator of an artery’s stiffness even though PWV is dependent on arterial dimensions and is often applied for small arteries, for which viscous friction is not negligible. Fig. 2 Predicted macrocirculation bifurcation reflection coefficient 4.3 Quantification of Wave Reflection at a Generalized as a function of the quotient of Stiffness Ratio (Branch/Trunk) divided 5/4 Macrocirculation Bifurcation by Area Ratio (Branch/Trunk) . Arrow indicates optimum bifurcation design (minimum reflection) The predicted reflection coefficient for bifurcations in the macrocirculation (i.e., for arterial trunk diameters greater than about 6 mm), is shown in the plot of Fig. 2 (calcu- lated using Appendix A Eq. 33). The reflection coefficient of Fig. 2 is a function of the bifurcation’s Stiffness Ratio/ 5/4 Area Ratio (as opposed to PWV Ratio/Area Ratio). The optimum design, which corresponds to the minimum in reflection, is indicated by an arrow in Fig. 2. The mini - mum reflection coefficient for bifurcations in the macro - circulation, representing optimum design, is predicted to be less than 0.1%. The optimum impedance match, or minimum in the reflection coefficient plot, as indicated in Fig. 2 for mac- rocirculation bifurcations, occurs for the condition expressed by Eq. (9): Fig. 3 The plot of the predicted microcirculation bifurcation reflection coefficient as a function of the quotient of branch-to-trunk Stiﬀness Ratio (Macro Branch/Trunk) 7/4 (Stiffness Ratio)/(Area Ratio) −1/4 5/4 =  ∗ [Area Ratio (Macro Branch/Trunk)] (9) Equation (9) predicts that, if the aortic trunk stiffens [Arterial Stiﬀness] relative to its branch arteries, then to maintain optimum = [Arterial Wall Thickness (7) homeostasis in pulse pressure and flow, the autonomic processes must increase the luminal area of the trunk rel- ∗Elastic Modulus ∗ Blood Density ∗ 2]/3 ative to that of the branches. In the macrocirculation, if a Defining the arterial stiffness using only the material central artery stiffens by 25%, (without significant reduc - parameters of the arterial wall’s thickness and elastic tion in autonomic regulation due to atheroma develop- modulus, and the blood’s density, provides a general ment or other vascular or neurovascular disease) then definition of the stiffness of a bifurcation’s arteries, the central artery’s luminal area should dilate by about which is independent of arterial diameters. This defini - 20% in order to maintain homeostasis in pulse pressure tion of arterial stiffness inherently includes the effect and flow. Atheroma development, or other arterial dis - of viscosity and is valid for arbitrarily small values of ease, which alters the optimum bifurcation design may luminal area (with the possible exception of arteries significantly increase both central and peripheral pulse sufficiently fine that the Fahraeus–Lindqvist effect is pressure and decrease peripheral pulse flow. significant. In the large arteries of the macrocirculation, the rela- 4.4 Quantification of Pulsatile Wave Reflection tionship between the Stiffness Ratio (Branch/Trunk) and at a Microcirculation Arterial Bifurcation the PWV Ratio (Branch/Trunk), is as shown in Eq. (8) The reflection coefficient for bifurcations in the micro - (see the Appendix A for the derivation), circulation (for which the diameter of the bifurcation’s trunk artery is less than about 1 mm), calculated using Tucker Artery Research Appendix A Eq. (38), is shown in the plot of Fig. 3 (as a 7/4 function of the bifurcation’s Stiffness Ratio/Area Ratio ). The optimum design or minimum reflection coefficient value, for a single microcirculation bifurcation, is pre- dicted to be less than 0.1%. The minimum, or optimally matched condition in the microcirculation’s reflection coefficient, as shown in Fig. 3, occurs for the condition identified by Eq. (10): Stiﬀness Ratio (Micro Branch/Trunk) −3/4 7/4 = 2 ∗ Area Ratio Micro Branch/Trunk [ ] [ ( )] (10) Fig. 4 Predicted reflection coefficient, near the center of the The plots of Figs. 2 and 3 are new and unique to this mesocirculation region (for a 3 mm diameter trunk artery), as a analysis yet are as fundamental as “Murray’s Scaling Law” function of the Stiffness Ratio, for three different values of Area Ratio. The arrow at stiffness ratio SR = 1.0 which is discussed in more detail in subsequent sections of the manuscript. An optimally matched bifurcation in the microcircu- lation means that, with an area ratio that is greater than transitions from a value of greater than unity (for bifurca- unity, the stiffness ratio may be less than unity. This prop - tions proximal to the macrocirculation region), to a value erty of the microcirculation’s bifurcations means that, that is less than unity, (for bifurcations proximal to the in proceeding through multiple generations of cascaded microcirculation) an average stiffness ratio of approxi - bifurcations, from the more central arteries into the arte- mately one is indicated. As shown in Fig. 4, the optimum rioles and capillaries, with the attendant increase in total reflection coefficient, near the middle of the mesocircu - arterial area, successive generations may also increase in lation region, has a finite value of approximately 0.05 or compliance (decrease in arterial stiffness). 5%. In the mesocirculation, wave reflection is predicted Although the optimum reflection from a single micro - even in the case of optimum bifurcation design. The circulation bifurcation may be small (less than 0.1%), physical cause of this finite reflection coefficient in the there are many generations of cascaded bifurcations mesocirculation is that the in-phase, resistive contribu- between the 1 mm diameter (largest microcirculation) tion to the reflection coefficient and the orthogonal, or artery to the smallest 7-micron diameter capillary. Com- out-of-phase, inertial/compliant contribution are both bined reflected waves from many generations of cascaded finite and not simultaneously matched. For a mesocircu - bifurcations may present a potentially significant aggre - lation bifurcation, this analysis predicts that the optimum gate reflection coefficient, particularly in the presence of arterial match does not reduce to the low levels of the small vessel disease. In youth (i.e., 20–30 years) and good optimum match for bifurcations in the macrocirculation health, however, (and with optimum design) the micro- and the microcirculation (which may optimally be less circulation’s contribution to central wave reflection and than 0.1%). The mesocirculation reflection coefficient, pulse pressure is predicted to be relatively small. With which is predicted to be in the 4–6% range, represents an small vessel disease, a relatively small increase in the optimum value, irrespective of autonomic vasodilation reflection coefficients of a number of cascaded bifurca - processes. Although there may be relatively few genera- tions, may result in significant combined wave reflection, tions of mesocirculation bifurcations, each is indicated to with accompanying central pulse pressure increase and make a significant contribution to the aggregate of reflec - peripheral pulse flow decrease. tions from all arterial regions, macrocirculation, mesocir- culation and microcirculation. 4.5 Q uantification of Pulsatile Wave Reflection The finite reflection coefficient for bifurcations in the at a Mesocirculation Bifurcation mesocirculation, results in an essential impedance mis- Near the centre of the mesocirculation region, with an match between the macrocirculation and microcircula- assumed mean stiffness ratio of near unity, the reflec - tion regions. This impedance mismatch between macro tion coefficient for a bifurcation is predicted to be a func - and microcirculation regions results in wave reflection in tion of area ratio and stiffness ratio, as shown in Fig. 4 the central arteries, including the aorta. Because the cen- (for a bifurcation with trunk diameter = 3 mm). In the tral pulse pressure is the sum of the forward and reflected mesocirculation, the region in which the stiffness ratio wave pressures the central (aortic) pulse pressure is T ucker Artery Research predicted to be substantially dependent on the amount of each successive bifurcation increments by 1.5% which mesocirculation bifurcation reflection. yields, by a similar calculation, an aggregate reflection If arterial stiffnesses or luminal areas change from opti - coefficient for the mesocirculation of about 31.4%. The mum values in youth, through aging, through atheroma prediction that the mesocirculation generates the great- development or through other vascular diseases, then the est pulse wave reflection in the central arteries is a funda - central pulse pressure is predicted to increase, potentially mental new assessment of an otherwise elusive effective significantly, particularly if a number of generations of reflection site. A maximum reflection coefficient of 34% arterial bifurcations are sclerotic or diseased. from arteries with diameters in the 3–4 mm range was also predicted by Pollock . Even in youth and good health, with assumed opti- 4.6 E stimation of Optimum Aggregate Mesocirculation mum arterial impedance matching, there is predicted to Reflection be finite wave reflection in the macrocirculation caused The minimum reflection coefficient plots for bifurcations by essential mismatches in mesocirculation bifurcations. in the macrocirculation, as shown in Figs. 1 and 2, are A minimum central artery reflection coefficient in the based on the load impedances at the distal end of each range of 22–31% is predicted from inherent mesocircu- of the bifurcation’s branches being matched to the char- lation mismatch. In optimal mesocirculation bifurcation acteristic impedance of each branch (i.e., if the branch design, this inherent mismatch represents a minimum in is well matched to its sub-branches). If, however, the central (aortic) pulse pressure and a maximum in periph- branch is mis-matched at its distal end, then that mis- eral (microcirculation) pulse flow. match is transformed along the branch to its proximal end at the bifurcation (see Appendix C). To a first order 4.7 Optimum Combinations of Arterial Dimensions of approximation, this sub-branch reflection transforms and Stiffnesses the reflection coefficient at a primary trunk-to-branch Figure 5 provides plots of the optimum combinations of bifurcation to that of the mismatch at the distal end of bifurcations’ stiffnesses and area ratios corresponding to the branch arteries. minimal pulse wave reflection in the two limiting cases of Figures 4 and 13 (Appendix B) indicate that near the the macrocirculation (see Eq. 9) and the microcirculation centre of the mesocirculation region, which corresponds (see Eq. 10). The circled area in Fig. 5, corresponds to the to an arterial diameter of about 2–4 mm, the optimum optimum conditions in which the stiffness ratio transi - reflection coefficient for that centre bifurcation is in the tions from greater than unity in the macrocirculation, to range of 4–6%. With eight symmetric bifurcations in the less than unity in the microcirculation, with the case of mesocirculation, to attain 4% in the centre of the meso- SR = 1.0 centered between the two limiting cases. circulation, implies, each successive bifurcation incre- In youth and good health, the aorta is more compliant ments the reflection by 1%. With each branch in the than the aortic branch arteries. However, also in youth mesocirculation sequentially mismatched by about 1%, and good health, in the smaller arteries, (i.e., the micro- the total mismatch, or aggregate reflection coefficient for circulation) the branch arteries must be more compliant eight bifurcations, is estimated to be about 21.7% (1.01* than their trunk arteries to ensure the arterioles are sub- 1.02*1.03*1.04*1.04*1.03*1.02*1.01 = 1.217). To attain 6% stantially more compliant than the more central arteries. reflection in the centre of the mesocirculation region, Fig. 5 a Optimum Stiffness Ratio as a function of Area Ratio for a minimum in bifurcation reflection coefficient for the two limiting cases of macrocirculation and microcirculation; b expanded plot of Fig. 10a with an optimum match in the centre of the mesocirculation region indicated Tucker Artery Research The circled area is centered on a stiffness ratio of unity Two generations of bifurcations are a small fraction of (SR = 1.0) which corresponds to the arterial stiffness ratio the estimated [15, 23, 24] twenty to thirty generations of near the centre of the mesocirculation region. cascaded bifurcations which occur in a single vascular Figure 5 shows that the area ratios in both the macro- tree extending from the central arteries to the capillaries. circulation and in the microcirculation have a value of For mesocirculation bifurcations, each with an approximately AR = 1.26 for the condition that the stiff - assumed area ratio of about 1.3 (for which the diame- ness ratio of SR = 1.0 lies at the centre of the mesocircula- ter of a branch artery is about 81% of the trunk artery), tion region. Figure 5 demonstrates that the stiffness ratio eight generations of bifurcations are required to reduce a (SR) in the macrocirculation region which corresponds 6 mm trunk into a 1 mm branch. In the mesocirculation to an area ratio of AR = 1.26, is approximately SR = 1.12. the stiffness ratio transitions from a value of about 1.12, In the macrocirculation, for which arterial diameters are proximal to the macrocirculation, to a value of about greater than about 6 mm, the arteries become stiffer (in 0.89, proximal to the microcirculation. youth at least) in progressing from the aorta into the aor- For microcirculation bifurcations, each with an tic branch arteries. In the microcirculation, the stiffness assumed area ratio of about 1.26, twenty generations ratio corresponding to AR = 1.26 is about SR = 0.89. In of bifurcations are required to reduce a 1 mm diameter the microcirculation, in proceeding from generation to trunk to a 9 micron diameter capillary. With a stiffness generation of cascaded microcirculation bifurcations, the ratio of about 0.89 for each microcirculation bifurcation, total area of the arterial bed increases while the arterial the stiffness of the capillaries is predicted to be a factor stiffness decreases. of about one tenth (0.89 ) that of the microcirculation The optimum area ratio in the mesocirculation is pre - arteries which are proximal to the mesocirculation. dicted to vary from the value of AR = 1.26 which is appli- Hence, with two generations of bifurcation in the mac- cable to the macro and microcirculation regions. In the rocirculation, eight generations in the mesocirculation mesocirculation, the stiffness ratio varies from SR = 1.12 and twenty generations in the microcirculation, there are proximal to macrocirculation to SR = 0.89 proximal to the estimated to be about thirty generations, (based on the microcirculation. At the stiffness ratio of unity (SR = 1.0) simplifying assumption of junction symmetry) of bifurca- the Area Ratio which corresponds to an optimum match tions between the aorta and the capillaries. is in the range of about AR = 1.26–1.31 (see Fig. 4). The optimum value of bifurcation reflection coeffi - An area ratio of AR = 1.26 corresponds to the diameter cient in the macrocirculation is less than 0.1%. Hence, in of each bifurcation’s branch being approximately 79% of two generations of optimally designed macrocirculation that of its trunk artery. In the microcirculation, at each bifurcations, wave reflection is negligible, and the pulse level, or generation, in a progression of cascaded bifur- wave flow out of the macrocirculation is approximately cations, while the diameter of individual branch arteries equal to that in the aorta. The value of bifurcation reflec - reduces, the stiffness of those arteries simultaneously tion coefficient in the mesocirculation varies between a reduces. Hence, in the microcirculation, the analysis pre- low of close to zero to a maximum in the range of about dicts that, while the total cross-sectional area of the arte- 4–6%. The aggregate reflection coefficient for the meso - rioles and capillaries may be substantially greater than circulation, assuming coherent summing of the reflec - that of the central arteries, they may also, simultaneously, tions, is predicted to be in the range of about 22–31%. be substantially more compliant. This prediction is con - The percentage of total antegrade pulse flow from the sistent with the human vascular physiology. mesocirculation into the microcirculation is, therefore, predicted to be in the range of about 69–78% of that in 4.8 Q uantification of the Number of Generations the aorta. of Cascaded Bifurcations The value of reflection coefficient for a single, optimally The number of generations of bifurcations which occur designed, bifurcation in the microcirculation is less than in the microcirculation and the mesocirculation regions 0.1%. Hence, in about twenty-two generations of opti- is much greater than in the macrocirculation. In the mac- mally designed bifurcations in the microcirculation, less rocirculation, between the brachiocephalic trunk artery than 3% (1–0.999 ) of the pulse wave emerging from the (diameter typically 14 mm) and the internal carotid artery mesocirculation is reflected and more than 97% flows (diameter typically 6–7 mm), there are typically two gen- into the capillaries. Hence, the optimum or maximum erations of (asymmetric) bifurcations. For an optimum total antegrade pulse wave flow into the capillaries is in macrocirculation bifurcation area ratio of 1.26, the opti- the range of about 66–75% of that from the aorta. The mum stiffness ratio is about 1.12. In two macrocircula - largest single contributors to wave reflections in a well- tion bifurcations the stiffness of a 6 mm diameter distal matched arterial tree are predicted to be those arteries in artery would be about 1.25 (1.12 ) times that of the aorta. T ucker Artery Research the mesocirculation whose diameters are in the range of the veins. If autonomic adaptation processes modulate 2–3 mm, and whose stiffness ratios are near unity. stiffness ratios and arterial area ratios to maintain flow Any bifurcation in the arterial tree in which imped- homeostasis, then in “n” generations of cascaded bifur- ance match is not maintained by autonomic processes, cations the total area of the capillary bed may be repre- perhaps as a result of atheroma development or a result sented by Eq. (11): of other vascular or neurovascular diseases, will impact Total Area Ratio Capillary Bed/Central wave flow downstream of such bifurcation mismatch, = Average Area Ratio (Branch/Trunk) ≈ 1000 [ ] flow into the distal capillaries in that entire arterial (11) branch. Figure 6 shows the predicted arterial diameter, arte- In addition, assuming the arterial stiffness decrease rial stiffness, and total arterial bed area, relative to the from the capillaries to the central veins is comparable to aorta, for thirty generations of arterial bifurcations. The that from the central arteries to the capillaries, the stiff - total arterial bed area is predicted to increase about one ness relationship between the central arteries and the thousand times (i.e., the total capillary bed area is about capillaries may be expressed by Eq. (12): one thousand times greater than that of the aorta), while Stiﬀness Ratio (Capillaries/Central) the individual arterial size is predicted to decrease by = [Average Stiﬀness Ratio (Branch/Trunk)] ≈ 1/15 about one thousand times, each relative to the aorta. (12) The stiffness ratio is predicted to increase slightly in the macrocirculation and until about the mid-point of the The above three Eqs. (10), (11) and (12) can be solved mesocirculation, at which point it begins to decrease. The for the three unknowns, “n”, “Average Stiffness Ratio stiffness of the first branch of the microcirculation, as (Branch/Trunk)” and “Average Area Ratio (Branch/ shown, in Fig. 6, is predicted to be about the same value Trunk)”. An Average Area Ratio = 1.26, which is bifur- as the stiffness of the aorta. The stiffness of the arteries cated 30 times produces a total arterial bed result of in the microcirculation decrement by about 11% for each about 1000 (1.26 = 1026). This method of estimating generation of bifurcation progressing into the capillar- average area ratio, as discussed by Zamir , produces ies. The stiffness of the capillaries is predicted to be about an almost identical result to that based on the minimal one tenth (0.89 ) that of the aorta. pulse wave reflection principle discussed above. The classic “Murray’s Law”  of arterial bifurcation 4.9 Bifur cation Cascades and “Murray’s Scaling Law” area ratios, which has been widely addressed in many An alternate means of estimating the number of cascaded of the standard texts on blood flow [12, 15–17], offers bifurcations between the aorta and the capillaries arrives a “scaling law” for the dimensions of arteries at sym- at a similar set of estimates (i.e., about 30 generations metric bifurcations for which the area ratio is given by 1/3 of bifurcations, with an average Area Ratio ≈ 1.26). The Area Ratio = 2 = 1.26. This analysis, which is based total arterial area of the capillaries is historically reported on the principle of minimizing the magnitude of pulsa- [15, 23, 24] to be about one thousand times that of the tile waves reflected from bifurcations, produces a scal - central arteries, while the central arteries are historically ing law apparently identical to Murray’s Law, which is reported [15, 29, 30] to be about thirty times stiffer than based on the principle of minimizing the work required to move nonpulsatile blood flow through bifurcations. Although Murray’s Law does not include the influence of arterial stiffnesses on the optimum bifurcation imped - ance match, this analysis indicates there is an influence of such arterial stiffness in minimizing wave reflection. This analysis indicates that an idealized impedance match can be maintained in bifurcations for which the Area Ratio is other than 1.26, if autonomic vasodilation processes adjust the arterial stiffnesses in compensation. Such arte - rial stiffness compensation processes are not predicted by the Murray analysis approach. Fundamental new “Scaling Laws” which incorporate the arterial stiffness ratios are developed in this analysis, one applicable to the macrocirculation and another appli- Fig. 6 The relative (to the aorta) arterial diameter, arterial stiffness, and total area of the arterial bed at each generation of bifurcation in cable to the microcirculation. For the macrocirculation, progressing from the aorta to the capillaries Eq. (9) can be rewritten as the scaling law of Eq. (13): Tucker Artery Research 1/5 4/5 than that for youthful patients in good health and is also Area Ratio = 2 ∗ Stiﬀness Ratio [ ] (13) expected to be heterogeneous. Under the specific macrocirculation condition that The minimum, or idealized, aggregate reflection coef - 1/6 `1/3 Stiffness Ratio = 2 = 1.12, the AreaRatio = 2 , which ficient from multiple bifurcations in the mesocircula - matches Murray’s Law. tion, as discussed in the Sect. “4.6” is predicted to be in For the microcirculation, Eq. (10) can be rewritten as the range of 21.7–31.4%. Measured values for reflection the scaling law of Eq. (14): factor that are greater than this idealized range may be interpreted as representing impedance mismatches asso- 3/7 4/7 Area Ratio = 2 ∗ [Stiﬀness Ratio] (14) ciated with arteriosclerotic, stenotic or aneurysm devel- opment or other vascular diseases. Values that are less Under the specific microcirculation condition that than this idealized range may be interpreted as either –1/6 1/3 Stiffness Ratio = 2 = 0.89, the AreaRatio = 2 , also measurements on a single bifurcation (as opposed to the matching Murray’s Law. aggregate of a cascade of bifurcations) or a consequence Two additional, slightly different, arterial area scaling of such influences as the turbulent mixing of forward laws, the Huo–Kassab Law  and the Finet Law  and reverse flow waves, and other measurement-related have also been identified. The scaling laws developed inaccuracies. As discussed in the Sect. “4.1” a variance in in this analysis are more general that the Murray, Finet PWV Ratio/Area Ratio from the ideal value results in a or Huo–Kassab Laws, in that the laws developed here direct increase in reflection coefficient in a single bifurca - explicitly include the influence of arterial stiffness. Each tion. Similarly, variance from the ideal impedance match of the previous three scaling laws represents a specific in cascaded bifurcations will accumulate in significant case of the new scaling laws for specific equivalent val - increases to the aggregate reflection coefficient. ues of stiffness ratio. Table 1 below identifies each of the The ratio of measured peak reverse-to-forward flows, three previous scaling laws (for the symmetric bifurca- as indicated in Table 2, tended to have lower reflection tion case) and the specific values of stiffness ratio in both factor values than similar ratios of pressure or velocity the macrocirculation and the microcirculation which measurements, potentially as a result of simultaneous provide equivalence to the new scaling laws developed forward and reverse flows with turbulent mixing across a here. luminal area. Such flow mixing will affect the interpreta - tion of forward and reverse flows, resulting in a reduction 4.10 Comparison of Predicted and Measured Wave in apparent net flow, leading to an underestimation of Reflection Coefficient reflection coefficient. The pulse wave separation (forward Table 2 summarizes reflection factor measurement data and reverse pressure waves) analysis technique used for taken using various measurement techniques. The term pressure ratio computations resulted in substantial heter- reflection factor is applied in Table 2 as a generalization ogeneity in the calculated reflection coefficient, perhaps of the term reflection coefficient. This generalization is as a consequence of the different measurement tech - intended to reflect the different measurement techniques niques used and the difficulty of accurately separating the and calculations applied, and for consistency with a num- contributions of the forward and reversed pressure waves ber of the references in the table. to the combined pressure wave. The average value of measured aggregate reflection The calculated reflection factor associated with the factor in Table 2, is 30.9%, with a standard deviation of ratio of measured peak reverse to forward velocities 10.8%. Given that the measured data includes that taken resulted in a more uniform set of measured results than in various arteries and in elderly patients with attendant either the flow or pressure ratio measurements. The atheroma and other vascular diseases, the average value reflection factors measured using the peak velocities of measured reflection factor is expected to be greater ratio, of middle-aged patients (45–55 years), as indicated Table 1 Values of Stiffness Ratios which provide equivalence of the scaling laws developed here to the previously developed Murray , Huo–Kassab  and Finet  scaling laws Scaling Law Macrocirculation Microcirculation AR Equivalent SR Equivalent Macro SR Micro 1/5 4/5 3/7 4/7 This analysis AR = 2 *[SR] AR = 2 *[SR] 1.26 1.12 0.89 1/3 1/3 Murray  AR = 2 AR = 2 1.26 1.12 0.89 1/7 1/7 Huo–Kassab  AR = 2 AR = 2 1.10 0.95 0.74 Finet  AR = 1.09 AR = 1.09 1.09 0.93 0.69 T ucker Artery Research Table 2 Measured reflection factor data. In vivo measurement methods included the tonometric measurement of pressure and the Doppler ultrasound, phase contrast MRI and 4D flow MRI measurement of peak velocity and peak flow ratios References Patient age and Artery Type Method Equation Reflection factor Comment gender RF ± SD (%) Greenwald et al. < 50 Abdominal aorta In vitro Intra-lumen 1-2AR/PWVR 10 ± 4 Iliac/aorta bifurca- (1990)  Mixed transducer 1 + 2AR/PWVR tion only Greenwald et al. > 50 Abdominal aorta In vitro Intra-lumen 1-2AR/PWVR 10 to 30 ± 8 Iliac/aorta only RF (1990)  Mixed transducer 1 + 2AR/PWVR increases with age Yamamoto et al. 48 ± 20 Renal In vivo Doppler ultra- Velocity 30 ± 10 Vortical, mixed PeakReverse (1996)  Mixed sound Velocity reverse and forward PeakForward Mitchell et al. 58 ± 9 Carotid In vivo Tonometry Pressure 13 ± 5 RF increases with PeakReverse (2003)  Male Pressure age PeakForward Mitchell et al. 57 ± 9 Carotid In vivo Tonometry Pressure 22 ± 8 RF increases with PeakReverse (2003)  Female Pressure age PeakForward Mitchell et al. 37 ± 7 Proximal aorta In vivo Tonometry, Dop- Pressure 34 ± 6 Healthy controls, RF PeakReverse (2010)  Mixed pler US Pressure increase with age PeakForward Hashimoto and Ito 56 ± 13 Femoral In vivo Tonometry, Dop- Velocity 28 ± 10 RF decreases with PeakReverse (2010)  Mixed pler US Velocity increased aortic PeakForward PWV Hashimoto et al. 56 ± 12 Femoral In vivo Tonometry, Dop- Velocity 30 ± 10 RF increase with PeakReverse (2011)  Mixed pler US Velocity increased PourceIot PeakForward index Mitchell et al. 76 ± 4 Carotid In vivo Tonometry Pressure 6 ± 3 Carotid/aorta bifur- PeakReverse (2011)  Mixed Pressure cation only PeakForward Hashimoto and Ito 54 ± 13 Thoracic aorta In vivo Tonometry, Dop- Velocity 35 ± 10 Reflection increases PeakReverse (2013)  Mixed pler US Velocity with PWV gradient PeakForward Coutinho (2013) 67 ± 9 Carotid In vivo Tonometry, Dop- Pressure 36 ± 13 cfPWV = 11.9 ± 3.8 PeakReverse  Male pler US Pressure PeakForward Coutinho et al. 65 ± 9 Carotid In vivo Tonometry, Dop- Pressure 37 ± 13 cfPWV = 10.5 ± 3.4 PeakReverse (2013)  Female pler US Pressure PeakForward Bensalah et al. 27 ± 6 Ascending aorta In vivo PC MRI Flow 11 ± 4 Vortical, mixed PeakReverse (2014)  Mixed Flow reverse and forward PeakForward Bensalah et al. 54 ± 9 Ascending aorta In vivo PC MRI Flow 18 ± 7 Vortical, mixed PeakReverse (2014)  Mixed Flow reverse and forward PeakForward Torjesen et al. 51 ± 15 Central (Aorta?) In vivo Tonometry, Dop- Pressure 34 ± 6 RF increases with PeakReverse (2014)  Male pler US Pressure age PeakForward Torjesen et al. 51 ± 16 Central (Aorta?) In vivo Tonometry, Dop- Pressure 36 ± 7 RF increases with PeakReverse (2014)  Female pler US Pressure age to 55, decreases PeakForward after age 55 Hashimoto and Ito 52 ± 12 Proximal aorta In vivo Doppler US Velocity 33 ± 10 Increased RF PeakReverse (2015)  Mixed eGFR > 60 Velocity decreases eGFR PeakForward Hashimoto and Ito 58 ± 13 Proximal aorta In vivo Doppler US Velocity 38 ± 10 Increased RF PeakReverse (2015)  Mixed eGFR < 60 Velocity decreases eGFR PeakForward Breton et al. (2016) 40 ± 10Mixed Brachial In vivo Tonometry, Dop- Velocity 24 RF and PWVR PeakReverse  pler US Velocity increase with age PeakForward Breton et al. (2016) 61 ± 9 Mixed Brachial In vivo Tonometry, Dop- Velocity 54 RF and PWVR PeakReverse  pler US Velocity increase with age PeakForward Kim et al. (2017) 59 ± 12 Descending aorta In vivo Tonometry, Dop- Velocity 40 ± 10 Pulse pressure (PP) PeakReverse  Mixed pler US Velocity PP < 71 mmHg PeakForward Kim et al. (2017) 65 ± 9 Descending aorta In vivo Tonometry, Dop- Velocity 45 ± 10 PP > 71 mmHg, RF PeakReverse  Mixed pler US Velocity and PP increase PeakForward with age Jue et al. (2019) 62 + 12 Male Carotid? In vivo Tonometry, Dop- Pressure 39 ± 3 Aortic Aneurysm PeakReverse  pler US Pressure (AA) PeakForward RF independent of AA Jue et al. (2019) 65 + 9 Female Carotid? In vivo Tonometry, Dop- Pressure 46 ± 10 RF increases with PeakReverse  pler US Pressure AA diameter PeakForward increase Tucker Artery Research Table 2 (continued) References Patient age and Artery Type Method Equation Reflection factor Comment gender RF ± SD (%) London et al. 54 ± 2 Carotid In vivo Tonometry Pressure 26 ± 2 Normotensive PeakReverse (2019)  Mixed Pressure controls PeakForward London et al. 54 ± 1 Carotid In vivo Tonometry Pressure 41 ± 1 Hypertensive, RF PeakReverse (2019)  Mixed Pressure increases with PP PeakForward Evdochim et al. 24, Brachial In vivo Tonometry Pressure 0 to 50 RF varies with mean PeakReverse (2020)  Single subject Pressure pressure, RF = 0 at PeakForward MAP = 100 mmHg Jarvis et al. (2020) 36 ± 9 Upper aorta In vivo 4D Flow MRI Flow 8 ± 3 Youthful controls, RF MeanReverse  Mixed Flow affected by mixed MeanForward reverse, forward flow Jarvis et al. (2020) 65 ± 8 Upper aorta In vivo 4D Flow MRI Flow 15 + 5 Age matched MeanReverse  Mixed Flow controls MeanForward RF increase with PWV Jarvis et al. (2020) 69 ± 9 Upper aorta In vivo 4D Flow MRI Flow 17 + 6 Stroke patients MeanReverse  Mixed Flow RF affected by MeanForward mixed reverse, forward flow Haidar et al. 75 ± 4 Carotid and others In vivo Tonometry, Dop- Flow 34 ± 10 Asymmetric carotid- PeakReverse (2021)  Mixed pler US Flow aorta bifurcation PeakForward well matched Haidar et al. 75 ± 4 Carotid and others In vivo Tonometry, Dop- Pressure 41 ± 11 Increased aorta stiff- PeakReverse (2021)  Mixed pler US Pressure ness decreases RF PeakForward Hashimoto et al. ( 55 ± 14 Femoral In vivo Doppler US Velocity 32 ± 10 Ischemic organ PeakReverse 2022)  Mixed Velocity damage with PeakForward increased Reflection Factor in Table 2, ranged from a low of about 30% ± 10% in designed, structured, and tuned to minimize central the renal artery to a high of about 45% ± 10% in the pulse pressure and to maximize peripheral and capillary descending aorta. These measured values of reflection pulse flow. The simultaneous minimization of central factor (average value of 30.9% ± 10%) compare reason- pulse pressure and maximization of peripheral pulse flow ably favorably with the predicted optimum (or idealized) is associated with optimizing arterial bifurcation design reflection coefficient range of 22–31% and with arterial and structure throughout the vasculature. variance in human physiology. In human physiology, The analysis also infers that arterial property changes, reverse to forward wave ratio measurements are pre- particularly changes in luminal areas or arterial wall stiff - dicted to increase as a consequence of atheroma devel- nesses, can significantly increase wave reflection at arte - opment or other vascular physiological irregularities. rial bifurcations, causing increased central pulse pressure The measurement data of Table 2 indicates an increase and decreased peripheral pulse flow. For cascaded bifur - in reflection factor with increasing age, irrespective of cations which are mismatched, the deleterious pressure the measurement technique applied. Such an increase is and flow effects can be cumulative, and hence, substan - consistent with increasing impedance mismatches which tial. There are many potential causes of such arterial occur with aging, as arterial luminal areas reduce and property changes, including age-related arteriosclerosis, vessel walls thicken with atheroma development, particu- obesity, smoking, diabetes, vascular diseases, and neu- larly at the ostia of macrocirculation and mesocirculation rovascular disorders. Autonomic regulatory processes, if bifurcations. unaffected by disease, would tend to mitigate the effects of such deleterious arterial changes through central pres- 5 Discussion sure homeostasis maintenance, endothelial dependent 5.1 M edical Significance of Optimal Arterial Design vasodilation and neurovascular baroreflex regulation. and Structure Clinical diagnoses and treatments traditionally focus on This analysis infers that in youth, and good health, systolic, diastolic and pulse pressures, most often meas- bifurcations throughout the arterial tree are optimally ured at the brachial artery. Pharmaceutical treatments T ucker Artery Research tend to emphasize the reduction of pulse pressure and gradient. This physics-based analysis predicts that the mean arterial pressure through modulating the renin– central-to-peripheral stiffness gradient is a better predic - angiotensin–aldosterone system (with such medications tor of increased pulse pressure, and decreased pulse flow, as angiotensin converting enzyme inhibitors, angiotensin than central arterial stiffness alone. This prediction has receptor blockers and aldosterone receptor antagonists). greater validity if the peripheral PWV measured is that Treatments also address atheroma development and arte- of the femoral-to-ankle rather than that of the carotid-to- rial stiffness change through the use of pharmaceuticals brachial arteries. When carotid-to-femoral PWV meas- such as statins, to reduce lipid deposition, and calcium urements are used to describe the aortic stiffness, then channel blockers, but may also involve exercise regimes the femoral-to-ankle PWV offers a better indication of with dietary and smoking regulation. central-to-peripheral gradient arterial mismatch, than Adverse cardiovascular events have been widely asso- carotid-to-brachial or carotid-to-radial measurements, ciated with stiffening of the central arteries, in particu - (since the relevant bifurcation in the pulse wave’s reflec - lar, the aorta. The carotid-to-femoral pulse wave velocity tion is the iliac-to-femoral bifurcation). The Stone study (cfPWV) has been identified in several expert consensus  that used the femoral-to-ankle PWV for periph- reports [53–56] as the “gold standard” surrogate measure eral artery stiffness, reported a greater correlation with of central, or aortic, arterial stiffness (which is difficult to adverse cardiovascular events than the Fortier study  directly measure in vivo). Elevated cfPWV, or aortic “stiff - which used the carotid-to-brachial PWV measurement ening”, has been widely associated with hypertension, as the peripheral stiffness measurement. atheroma development, adverse cardiovascular events, This analysis predicts that the quotient of the ratio and target organ damage. of peripheral to central pulse wave velocities divided Various reports [39, 56–59] have suggested that, with by the ratio of peripheral to central luminal areas (i.e., aging, the inversion of the central-to-peripheral arte- Pulse Wave Velocity Ratio/Area Ratio) is a better predic- rial stiffness gradient increases the pulse pressure trans - tor of reflection coefficient, and hence, increased cen - mitted into target organs, causing organ damage. This tral pulse pressure and decreased peripheral pulse flow, analysis offers a prediction that is at some variance with than either central pulse wave velocity alone or PWV the perception that the central artery’s stiffness exceed - gradient. The implication of this prediction is that, in ing that of the peripheral arteries causes increased pulse clinical measurements of arterial stiffness, arterial diam - pressure in the peripheral arteries. This analysis indi - eter measurements should also be taken, if feasible. This cates that a bifurcation impedance mismatch, involv- analysis predicts that adverse cardiovascular events are ing both arterial stiffnesses and luminal areas, results in associated, not only with arterial hardening, but also increased central pulse pressure (with a simultaneous with arterial dimensional changes. Although the previous decrease in peripheral pulse flow). The optimum combi - literature identifies arterial stiffness is a factor in arte - nation of arterial stiffnesses and luminal areas is different rial wave reflection and luminal area is also a factor, this in the three different regions of the vasculature. Although analysis offers the unique combination of stiffnesses and an arterial stiffness gradient that is associated with the luminal areas together, in each of the macrocirculation central arteries being stiffer than the peripheral arter - and mesocirculation, which are determinants in pulse ies may result in increased peripheral pulse pressure, wave reflection. the converse may also be true. Peripheral arteries which The finding that the value of area ratio of 1.26 in both are much stiffer than central arteries can also result in the macrocirculation and the microcirculation (which increased pulse reflection and peripheral pulse pressure. is associated with a minimum in pulse wave reflection), The relative luminal areas of the central and peripheral is apparently identical to the value identified by Mur - arteries also affect the impedance mismatch at bifurca - ray’s Law (which is associated with minimum work in tions and hence, peripheral pulse pressure. moving steady flow through the arterial tree), was unex - pected. The potential for these two minimized conditions 5.2 A P rospective New “Gold Standard” for Arterial being potentially physically identical is worthy of further Stiffness Measurement fundamental research. In addition, the reservoir-wave Some recent longitudinal studies [60–62] suggest that the analytic approach of Parker et al. [64–66], which is com- ratio of central to peripheral pulse wave velocities may be plementary to this impedance matching, wave-propaga- a better predictor of adverse cardiovascular events than tion approach, identifies the reservoir waveform as that cfPWV alone and has also been suggested  as a pos- associated with minimum work. The reservoir wave ana - sible “new gold standard” for the measurement of arterial lytic approach includes an “excess waveform” component stiffness. The ratio of central-to-peripheral pulse wave that identifies separate backward and forward waves. velocities is also often referred to as the arterial stiffness The physical relationships between the minimal pulse Tucker Artery Research wave reflection and minimal work and between the res - reflection which occurs at arterial bifurcations. Although ervoir wave and impedance-matching wave propagation human arterial morphology has tapered arteries, mainly approaches also merit further basic research. with a slowly decreasing luminal area, wave reflection is predicted to be a function of both luminal area and wall 6 Study Limitations stiffness. Generally, arterial walls decrease in thickness This analysis of pulse wave reflections at arterial bifur - with decreasing luminal area, thereby minimizing wave cations, with their related pressure and flow linkages reflection in tapered arteries. In silico and phantom arte - between the macrocirculation and microcirculation rial models  indicate that the reflection site associ - regions, is limited to the consideration of symmetric ated with arterial taper is distributed along the length of bifurcations. The results of the analysis are compared the arterial segment and is superimposed on reflections with the Womersley [10, 11] results, which were devel- from individual major bifurcation sites. oped for symmetric bifurcations only. The human In developing the closed form reflection coefficient vasculature includes asymmetric junctions, including tri- Eqs. (44–48), linear approximations to the blood viscosity furcations and quadfurcations. The additional complexity and the elastic modulus parameters were applied. These of analyzing asymmetric bifurcations may obscure the are the same linearization approximations used by Wom- physical and medical implications of the analysis. While ersley [10, 11] and others [15, 22]. As discussed by Nich- analyses of impedance mismatch at asymmetric bifur- ols , the impact of nonlinearities has been assessed cations have been reported [15, 22], such analyses draw [11, 68–70] to be relatively insignificant, particularly in on the Womersley approach involving the asymptotic relation to the potential for the nonlinearities giving rise expansion of Bessel functions which may also obscure the to inter-modulation products of the Fourier harmonics of physical implications of the analysis. One consequence of the heart rate. the symmetric bifurcation assumption is that the result- With the occurrence of cyclically reversing flow, indi - ing estimate of thirty generations of bifurcations between vidual cells must stop and reverse direction at select arte- the aorta and the capillaries, as discussed in Sect. “4.8”, is rial points. With the blood’s viscosity having a strong probably an overestimation. dependence on its flow velocity (at very low flow veloci - One basic purpose of the study was to extend Womers- ties the blood’s viscosity may be more than an order of ley’s seminal physics-based analysis of wave reflections at magnitude greater than at normal systolic flow rates [71, symmetric arterial bifurcations, to explicitly include arte- 72]). Such viscosity non-linearity is anticipated to be sig- rial stiffnesses in both large and small arteries. The equa - nificant in the narrower arteries in which the effect of tions of reflection coefficient, (as a function of arterial viscosity is important. With arterial impedance in the stiffness and area ratios) developed in this analysis are macrocirculation being independent of viscosity (Eq. 29), closed form equations, which are relatively easily com- the impact of flow velocity on macrocirulation imped - puted. The equations do not involve expansions of Bessel ance mismatch is predicted to be relatively minor, (as dis- functions as provided in tabular form by Womersley. cussed by Nichols ). In the microcirculation, however, The use of symmetric bifurcations in this analysis does arterial impedance is dependent on the blood’s viscosity not limit the generality of the principle which the analysis (through the dependence of impedance on Womersley’s offers, that pulse wave reflections at arterial bifurcations number “α”, as identified in Eqs. (23) and (31). The impli - are linked to both increased aortic pulse pressure and cation of an increase in viscosity would be an apparent decreased peripheral pulse flow. The symmetric bifur - decrease in the luminal area and the potential need for cation focus also does not affect the principle that, with the introduction of a correction factor in the determina- well matched arterial bifurcations, the central arteries are tion of optimum Pulse Wave Velocity Ratio/Area Ratio more compliant than the first few generations of branch in the mesocirculation and microcirculation. With the arteries. For smaller arteries (less than about 3 mm diam- focus of this analysis on the ratio of luminal areas, any eter) however, the arteries soften, while the total arte- flow velocity dependent correction factor which may be rial bed area increases with each generation of cascaded applied to the luminal area on both sides of a bifurcation bifurcation. is likely to be somewhat self-correcting in the determina- The analysis and equations do not consider pulse wave tion of the equivalent luminal area ratio. reflections which may occur from arterial taper. The With the pulse wave’s amplitude attenuating as the basic assumption of uniform wall thickness and stiffness wave propagates, the magnitude of the reverse flow in each arterial segment does not affect the pulse wave component similarly decreases in progressing into T ucker Artery Research the microcirculation. Low and reversing flow veloci- pulsatile (i.e., more sinusoidal at the fundamental fre- ties, with viscous dependence on flow velocity, will quency) and, with the low pass filtering of each bifur- influence arteries’ impedances and bifurcation wave cation generation, also becomes reduced in amplitude. reflections, particularly in the large reflection coef- ficient region of the mesocirculation. The current lit-7 Conclusions erature is relatively silent on pulsatile wave reflection This wave propagation-based analysis extends the semi - in the mesocirculation, indicating the need for addi- nal physics-based analyses offered by both Murray  tional research related to wave reflection and flow in 1928 and by Womersley  in 1958. This analysis reversal flow in this region of the vasculature. The produces reflection coefficient plots for pulse waves potential impact of cyclically reversing flow on arte- introduced by impedance mismatches in flow through rial wall shear stresses, with high blood viscosities, as bifurcations of the macrocirculation, reflection coeffi - associated with low flow rates, and the impact of such cient plots which match Womersley’s plots. The analysis reversing flow on endothelial layer continuity and predicts that in both the macrocirculation and microcir- function, merits further clarification. culation the optimum pulse wave antegrade flow condi - For arterioles (less than about 100 microns in diam- tions occur when the luminal areas of bifurcation trunk eter), the Fahraeus–Lindqvist [73, 74] effect will also and branch arteries are as described by Murray’s Scaling affect blood viscosity. With relatively low pulsatility at Law, but for specific values of arterial stiffness. For Mur - such small arterial dimensions and the error-correct- ray’s optimum area ratio value of 1.26 the optimum ratio ing effect which the ratio of luminal areas imparts, the of branch to trunk stiffness is 1.12 in the macrocircula - applicability of the reflection coefficient Eqs. (44–48) tion and 0.89 in the microcirculation. This analysis, there - for flow in the arterioles is indeterminate. fore, offers a physics-based linkage between the classical The analysis considers only the fundamental Fou- analyses of Murray and Womersley. The analysis also pre - rier component of pulsatile waveforms. The reflection dicts that if the luminal area ratio for a bifurcation does coefficient for each of the Fourier harmonic compo- not satisfy Murray’s Scaling Law, optimal antegrade pulse nents of a pulsatile wave will display similar V-shaped flow can still occur if the stiffness ratio for the bifurcation plots (each as a function of Stiffness Ratio and Area is adjusted to offset the nonoptimal area ratio. Ratio) as displayed by the fundamental harmonic com- The analysis predicts that the mesocirculation region, ponent, but for somewhat different arterial diameters. the region of the vasculature with arterial diameters In the limits of the macrocirculation and the micro- between one and six millimeters, is the greatest pulse circulation the reflection coefficient, as described by wave reflection region of the arterial tree, hence pre - Eqs. (33) and (42), are dependent only on Stiffness sents the greatest reduction to antegrade pulse wave Ratio (SR) and Area Ratio (AR). Neither Stiffness Ratio flow. The optimum reflection coefficient predicted by nor Area Ratio, by their definitions, are dependent on this analysis is in the range 22–31%, which compares frequency. However, the reflection coefficient in the favourably with the value of clinically measured reflec - mesocirculation is dependent on frequency, through tion factors of 30.9%, the averaged of 18 different studies, the Womersley number (α). The Womersley num- involving patients of all ages with various cardiovascular ber is dependent linearly on the arterial diameter and conditions. on the square root of the frequency. Hence, the first Most of the current focus on arterial stiffness in hyper - Fourier harmonic component (i.e., double the funda- tension relates to the aorta’s stiffening with age and with mental frequency) has the same reflection coefficient cardiovascular diseases. This analysis indicates that any plot as the harmonic, but at a value of diameter that is change in arterial stiffness or luminal area in any artery, −1/2 0.707 (2 ) of that of the fundamental frequency. The either central or peripheral, which results in increased implication is that the luminal diameters which define bifurcation impedance mismatch, can increase central the mesocirculation for the first Fourier harmonic are and peripheral pulsatile pressure. Increased impedance not 1–6 mm, but rather are 0.7–4.2 mm and for the mismatch also decreases pulse wave flow in downstream second harmonic are 0.6 and 3.5 mm, etc. The practi- segments of the arterial tree, thereby influencing the per - cal implication of this is that the shape of the pulsatile fusion of target organs. The analysis indicates that the wave will change slightly as is progresses through each recently proposed use of central to peripheral arterial bifurcation. The specific shape change will be depend- stiffness gradient, as a predictor of adverse cardiovascu - ent on the values of specific Fourier coefficients of the lar events, potentially offers sufficient improvement over pulsatile waveform. However, in general, the shape of currently used predictors to merit further research. the pulse as it progresses through the mesocirculation From the points of view of medical research and clini- into the microcirculation is predicted to become less cal practice, the predictions offered by this analysis Tucker Artery Research are potentially far reaching. The analysis predicts that Appendix A increased pulse pressure which is a consequence of bifur- Impedance Matching for Wave Propagation in a Compliant cation impedance mismatch will affect both measures of Vessel pulse pressure amplification and augmentation index (as Derivation of the Generalized Bifurcation Reflection indicators of cardiovascular health). In various hyperten- Coefficient Equation sive conditions such as isolated systolic hypertension in For pulsatile pressure waves propagating in a fluid- youth, elevated brachial pulse pressure may be a conse- filled, compliant vessel the relationship between flow quence of pulse reflection from mesocirculation bifurca - and pressure is determined by the vessel’s character- tions for which the stiffness ratio is too low, rather than istic impedance (Z ). The characteristic impedance is too high. defined by the quotient of the wave’s pressure divided High flow demand organs, such as the heart, brain by its flow. The characteristic impedance is determined and kidneys are likely to be most affected by increased by [16, 19] the vessel’s longitudinal impedance (Z ), and pulse wave reflection from mis-matched mesocircula - transverse impedance (Z ), as given by Eq. (15): tion bifurcations. In chronic kidney disease, the Dop- Characteristic Impedance : Z = Z Z (15) C L T pler measurement of the Pourcelot “Resistive” Index in the renal arteries provides some measure of renal pulse The longitudinal impedance, in turn, is related to flow anomaly. However, the Pourcelot Index uses the the viscous resistance (R) to flow presented by the ves - maximum value of diastolic Doppler, indicative only of sel’s walls and the inertial impedance (L) of the blood’s the maximum antegrade flow velocity at diastole. The mass, as given by Eq. (16): impact of reversing flow (with accompanying increased blood viscosity) on endothelial function is not captured Longitudinal Impedance : Z = R + jωL L (16) by the measurement of maximum antegrade diastolic The transverse impedance (Z ) is related to the stiff - flow. Given that reverse renal artery flow is profoundly T ness, or its inverse, compliance, of the vessel (often symptomatic of end-stage renal disease, measurement of described as capacitive impedance (C)), as given by Eq. the maximum retrograde pulse flow in the renal arteries (17): is required. This implies the need for a fundamental new Doppler ratio measure (using the existing Doppler ultra- 1 j Transverse Impedance : Z = =− sound techniques) which is defined by the increment T (17) jωC ωC between maximum antegrade flow velocity and maxi - mum retrograde flow velocity. This analysis indicates sig - In Eqs. (16) and (17), “ω” is the frequency of the nificant diagnostic value in greater clinical use of Doppler heart rate, expressed in radians/sec and the “j” opera- ultrasound measurements of the diastolic flow velocities tor represents the out of phase (or orthogonal compo- into target organs, particularly in quantifying the amount nent) of the pressure wave relative to that of the flow, of reverse or retrograde flow, with its associated blood with + j representing the pressure wave leading that of viscosity increase and endothelial function decline. The the flow wave and − j representing the pressure wave’s measurement of brachial pulse pressure is not a measure phase lagging the flow wave. This analysis of pulsatile of the pulse flow and perfusion of target organs. blood flow focuses on the pressure and flow relation - Most current pharmaceutical treatments for hyperten- ships of the fundamental harmonic of the pulsatile sion in cardiovascular diseases are designed to reduce wave, the largest amplitude harmonic, which is also the or control macrocirulation pressures, not mesocircula- frequency of the heart rate. The three constituent com - tion elevated pulse pressures associated with pulse wave ponents, “R”, “L” and “C”, which comprise the charac- reflections. This analysis indicates the need for research teristic impedance, are given approximately by [16, 19] on treatments which reduce central and peripheral pulse Eqs. (18)–(20): pressures through the minimization of pulse wave reflec - 8μ tions from arterial bifurcations for which the arterial R = , (18) πr luminal areas and stiffnesses are not optimally matched. L = , (19) πr and T ucker Artery Research 3πr 2 (26) A = πr C = . (20) 2Eh Although the above equations are based on the previ- In Eqs. (18)–(20), each of the constituent impedance ously established analyses, the majority of the analysis components is dependent on the radius of the vessel below, and the associated plots, are fundamental and “r”. In Eq. (18), the resistance to flow “R ” also depends new developments. on the viscosity of the blood “μ”. In Eq. (19) the iner- The characteristic impedance of compliant vessels, as tial component “L” also depends on the density of the described  by Eq. (25), for arbitrarily large or small blood “ρ”. In Eq. (20), the vessel wall’s compliant com- values of α, is solvable through application of the iden- ponent “C” also depends on the vessel wall’s elastic tity of Eq. (27): modulus “E”, and thickness “h” (assumes h < < r). 1 1 1 1 2 2 The impedance to flow, as a function of the three 2 2 2 2 8 8 1 + + 1 1 + − 1 1 2 2 ∝ ∝ impedance components of “R”, “L” and “C”, is obtained 2 j8 1 − = √ − j √ by combining Eqs. (15)–(17), yielding Eq. (21): α 2 2 (27) 1 1 2 2 L jR Substituting Eq. (27) into Eq. (25) results in a general- Z = 1 − (21) C ωL ized expression for the characteristic impedance of a compliant artery, as shown in Eq. (28): Substituting Eqs. (18)–(20) into Eq. (21) yields the 1 1 relationship between the characteristic impedance and 1 1 2 2 2 2 2 2 S 8 8 the vessel’s and the blood’s parameters, as given in Eq. Z = √ 1 + + 1 − j 1 + − 1 2 2 A 2r ∝ ∝ (22): (28) 1 1 2 2 2ρEh j8μ 1 (22) Equation (28) is simplified considerably in the two lim - Z = 1 − 2 2 3r ωρr πr iting cases of large arteries (diameter > 6 mm), and small arteries (diameter < 1 mm). Two parameters are defined, the Womersley  In the large artery case (diameter > 6 mm), the char- number, “α”, and a stiffness factor, “S ”, as shown in Eqs. acteristic impedance of Eq. (28) is approximated by Eq. (23) and (24) respectively: (29): ωρr α ≡ (23) Z = √ C (29) A r and In the small artery case (diameter < 1 mm), the charac- 2ρEh teristic impedance is approximated by Eq. (30): S ≡ (24) 2S Z = √ [1 − j] C (30) The stiffness factor “S ”, as defined here, includes the Aα r elastic modulus “E” of the vessel wall, the thickness In Eq. (30), the [1 − j] term indicates a 45° phase dif- “h” of the vessel wall, and the blood’s density “ρ”. This ference between the pressure and the flow waves. The definition of arterial stiffness depends only on wall and magnitude of the [1 − j] term is √2, so the magnitude of blood material parameters and combines the effect of the small artery’s characteristic impedance is given by Eq. the vessel wall’s thickness and elasticity. Substitut- (31): ing Eqs. (23) and (24) into Eq. (22) results in a general equation for the characteristic impedance “Z ” of the 2 2S vessel, as a function of vessel wall stiffness “S ”, luminal |Z | = √ (31) Aα r cross-sectional area “A”, luminal radius “r”, and Womer- sley number “α”, as given in Eq. (A11): Arterial bifurcations, in general, present abrupt changes to the artery’s characteristic impedance. Propa- 1/2 S 8 gating pressure waves which impinge on changes in Z = 1 − j (25) A r α the characteristic impedance of the artery are partially reflected in retrograde wave flow. In the development where the cross-sectional area “A” of a cylindrical vessel of the wave reflection equation for the impedance mis - is given by: match which may occur at a bifurcation, a symmetrical Tucker Artery Research Fig. 8 Reflection Coefficient for a Large Artery (Macrocirculation) Fig. 7 Representation of a symmetric arterial bifurcation showing Bifurcation as a Function of the Bifurcation’s (Stiffness Ratio/(Area arterial dimensions and characteristic impedances (d = diameter, 5/4 Ratio) r = radius, Z = characteristic impedance, 1 = trunk artery, 2 = branch artery) sub-branches at the distal end of the branch artery (see Appendix C). bifurcation is assumed, as shown in Fig. 7. The assump - In the large artery (macrocirculation, d > 6 mm) case, by tion of bifurcation symmetry is primarily to simplify substituting the branch and trunk impedances of (Eq. 29) the equations as an assist in reader interpretation of the into Eq. (32), yields the relatively simple reflection coeffi - results. Although in the human body a relatively small cient Eq. (33): percentage of arterial junctions would normally be described as symmetric, in many junctions, particularly −5/4 1/4 2 SR AR − 1 [ ] in the microcirculation which possesses the majority of RC = (33) the vascular bifurcations, asymmetry is relatively minor. −5/4 1/4 2 SR[AR] + 1 The assumption of bifurcation symmetry in this analy - sis does not affect the validity of the physics principles where SR is the ratio of arterial stiffnesses, branch-to- affecting the cross-linkages between macrocirculation trunk, (which may also be called the stiffness gradient), as and microcirculation. References [15, 22] offer analyses defined by Eq. (24), and as given by Eq. (34): of the impact of asymmetric bifurcations on pressure and flow in the macrocirculation. 1/2 S E h 2 2 2 SR = = (34) S E h 1 1 1 Under the assumption that each branch artery is well matched at its distal end, the reflection coefficient “RC” In addition where AR is the branch-to-trunk luminal associated with the bifurcation is determined by the ratio cross-sectional area ratio as defined by Eq. (35): (or gradient) of characteristic impedances of the branch 2A 2r and trunk arteries, as given [15–20] by Eq. (32): AR = = (35) Z Z 2 2 − Z − 1 2 2Z RC = = (32) A plot of Eq. (33), the reflection coefficient for a large Z Z 2 2 + Z + 1 2 2Z artery bifurcation, as a function of the (Stiffness Ratio)/ 5/4 (Area Ratio) , is shown in Fig. 8. The macrocirculation’s where Z is the characteristic impedance of the trunk 1 bifurcation reflection coefficient plot of Fig. 8 shows that, artery and Z is the impedance at the bifurcation of one 2 for optimally matched bifurcations, there is negligible of the branch arteries. reflection under the specific condition that: Under the condition that the branch artery is not well Stiﬀness Ratio matched at its distal end, then the impedance at that dis- −1/4 5/4 tal end should be transformed along the length of the = 2 ∗ Area Ration (36) branch artery to the proximal (bifurcation) end. If the 5/4 = 0.841 ∗ Area Ratio . arteries are short in comparison to the inverse of the wave propagation’s attenuation coefficient and the impedance transformation is small. In this case the “effective” imped - ance [15, 44] of the branch artery approximates that of the T ucker Artery Research If a matched macrocirculation bifurcation stiffness gra - In the case of small arteries (d < 1 mm), the micro- dient (ratio) increases by 10%, in order that autonomic circulation case, the reflection coefficient assumes the flow regulatory processes maintain a minimum in the relatively simple form of Eq. (42): reflection coefficient, and thereby maintain homeostasis 3/4 −7/4 SR 2 [AR] − 1 in the branch arteries’ pressures and flows, the bifurca - RC = (42) tion’s area ratio must increase (vasodilate) by about 8%. −7/4 3/4 SR 2 AR + 1 [ ] In the macrocirculation, if a bifurcation’s area ratio is greater than about 1.15, then the stiffness ratio is greater A plot of Eq. (42), the reflection coefficient for a small than 1.0. for a well-matched bifurcation. In other words, artery bifurcation, as a function of the (Stiffness Ratio)/ for optimal match at a macrocirculation bifurcation, if 7/4 (Area Ratio) , is shown in Fig. 9. the area ratio is greater than about 1.15, then the branch artery is predicted to be stiffer than the trunk artery. The small artery (microcirculation) bifurcation reflec - For macrocirculation arteries the relationship tion coefficient plot of Fig. 9 shows that, for matched between arterial stiffness, as defined here, and the more microcirculation bifurcations, there is negligible reflec - readily measurable pulse wave velocity (PWV) of a pres- tion under the condition (Eq. 43) that: sure wave, in a very thin-walled artery (h < < r), is given by the Moens–Korteweg [12–17] Eq. (37): Stiﬀness Ratio −3/4 7/4 1/2 = 2 ∗ Area Ratio Eh (43) PWV = (37) 7/4 2rρ = 0.594 ∗ Area Ratio . For arteries in which the thin wall criteria of h < < r If the matched microcirculation bifurcation’s stiffness does not hold, the equation for pulse wave velocity ratio, increases by 10%, then for the autonomic flow reg - becomes  Eq. (38): ulatory processes to maintain a minimum in the reflec - tion coefficient and thereby maintain homeostasis in 1/2 Eh peripheral pulse pressure and flow, Eq. (43) requires that PWV = (38) 2rρ(1 − σ ) the area ratio must increase, or vasodilate, by about 6%. The circled area in Fig. 10, corresponds to the opti- where σ is Poisson’s ratio for the artery wall. mum conditions in which the stiffness ratio transitions The pulse wave velocity ratio (PWVR), or gradient, for from greater than unity in the macrocirculation to less a bifurcation is defined by Eq. (39): than unity in the microcirculation. In youth the aorta is PWV more compliant than its branch arteries. However, also in PWVR = (39) youth, in the smaller arteries, (i.e., the microcirculation) PWV the branch arteries must be more compliant than their Combining Eqs. (35, 37 and 39) results in the rela- trunk arteries to ensure the arterioles are substantially tionship between the stiffness ratio and the pulse wave more compliant than the more central arteries. The cir - velocity ratio as shown by Eq. (40): cled area is centered on a stiffness ratio of unity (SR = 1.0) SR PWVR = (40) 1/4 AR For large arteries, substituting (40) into (33) yields the bifurcation’s branch-to-trunk reflection coefficient as shown by Eq. (41): PWVR − 1 AR RC = (41) PWVR + 1 AR A plot of the reflection coefficient as a function of [Pulse Wave Velocity Ratio/Area Ratio], applicable to the macrocirculation, is provided in Fig. 2 in the body of the text. Fig. 9 Reflection Coefficient for a Small Artery (Microcirculation) 7/4 Bifurcation, as a Function of (Stiffness Ratio/(Area Ratio) Tucker Artery Research Fig. 10 a Optimum Stiffness Ratio as a function of Area Ratio for a minimum in bifurcation reflection coefficient for the two limiting cases of macrocirculation and microcirculation; b expanded plot of (a) with an optimum match in the centre of the mesocirculation region indicated 1/4 −5/4 1/4 −5/4 which corresponds to the arterial stiffness ratio in the 2 a SR AR − c + j 2 b SR AR − d ( ) ( ) RC = −5/4 −5/4 mesocirculation transition from stiffer branch arteries 1/4 1/4 2 a(SR)AR + c − j 2 b(SR)AR + d to more compliant branch arteries. The area ratio which (44) corresponds to a stiffness ratio of unity in the mesocir - where: culation is in the range of about AR = 1.26–1.31. The stiffness ratio in both the macrocirculation and microcir - 2 2 culation which corresponds to the optimum match is also 1 + + 1 AR∗α (45) approximately AR = 1.26. a ≡ √ The stiffness ratio in the macrocirculation which corre - sponds to AR = 1.26, is approximately SR = 1.12. In other 2 2 words, in the macrocirculation, for which arterial diam- 1 + − 1 eters are greater than about 6 mm, the arteries become AR∗α (46) stiffer (in youth) in progressing from the aorta into its b ≡ √ branch arteries. However, in the microcirculation the stiffness ratio which corresponds to AR = 1.26 is about SR = 0.89. In other 2 2 words, in the microcirculation in proceeding from genera- 1 + + 1 (47) tion to generation of cascaded bifurcations, the area ratio c ≡ increases while the stiffness ratio decreases. An area ratio of 1.26 corresponds to the diameter of each bifurcation branch being approximately 79% of that of its trunk artery. In other words, in the microcirculation, 1 + − 1 at each level, or generation, in a progression of cascaded α (48) d ≡ √ bifurcations, while the diameter of individual branch arter- ies reduces, the stiffness of those arteries simultaneously also reduces. Hence, in the microcirculation, the analysis Equations (44) through (48) provide a general solution predicts that, while the total cross-sectional area of the for pulsatile wave reflection at bifurcations, and is appli - arterioles and capillaries may be substantially greater than cable in all arterial segments, including the mesocircu- that of the central arteries, they may also, simultaneously, lation segment (i.e., for all combination of stiffness and be substantially more compliant. area ratios). The equation for the generalized reflection coefficient for Figure 11 shows plots of reflection coefficient for bifur - a symmetric bifurcation, applicable to all values of Wom- cations, (as a function of Area Ratio and Stiffness Ratio) ersley number (i.e., all arterial diameters), is given by Eqs. for four different values of arterial diameter (including: (44–48): (a) the macrocirculation; (b) and (c) the mesocirculation; and (d) the microcirculation) and for three different val - ues of branch-to-trunk stiffness ratio (SR = 1.1, SR = 1.0 T ucker Artery Research Fig. 11 Plots of Reflection Coefficient as a function of Area Ratio for four different values of trunk artery diameter (a α = 8, trunk diameter = 10 mm. b α = 4, trunk diameter = 5 mm. c α = 2, trunk diameter = 2.5 mm. d α = 0.8, trunk diameter = 1 mm. In each of the four graphs above there are three different plots of stiffness ratio shown: (SR = 1.1—solid line; SR = 1.0—short dashes; SR = 0.9—long dashes). The arrows indicate the point at which area ratio AR = 1.26 and SR = 0.9). The arrows in Fig. 11 indicate the stiffness Appendix B match corresponding to an area ratio of approximately Definitions of Macrocirculation, Microcirculation 1.26 for each of the four arterial diameters shown. and Mesocirculation Standardized definitions of macrocirculation and micro - The minimum in the reflection coefficient plots (cor - circulation are somewhat elusive . For the purposes responding to the optimum impedance match) predicted of this analysis, microcirculation arteries are defined for bifurcations in the mesocirculation, the transition as those whose diameters are less than 1000 microns region between the macrocirculation and the microcircu- (d < 1.0 mm). Arteries of the macrocirculation are defined lation, as indicated in Fig. 11c and d, lies typically in the as those whose diameters are greater than 6000 microns range of 4–6%. Compared with the minimum, or opti- (d > 6.0 mm). In addition, a transitional circulatory region mum, reflection coefficient predicted for bifurcations in between the macrocirculation and the microcirculation, both the microcirculation and macrocirculation (which described here as the mesocirculation, applies to arteries are near zero) the predicted (relatively large value of that are between 1.0 and 6.0 mm in diameter. reflection coefficient in the transition region) represents The reason for selecting these specific circulation a potentially significant contribution to the total reflec - boundary values is demonstrated in the plots of Fig. 12. tion coefficient associated with the extended arterial tree. With a major focus of this analysis of bifurcation reflec - tion coefficients, if the arterial diameter is less than about 1.0 mm, then the reflection coefficient plot is Tucker Artery Research Fig. 12 Plots of reflection coefficient, as a function of branch to trunk area ratios for six arterial diameters a branch to trunk stiffness ratio = 0.9 (i.e., the branch artery are softer than the trunk artery, and b branch to trunk stiffness ratio = 1.1 (i.e., the branch arteries are stiffer than the trunk) independent of the absolute value of arterial diameter femoral, brachial, and internal carotid arteries. An artery and the minimum (optimum) reflection coefficient is of 1 mm diameter, which represents the largest of the close to zero. For these small diameter microcircula- microcirculation arteries, is comparable to many small tion arteries, the impedance is dominated by viscous arteries such the ophthalmic artery. resistance. For the case in which branch arteries are stiffer than trunk arteries, as is the case in youth and good health, for central bifurcations (such as the macrocirculation’s On the other hand, if the arterial diameter is greater aortic/iliac bifurcation), with the plots of Fig. 12b repre- than about 6 mm the reflection coefficient plot is again sentative of branch arteries stiffer than trunk arteries (SR independent of the absolute value of the diameter, and the optimum (minimum) reflection coefficient is close = 1.1), the optimum area ratio is about 1.26. In the mac- to zero. In this large artery macrocirculation case, the rocirculation the minimum reflection coefficient is close viscous resistance is negligible, and the reflection coeffi to zero For the case in which the branch arteries are softer cient is dominated by the balance between the compliant than the trunk arteries, as occurs in the continuous sof- response of the arterial wall and the inertial response of tening of arteries in progressing through the cascaded the stroke (or mass) of blood in the artery. generations of bifurcations of the microcirculation, for a A macrocirculation artery of 20 mm diameter, (as plot- stiffness ratio of 0.9 the optimum area ratio is also about ted in Fig. 12a and b) is comparable to the abdominal 1.26 (see Fig. 12a). In the microcirculation the minimum aorta. An artery of 6 mm diameter, which represents the reflection coefficient is also close to zero. smallest macrocirculation artery and, also represents the beginning of the mesocirculation, is comparable to the Fig. 13 Plots of Reflection Coefficient as a function of branch to trunk stiffness ratio for various values of branch to trunk area ratio for two different trunk arterial diameters in the mesocirculation region (a trunk diameter = 3 mm, and b trunk diameter = 4 mm.) T ucker Artery Research In the mesocirculation (arterial diameters between 1.0 Z ≈ Z ∗ [1 + 2RC ] 2mis 2 2mis (50) and 6.0 mm) the minimum reflection coefficient, is finite, in the range of, typically 4–6%, (as shown in Fig. 13), as The impedance of each branch artery is, therefore, contrasted with the low values of minimum reflection increased by a factor of [1 + 2RC ] which affects the 2mis coefficient for optimally matched bifurcations of the reflection coefficient (see Eq. 15) at the trunk-to-branch macrocirculation and microcirculation. The optimum bifurcation as shown in Eq. (51): area ratio in the mesocirculation varies with arterial 2mis − 1 diameter. In the middle of the mesocirculation region, 2Z RC = (51) assuming equal arterial stiffnesses on either side of the 2mis + 1 2Z bifurcation (arrows in Fig. 13 at SR = 1.0) the optimum 1 area ratio is about 1.3, slightly greater than that in the Substituting Eq. (50) into (51) results in Eq. (52): microcirculation and macrocirculation. Of fundamental importance is that the minimum in reflection coefficient Z [1+2RC ] 2 2mis − 1 2Z for the bifurcations in the middle of the mesocirculation 1 RC = (52) Z [1+2RC ] region is not near zero, but rather about 4–6%, repre- 2 2mis + 1 2Z senting the most significant individual contributors to reflection in a wave’s propagation through cascaded gen - Under the assumption that the characteristic imped- erations of bifurcations. ances of the trunk and its branches are, themselves well matched (i.e., Z /2Z ≈ 1) then the value of the reflec - 2 1 tion coefficient for the trunk-to-branch bifurcation is Appendix C given by Eq. (53): Estimation of Optimum Aggregate Mesocirculation RC ≈ RC Reflection Coefficient 2mis (53) The minimum reflection coefficient for bifurcations in To a first order of approximation, therefore, the mis - the macrocirculation is near zero if the characteristic match at the distal end of a mesocirculation branch impedances on either side of the bifurcation are equal artery is transferred to mismatch the primary trunk-to- and if the load impedances at the distal end of each of branch bifurcation. the bifurcation’s branches is matched to the characteris- Figures 4 and 13 (Appendix B) indicate that near the tic impedance of each branch (i.e., if the branch is well centre of the mesocirculation, which corresponds to an matched to its sub-branches). If, however, the branch is arterial diameter of about 3 mm, the optimum reflec - mis-matched at its distal end, then that mismatch (with tion coefficient for that centre bifurcation of the meso - its reflection coefficient ) is transformed along the RC2mis circulation is in the range of 4–6%. With 8 bifurcations branch to its proximal end at the bifurcation. This sub- in the mesocirculation, to attain the maximum reflec - branch reflection transforms the branches’ impedances at tion coefficient of 4% in the centre of mesocirculation, the bifurcation, from the characteristic Z to a value of through about four generations of bifurcations, implies Z as given [65, 66, 76] by Eq. (49): 2mis each successive bifurcation increments the reflection by about 1%. With each branch in the mesocircula- Z = Z ∗ [1 + RC ]/[1 − RC ] 2(mis) 2 2mis 2mis (49) tion sequentially mismatched by increments of 1%, the where Eq. (49) assumes that the length of the branch is total mismatch, corresponding to minimum aggregate sufficiently short that the wave is not appreciably attenu - reflection coefficient, is estimated to be about 21.7% (1. ated in transit along the branch’s length. The attenua - 01*1.02*1.03*1.04*1.04*1.03*1.02*1.01 = 1.217). Hence, tion coefficient of each artery is a function of the artery’s even in youth, with assumed optimum arterial imped- diameter. Reported  measured values of attenua- ance matching, there is predicted to be finite and sig - tion coefficient (σ) for various arteries are as follows: nificant wave reflection in the macrocirculation caused −1 −1 −1 σ = 0.5 m , σ = 1.0 m , σ = 1.7 m , AbdominalAorta Iliac Femoral by essential mismatches in the mesocirculation. −1 and σ = 1.2 m . For artery lengths that are small Carotid This analysis indicates that an optimally designed relative to the reciprocal of the attenuation coefficient, mesocirculation bifurcation presents an inherent the assumption is valid. impedance mismatch with finite wave reflection into In the center of the mesocirculation region the opti- the macrocirculation, and with an attendant central mum (minimum) value of RC is about 4–6% which 2mis pulse pressure increase. implies that Z can be approximated by Eq. (50): 2mis Tucker Artery Research Author contributions 14. Caro CG, Pedley TJ, Schroter RC, Seed WA (2012) The mechanics A single author only. of the circulation, 2nd edn. Cambridge University Press, UK. ISBN 978-0-521-15177-1 Funding 15. Zamir M (2016) Hemo-dynamics. Springer International, Switzerland. ISBN Self funded. No outside funding to declare. 978-3-319-24101-2. https:// doi. org/ 10. 1007/ 978-3- 319- 24103-6 16. Westerhof N, Stergiopulos N, Noble M, Westerhof B (2019) Snapshots of Availability of data and materials hemodynamics—an aid for clinical research and graduate education, 3rd Analysis software available on request. edn. Springer Nature, Switzerland. ISBN 978-3-319-91931-7. https:// doi. org/ 10. 1007/ 978-3- 319- 91932-4 17. Chirinos JA (2022) Textbook of arterial stiffness and pulsatile hemody- Declarations namics in health and disease. Editor, Academic Press, London. ISBN: Conflict of interest 18. Segers P, Chirinos JA (2022) Essential principles of pulsatile pressure-flow No competing or conflicting interest exist. relations in the arterial tree. In: Chirinos JA (ed) Textbook of arterial stiff- ness and pulsatile hemodynamics in health and disease, vol 1, Chapter 3. Ethics approval and consent to participate Academic Press, London. ISBN: 9780323913911 None required. 19. Tucker T. Arterial stiffness as a vascular contribution to cognitive impairment: a fluid dynamics perspective. 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Artery Research – Springer Journals
Published: Jun 1, 2023
Keywords: Arterial bifurcation; Impedance; Stiffness gradient; Area ratio; Pulse wave velocity; Reflection coefficient; Target organ ischemia; Hypertension; Womersley; Murray’s Law
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