## Abstract

Congenital heart disease with increased blood flow commonly leads to the development of increased pulmonary vascular reactivity and pulmonary arterial hypertension by mechanisms that remain unclear. We hypothesized a shear stress paradigm of hemodynamic reactivity and network remodeling via the persistence and/or exacerbation of a fetal diameter bifurcation phenotype [parent diameter *d*_{0} and daughters *d*_{1} ≥ *d*_{2} with α < 2 in (*d*_{1}/*d*_{0})^{α} + (*d*_{2}/*d*_{0})^{α} and area ratio β < 1 in β = (*d*_{1}^{2}+ *d*_{2}^{2})/ *d*_{0}^{2}] that mechanically acts as a high resistance magnifier/shear stress amplifier to blood flow. Evidence of a hemodynamic influence on network remodeling was assessed with a lamb model of high-flow-induced secondary pulmonary hypertension in which an aortopulmonary graft was surgically placed in one twin in utero (Shunt twin) but not in the other (Control twin). Eight weeks after birth arterial casts were made of the left pulmonary arterial circulation. Bifurcation diameter measurements down to 0.010 mm in the Shunt and Control twins were then compared with those of an unoperated fetal cast. Network organization, cumulative resistance, and pressure/shear stress distributions were evaluated via a fractal model whose dimension *D*_{0} ≈ α delineates hemodynamic reactivity. Fetus and Control twin *D*_{0} differed: fetus *D*_{0} = 1.72, a high-resistance/shear stress amplifying condition; control twin *D*_{0} = 2.02, an area-preserving transport configuration. The Shunt twin (*D*_{0} = 1.72) maintained a fetal design but paradoxically remodeled diameter geometry to decrease cumulative resistance relative to the Control twin. Our results indicate that fetal/neonatal pulmonary hemodynamic reactivity remodels in response to shear stress, but the response to elevated blood flow and pulmonary hypertension involves the persistence and exacerbation of a fetal diameter bifurcation phenotype that facilitates endothelial dysfunction/injury.

- pulmonary arterial morphometry
- branching complexity
- fractals

the pulmonary circulation undergoes a dramatic hemodynamic change at birth, from a state of high pressure/low flow to one of low pressure/high flow (50). This change occurs in conjunction with a decrease in pulmonary arterial resistance (8) and a transformation in the complexity of arterial diameter network organization (3). However, in cases of congenital heart disease with a systemic-to-pulmonary communication, an abnormally increased flow is superimposed on the usual decrease in resistance and commonly leads to development of pulmonary hypertension with functional and structural complications (25). While the mechanisms are not completely understood, the present consensus is that endothelial injury and dysfunction increase vascular tone and reactivity, leading to an increase in pulmonary artery pressure that is initially considered reversible (20). Pulmonary hypertension becomes problematic when vascular wall remodeling leads to the appearance of smooth muscle and alterations of medial wall thickness that encroach upon the lumen diameter (20). Since the forces of pressure and shear stress are known to be transduced and translated into molecular mechanisms that are instrumental in regulating diameter, vascular tone, and the remodeling of the vascular wall (11), further understanding of the factors aggravating these forces within the pulmonary arterial circulation should aid in understanding patterns of reactivity and the demarcation of reversibility in the disease process.

In general, mammalian vascular networks are thought to possess a universal fractal pattern of diameter network organization that appears independent of organ system, body size, and hemodynamic state (32, 33, 55–57). However, the fetal pulmonary circulation is extraordinary. Fetal arterial bifurcations appear to possess a hemodynamic phenotype that is distinguished by an area ratio <1 (3), not >1 as typically found in the adult pulmonary circulation (28). This curious vessel diameter topology has a design that mechanically aggravates hemodynamic reactivity per unit blood flow, leading to an elevated driving pressure, a magnified resistance, and the concomitant amplification of shear stress in the course of branching (3). Although intended to be rapidly adapting to an adult pattern via perturbations in blood flow and shear stress in the fetal-neonatal transition after birth, the shear stress-amplifying characteristics of the fetal network design may be precarious to the pulmonary circulation in the presence of an elevated blood flow. Shear stress is a force known to influence the remodeling of vascular diameter and/or vascular wall properties in response to transient and chronic changes in blood flow (35). However, acute elevations beyond critical levels can induce endothelial dysfunction and/or injury (19, 35, 54). Thus the interaction of the fetal phenotype with amplified shear stress secondary to elevated blood flow, or any other source of fetal induced endothelial injury and/or pattern of dysfunction (7), may cause the fetal phenotype to persist or become exacerbated in the presence of elevated blood flow after birth (25).

While the changes in vascular diameter geometry and vascular wall morphology of the pulmonary circulation are well understood, the structural remodeling patterns of arterial network organization in response to the hemodynamic forces of pressure and shear stress are not. We hypothesized that hemodynamic reactivity and network remodeling via the persistence and/or exacerbation of a fetal diameter bifurcation phenotype might mechanically act as a high resistance magnifier/shear stress amplifier to blood flow. Therefore, one purpose of this study was to test the hypothesis that a fetal phenotype of arterial branching persists and/or is exacerbated after birth. We utilized a well-established lamb model of congenital heart disease with increased pulmonary blood flow, in which an aortopulmonary graft is surgically placed in one twin in utero (Shunt twin) but not in the other twin (Control twin) (40, 44). Diameter measurements in lung casts were made to characterize longitudinal branching patterns and design of bifurcations from the left pulmonary artery (LPA) in a Shunt twin and its Control twin, relative to an unoperated fetal lung (3). In addition, as the hemodynamic forces of pressure and shear stress are linked to multiple mechanisms of vascular diameter and wall remodeling, our second objective was to evaluate the fractal geometry, topology, and design of the fetal, Shunt, and Control pulmonary arterial networks as a means of predicting the distribution of pressure and shear stress, and to determine whether or not their respective fractal dimensions conform to a universal design independent of hemodynamic forces.

## METHODS

#### Cast preparation.

The respective studies were approved by animal use protocols issued by the University of California Davis (3) and the University of California San Francisco (40). Lung casts of 8-wk-old twin lambs and a term fetus (140 days gestation, 145 days term) were made from experimental animals originating from previous studies (3, 40). In the aortopulmonary shunt study, mixed-breed pregnant Western ewes with twin lambs (between 137 and 141 days gestation, term = 145 days) were operated on under sterile conditions (44). One twin had an aortopulmonary shunt surgically implanted as previously described (40) (Shunt twin). The other lamb was not operated on (Control twin). As previously described (40), after spontaneous birth the lambs were kept with their mothers and weighed daily, and respiratory rate and heart rate were measured. In Shunt lambs furosemide (1 mg/kg im) was administered daily and elemental iron (50 mg im) was given weekly. Eight weeks after delivery the lambs were euthanized with an intravenous injection of pentobarbital sodium (Euthanasia CII; Central City Medical, Union City, CA) and subjected to bilateral thoracotomy. In one set of twins, lung casts were made of the left pulmonary arterial circulation, while the right pulmonary arterial circulation was harvested for tissue. Lung casts were made as follows: a 4.5-Fr cannula was inserted into the proximal LPA. The airways were expanded with saline under a hydrostatic pressure gradient of 20 cmH_{2}O via a tracheal cannula. The pulmonary arterial circulation was washed free of blood by saline perfusion. To remain consistent with the morphometric diameter measurement methods utilizing gelatin-barium-filled arteries, a high-viscosity methyl methacrylate plastic (Coe Tray Plastic, GC America, Chicago, IL) was injected into the pulmonary arterial system slowly with a syringe over a 2-min period under a maximum pressure of 75 mmHg. The plastic was allowed to polymerize overnight, whereupon the lung tissue was macerated in a 20% KOH bath for 3–5 days. The remaining tissue was washed away gently with distilled water, and the cast was allowed to dry before bifurcation diameter measurements.

#### Diameter measurement.

A large number of diameters within bifurcations of the left pulmonary arterial tree were sampled according to previous methods (3). In the Shunt and Control twins, the samples consisted of diameters of bifurcations along the LPA branch, plus an extensive sampling of bifurcations within the upper cranial lobe. Bifurcation diameters were measured with a methodology described previously (3). Briefly, a video microscope system (Infinivar Video Microscope with Zoom, Infinity Photo-optical, Boulder, CO) connected to a Cohu model 2600 solid-state camera (Cohu, San Diego, CA) and to a Macintosh computer (Cupertino, CA) was used to acquire and analyze images via a program (Object Image) developed by Norbert Vischer of the Faculty of Science, University of Amsterdam, Amsterdam, The Netherlands (http://simon.bio.uva.nl). Object Image is an extended version of the program NIH Image (http://rsb.info.nih.gov/nih-image), which allows diameter measurements and derived calculations to be recorded into an extensive database. To measure diameters, the three-dimensional branching aspect of each cast was broken into pieces to facilitate placement of bifurcations onto a plane. Assuming vessels were circular, vessel diameter was calculated as the average of two diameter measurements per vessel segment. The three averaged diameter measurements, consisting of a parent vessel, *d*_{0}, and two daughter vessels, *d*_{1} and *d*_{2}, where *d*_{0} > *d*_{1} ≥ *d*_{2}, were stored in a database and used to make functional morphometric calculations within bifurcations.

#### Functional morphometric relationships.

The measurements *d*_{0}, *d*_{1}, and *d*_{2} were utilized to calculate functional morphometric parameters that characterize local hemodynamic reactivity, via diameter geometry, topology, and design of bifurcations within the pulmonary arterial tree (appendix; Ref. 3). Diameter branching geometry is defined by the diameter ratio (1) along with the asymmetry ratio γ = *d*_{2}/*d*_{1} ≤ 1. The area ratio (2) designates the topology between parent and daughter diameters in terms of the relative cross-sectional area available for perfusion. The design parameter α was iteratively solved (28) via the equation (3) The values of α and/or β have hemodynamic significance conferring reactivity to bifurcation network organization (Fig. 1; appendix): a high-resistance/shear stress-amplifying fetal design (3) occurs when α < 2 and/or β < 1; a lower resistance transport design (57) is evident when α ≥ 2 and/or β ≥ 1.

#### Diameter ordering/analysis of arterial tree.

We categorized the parent diameters (*d*_{0}) of bifurcations into orders within the arterial tree via a diameter ranking method that calculates an average diameter per order, *d̄*_{k}, as previously described (3). The diameter data set for each bifurcation (*d*_{0}, *d*_{1}, *d*_{2}, α, and β) was first ranked from the largest parent diameter *d̄*_{0} (designated order 0) down to the smallest diameter, by orders *k* = 0, 1… *n* categorized by the overall average diameter ratio R̄_{d} according to R̄_{d}^{−k} ≥ (*d*_{i}/*d̄*_{0}) > R̄_{d}^{−(k+1)}. Here, *d*_{i} is a parent diameter of a bifurcation in the sorted list, *d*_{0} is the diameter of the largest (main or left) pulmonary artery, and R̄_{d} represents the average diameter ratio, computed from the pool of R_{d} calculated from bifurcations. Once orders were assigned to each parent diameter, the average values of diameter *d̄*_{k}, logᾱ_{k}, and logβ̄_{k} of each order in the arterial tree were calculated. The ranking procedure results in a nonoverlapping range of average parent diameters for each vessel diameter order, where each order is assumed to be statistically independent from another but where the log-transformed values of α_{k} and β_{k} are assumed to possess a statistically self-similar Gaussian distribution with bifurcation level independent of the average value of α or β (3).

#### Fractal model.

The average values of *d̄*_{k} and ᾱ_{k} for parent diameters within orders, from the largest diameter artery *d̄*_{0} down to terminal arteries ∼0.010 mm, were used to synthesize an equivalent homogeneous fractal branching tree to summarize and compare the hemodynamic reactive properties of arterial network organization (appendix). The fractal properties of the arterial tree are summarized by global ratios: a network branching ratio R̂_{b}, a network diameter ratio R̂_{d}, and a fractal dimension *D*_{0}, derived from the distribution of average *d̄*_{k} and ᾱ_{k} in the following way. To determine the number of vessels with each order of branching, a local branching ratio for each order was determined by (4) where R_{dk} is the local diameter ratio between orders derived from (5) and ᾱ_{k} is the average value of α for the order *k*. The number of vessels per order was calculated according to *N*_{d} = *N*_{0}R_{b1}R_{b2}… R_{bd}. The global network branching ratio R̂_{b} was determined from the numbers of vessels per order via *N*_{k} = *N*_{0}R̂_{b}^{k}, computed via the slope of the linear regression relationship between vessel number *N* and order (6) where R̂_{b} is the antilog of the slope and the intercept was comprised of the main parent *N*_{0} = 1. The global network diameter ratio R̂_{d}, determining the decrement in average diameter with order *d̄*_{k} = *d̄*_{0}R̂_{d}^{−k} and representative of an average of all local diameter ratios R_{dk} between orders (*Eq. 5*), was evaluated by linear regression of the logarithm of average diameter *d̄*_{k} vs. order *k*, (7) where R̂_{d} is the negative of the antilog of the regression slope. The fractal dimension of the model, *D*_{0}, designating the design of the arterial tree, was then evaluated from the power law relationship between vessel number *N*_{k} = *N*_{0}R̂_{d}^{−kD0} and average diameter *d̄*_{k} = *d̄*_{0}R̂_{d}^{−k} via the slope of the linear regression of (8) where the fractal dimension *D*_{0} is equal to the negative of the regression slope. The values of the fractal dimension *D*_{0}, the network diameter ratio R̂_{d}, and the network branching ratio R̂_{b} in the fetal, Control twin, and Shunt twin lung casts were then used to calculate their respective cumulative resistance and shear stress distribution (appendix; Fig. 1). The cumulative resistance (*R*_{cum}) for each order *k* (0–*k*+1) down to vessels ∼0.020 mm was calculated according to (9)

The hydraulic resistance of the LPA to blood flow was computed via Poiseuille's law, (10) where μ is blood viscosity (assumed to be 3.0 cP), *l*_{0} is vessel length, and *d̄*_{0} is the LPA vessel diameter. *Equation 9* assumes the presence of a length-diameter scaling relationship *l̄*_{k} = *a**d̄*_{k}^{α2} where *a* and α_{2} are constants (30). We assumed that α_{2} = 1 (30) and that *a* = 1.7, where the constant *a* was evaluated on the basis of human pulmonary data (52) ordered according to the Horsfield method (27). The rectilinear pressure response to an increase in flow (13) was calculated according to (11) where *q* is pulmonary arterial flow in the parent vessel normalized to body weight and p_{int} is the intercept pressure, set to 25 mmHg in the fetal state (46) and 6 mmHg in the postnatal state. The associated pressure distribution in the fractal continuum equivalent tree at a steady-state flow *q*_{ss} was evaluated according to (12) The distribution of shear stress with branching order *k* was calculated according to (13) where (14) and *q*_{ss} is the absolute steady-state blood flow (ml/min) with *d̄*_{0} the diameter of the LPA. Steady-state values for flow *q* used in the calculations were taken from previously published studies in which LPA flow was measured: 139-day fetal lambs/145 days term (with average body wt 5.0 kg), where *q* = 9.5 ml·min^{−1}·kg^{−1} or *q* = 38 ml/min (46); 8-wk-old Control twins (average body wt 22.6 kg), where *q* = 33 ml·min^{−1}·kg^{−1} or 743 ml/min; and 8-wk-old Shunt twins (Ref. 40; average body wt 18.9 kg), where shunt open *q* =115 ml·min^{−1}·kg^{−1} or 2,200 ml/min. The Reynolds number per order *k* associated with a given flow to the network was calculated according to Re(*k*) = Re(0)R_{d}^{(1−D0)k}, where Re(0) = *q*ρ/μ*d*_{0}, with the density of blood ρ = 1.05 g/cm and blood viscosity μ = 0.03 g·cm^{−1}·s^{−1} (appendix).

#### Statistical analysis.

The distributions of α and β exhibit log-normal behavior; therefore, a two-way ANOVA was performed on log-transformed values of α and β to evaluate differences between the fetal, Shunt twin, and Control twin data. One factor was the experimental group (Control twin, Shunt twin, fetus), and the other factor was branching order *k*. Our null hypothesis was that the mean values of α and β, over all orders, are equal to a transport design (α = 2 and β = 1), considered to be a universal theoretical pattern of arterial vascular network organization (57). Results of the analysis of the log-transformed values of α and β are reported as means ± 95% confidence intervals (CI) (58), expressed in terms of their geometric mean adjusted values (37). The regression slopes *D*_{0} of *Eq. 8* derived from the lung casts are also reported in terms of 95% CI. Significant differences comparing branching order design, topology, and group fractal dimensions were distinguished by nonoverlapping CI, indicative of a *P* > 0.05 level of significance. Statistical tests were performed with JMP 6.0.

## RESULTS

A total of *n* = 46,673 bifurcation diameter measurements for α and β were evaluated in the three lung casts (Control twin *n* = 16,443 bifurcations, Shunt twin *n* = 19,816 bifurcations, and fetus *n* = 10,414 bifurcations), where 1 bifurcation = 1 parent diameter connected to 2 daughter diameters.

#### Distributions of arterial design (α) and topology (β) with diameter and daughter asymmetry (γ).

ANOVA revealed significant differences in the averages between experimental groups and within orders for (ᾱ, β̄) at a *P* < 0.0001. In the fetal arterial cast ᾱ =1.736 (CI 1.66–1.815) and β̄ = 0.846 (CI 0.816–0.878); in the Control twin cast ᾱ =1.94 (CI 1.88–2.00) and β̄ = 0.934 (CI 0.911–0.957); and in the Shunt twin cast ᾱ = 1.671 (CI 1.60–1.744) and β̄ = 0.850 (CI 0.821–0.881). These results indicate that, relative to the fetal cast, the control cast altered its bifurcation design and topology toward a West-Brown-Enquist design. However, the Shunt twin demonstrated the persistence of a fetal design and topology.

Figure 2 compares conventional histologically derived indices of vascular-diameter reactivity [percentage of smooth muscle and medial thickness-diameter ratio measured previously in this experimental model (40)] relative to indices of hemodynamic reactivity, average ᾱ and β̄, relative to average diameter and branching order. The fetal-reactive regions (β < 1, α < 2) are colored gray in Fig. 2, while the white area (β ≥ 1, α ≥ 2) indicates a hemodynamically low-reactive adult transport criterion of bifurcation branching. Statistically significant differences of ᾱ and β̄ between groups within orders according to ANOVA are earmarked within the Shunt twin markers as a white circle within a black circle in Fig. 2. Significant differences between any two groups within an order are characterized by nonoverlapping CI. Overlapping CI were observed for orders 0–6 in the Control twin and Shunt twin and orders 0–2 in the fetus, and they are not shown in Fig. 2. Figure 2 illustrates that the fetal pulmonary circulation exhibits a regional mechanical source of hemodynamic reactivity in intra-acinar arteries [0.020-mm to 0.100-mm parent inner diameter (ID)], a region we previously showed to be devoid of smooth muscle(40). The Shunt twin lamb demonstrated a similar fetuslike design profile of α < 2 and β < 1 in preacinar arteries >0.150 mm. In addition, intra-acinar Shunt twin arteries (0.020- to 0.040-mm ID), which we previously showed to demonstrate the appearance of increased smooth muscle and medial wall thickness (40), were associated with significant differences in hemodynamic reactivity (smaller α and β) relative to both the Control twin and the fetus.

The fetal cast demonstrated significantly different average values of diameter asymmetry γ̄ relative to the Control twin and the Shunt twin: Fetus 0.62 (SD 0.20), Control twin 0.71 (SD 0.18); Shunt twin 0.70 (SD 0.19). Figure 3 shows the interrelationship between area ratio β, bifurcation design exponent α, and diameter asymmetry γ as a bivariate frequency distribution. The overall range and spread of β between the casts are similar and substantial. The fetal lung cast bifurcations show a predominant frequency toward acute asymmetry (γ ≈ 0.5), with area ratios favoring 0.8 < β < 1 rather than β > 1. In the Control twin and Shunt twin casts, the greatest frequency of asymmetry occurred in a higher range, 0.6 < γ < 0.8. The area ratios of the Control twin demonstrate their greatest frequencies about 0.8 < β < 1.2, while the area ratios in the Shunt twin included relatively larger numbers of bifurcations 0.6 < β < 0.8.

#### Fractal power law properties.

The power law behavior synthesized via *Eqs. 4*–*7* for the fetal, Control twin, and Shunt twin lung casts is shown in Fig. 4.

In Fig. 4*A*, the diameter ratio (R̂_{d}) of the Shunt twin was significantly different from that of the fetal and Control twin casts: Shunt twin R̂_{d} = 1.41 ± 0.015 (CI), fetus R̂_{d} = 1.30 ± 0.003 (CI), and Control twin R̂_{d} = 1.31 ± 0.001 (CI). While the difference between the fetus and the Control twin was statistically significant, it was small. The relatively small difference between the fetus and Control twin cast R̂_{d} values, plus the parallel lines in average diameter with order in Fig. 4*A*, indicate that growth and development scaled diameters upward with branching order. However, in the Shunt twin the average arterial diameters for orders 0–8 were larger relative to the Control twin or fetus cast, while orders 9–10 of the Shunt and Control twins were similar. Diameters of vessel orders 11 and larger were smaller in the Shunt twin compared with the Control twin. Alternatively, the fractal regression indicates that the intersection between Control twin and Shunt twin occurs at order 11, at a diameter of 0.5 mm. Thus the differences between the Shunt twin and other casts indicate that the Shunt twin remodeled and rescaled diameter geometry about a diameter pivot region of 0.5–1.0 mm, leading to smaller resistances in larger arteries and elevated resistances in smaller arteries.

In Fig. 4*B*, the branching ratios R̂_{b} and their respective 95% CI were statistically different from one another: fetus R̂_{b} = 1.572 ± 0.03 (CI), Shunt twin R̂_{b} = 1.79 ± 0.03 (CI), and Control twin R̂_{b} = 1.73 ± 0.002 (CI).

In Fig. 4*C*, the fractal dimension ± CI of the fetus was *D*_{0} = 1.72 ± 0.069, Shunt twin *D*_{0} =1.70 ± 0.038, and Control twin *D*_{0} = 2.02 ± 0.021. Thus the Shunt twin was not statistically different from the fetus, but both were both significantly different from the Control twin, indicating the persistence of a fetal fractal network design in the arterial tree at 8 wk of shunting.

Together these results indicate that the influence of a chronic shunt interrupts an apparent transition in arterial network organization at birth: the fetal-Control transition maintained the scaling of arterial diameter geometry uniformly increasing diameters but remodeled arterial design and topology to an adult transport design. Shunting remodels diameter geometry by rescaling large and small diameters to where large arterial vessels increase diameter to decrease resistance and small arterial vessels decrease diameter to increase resistance, while leaving the topology and fractal design of the pulmonary circulation in a hyperreactive fetuslike configuration.

#### Cumulative resistance distribution.

Figure 5 illustrates the influence of fractal network organization in the fetal, Control twin, and Shunt twin casts on the cumulative resistance distribution versus diameter, as calculated from the regression values of R̂_{d} and *D*_{0} according to *Eqs. 9* and *10*. In general, the smaller fractal dimension (*D*_{0} = 1.7) of the fetus and the Shunt twin demonstrates greater resistance reactivity with arterial branching, evidenced by a larger slope of cumulative resistance with decreasing diameter. However, the Shunt twin demonstrates an absolute resistance that is less than the fetus and the Control twin: the remodeling of arterial geometry (*R*_{0}, R̂_{b}) counteracts the increased hemodynamic reactivity of a smaller fractal dimension by decreasing the absolute resistance per order, leaving a smaller cumulative arterial resistance.

#### Pressure-flow curves/pressure distribution.

Figure 6 illustrates the influence of *R*_{cum} on the predicted pressure-flow curves (Fig. 6*A*) of the LPA circulation in the fetal, Control twin, and Shunt twin fractal configurations and (Fig. 6*B*) the corresponding distribution of pressure versus diameter at a given steady states of flow. The fetal state is the most pressure reactive, yielding hypertensive pressures in all arterial regions over a small flow range. Fractal network organization in the Shunt twin configuration is the least pressure reactive to flow of the three states, with hypertension arising secondary to the increased steady-state blood flow. As Fig. 6 indicates, the Control twin state also predicts pulmonary hypertension at a flow comparable to the Shunt twin, but with higher driving pressures and higher values in vessels >0.060 mm. Conversely, arterial network remodeling in the shunt condition predicts that shunt closure will result in nonhypertensive pressures with smaller pulmonary vascular resistances than the corresponding Control twin.

#### Shear stress distribution.

Figure 7 illustrates the influence of diameter network organization on the amplification of shear stress at steady-state blood flows in the fractal model. The calculated values for τ_{0} shear stress and Reynolds numbers in the LPA for the different states were as follows: fetus, LPA *d*_{0} = 5.4 mm at a flow of *q* = 38 ml/min yields τ_{0} = 1.25 dyn-cm^{−2}, Re(0) = 82; Shunt twin with the shunt-open condition, LPA *d*_{0} = 21.2 mm and flow *q*_{0} = 2,200 ml/min yields τ_{0} = 1.16 dyn-cm^{−2} at Re(0) = 774; Control twin, LPA *d*_{0} = 11.5 mm and *q*_{0} =743 ml/min yields τ_{0} = 2.51 dyn-cm^{−2}, Re(0) = 486. The common feature in all groups is that extremely large shear stress avalanches are developed with low Reynolds numbers (Re < 10) in arteries of 0.020 mm. For comparison purposes, the gray area in Fig. 7 represents the reported ranges of critical and/or erosion yield shear stresses (165–8,000 dyn-cm^{−2}) of endothelial dysfunction/damage to an acute increase in shear stress as it occurs in large-diameter conduit vessels over a range of Reynolds numbers (0.21–2,300) (19, 35, 54). Interestingly, the fetal fractal model (τ_{0.020} = 1,200 dyn-cm^{−2} at Re_{0.020} = 1.4) and Control twin model (τ_{0.020} = 1,100 dyn-cm^{−2} at Re_{0.020} = 0.43) demonstrated similar magnitudes of shear stress experienced in arterial vessels of 0.020-mm ID as derived from their respective steady-state blood flows. However, the Shunt twin steady-state values of blood flow yielded a magnitude of shear stress at τ_{0.010} =7,100 dyn-cm^{−2} at Re_{0.020} = 2.6, which extends into the upper region of thresholds reported to induce endothelial injury.

## DISCUSSION

The present study is the first to demonstrate the existence of an unusual fetuslike pattern of arterial diameter network organization after birth in an experimental model of pulmonary hypertension with increased pulmonary blood flow. Persistence and/or exacerbation of the fetal pattern after birth suggests a mechanical basis of hemodynamic reactivity that may be linked to the origin of the disease and implicates a shear stress paradigm of structural remodeling in the pathology and progression of pulmonary hypertension in high-flow lesions. Although this study is limited to a morphometric analysis based on a sample of three lung casts only and not a population, our results indicate that a fetal phenotype may be a major factor that influences the state of structural remodeling in intra-acinar vessels when viewed in the context of classic grades of vascular remodeling determined by vessel geometry alone and is involved in two divergent patterns of adaptation in arterial network organization in a model of neonatal congenital heart disease with increased pulmonary blood flow.

Previous work regarding pulmonary vascular abnormalities in patients with pulmonary hypertension has focused primarily on histological appearance of the arterial wall. In 1958, Heath and Edwards (24a) described a classification sequence that ranged from reversible medial hypertrophy to terminal changes such as angiomatoid formation and fibrinoid necrosis. Rabinovitch et al. in 1978 (42a) described a modification that featured alterations in normal remodeling and growth in three progressive stages (A–C), characterized by progressive medial hypertrophy, abnormal muscle extension, and reduced artery size and concentration, that have been correlated with perioperative hemodynamics. The histological pattern of vascular remodeling in the Shunt twin at 8 wk of age occurring in this experimental model has been previously characterized as grade B alterations, with medial hypertrophy and abnormal appearance of muscle in the walls of intra-acinar arteries (40). A fundamental limitation of histologically derived indices of vascular reactivity is that it is not clear whether their impact on arterial diameter comprises a mechanical cause of pulmonary hypertension or represents an adaptation in response to it (47). Alternatively, our study was concerned with further characterization of the high-flow secondary hypertension experimental model with specific regard to the influence of elevated flow complicating the structural remodeling of a preexisting fetal phenotype of arterial diameter network organization to a postnatal adult pattern in the Shunt twin, but not in the Control twin. In this study we show that a fetal phenotypic source of hemodynamic reactivity (α < 2, β < 1) is evident in arterial diameters of fetal and Shunt twin intra-acinar arteries. We note that fetal intra-acinar arteries are typically devoid of smooth muscle (2, 41), in which case the fetal phenotype represents a preexisting mechanical source and state of aggravated hemodynamic reactivity contributing to the high-resistance, hypertensive condition normally seen in the late-term fetal pulmonary circulation (50). Consequently, as Fig. 1 indicates, we would expect that the structural remodeling of the fetal phenotype to smaller area ratios would be of greater hemodynamic significance to the progression and timing of hypertension than that expected from the structural remodeling and encroachment of vessel diameter alone fixed at a given area ratio (47). Moreover, because the unusual fetal topological pattern between diameters represents a source of hemodynamic reactivity not considered in the classic vascular remodeling perspective, it is likely that changes in the topological form of hemodynamic reactivity in the fetal-neonatal state may precede the timing and appearance of more complex lesions after birth (43). Indeed, preferential patterns of structural remodeling of bifurcation topology and design in relationship to the appearance of smooth muscle and wall thickness are seen in our data in two ways. First, as seen in the 8-wk Control twin lung cast (Fig. 3), the appearance of a normal pattern of smooth muscle is related to the timing of significant increases in the area ratio. In this case, the migration of α and β toward values of α ≈ 2 and β ≈ 1, and to the fractal dimension of *D*_{0} ≈ 2, indicates that the degree of hemodynamic reactivity is significantly diminished relative to that of the fetus. Under these circumstances, the structural and functional impact of the residual fetal configuration in the Control twin on overall hemodynamic reactivity at this point may be comparatively minimal relative to the emerging influence of smooth muscle on vascular reactivity and tone. Second, Fig. 3 also illustrates that the abnormally increased muscle mass in Shunt twin intra-acinar vessels (0.020- to 0.040-mm ID) relative to the Control twin is associated with diameters of parent/daughter vessels with significantly smaller α and β, relative to both the fetal and Control twin lungs. Under these circumstances, the increased muscle mass and the aggravated hemodynamic reactive design and topology indicate a preferential action of smooth muscle vascular remodeling that can exacerbate the fetal phenotype in the course of hypertension disease. Thus the regional pulmonary patterns of α and β we observed in this study emphasize that there is a preexisting mechanical source of hemodynamic reactivity and remodeling in the fetal pulmonary circulation that extends beyond the apparently latent remodeling patterns of vascular wall properties at a vessel level of organization.

The conceptual advantage of the fractal model is that it simplifies the complexity of arterial network organization into a simple static parametric characterization of arterial geometry (R_{d}), topology (area ratio β), and design (fractal dimension *D*_{0}), whose impact on hemodynamics can be inferred from an equivalent symmetric bifurcation-branching arterial network (24, 30). The fractal dimension and/or area ratio have structural and functional significance in that their values define the global hemodynamic reactivity of the arterial tree, along with the pattern of transformation of pressure and shear stress with arterial branching (Fig. 1; appendix) (4, 30). Our morphometric results demonstrated differences in the fractal dimension between the fetus (*D*_{0} = 1.72) and the Control twin (*D*_{0} = 2.02), suggesting a transformation in fractal arterial network organization at birth; furthermore, our results indicate that this transformation did not take place in the Shunt twin (*D*_{0} = 1.70). However, we emphasize that our study is limited in the sense that it is based on the results of three lung casts only, which are isolated samples of a term lamb fetus and a postnatal state at 8 wk of birth. Unfortunately, the limited number of lung casts we studied does not identify the exact timing of when the adult bifurcation transport design is achieved after birth. Also, the limited number of lung casts we utilized may not necessarily reflect the developmental dynamics of change in network properties representative of the population of human subjects or other species (39) experiencing the same hemodynamic conditions. Here, genotypic variation and heterogeneity in patterns of vascular wall remodeling may exist within a population of individuals and between species that also influences the distribution and reactivity of the fetal phenotype of arterial branching and its responsiveness to hemodynamic forces for arterial network structural remodeling (53). In addition, at a given developmental time period lung casts are static pictures of network organization that neglect the dynamic information about the influence of vasomotor activity on bifurcation constriction or dilation. Here it should be appreciated that the actions of vasomotion and vascular tone are now more complex beyond their influence on vascular diameter (12, 17), because they also modulate bifurcation design and topology, which can also influence the fractal dimension of design. Thus a spectrum of hemodynamic reactivity in network organization may be present at different times under different hemodynamic circumstances in the perinatal period than those observed in this study. Despite limitations, the pervasive nature of the unusual fetal design phenotype and its divergent directions of structural remodeling seen here after birth indicate that persistence and/or aggravation of the fetal phenotype after birth may be important in developing further understanding of the spectrum of abnormal patterns in hemodynamic reactivity and vascular wall remodeling seen in patients with pulmonary hypertension secondary to elevated blood flow (43).

The fractal dimensions observed in the fetus and the Shunt twin are curious because they are at odds with what is predicted to be universal for arterial network organization in mammalian vascular systems in general (32, 33, 55–57). While hemodynamic forces, such as cyclic or mean pressure and shear stress secondary to increased flow, are stimuli known to influence endothelial dysfunction and vascular wall remodeling (10), various theories predict that arterial vascular network organization maintains a universal design, or a target configuration, with arterial branching ratios always β ≥ 1, with the condition of β < 1 improbable. An assertion made by West-Brown-Enquist (57) is that arterial network organization has a universal design based on a widely known metabolic scaling principle in adult species, which uniquely links the fractal dimensions of arterial network organization to the exponent of the 3/4 law metabolic scaling law [Kleiber's relationship (29)]. Their theory maintains that a fractal design of *D*_{0} = 2 and a diameter topology of β = 1 exist for all arterial transport and capillary exchange networks, and that this design remains independent of animal size, ontogenic state, and hemodynamic condition (56, 57). Another classic design, called Murray's law, dictates that arterial network organization has a universal configuration of *D*_{0} = 3.0, a condition optimized for constant shear stress and resistance with arterial branching (33). Alternatively, Kurz and Sandau (32) predicted that the bifurcation scaling exponents of arterial network organization remodel during ontogenesis and converge to a target configuration of *D*_{0} = 2.7, a design indicative of a vascular system of constant wall tension. Whatever the target, the assumption of a fixed, universal design independent of hemodynamic conditions is relevant and crucial to the study of structural remodeling in pulmonary hypertension. Design invariance with respect to hemodynamic conditions is an operational premise that serves to justify the idea that the complexity of network organization is not a precursor to the generation of hemodynamic forces that are instrumental to the remodeling process of pulmonary hypertension disease. In addition, network design invariance provides a necessary theoretical justification that the complexity of the pulmonary arterial system can be exclusively reduced to a vessel level of organization to elucidate mechanisms controlling diameter restriction in pulmonary hypertension. What the theories of arterial network organization overlook is that the fetal pulmonary circulation does not function as a generalized transport/exchange organ system in utero. In addition, hemodynamic conditions in the fetal pulmonary circulation are also much different from those in the target adult pulmonary circulation: the fetal pulmonary bed is hypertensive in relation to the systemic bed and is subject to a relatively small blood flow (50). At birth the fetal configuration switches in response to an increase in blood flow, via the removal of the placenta and the closure of the ductus arteriosis and foramen ovale, in conjunction with a decrease in capillary resistance with airway expansion. Our morphometric results indicate that the hemodynamic transition at birth is associated with an important, necessary transformation in arterial network organization, such that the fetal configuration functions as a rapidly adapting high-impedance transformer, which is capable of dynamically altering its hemodynamic reactivity in response to shear stress perturbations via changes in blood flow.

Although our data are limited in not allowing us to generalize to the fractal state of arterial network organization in other fetal and neonatal states as a population or conditions of hypertension, we can address the general question of whether fractal network organization is independent of hemodynamic conditions in another way, via the influence of fractal network organization on the scaling of cumulative arterial resistance and slopes of pressure-flow curves relative to body weight. As *Eq. 12* indicates, *R*_{cum} functionally dictates the slope of the pressure-flow curve (13), with increases in *R*_{cum} (*Eq. 10*) accomplished via smaller fractal dimensions and larger diameter ratios R_{d}, while a compensating decrease in *R*_{cum} can be accomplished via a smaller *R*_{0}. However, arterial resistance *R*_{cum} is predicted to scale with body weight independent of species, state of development, and hemodynamic condition via a fixed fractal dimension according to different patterns: the West-Brown-Enquist law predicts resistance to universally scale according to *R*_{cum} ∝ *M*^{−3/4} with arterial *D*_{0} = 2 (57), while Murray's law predicts R_{cum} ∝ *M*^{−1} with arterial *D*_{0} = 3 (51). Consequently, if vascular network organization is independent of hemodynamic conditions and state of pulmonary development, then the slopes of pulmonary pressure-flow curves sampled from different species, ages, and conditions of pulmonary hypertension would be anticipated to be devoid of outliers and invariant about a given power law. Figure 8 summarizes the scaling of *R*_{cum} estimated via the slope of pulmonary pressure-flow curves drawn from the literature (1, 15, 16, 22, 31, 36, 38, 42, 45) and indicates that the resistance scaling law for the pulmonary circulation of the normotensive adult does not adhere to a *R*_{cum} ∝ *M*^{−3/4} as predicted by the West-Brown-Enquist theory, but instead migrates to an *M*^{−1} power law only in the adult case free of pulmonary hypertension. At the same time, under conditions of pulmonary hypertension in the fetal-neonatal state (8, 46) and the adult state with primary pulmonary hypertension (36), the deviations about the *M*^{−1} power law indicate that resistance does not adhere to any particular law at all. Under these circumstances, the degrees of freedom possible in a fractal pattern of network organization to influence resistance suggest that the elevated cumulative resistance seen in the fetus and adult with pulmonary hypertension may possibly share a common topology (β < 1) and design (*D*_{0} < 2) in arterial network organization. At the same time, if arterial network organization is, in general, not independent of hemodynamic conditions, then there must be a common hemodynamic feedback signal to condition arterial design to target the fractal dimensions of arterial network organization from one resistance state of organization to another. As for targets and their feedback signals, LaBarbera (33) proposed constant shear stress for a target of *D*_{0} = 3 while Kurz and Sandau (32) theorized constant wall tension for a target of *D*_{0} = 2.7. Regardless of the signal and the target design, intermediate states of *R*_{cum} and its associated fractal dimension seen in Fig. 8 can be understood exclusively in terms of a network design that is dynamic, with a capacity to change its organization to act as a variable hemodynamic transformer (3, 4) that changes its reactivity under different hemodynamic conditions.

One major pattern of structural remodeling observed in this study was the transformation in fractal dimension between the fetal and postnatal states (fetus and Control twin), which was not seen in the Shunt twin. There is likely more than one contribution responsible for the structural remodeling of arterial network organization. As the hemodynamic transition at birth is associated with changes in vascular tone (17), we suspect that vascular tone is involved in remodeling of bifurcation design. Indeed, Griffith et al. (23) demonstrated in isolated bifurcations that vasodilation selectively increases the area ratio in bifurcations from a baseline state of α = 2 to α → 3, a constant shear stress design, via a nitric oxide (NO) mechanism, while vasoconstriction from a baseline state to a fetal design of α → 1.25 and β < 1 is induced by endothelin (ET)-1. Another factor expected to influence the structural remodeling of arterial design and topology is the influence of shear stress. In general, the endothelium responds to shear stress by altering its functional phenotype and endothelial gene expression via several thousand genes over different timescales (21). The endothelium incorporates machinery for a rapid transduction-translation response in function and geometry coupled to shear stress (11), which includes potassium channels (9), vasodilation via NO and ET-1 (5, 18), and possibly lung Krüppel-like transcription factors to gradually remodel the state of tone (14). Over hours to days shear stress is capable of signaling a spectrum of remodeling changes within the vessel wall (endothelium, intima, smooth muscle, and adventitia) working as a syncytium that add or detract from vessel wall mass to either encroach or increase diameter (21, 34). Thus, insofar as the unusual configuration of the term fetal state is already preconditioned to hypertension, and small increments in flow lead to large transient avalanches of shear stress, causal reasoning implicates that it is the transduction of shear stress via the endothelium and its translation of endothelial/smooth muscle function to reorganize bifurcation topology, design, and hemodynamic reactivity that is a principal hemodynamic stimulus for remodeling to an adult configuration at birth, rather than the vasodilation of vascular diameter alone (17). We also suspect that shear stress is implicated in the abnormal structural remodeling of arterial diameter geometry, topology, and design in the Shunt twin. In postnatal experimental models of congenital heart disease, where the reduction in resistance has occurred in conjunction with an adult pulmonary arterial configuration, we would have anticipated that the chronically elevated pressure would have increased pulmonary vascular resistance (6, 48). However, while the reactive branching pattern persisted and was aggravated in smaller arterial vessels, the Shunt twin demonstrated a paradoxical decrease in absolute cumulative resistance relative to the Control twin, which is consistent with our previous findings (40, 44). In this example, the decrease in resistance can be attributed to geometric remodeling leading to a significant increase in larger diameter vessels. Figure 5 illustrates the regional influence of arterial diameter scaling in the Shunt twin relative to the fetus and the Control twin: about a pivot point of order 11, orders 0–8 possess larger diameters and orders 9–11 approximate Control twin diameters, while orders 12–22 are smaller than Control twin diameters. The impact of the increased diameters in the larger arteries is to increase arterial volume, relative to control, a common finding in this model and in patients with chronic high-flow hypertension (44). This geometric feature of structural remodeling may have arisen from the increased flow, rather than the pressure alone, via a negative feedback mechanism of diameter structural remodeling dependent on an endothelial shear stress mechanism, mediated by NO (34). However, the persistent pattern of fetal arterial network organization in the Shunt twin in combination with elevated blood flow suggests a potential initiating physical-mechanical cause and site of action of endothelial injury/dysfunction due to amplified shear stress (25). In Fig. 7 the flow in the Shunt twin may be elevated to a sufficient extent to lead to excessive chronic levels of shear stress known to transiently induce permeability changes and mechanical injury (19, 35, 54). Unfortunately, our study is limited in that it does not identify whether the amplitude of shear stress alone is an effective cause of the persistence of the fetal bifurcation design. Alternatively, other factors operating within the fetal state besides shear stress-induced endothelial dysfunction and/or injury, such as the disruption of endothelial/smooth muscle function and communication (10), may be instrumental effective causes interrupting the fetal-neonatal transformation to a low-impedance transport design in which elevated shear stress is a complication. Under these conditions, amplified shear stress itself may represent an additional complication factor that interferes with the structural remodeling of the fetal bifurcation design to an adult configuration. We submit that the nature and direction of structural remodeling in arterial networks with area ratios <1 under a shear stress paradigm may have implications for understanding the origins and course of vascular reactivity in pulmonary hypertension in the perinatal period.

In summary, our results indicate that the unusual hemodynamic reactivity of the perinatal pulmonary arterial circulation in response to elevated blood flow leading to pulmonary hypertension can be potentially explained via a shear stress paradigm of arterial diameter network remodeling. In an elevated pulmonary blood flow lamb model, the increased resistance and reactivity predicted in pulmonary hypertension has a preexisting mechanical basis evidenced by a fetal diameter branching phenotype that acts as a transformer/amplifier of hemodynamic forces that persists and/or becomes exacerbated after birth. This fetal phenotypic pattern appears before the emergence of abnormal vascular histological patterns of remodeling seen in the course of secondary pulmonary hypertension and serves to maintain and/or exacerbate the hemodynamic reactivity of the fetal phenotype. The levels of shear stress predicted in preacinar and intra-acinar arterial vessels in our model indicate that the critical levels of shear stress leading to endothelial dysfunction/injury may arise mechanically from the interaction of blood flow and the fetal phenotype of arterial network organization itself. While the actual mechanisms responsible for arterial network remodeling and adaptation in the perinatal period are not yet identified, our shear stress paradigm suggests possible remodeling pathways of arterial network diameter topology and design that extend beyond present understanding of a stationary pattern of arterial diameter geometry under a transmural pressure paradigm of vascular remodeling diameter and wall properties. Given the pressure-flow properties of the pulmonary circulation in patients with other forms of pulmonary hypertension, and the theoretical dependence of the area ratio on hemodynamic reactivity, we speculate that diverse forms of pulmonary arterial hypertension may share a common origin and formal cause embedded in replication of a high-impedance fetal pattern of branching complexity.

## APPENDIX: ROLE OF ARTERIAL FRACTAL NETWORK ORGANIZATION ON HEMODYNAMIC REACTIVITY

Our present concept of vascular reactivity is based primarily on factors that influence vascular diameter impacting vessel hemodynamic resistance via Poiseuille's law via *Eq. 10*. However, an additional source of hemodynamic reactivity arises from an influence of arterial network topology and design, which affects the distribution and scaling of hemodynamic forces via the manner in which diameters are interconnected. The purpose of this appendix is to summarize the functional morphometric relationships of topology and design used to characterize hemodynamic reactivity in terms of dimensionless constants, such as the area ratio β, the bifurcation scaling exponent α, the fractal dimension *D*_{0}, and the resistance/shear stress reactivity constant κ with arterial branching.

#### Bifurcation topology.

The area ratio expresses the cross-sectional area of perfusion with arterial branching that can be approximated in terms of network topology and geometry (A1) This relationship states that the change in cross-sectional area of perfusion to blood flow is determined by the product of the relative increase in the number of vessels with arterial branching (via the branching ratio R̂_{b} = *N*_{k+1}/*N*_{k}) and the relative decrement in average diameter (R̂_{d} = *d̄*_{k+1}/*d̄*_{k}), where the subscripts *k* and *k*+1 refer to the order of branching, with *k* the parent to daughters *k*+1. A bifurcation is the simplest network topology in which the area ratio can be explicitly expressed in terms of diameter geometry. In general, arterial networks branch asymmetrically: branching typically consists of a parent diameter *d*_{0} and daughter diameters *d*_{1} and *d*_{2} such that *d*_{0} > *d*_{1} ≥ *d*_{2}. Here bifurcation summarizes the bifurcation's topology via a geometric relationship (A2) where the local bifurcation diameter ratio is defined as R_{d} = *d*_{1}/*d*_{0} and the asymmetry ratio is defined as γ = *d*_{2}/*d*_{1}. Thus, in the context of the definition in *Eq. A1*, the diameter ratio and daughter asymmetry of the bifurcation are associated with hemodynamically equivalent average branching ratio and diameter ratio. In the case of a symmetric bifurcation γ = 1 the area ratio has a simple topological-geometric interpretation, as (A3) where R̂_{b} = 2 and R̂_{d} = R_{d}. Alone, the area ratio alone says nothing about its hemodynamic impact in response to flow.

#### Bifurcation design.

A structure-function scaling relationship between flow and diameter within a vessel leads to the concept of a hemodynamic design for a given diameter network topology that can be used for hemodynamic analysis and prediction. Roy and Woldenberg (49) argued that the steady-state flow (*q*) in the vasculature has a general scaling relationship with diameter (*d*) according to (A4) where *k* is a constant. The value of the scaling constant α has hemodynamic significance, imparting a design or plan for the steady components of flow and pressure that are optimal in some sense (3). In the case of a bifurcation, the conservation of flow dictates that *q*_{0} = *q*_{1} + *q*_{2}. With the fraction of flow distributed in daughter vessels according to (*p*_{1} = *q*_{1}/*q*_{0}, *p*_{2} = *q*_{2}/*q*_{0} = 1 − *p*_{1}) the flow-diameter scaling relationship in the bifurcation is (A5) In the case of an asymmetric bifurcation the area ratio is related to the diameter (49) (A6) where the value for a symmetric bifurcation network is noted by γ = 1.

The value of α, although dimensionless, has hemodynamic significance that reflects an underlying law or principle associated with the influence of topology of network organization on the distribution of blood flow (3): α = 4, β > 1 confers constant resistance with branching; α = 3, β > 1 constant shear stress (Murray's law); α = 2.7, β > 1 constant wall tension (Kurz-Sandau law); and α = 2, β = 1 constant velocity (West-Brown-Enquist law). The hemodynamic influence of a given value of α also impacts the scaling of hemodynamic properties associated with flow *q*, such as the Reynolds number (Re = *q*ρ/μ*d*), flow velocity (ν = *q*/π*d*^{2}), and/or shear stress (τ = 32*q*/π*d*^{2}). In essence, arterial network organization can influence a given hemodynamic property depending on the value of the design parameter α. In a symmetric bifurcation (R_{b} = 2 and R_{d} = 2^{1/α}): (A7) (A8) (A9) These indices ignore the development of flow, the formation of fluid dynamic boundary layers, and flow-diameter asymmetry. However, what these scaling relationships have in common is that there is a critical value of α_{crit} that influences whether network topology either attenuates or amplifies a given hemodynamic property.

#### Network topology and design.

Horsfield and Woldenberg (28) argued that, akin to flow-diameter scaling in a bifurcation, there is a fractal flow-diameter scaling relationship in arterial trees. The number of branches in successive orders down a fractal tree, such as the pulmonary circulation, increases by a constant proportion (R̂_{b} = *N*_{k+1}/*N*_{k}), while the mean flow in branches in successive orders will decrease by a proportion *p̄* = *q̄*_{k+1}/*q̄*_{k} that is inversely proportional to the branching ratio, R̂_{b} = *p̄*^{−1}. The scaling relationship between the average decrease in diameter with branching, R̂_{d} = *d̄*_{k+1}/*d̄*_{k} in conjunction with the vessel number or inverse fraction of flow distributed, leads to the seed of a fractal power law scaling relationship, (A10) If R_{b} and R_{d} are constant with branching order *k*, then the arterial network can be formally considered as self-similar, with the scaling constant α approximately equal to a fractal dimension α ≈ *D*_{0} (30). Under these circumstances, the topology network organization via the area ratio is related to the fractal network scaling property according to (A11) Consequently, network organization and design has an exponential scaling influence with branching order *k* on the attenuation/amplification properties according to (A12) (A13) (A14) The idea that network organization influences hemodynamic properties in response to blood flow suggests that the diameter scaling properties can be summarized in terms of a reactivity parameter κ that demonstrates its dependence on arterial topology and design with arterial branching. Thus, for shear stress reactivity, the relative increment of shear stress from the first order 0 to the first branching level 1 is given by (A15) Figure 1 illustrates the relative influence of β and *D*_{0} for κ relative to one level of branching, as well as the amplification effects over several orders of branching.

#### Resistance reactivity.

Under the assumptions that Poiseuille's law applies (*Eq. 10*), the cumulative resistance of a complex network depends on how resistance changes with arterial branching (A16) where *R̄*_{k} is the average resistance (*Eq. 10*) of the vessels of order *k* with average diameter *d̄*_{k} and length *l̄*_{k}, and *N*_{k} is the number of vessels within order *k*. Expressing *Eq. A16* in terms of the presence of self-similar fractal scaling relationships, a constant diameter ratio R̂_{d} = *d̄*_{k+1}/*d̄*_{k} and a constant length ratio R̂_{l} = *l̄*_{k+1}/*l̄*_{k}, *R*_{cum} becomes (A17) In asymmetric trees, the relative values of R̂_{d} and R̂_{b} depend on the methods of diameter ordering: Horsfield ordering yields R̂_{b}^{H}≤ 2, and a smaller R̂_{d}^{H}(26, 27) with more orders for the same hemodynamically equivalent tree than that ordered by a Strahler method with R̂_{b}^{S}≤ 2 (30). The branching ratios are related by 1 = (R_{d}^{H})^{−1} + (R_{d}^{S})^{−1} = (R_{d}^{H})^{−1} + (R_{d}^{H})^{−(1+Δ)} via a symmetry factor Δ (26, 27). If an additional scaling relationship for vessel length and diameter is taken into account, assuming that arterial network organization has a common fractal length-diameter scaling relationship *l̄*_{k} = *a**d̄*_{k}^{α2}(α_{2} = 1 and *a* is an arbitrary scaling constant that depends on the method of vessel ordering) and fractal scaling condition R̂_{b} = R̂_{d}^{D0}, then *R*_{cum} can be expressed more simply as (A18) In effect, the resistance of order *k* is *R*(*k*) = *R*_{0}R̂_{d}^{(3−α)k}, in which the fractal network organization imparts a contributing reactive factor κ_{R} = R̂_{d}^{3−D0} within each order. Thus, under the assumptions of network organization in this fractal model, the resistance reactivity is equal to the shear stress reactivity κ = κ_{τ} of *Eq. A15*. (A19) Figure 1 illustrates the impact of hemodynamic reactivity with branching on normalized values of *R*_{cum} and τ for the simplest case of a symmetrically bifurcating network where R̂_{b} = 2 and R̂_{d} = 2^{1/D0}.

## Footnotes

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