Intraspecific scaling laws of vascular trees

Abstract

A fundamental physics-based derivation of intraspecific scaling laws of vascular trees has not been previously realized. Here, we provide such a theoretical derivation for the volume–diameter and flow–length scaling laws of intraspecific vascular trees. In conjunction with the minimum energy hypothesis, this formulation also results in diameter–length, flow–diameter and flow–volume scaling laws. The intraspecific scaling predicts the volume–diameter power relation with a theoretical exponent of 3, which is validated by the experimental measurements for the three major coronary arterial trees in swine (where a least-squares fit of these measurements has exponents of 2.96, 3 and 2.98 for the left anterior descending artery, left circumflex artery and right coronary artery trees, respectively). This scaling law as well as others agrees very well with the measured morphometric data of vascular trees in various other organs and species. This study is fundamental to the understanding of morphological and haemodynamic features in a biological vascular tree and has implications for vascular disease.

1. Introduction

The fractal dimension defined by Mandelbrot [1] has been shown to characterize natural phenomena with remarkable simplicity. The vascular [2–19], bronchial [20,21] and botanical trees [22–25] have been found to have fractal-like features. For interspecific scaling, based on three major assumptions in a fractal-like cardiovascular system (i) area-preservation for branching patterns of vessel diameter greater than 1 mm and cubed-law for smaller vessels as well as space-filling branching pattern; (ii) size-invariant capillaries; and (iii) minimum energy loss, West et al. [26] proposed a mathematical model referred to as the WBE (West, Brown, and Enquist) model to support the allometric 3/4 scaling law of metabolism. In the process, they assumed that the fractal-like networks in life have a fourth spatial dimension [27], which served as the basis of the WBE model [26]. There is much debate on the WBE model, however, especially for the assumption of the space-filling branching pattern in the cardiovascular system (i.e. LR = BR^1/3, where BR and LR are the branching ratio and length ratio, respectively) [28,29].

For scaling laws within a species (intraspecific scaling), several power-law relations of vascular trees have been proposed with an empirical parameter determined experimentally in the exponent of equivalent tree resistance [30]. The objective of this study is to provide an alternative derivation that does not invoke an empirical parameter and to critically validate the model using detailed morphometric and haemodynamic measurements. In the process, we first derive volume–diameter and flow–length scaling laws from conservation of mass in a vascular tree without invoking the controversial space-filling assumption disputed in the WBE model. These two scaling laws, which agree well with the morphometric measurements of coronary and other vascular trees, lead to the diameter–length, flow–diameter and flow–volume scaling power-laws as a result of the minimum energy hypothesis. The physiological basis of these scaling laws is demonstrated and the implications on coronary heart disease (CHD) are discussed.

2. Methods

2.1. Scaling laws in a vascular tree

A proximal vessel segment is defined as a stem and the tree distal to the stem (down to the smallest arterioles or venules) is defined as a crown, as shown in figure 1. An entire tree consists of many stem-crown units. The capillary network (vessel diameter less than 8 µm) is excluded from the present analysis because it is not tree-like in structure [31]. The vessel segment is assumed to be a cylindrical tube and other nonlinear effects (e.g. vessel compliance, turbulence, variation of viscosity in different vessel segments, etc.) are neglected because of their relatively small contribution to the haemodynamics of an entire integrated tree (millions of blood vessel segments) [26]. In an integrated system of stem-crown units, the crown volume (V_c(ml)) is defined as the sum of the intravascular volume of each vessel segment and the crown length (L_c(cm)) is defined as the sum of the lengths of each vessel segment in the entire crown from the stem to the most distal vessels.

Figure 1.

A schematic illustration of the definition of stem-crown units and the corresponding parameters. D, L, Q, V and R are the diameter, length, flow rate, blood volume and flow resistance with subscripts ‘s’ and ‘c’ corresponding to stem and crown, respectively, in a stem-crown unit. Subscripts ‘max’ or ‘0’ represent the most proximal stem-crown unit in a vascular tree, while subscript ‘N_total’ refers to the pre-capillary (the most distal vessel).

To determine scaling laws of a vascular tree, the branching ratio, diameter ratio and length ratio in a fractal-like tree structure are defined as: BR = n_i/n_i−1, DR = D_i/D_i−1 and LR = L_i/L_i−1, where n_i, D_i and L_i are the number, diameter and length of vessels in level i, i = 1, …, N_total. Level 0 is the most proximal stem for an entire tree or a stem for the stem-crown unit, and level N_total refers to the smallest arterioles or venules. Based on the assumptions of LR = BR^{−1/(3−γ)} and DR = BR^−1/(2+ɛ) (where γ = 0 represents space-filling, γ = 1 area-filling and γ = 2 length-preservation; ɛ = 0 represents area-preservation and ɛ = 1 Murray's law) for a tree structure, we provide a fundamental derivation of the volume–diameter and flow–length scaling laws for arbitrary γ and ɛ (i.e. do not invoke the space-filing assumption) as shown in appendices A and B, respectively.

As shown in appendix A, the crown volume is found to scale with the stem diameter in a stem-crown unit as:

2.1

where D_s is the stem diameter (cm) in a stem-crown unit. (V_c)_max and (D_s)_max refer to the cumulative vascular volume and the most proximal stem diameter in an entire tree, respectively.

In appendix B, we derive a scaling relation between stem flow rate and crown length in a stem-crown unit expressed as:

2.2

where Q_s is the flow rate through the stem (ml s⁻¹), (Q_s)_max is the flow rate through the most proximal stem of the entire tree and (L_c)_max is the cumulative vascular length of the entire tree. The physical basis of this scaling law (equation (2.2)) is the conservation of mass in a fractal-like tree structure.

In appendix C, we derive the diameter–length scaling law from the minimum energy hypothesis and equations (2.1) and (2.2), which can be expressed as:

2.3

A combination of equations (2.2) and (2.3) leads to the flow–diameter scaling law (similar to Murray's law but with a different exponent) as:

2.4

If equation (2.4) is combined with equation (2.1), we obtain:

2.5

Equations (2.1)–(2.5) refer to the volume–diameter, flow–length, diameter–length, flow–diameter and flow–volume scaling laws within a vascular tree, respectively.

2.2. Morphometric vascular trees

The scaling power-laws (equations (2.1)–(2.5)) are valid in the entire coronary arterial tree in pig hearts, based on the measured morphometric data [6]. Similarly, vascular trees of many organs down to the pre-capillary vessels were also used to verify the scaling power-laws, which are constructed in the Strahler system [30], based on the available literature [6–19]. The pulmonary arterial tree of rats was obtained from the study of Jiang et al. [7]; the pulmonary arterial/venous trees of cats from Yen et al. [8,9]; the pulmonary venous trees of dogs from Gan et al. [10]; the pulmonary arterial trees of humans from Singhal et al. [11,12] and Huang et al. [13]; the pulmonary venous trees of humans from Horsfield & Gordon [14] and Huang et al. [13]; the skin muscle arterial tree of hamsters from Bertuglia et al. [15]; the retractor muscle arterial tree of hamsters from Ellsworth et al. [16]; the mesentery arterial tree of rats from Ley et al. [17]; the sartorius muscle arterial tree of cats from Koller et al. [18]; the bulbular conjunctiva arterial/venous trees of humans and the omentum arterial tree of rabbits from Fenton & Zweifach [19].

2.3. Data analysis

A least-squares fit of morphometric data was used to determine the exponents in the scaling power-laws of vascular trees. In particular, a least-squares fit of morphometric data (D_i versus n_i and L_i versus n_i, i = 0, …, N_total) was made to determine the parameters ɛ and γ in DR = BR^−1/(2+ɛ) and LR = BR^{−1/(3−γ)}, respectively. The relative error was used to quantify the discrepancy of exponents fitted by a least-squares method and theoretical exponents. The relative error expressed as

was also calculated in each generation of vascular trees. A two-sample t-test was used to compare the theoretical prediction with the anatomical data, where p < 0.05 represented statistically significant differences.

3. Results

3.1. Volume–diameter scaling law

Figure 2 shows a log–log plot of normalized crown volume as a function of normalized stem diameter,

for the entire pig left anterior descending artery (LAD), left circumflex artery (LCx) and right coronary artery (RCA) trees. Using a least-squares fit, the log–log plots have exponents of 2.96 (r² = 0.999), 3.00 (r² = 0.999) and 2.98 (r² = 0.999) for LAD, LCx and RCA trees, respectively, which are in excellent agreement with the model prediction.

Figure 2.

Relation between normalized stem diameter (D_s/(D_s)_max) and normalized crown volume (V_c/(V_c)_max) in the entire (a) LAD, (b) LCx, and (c) RCA trees of pig, which include 946 937, 571 383 and 836 712 vessel segments, respectively. The entire tree data are presented as log–log density plots showing the frequency of data because of the enormity of data points, i.e. darkest shade reflects highest frequency or density and the lightest shade reflects the lowest frequency [32]. The solid lines represent the least-squares fit of the experimental measurements in LAD, LCx and RCA trees, which have exponents of 2.96 (r² = 0.999), 3.00 (r² = 0.999) and 2.98 (r² = 0.999), respectively.

Figure 3 shows a log–log plot of normalized crown volume as a function of normalized stem diameter in the vascular trees of various organs and species, where the solid line represents the least-squares fit of all the experimental measurements (exponent of 2.91, r² = 0.966). Accordingly, we identified the exponent by a least-squares fit of morphometric measurements in each vascular tree, as shown in table 1. The relative errors as were calculated in each tree as

and

Table 1 shows the mean value of relative errors where

was averaged over the total number of generations, N_total + 1. The prediction of the present model agrees very well with the measured morphometric data.

Figure 3.

Relation between normalized stem diameter (D_s/(D_s)_max) and normalized crown volume (V_c/(V_c)_max) for vascular trees of various organs and species corresponding to those trees in table 1. The solid line represents the least-squares fit of all the experimental measurements (exponent of 2.91, r² = 0.966).

Table 1.

A least-squares fit of exponent β in (V_c/(V_c)_max) = (D_s/(D_s)_max)^β (the equation of volume–diameter scaling law) for vascular trees in various organs and species. RCA, right coronary artery; LAD, left anterior descending artery; LCx, left circumflex artery; PA, pulmonary artery; PV, pulmonary vein; SKMA, skin muscle arteries; SMA, sartorius muscle arteries; MA, mesentery arteries; OV, omentum veins; BCA, bulbular conjunctiva arteries; RMA, retractor muscle artery; BCV, bulbular conjunctiva vein. Error_β = ((|β − 3|)/3) × 100%; error in levels = ; N_total + 1, total no. of levels or generations in the respective vascular tree.

		anatomical data		present model
	species (Ntotal + 1)	β	r2	errorβ (%)	error in levels
large number of orders	pig RCA (11)	3.11	0.998	3.67	0.88
	pig LAD (11)	3.09	0.997	3	0.81
	pig LCx (10)	3.10	0.998	3.33	0.54
	rat PA (11)	3.19	0.995	6.33	1.02
	cat PA (10)	3.26	0.998	8.67	2.08
	cat PV (10)	3.30	0.998	10	2.09
	dog PV (11)	3.11	0.998	3.67	0.24
	human PA (17)	2.91	0.990	3	0.73
	human PA (15)	3.20	0.998	6.67	0.64
	human PA (17)	2.84	0.995	5.33	0.68
	human PV (15)	3.24	0.997	8	5.01
	human PV (15)	2.99	0.992	0.33	0.67
	mean ± s.d.	3.11 ± 0.15		5.30 ± 2.92	1.38 ± 1.32
small number of orders	hamster SKMA (4)	3.19	0.993	6.33	1.23
	rat MA (4)	4.45	0.997	48.3	2.31
	rabbit OV (4)	3.55	0.999	18.3	0.74
	human BCA (5)	5.05	0.985	68.3	2.7
	human BCV (4)	3.11	0.983	3.67	0.69
	hamster RMA (4)	3.15	0.999	5	1
	cat SMA (4)	4.78	0.940	59.33	1.99
	mean ± s.d.	3.90 ± 0.84		29.9 ± 27.9	1.52 ± 0.81

3.2. Scaling laws in a vascular tree

Table 2 shows the values of parameters ɛ and γ in DR = BR^−1/(2+ɛ) and LR = BR^{−1/(3−γ)} for various vascular trees corresponding to table 1, which have mean ± s.d. of 0.64 ± 0.64 and 0.45 ± 0.49 over these trees, respectively. Table 3 shows a comparison of scaling laws (i.e. volume–diameter, flow–length, diameter–length, flow–diameter and flow–volume scaling power-laws) between the present theoretical models and experimental measurements. The mean (±1 s.d.) of exponents was determined by a least-squares fit of morphometric measurements in the vascular trees of various organs and species in correspondence to table 1.

Table 2.

A least-squares fit of parameters ɛ and γ in DR = BR^−1/(2+ɛ) and LR = BR^{−1/(3−γ)} for vascular trees in various organs and species. A least-squares fit of morphometric data (D_i versus n_i and L_i versus n_i, i = 0, …, N_total) was made to determine the parameters ɛ and γ in DR = BR^−1/(2+ɛ) and LR = BR^−1/(3+γ), respectively.

	species (Ntotal + 1)	ɛ	r2	γ	r2
large number of orders	pig RCA (11)	0.11	0.996	1.08	0.988
	pig LAD (11)	0.07	0.993	1.02	0.990
	pig LCx (10)	0.04	0.994	1.20	0.987
	rat PA (11)	0.22	0.998	0.86	0.956
	cat PA (10)	0.37	0.997	0.67	0.975
	cat PV (10)	0.30	0.993	0.82	0.954
	dog PV (11)	0.5	0.998	0	0.995
	human PA (17)	0.65	0.991	−0.16	0.983
	human PA (15)	0.73	0.994	−0.04	0.978
	human PA (17)	0.44	0.992	−0.04	0.974
	human PV (15)	0.65	0.998	0.08	0.982
	human PV (15)	0.49	0.994	0.24	0.986
	mean ± s.d.	0.38 ± 0.24		0.48 ± 0.51
small number of orders	hamster SKMA (4)	0.33	0.992	0.35	0.870
	rat MA (4)	1.79	0.990	0.34	0.924
	rabbit OV (4)	0.74	0.933	0.45	0.836
	human BCA (5)	2.18	0.991	−0.32	0.918
	human BCV (4)	0.43	0.971	0.09	0.955
	hamster RMA (4)	0.05	0.991	1.29	0.968
	cat SMA (4)	1.98	0.938	0.72	0.954
	mean ± s.d.	1.07 ± 0.89		0.42 ± 0.5

Table 3.

A comparison of scaling laws between anatomical data of various organs and theoretical model. , volume–diameter scaling law; , flow–length scaling law; , diameter–length scaling law; , flow–diameter scaling law (i.e. Murray's law if X = 3.0); and , flow–volume scaling law. The exponents (mean ± 1 s.d.: averaged over the vascular trees for various organs and species) are determined by a least-squares fit of anatomical measurements in corresponding to those trees in table 1.

	exponent X in power-law scaling relations
scaling relations	anatomical data	present model
	3.38 ± 0.63	3
	0.94 ± 0.28	1
	0.37 ± 0.08	0.43
	2.62 ± 0.65	2.33
	0.79 ± 0.055	0.78

4. Discussion

We derived a volume–diameter scaling power-law (equation (2.1)), which agrees well with the measured morphometric vascular trees. Moreover, based on conservation of mass, we theoretically derived the flow–length scaling law (equation (2.2)) previously confirmed by experimental observations [30,33]. A combination of the two laws results in the diameter–length (), flow–diameter () and flow–volume () scaling power-laws in a fractal-like vascular tree as a result of minimum energy hypothesis. These laws provide a quantitative integration of haemodynamic and morphometric features in a vascular tree, as discussed below.

4.1. Scaling relations of length, diameter and branching ratios

The space-filling assumption in the WBE model [26] to express the length ratio to the branching ratio (LR = BR^−1/3) has been the subject of significant criticism [28,29]. In the present derivation of the forgoing scaling laws, we assumed that LR = BR^{−1/(3−γ)} (where γ = 0 represents space-filling, γ = 1 area-filling and γ = 2 length-preservation) for a tree structure without invoking the space-filling assumption. It is found that LR = BR^{−1/(2.55±0.49)} for vascular trees of various organs and species so that the mean value (±s.d.) of γ equals 0.45 (±0.49). As shown in table 2, the value of γ for most vascular trees in different organs varies in the range of zero to unity (i.e. in the range between space-filling and area-filling), but is significantly different from zero or unity when all experimental measurements of various vascular trees are statistically analysed (p ≪ 0.05). The large variation of γ may be attributed to the structure and metabolic supply–demand balance of each organ, which requires further investigation.

In a vascular tree structure, the area-preservation for large vessels (i.e. DR = BR^−1/2) and Murray's cubed-law for small vessels (i.e. DR = BR^−1/3) were assumed in the WBE model [26]. Based on the morphometric measurements, we reported an exponent of approximately −1/2.1 for D_s ≥ 200 µm, an exponent of approximately −1/1.7 for 70 ≤ D_s < 200 µm, and a monotonic increase to approximately −1/3 as D_s decreased from 70 µm to the pre-capillary arterioles [34], which was consistent with other coronary data [35]. The trees of small orders with (D_s)_max < 70 µm in table 2 have an exponent of −1/(3.07 ± 0.89) (mean ± s.d.). The morphometric measurements support the area-preservation/cubed-law assumption in the WBE model to some extent. The exponents, however, vary greatly in different vascular trees of small orders in table 2, which are close to −1/4 (e.g. rat MA, human BCA and cat SMA) or −1/2 (e.g. hamster SKMA, hamster RMA and human BCV) albeit the mean value is 3.1. Hence, a more general form of DR = BR^−1/(2+ɛ) was considered for various vascular trees regardless of the size of vessel segments. As shown in table 2, there is a good least-squares fit (r² > 0.93) of DR = BR^−1/(2+ɛ) to the experimental measurements in vascular trees of various organs and species, which results in the value of ɛ equal to 0.64 ± 0.64 (mean ± s.d.).

4.2. Volume–diameter scaling law

Although parameters ɛ and γ have large variation in different vascular trees, the volume–diameter scaling law has the exponent of 3 (equation (2.1)). Figure 3 shows good agreement between the present model of equation (2.1) and the least-squares fit of all the experimental measurements in various vascular trees (i.e. 3 versus 2.91). Since scaling laws are generalities, it is natural that estimates of the scaling properties should be somewhat noisy. In this context, the variation seen in figure 3 is remarkably small (r² = 0.966), tending to validate the conclusions of the present study. Moreover, the exponents for all of vascular trees in table 1 have a mean value (±s.d) of 3.38 (±0.63), which also supports the prediction of the volume–diameter scaling law in equation (2.1). The perfusion trees with a small number of orders of branching in rat MA, human BCA and cat SMA have a larger exponent of 4.45, 5.05 and 4.78, respectively, which is caused by

From equation (A 3) in appendix A,

implies an increase in total blood volume of vessels in each generation from proximal to distal, which clearly does not occur over the entire organ level, but may exist in the smaller generations of microcirculation in some organs (D_s < 15 µm).

It should be noted that the WBE model is intended to express an allometric scaling law of metabolism between species of varying size. In future studies, the present scaling derivations need to be extended to encompass the metabolic scaling law at the organ–tissue level [36] and examined across species.

4.3. Physiological basis of scaling laws in a vascular tree

Physiological trees provide flow transport to the capillary network to support tissue demands. Vascular development is generally guided by tissue's metabolic needs and by the minimization of specific costs for growth and maintenance of the delivery of substrate materials and elimination of waste products. Here, we summarize common scaling features in various vascular trees.

First, two fundamentally constructed structure–structure scaling laws, volume–diameter (equation (2.1)) and resistance (equation (C 2)), were validated for measured morphometric vascular trees (here and in the study of Huo & Kassab [37]). The premise for the derivation of these scaling laws is that morphometric vascular trees are fractal-like, which obey self-similarity of form, as confirmed by experimental observations from earlier studies [2–19] (i.e. similar branching ratios in each generation). The combination of equations (2.1) and (C 2) leads to R_c ∝ L_c/V_cD_s, which suggests that the flow resistance of a crown is affected by the interplay of crown length, crown volume and stem diameter.

Second, a flow–length scaling law (equation (2.2)) in mammals was fundamentally derived from conservation of mass in a fractal-like vascular tree structure, which agrees very well with the experimental observations [33]. This relation is an expression of balance between supply (perfusion or flow) and demand (vascularity).

Finally, a diameter–length scaling law (equation (2.3)) was deduced by applying equations (2.1) and (C 2) in conjunction with equation (2.2) and the minimum energy hypothesis (equation (C 1)). The volume–diameter (equation (2.1)) and diameter–length (equation (2.3)) scaling laws show that the distal crown volume and crown length are power-law functions of the stem diameter. Similarly, the flow–diameter (equation (2.4)) and flow–volume (equation (2.5)) scaling laws show a power-law scaling between stem flow and stem diameter and crown volume, respectively. The flow–diameter scaling law is identical to Murray's law in form but has a smaller exponent (equal to 3.0 for Murray's law). Although an agreement between experiments and Murray's cubed-law was found in small arteries and arterioles of vascular trees [38,39], table 3 shows an exponent of 2.64 ± 0.64 (mean ± s.d.) for all trees in table 1 and 2.32 ± 0.18 (mean ± s.d.) for large-order trees in table 1. The present theoretical value of 2.33 agrees better with the measurements over many generations of vascular tree than Murray's law. The structure–structure (equation (2.3)) and structure–function (equations (2.4) and (2.5)) scaling laws reflect the design of biological trees for flow transport under the principle of minimum energy.

4.4. Comparison with Huo & Kassab's model

Huo & Kassab [40] previously proposed a volume–diameter–length scaling relation () based on experimental observation of V_c ∝ M [41] and two assumptions of D_s ∝ M^3/8 and L_c ∝ M^3/4. Equations (2.1) and equations (2.3)–(2.5) of the present study were also obtained from the volume–diameter–length scaling relation coupled with the hypotheses of minimum energy loss and flow–length scaling law (see appendix in [40]). Equations (2.1) and (2.3) resulting from the volume–diameter–length scaling relation, however, suggest that D_s ∝ M^1/3 and L_c ∝ M^7/9, respectively, which slightly differ from previous exponents (i.e. D_s ∝ M^1/3 versus D_s ∝ M^3/8 and L_c ∝ M^7/9 versus L_c ∝ M^3/4); i.e. 0.33 versus 0.37 and 0.78 versus 0.75, respectively. It is experimentally difficult to discriminate between such similar exponents based on the least-squares fit. In conclusion, this formulation provides the same volume–diameter–length scaling relation () [40].

In the present study, we provide a more fundamental derivation of the volume–diameter scaling law (equation (2.1)) and the flow–length scaling law (equation (2.2)) from conservation of mass in a fractal-like tree structure (see appendices A and B), which results in the volume–diameter–length scaling relation without invoking the previous assumptions of D_s ∝ M^3/8 and L_c ∝ M^3/4. Therefore, the present study substantiates the previous conclusion [40] and confirms the scaling power-laws of vascular tree (equations (2.1)–(2.5)). In table 3, the mean values (±1 s.d.) of exponents for various morphometric vascular trees agree well with the scaling power-laws of vascular tree.

4.5. Implications for coronary heart disease

The coronary blood volume and flow rate are important parameters for the assessment of CHD. The volume–diameter (equation (2.1)) and flow–diameter (equation (2.4)) scaling power-laws can provide estimates of blood volume and flow rate from medical imaging (e.g. digital subtraction angiography, computed tomography, magnetic resonance imaging, etc.). Moreover, regional values of myocardial blood volume (MBV) and blood flow (MBF) can also be quantified by using fast mapping techniques of MRI [42]. A comparison of MBV and MBF between normal and CHD populations can provide a severity index for the extent of myocardial ischaemia. The validation of this rationale requires future studies.

5. Summary and conclusion

The power-law scaling relation has been assumed to be ubiquitous in biology and is relevant to medicine, nutrition and ecology. For the intraspecific scaling laws within an organ of a given species, we derived and validated the volume–diameter and flow–length scaling laws (equations (2.1) and (2.2), respectively) using conservation of mass in a fractal-like and the minimum energy hypothesis. The fundamentally derived scaling laws are in very good agreement with the available morphometric and haemodynamic data in various vascular trees. These scaling laws have significant merit in basic studies and ultimately in clinical diagnosis and therapy.

Appendix A

An idealized symmetric tree is used to obtain the relationship between crown volume and stem diameter. The crown distal to a stem is composed of N_i levels (or generations) from the stem (level zero of a crown) to each terminal (the smallest arterioles or venules, level N_i of a crown). The volume of a crown, V_c, can be written as:

A 1

where

V_s, L_s and D_s are the volume, length and diameter of the stem, respectively. Similarly, V_i, L_i and D_i are the volume, length and diameter of a vessel in level i, respectively, and n_i is the total number of vessels in level i. Equation (A 1) can be written as:

A 2

In the following derivation, we introduce three definitions:

A.1. Branching ratio

The branching ratios (BR = n_i/n_i−1, i = 1, …, N_i) are relatively constant in each level from the stem (level 0) to the smallest arterioles or venules (level N_i), such that n_i = BRⁱ in a crown.

A.2. Diameter ratio

The diameter ratio is defined as: DR = D_i/D_i−1, i = 1, …, N_i. It can be shown that where ɛ = 0 represents area-preservation from one level to the next. Conversely, ɛ = 1 represents Murray's law; i.e. This provides the relation:

Therefore, the diameter ratio relates to the branching ratio as: DR = BR^−1/(2+ɛ) or D_i = BR^−i/(2+ɛ)D_s in a crown.

A.3. Length ratio

The length ratio is defined as: LR = L_i/L_i−1, i = 1, …, N_i. West et al. [26] proposed that the perfused volume from one level to the next was constant, so that

which has been highly disputed and for which there are no supporting experimental data [28,29]. Here, we assume a more general relation that has an experimental basis; namely:

where γ = 0 represent space-filling, γ = 1 represent area-filling and γ = 2 represent length-preservation. This leads to

The length ratio relates to the branching ratio as LR = BR^{−1/(3−γ)} or L_i = BR^{− i/(3−γ)} L_s in a crown.

A.4. Parameters ɛ and γ in vascular trees

Table 2 shows the relation between the branching and diameter ratios and the length ratio in vascular trees for various organs and species. It is found that DR = BR^{−1/(2.64±0.64)} and LR = BR^{−1/(2.55±0.49)}. Parameters ɛ and γ are equal to 0.64 ± 0.64 and 0.45 ± 0.49, respectively, which are significantly different from zero (p ≪ 0.05).

A.5. Volume–diameter scaling law

From n_i = BRⁱ, D_i = BR^−i(2+ɛ)D_s, L_i = BR^{−i/(3−γ)}L_s, and equation (A 2), we obtain the following equation:

A 3

Equation (A 3) relates the crown volume to the branching ratio of the vascular tree. Since (2 − γ)/(3 − γ) − 2/(2 + ɛ) < 0 for most vascular trees (mean of approx. −0.15 for vascular trees of various organs), the last term in Equation (A 3) has 0 < BR^{i((2−γ)/(3−γ)−2/(2+ɛ))} < 1. This implies a decrease in total blood volume of vessels in each level from the stem (level zero) to the terminal (level N_i) as supported by the experimental data [6]. Equation (A 3) is written as:

A 4

Since D_i = BR^−i/(2+ɛ)D_s and L_i = BR^{−i/(3−γ)}L_s, equation (A 4) can be written as:

A 5

where (L_s)_Ni and (D_s)_Ni are the length and diameter of a pre-capillary vessel segment (i.e. the smallest arterioles or venules). Given that

(ɛ−1)/(2 + ɛ) ≈ −0.14, 0 ≤ N_i ≤ 16, and 2 ≤ BR ≤ 4, there is

for different crowns in various vascular trees (negligible variation given the range of variables is very large, 10 decades on the y-axis in figure 3).

To be exhaustive, we consider two alternate scenarios: (i) it may be that (2 − γ)/(3 − γ) − 2/(2 + ɛ) = 0 (unchanged total blood volume of vessels in each level) or (ii) (2 − γ)/(3 − γ) − 2/(2+ɛ) > 0 (increase in total blood volume of vessels in each level from the stem to the terminal), which leads to:

A 6

and

A 7

For a symmetric tree, the ratio of vessel length to diameter, (L_s)_Ni/(D_s)_Ni, is constant in pre-capillary vessels. If we define that

and

equations (A 5)–(A 7) can be written as

A 8

When V_c = (V_c)_max and D_s = (D_s)_max, where (V_c)_max and (D_s)_max refer to the cumulative vascular volume and the most proximal stem diameter in the entire tree, respectively, equation (A 8) is written as:

A 9

where depends on the branching ratio, diameter ratio and total number of tree generations in an entire tree, and the ratio of vessel length to diameter in the pre-capillary vessel segment. From equations (A 8) and (A 9), we obtain:

A 10

Equations (A 8) and (A 10) are the volume–diameter scaling relation in a vascular tree.

Appendix B

The crown length, L_c, can be written as:

B 1

where N_i is the same as defined in appendix A. From n_i = BRⁱ, L_i = BR^{−i/(3−γ)}L_s and equation (B 1), we obtain the following equation:

B 2

Since L_s = (L_s)_Ni · BR^N_i/(3−γ), where (L_s)_Ni is the length of pre-capillary vessel segment, the crown length can be expressed as:

B 3

From conservation of mass, the flow rate at a stem vessel of a crown, Q_s, can be expressed by the flow rate at the pre-capillary vessel segment, (Q_s)_Ni, as:

B 4

Equation (B 4) divided by equation (B 3) results in the following expression:

B 5

For a symmetric tree, the ratio of flow rate to vessel length, (Q_s)_Ni/(L_s)_Ni, is constant in the pre-capillary vessel segment. If we define

(ml cm⁻¹), equation (B 5) can be written as:

B 6

When Q_s = (Q_s)_max and L_c = (L_c)_max, where (Q_s)_max and (L_c)_max refer to the maximal flow rate at the most proximal stem and the cumulative vascular length in the entire tree, respectively, equation (B 6) can be written as:

B 7

where K_Q = (Q_s)_max/(L_c)_max depends on the branching ratio, the total number of tree generations in an entire tree and the ratio of vessel length to flow rate in the pre-capillary vessel segment. From equations (B 6) and (B 7), we obtain:

B 8

Equations (B 6) and (B 8) are the flow–length relation in a vascular tree.

Appendix C

Similar to Murray's law, a cost function for an integrated system of stem-crown units is proposed [30], which consists of two terms: viscous and metabolic power dissipation. The cost function, F_c (erg), is written as:

C 1

where Q_s, Δp_c = Q_sR_c and V_c are the flow rate through the stem (ml s⁻¹), pressure drop in the distal crown (dynes cm⁻²) and crown volume (ml) (defined as the sum of the intravascular volume of each vessel segment in the entire crown from the stem to the most distal vessels), respectively. K_m is a metabolic constant of blood in a crown (dynes cm² s⁻¹). Two important structure–structure scaling laws are needed to implement the minimum energy analysis in the cost function (equation (C 1)). First, we have shown that the resistance of a crown (dynes s cm⁻⁵) has the following form:

C 2

where L_c is the crown length (cm) (defined as the sum of the lengths of each vessel segment in the entire crown) and D_s is the stem diameter (cm) [40]. (D_s)_max, (L_c)_max and (R_c)₀ correspond to the most proximal stem diameter of the entire tree, the cumulative vascular length of the entire tree and the total resistance of the entire tree, respectively. is a flow-resistance constant in a crown (dynes · s cm⁻²), which depends on the branching ratio and the total number of tree generation in a crown [40]. Second, the crown volume is found to scale with the stem diameter (equation (A 11)).

When resistance (equation (C 2)) and volume–diameter (equation (A 11)) scaling laws are substituted into the energy cost function, equation (C 1) can be written as:

C 3

Equation (C 3) can be normalized by the metabolic power requirements of the entire tree of interest, , to obtain the non-dimensional cost function (f_c). When the flow–length scaling law (equation (B 7)) is applied to the non-dimensional equation (C 3), the dimensionless cost function can be written as:

C 4

Similar to Murray's approach, we minimize the cost function with respect to diameter at a fixed crown length to obtain the following:

C 5

Equation (C 5) applies to any stem-crown unit. When

in equation (C 5), we find that

such that equation (C 5) can be written as:

C 6

This equation provides the diameter–length scaling law, which forms the basis for the formulation presented in this study.