Scaling Laws of Coronary Vasculature

Abstract

As demonstrated in Chap. 2, the coronary vasculature is complex. A natural question arises as to whether the construction of the vasculature is random or if it follows some design principles. This question has been pondered for nearly a century and numerous investigations have attempted to explain the design of vascular trees based on various principles, including cost functions of energy (Murray, 1926; Rosen, 1967), surface area and volume (Kamiya & Togawa, 1972), tensile stress (Kurz & Sandau, 1997), and shear force (Zamir, 1977). The major impetus for these investigations was the desire to understand the structure/function of the coronary vasculature relation under homeostatic conditions. Pathological states of the vasculature may then be understood in relation to perturbations in homeostasis.

Appendices

Appendix 1: Murray Formulation

Murray (1926) proposed a cost function, C, that is the sum of the rate at which work is done (product of flow and pressure drop) on the blood and the metabolic rate (assumed to be proportional to the volume of blood) on a vessel of length l, namely:

$$ C\left(Q,r\right)=\left(\frac{8\mu l}{\pi {r}^4}\right){Q}^2+{k}_{\mathrm{m}}\pi {r}^2l $$

(7.20)

where r, μ, and k_m represent the radius, viscosity, and a metabolic blood constant, respectively. The first term represents the work done by a Poiseuille flow and the second term represents the metabolic cost proportional to the volume of the blood in the vessel. Next, the cost function is minimized with respect to blood radius for a given vessel length, i.e., $\frac{\partial C}{\partial r}=0$, to yield:

$$ \frac{\partial C}{\partial r}=0=-\left(\frac{32\mu l}{\pi {r}^5}\right){Q}^2+2{k}_{\mathrm{m}}\pi rl $$

(7.21)

A rearrangement of Eq. (7.21) shows that for a given vessel of length l and flow rate Q, there is an optimal radius given as:

$$ Q={\left(\frac{\pi^2{K}_{\mathrm{m}}}{16\mu}\right)}^{1/2}{r}^3 $$

(7.22a)

Hence, Murray’s law states that flow is proportional to the cube of vessel radius or diameter as given by:

$$ Q={k}_{\mathrm{b}}{D}^3 $$

(7.22b)

where the proportionality constant depends on blood metabolism and viscosity given by: ${k}_{\mathrm{b}}=\frac{\pi }{32}{\left(\frac{K_{\mathrm{m}}}{\mu}\right)}^{1/2}$.

Appendix 2: ZKM (Zhou, Kassab, Molloi) Formulation

In the Zhou, Kassab, and Molloi (ZKM) model (Zhou et al., 1999), the cost function analogous with Murray (Appendix 1) consists of two terms: viscous flow and blood metabolic power dissipation. For a stem-crown unit, the cost function has the form:

$$ F\left(L,V\right)={P}_{\mathrm{vis}}\left(L,V\right)+{P}_{\mathrm{meta}}(V) $$

(7.23a)

where P_vis and P_meta correspond to the power dissipation due to viscosity and blood metabolism, respectively. Under the same assumptions as in Murray’s formulation, the cost function takes on the form:

$$ F\left(L,V\right)={Q}^2(L){R}_{\mathrm{c}}\left(L,V\right)+{K}_{\mathrm{m}}V $$

(7.23b)

where R_c is the crown resistance defined as the ratio of pressure difference (between inlet and outlet of crown) and flow rate into the crown; K_m is a metabolic constant for maintenance of blood. The length (L) and volume (V) of the crown are defined as the sum of all the individual vessel segments in the crown as:

$$ L=\sum \limits_{i=1}^N{l}_i $$

(7.24a)

and

$$ V=\sum \limits_{i=1}^N\frac{\pi {D}_i^2}{4}{l}_i $$

(7.24b)

where D and l are the diameter and length, respectively; and N represents the total number of segments i in the respective crown. Obviously, the blood vessels are assumed to be cylindrical in geometry.

Although it is easy to define the resistance of a single vessel segment in Murray’s formulation (i.e., $R=\frac{8\mu l}{\pi {r}^4}$), it is much more difficult to analytically express the equivalent resistance of the entire tree in the ZKM model because the vascular system is composed of millions of vascular segments: some coupled in series while others in parallel. To address this issue, a scaling relationship between the equivalent crown resistance, R_c, the crown volume, V, and crown length, L, is introduced as:

$$ \frac{R_{\mathrm{c}}}{R_{\mathrm{max}}}=\frac{{\left(L/{L}_{\mathrm{max}}\right)}^3}{{\left(V/{V}_{\mathrm{max}}\right)}^{\varepsilon^{\prime }}} $$

(7.25)

where R_max, L_max, and V_max are the resistance, length, and volume of the entire tree, respectively. The parameter ε′ is determined empirically by fitting Eq. (7.25) to the experimental data on morphometry of the vascular tree (length and volume) and a network analysis of flow distribution (resistance). The physical significance of this parameter will become apparent below.

In order to eliminate the dependence of cost function on flow rate, a proportional relationship between stem flow rate and crown length is introduced as:

$$ \frac{Q}{Q_{\mathrm{max}}}=\frac{L}{L_{\mathrm{max}}} $$

(7.26)

Equation (7.26) is validated using hemodynamic simulations (Zhou et al., 1999), in vivo experimental measurements (Zhou et al., 2002), and fractural arguments (Huo & Kassab, 2012). If Eqs. (7.23a)–(7.26) are combined and divided by the maximum metabolic power consumption, a nondimensional cost function can be obtained as:

$$ f=\frac{F\left(L,V\right)}{K_{\mathrm{m}}{V}_{\mathrm{m}\mathrm{ax}}}=\frac{Q_{\mathrm{m}\mathrm{ax}}^2{R}_{\mathrm{m}\mathrm{ax}}}{K_{\mathrm{m}}{V}_{\mathrm{m}\mathrm{ax}}}\frac{{\left(\frac{L}{L_{\mathrm{m}\mathrm{ax}}}\right)}^5}{{\left(\frac{V}{V_{\mathrm{m}\mathrm{ax}}}\right)}^{\varepsilon^{\prime }}}+\frac{V}{V_{\mathrm{m}\mathrm{ax}}} $$

(7.27)

Next, the cost function is minimized with respect to blood volume for a given crown length, i.e., $\frac{\partial f}{\partial \left(\frac{V}{V_{\mathrm{max}}}\right)}=0$, to yield the desired relation:

$$ \frac{{\left(\frac{L}{L_{\mathrm{m}\mathrm{ax}}}\right)}^5}{{\left(\frac{V}{V_{\mathrm{m}\mathrm{ax}}}\right)}^{\varepsilon^{\prime }+1}}\left(\frac{Q_{\mathrm{m}\mathrm{ax}}^2{R}_{\mathrm{m}\mathrm{ax}}}{K_{\mathrm{m}}{V}_{\mathrm{m}\mathrm{ax}}}\right){\varepsilon}^{\prime }=1 $$

(7.28)

Two important results follow from Eq. (7.28); the crown volume–length relation as expressed by:

$$ \frac{V}{V_{\mathrm{max}}}={\left(\frac{L}{L_{\mathrm{max}}}\right)}^{\frac{5}{\varepsilon^{\prime }+1}} $$

(7.29)

and the following equation for the crown resistance parameter as:

$$ {\varepsilon}^{\prime }=\frac{K_{\mathrm{m}}{V}_{\mathrm{m}\mathrm{ax}}}{Q_{\mathrm{m}\mathrm{ax}}^2{R}_{\mathrm{m}\mathrm{ax}}} $$

(7.30)

Hence, ε′ represents the ratio of maximum metabolic-to-viscous power dissipation for a given tree.

In the ZKM model (Zhou et al., 1999), conservation of energy imposed on a stem-crown system in conjunction with the results obtained above can yield stem-crown relationships. Conservation of energy requires that:

$$ \mathrm{Energy}\kern0.17em \mathrm{input}+\mathrm{Energy}\kern0.17em \mathrm{production}=\mathrm{Energy}\kern0.17em \mathrm{output}+\mathrm{Energy}\kern0.17em \mathrm{consumption} $$

(7.31a)

On the left hand side, there is work done by pressure, kinetic energy flowing into the system, and chemical energy into the system. The second term vanishes as no energy is produced in the system since the chemical energy is produced elsewhere in the body. On the right, pressure does work to the system, kinetic and chemical energy flows out, and mechanical and chemical energy is dissipated.

The inlet of the crown is the stem whose cross-sectional area is denoted by A_s. The outlet is the crown’s terminal vessels that have a mean area A_o. If the heat input from the stem to the crown, the work done through the vessel wall, the kinetic energy across the vessel wall, and the gravitational potential energy are assumed to be negligible, the conservation of energy can be written in terms of work rate and power dissipation as:

$$ {\displaystyle \begin{array}{ll}{\int}_{A_{\mathrm{S}}} pu\; dA-{\int}_{A_{\mathrm{O}}} pu\; dA+{C}_{\mathrm{meta}}=& {\int}_{A_{\mathrm{O}}}\frac{1}{2}\rho {q}^2u\; dA\\ {}& -{\int}_{A_{\mathrm{S}}}\frac{1}{2}\rho {q}^2u\; dA+{\int}_V\frac{\partial }{\partial t}\left[\frac{1}{2}\rho {q}^2\right] dV+{P}_{\mathrm{vis}}+{P}_{\mathrm{meta}}\end{array}} $$

(7.31b)

where p is the pressure, u the axial flow velocity, V the arterial volume of the crown, ρ the blood density in the coronary arteries, and q the flow speed. The flow speed q is the magnitude of the flow velocity which is in general a vector with three components u, v, and w in the x, y, and z directions, respectively, i.e., q = (u² + v² + w²)^1/2. The power dissipation due to viscosity and blood metabolism are denoted by P_vis and P_meta, respectively, as expressed in Eq. (7.23a). C_meta is the net (input–output) metabolic chemical power into the system which is related to the Gibb’s free energy and to the change of chemical potential. This term stems from the energy production (e.g., adenosine triphosphate, ATP) from various food sources (e.g., carbohydrates, fats, and proteins).

The first two terms on the right side of Eq. (7.31b) represent the rate of change of kinetic energy flowing in and out of the crown while the third term on the right side is the rate of change of kinetic energy in the crown volume. For a steady-state flow, assuming that the rate of change of kinetic energy flowing into the stem is much larger than that flowing out of the segment of crown, Eq. (7.31b) reduces to:

$$ {\int}_{A_{\mathrm{S}}} pu\; dA-{\int}_{A_{\mathrm{O}}} pu\; dA+{C}_{\mathrm{meta}}=-{\int}_{A_{\mathrm{S}}}\frac{1}{2}\rho {q}^2u\; dA+{P}_{\mathrm{vis}}+{P}_{\mathrm{meta}} $$

(7.32)

Using the condition that the inlet stem pressure and outlet crown pressure are uniform; and conservation of mass (flow in stem equals total flow out of crown), one obtains:

$$ \Delta pQ+{C}_{\mathrm{m}\mathrm{eta}}=-{\int}_{A_{\mathrm{S}}}\frac{1}{2}\rho {q}^2u\; dA+{Q}^2{R}_{\mathrm{c}}+{K}_{\mathrm{m}}V $$

(7.33)

where Δp is the pressure difference between the inlet (stem) and outlet of the crown (p_s − p_o). The second term on the left-hand side, representing the net chemical energy into the system, may typically be represented by energy or power per mole but can be converted into power/volume via the molecular weight and density of various energy sources. Since this term may be a very complex function (may vary with time and space), it is assumed that the total net chemical power (under steady-state conditions) into the system is constant and proportional to vascular volume (C_meta = K_cV). This assumption implies that the volumetric chemical reactions dominate the surface reactions. Under this assumption, Eq. (7.33) becomes

$$ \Delta pQ-{Q}^2{R}_{\mathrm{c}}=-{\int}_{A_{\mathrm{S}}}\frac{1}{2}\rho {q}^2u\; dA+{K}_{\mathrm{m}}^{\prime }V $$

(7.34a)

where ${K}_{\mathrm{m}}^{\prime }$ is K_m − K_c. The two terms on the left-hand side of Eq. (7.34a) are equal under the present assumption (Poiseuille flow). Hence, Eq. (7.34a) reduces to the following:

$$ {\int}_{A_{\mathrm{S}}}\frac{1}{2}\rho {q}^2u\; dA={K}_{\mathrm{m}}^{\prime }V $$

(7.34b)

For an incompressible, uniaxial laminar flow, q and u are equivalent and the integral in Eq. (7.34b) reduces to:

$$ {\int}_{A_{\mathrm{S}}}\frac{1}{2}\rho {q}^2u\; dA= a\rho \frac{Q^3}{A_{\mathrm{s}}^2} $$

(7.35)

or

$$ a\rho \frac{Q^3}{A_{\mathrm{s}}^2}={K}_{\mathrm{m}}V $$

(7.36)

where a is a numerical constant which depends on the shape of the velocity profile. Equations (7.26), (7.29), and (7.36) can be combined to yield:

$$ \frac{A}{A_{\mathrm{max}}}={\left(\frac{L}{L_{\mathrm{max}}}\right)}^{\frac{3{\varepsilon}^{\prime }-2}{2\left({\varepsilon}^{\prime }+1\right)}} $$

(7.37a)

or

$$ \frac{D}{D_{\mathrm{max}}}={\left(\frac{L}{L_{\mathrm{max}}}\right)}^{\frac{3{\varepsilon}^{\prime }-2}{4\left({\varepsilon}^{\prime }+1\right)}} $$

(7.37b)

Finally, the flow–diameter relation can be obtained by simply combining Eqs. (7.26) and (7.37a, 7.37b) as:

$$ \frac{Q}{Q_{\mathrm{max}}}={\left(\frac{D}{D_{\mathrm{max}}}\right)}^{\frac{4\left({\varepsilon}^{\prime }+1\right)}{3{\varepsilon}^{\prime }-2}} $$

(7.38)

The power-law is equivalent to Murray’s form (Eqs. (7.22a, 7.22b) in Appendix 1) but the exponent is not necessarily equal to 3, and instead depends on the ratio of metabolic-to-viscous power dissipation (Eq. 7.30).

Finally, Eqs. (7.27) and (7.29) lead to an analytical expression for the nondimensional cost function under optimal conditions, i.e., minimum power, as

$$ {f}_{\mathrm{min}}=\left(\frac{\varepsilon^{\prime }+1}{\varepsilon^{\prime }}\right){\left(\frac{L}{L_{\mathrm{max}}}\right)}^{\frac{5}{\varepsilon^{\prime }+1}} $$

(7.39)

It can be verified that the second derivative of the cost function is positive; thus, Eq. (7.39) represents a local minimum for the power dissipation.

Appendix 3: Validity of Scaling Laws in Various Organs and Species

Table 7.1 The parameter ε′ given by Eq. (7.25) and the parameters β, χ, and δ for the relationships V/V_max = (L/L_max)^β (Eq. 7.29), D/D_max = (L/L_max)^χ (Eq. 7.37b), and Q/Q_max = (D/D_max)^δ (Eq. 7.38), respectively

Appendix 4: Blood Vessel Wall Metabolism (Liu & Kassab, 2007a)

Murray’s law considers only one source of vascular metabolism, i.e., the power required to maintain blood volume denoted as P_b. This term is assumed to be proportional to blood volume V of a vessel segment (Murray, 1926), as

$$ {P}_{\mathrm{b}}={k}_{\mathrm{b}}V $$

(7.40)

where k_b is a blood metabolic coefficient. Zhou et al. (1999) extended this proportionality relationship for the blood metabolism in the stem-crown unit (SCU) of a vascular tree. The extension is based on the assumption that the metabolic rate of unit volume of blood is independent of its spatial position in a vascular tree. Therefore, V in Eq. (7.40) becomes the cumulative crown volume of a SCU.

7.1.1 Metabolism in Vessel Wall

There are two major sources of energy cost to maintain the vessel wall (Taber, 1998). For a vessel segment with radius R, length L, and wall thickness H, the basal metabolic energy is assumed proportional to the volume of vessel wall, i.e., P_wp = α ⋅ 2πRLH, where α is a passive metabolic parameter. There are also smooth muscle cells which generate active contraction stress σ_a to balance the blood pressure p (Paul, 1980; Taber, 1998). According to Laplace’s law and assuming σ_a contributes f(0 < f ≤ 1) of the total wall stress, Taber (1998) formulated the cost due to vasomotor tone as P_wa = fβp ⋅ 2πR²L, where β is an active metabolic parameter. Thus, the metabolic consumption in a vessel wall segment is given by:

$$ {P}_{\mathrm{w}}={P}_{\mathrm{w}\mathrm{p}}+{P}_{\mathrm{w}\mathrm{a}}=\left(\overline{H}\alpha + f\beta p\right)\cdot 2\pi {R}^2L $$

(7.41)

in which $\overline{H}=H/R$ denotes the thickness-to-radius ratio.

For the ZKM, the total metabolism of a SCU with N vessel segments is thus the sum over all the segments as:

$$ {P}_{\mathrm{w}}=\sum \limits_{n=1,N}\left({\overline{H}}_n{\alpha}_n+{f}_n{\beta}_n{p}_n\right)\cdot 2\pi {R}_n^2{L}_n $$

(7.42)

where subscript “n” denotes quantities of the nth segments. Note that p_n should be considered as the mean pressure of the nth segment. Metabolic parameters α_n, β_n, and f_n may vary among all the vessel segments in a SCU.

7.1.2 Metabolism in Stem-Crown Unit

The total metabolic cost in a SCU is thus the sum of P_b and P_w. While P_b (Eq. 7.40) is proportional to the crown volume V, the form of P_w (Eq. 7.42) is more complex and does not necessarily obey proportionality. For the passive basal metabolism, the wall thickness has been found to depend linearly on the radius along a vascular tree. For instance, H = 8.22 × 10⁻³R + 3.2 (in μm) is reported by Guo and Kassab (2004) for pig left anterior descending (LAD). Metabolic parameter α depends on the composition of the wall and may change along the tree. In the absence of experimental data, it is assumed that α_n in Eq. (7.42) is relatively uniform over the three order of magnitude variation in diameter over the vascular tree, i.e., α_n ≈ α. Thus, the basal metabolic cost of a SCU is rewritten as:

$$ {P}_{\mathrm{wp}}=\sum \limits_{n=1,N}{\overline{H}}_n{\alpha}_n\cdot 2\pi {R}_n^2{L}_n=\left(\frac{1}{V}2\alpha \cdot \sum \limits_{n=1,N}\overline{H}\pi {R}_n^2{L}_n\right)V={k}_{\mathrm{wp}}V $$

(7.43)

where $V={\sum}_{n=1,N}\pi {R}_n^2{L}_n$. Note that Taber (1998) approximated that $\overline{H}$ is constant, and thus ${k}_{\mathrm{wp}}=2\overline{H}\alpha$ is also a constant, showing that P_wp is approximately proportional to the arterial volume V. In Eq. (7.43), k_wp may vary along a vascular tree, as the relation between thickness and radius is no longer proportional.

Next, P_wa due to active wall stress is considered. Similarly to Eq. (7.43), it is rewritten as:

$$ {P}_{\mathrm{wa}}=\sum \limits_{n=1,N}\left(2{f}_n{\beta}_n{p}_n\right)\pi {R}_n^2{L}_n=\left(\frac{1}{V}\sum \limits_{n=1,N}\left(2{f}_n{\beta}_n{p}_n\right)\pi {R}_n^2{L}_n\right)V={k}_{\mathrm{wa}}V $$

(7.44)

where k_wa depends on the exact profile of blood pressure and parameters β_n and f_n in a SCU. Therefore, Eq. (7.44) does not indicate that P_wa is proportional to the arterial volume V. In the following experimentally based calculations, however, it is found that k_wa is nearly constant for all SCUs in a vascular tree under certain assumptions. Thus, the total metabolic consumption in a SCU is formulated as:

$$ P=\left({k}_{\mathrm{b}}+{k}_{\mathrm{w}\mathrm{p}}+{k}_{\mathrm{w}\mathrm{a}}\right)V=\left({k}_{\mathrm{b}}+{k}_{\mathrm{w}}\right)V={k}_{\mathrm{meta}}V $$

(7.45)

where k_meta is approximately a constant only if the wall metabolic coefficient k_w = k_wa + k_wp does not change significantly over various SCUs in a vascular tree as observed in Table 7.2, where the coefficient of variation ranges from approximately 2–20% over a six-fold difference in α.

Table 7.2 Wall metabolic coefficients k_wa, k_wp, and k_w of stem-crown units in pig left common coronary artery

Appendix 5: Scaling Law of Crown Resistance (Huo & Kassab, 2009b)

An idealized symmetric crown distal to the stem is composed of N_total levels or generations from the stem (level zero) to each terminal (the smallest arteriolar bifurcation, level N_total). The resistance of a symmetric crown, R_c, can be written as:

$$ {R}_{\mathrm{c}}={R}_{\mathrm{s}}+\sum \limits_{i=1}^{N_{\mathrm{total}}}\frac{R_i}{N_i};\kern1em {R}_i=\frac{128\mu {L}_i}{\pi {D}_i^4}={K}_{\mathrm{s}}\frac{L_i}{D_i^4}\kern1em i=1,\dots, {N}_{\mathrm{total}} $$

(7.46)

where

$$ {R}_{\mathrm{s}}=\frac{128\mu {L}_{\mathrm{s}}}{\pi {D}_{\mathrm{s}}^4}={K}_{\mathrm{s}}\frac{L_{\mathrm{s}}}{D_{\mathrm{s}}^4}\kern1em \mathrm{and}\kern1em {K}_{\mathrm{s}}=\frac{128\mu }{\pi } $$

R_s, L_s, and D_s are the resistance, length, and diameter of the stem, respectively, and K_s is a constant. Similarly, R_i, L_i, and D_i are the resistance, length, and diameter of a vessel in level i, respectively, and N_i is the total number of vessels in level i. Here, the effect of viscosity is neglected because the capillary network is not included in the analysis, where the viscosity has a significant effect. Equation (7.46) can be written as:

$$ {R}_{\mathrm{c}}={K}_{\mathrm{s}}\left(\frac{L_{\mathrm{s}}}{{\left({D}_{\mathrm{s}}^2\right)}^2}+\sum \limits_{i=1}^{N_{\mathrm{total}}}\frac{N_i{L}_i}{{\left({N}_i{D}_i^2\right)}^2}\right)={K}_{\mathrm{s}}\left(\frac{L_{\mathrm{s}}}{{\left({D}_{\mathrm{s}}^2\right)}^2}+\sum \limits_{i=1}^{N_{\mathrm{total}}}\frac{N_i{L}_i}{{\left({D}_{\mathrm{s}}^2\right)}^2\cdot {\left(\frac{N_i{D}_i^2}{D_{\mathrm{s}}^2}\right)}^2}\right) $$

(7.47)

To complete the derivation, the following three definitions are introduced:

• 1. Branching Ratio: The branching ratios (BR = N_i/N_i−1, i = 1, … , N_total) are relatively constant in each level from the stem (level 0) to the smallest arterioles or venules (level N_total) within an organ of a given species, such that N_i = BRⁱ.

• 2. Diameter Ratio: The diameter ratio is defined as: DR = D_i/D_i−1, i = 1, … , N_total. It can be shown that ${N}_i\pi {D}_i^{2+\varepsilon }={N}_{i-1}\pi {D}_{i-1}^{2+\varepsilon }$, where ε = 0 represents ${N}_i\pi {D}_i^2={N}_{i-1}\pi {D}_{i-1}^2$ area-preservation from one level to the next. Conversely, ε = 1 represents Murray’s law, i.e., ${N}_i\pi {D}_i^3={N}_{i-1}\pi {D}_{i-1}^3$. This provides the relation: $\left(\frac{D_i}{D_{i-1}}\right)={\left(\frac{N_i}{N_{i-1}}\right)}^{\frac{1}{2+\varepsilon }}$. Therefore, the diameter ratio relates to the branching ratio as: $\mathrm{DR}={\mathrm{BR}}^{-\frac{1}{2+\varepsilon }}$ or ${D}_i={\mathrm{BR}}^{\frac{i}{2+\mathrm{\mathcal{E}}}}{D}_{\mathrm{s}}$.

• 3. Length Ratio: The length ratio is defined as: LR = L_i/L_i−1, i = 1, … , N_total. West et al. (1997) proposed that the perfused volume from one level to the next is approximately unchanged, so that $\frac{4}{3}\pi {\left(\frac{L_i}{2}\right)}^3{N}_i=\frac{4}{3}\pi {\left(\frac{L_{i-1}}{2}\right)}^3{N}_{i-1}$, which leads to the relation: $\left(\frac{L_i}{L_{i-1}}\right)={\left(\frac{N_i}{N_{i-1}}\right)}^{\frac{1}{3}}$. Therefore, the relation between length ratio and branching ratio can be expressed as $\mathrm{LR}={\mathrm{BR}}^{-\frac{1}{3}}$ or ${L}_i={\mathrm{BR}}^{-\frac{1}{3}}{L}_{\mathrm{s}}$.

From Equations N_i = BRⁱ, ${D}_i={\mathrm{BR}}^{-\frac{i}{2+\mathrm{\mathcal{E}}}}{D}_{\mathrm{s}}$, ${L}_i={\mathrm{BR}}^{-\frac{i}{3}}{L}_{\mathrm{s}}$, and (7.47), the following can be obtained as follows:

$$ {\displaystyle \begin{array}{ll}{R}_{\mathrm{c}}& =\frac{K_{\mathrm{s}}}{{\left({D}_{\mathrm{s}}^2\right)}^2}\left({L}_{\mathrm{s}}+{L}_{\mathrm{s}}\cdot \sum \limits_{i=1}^{N_{\mathrm{total}}}\frac{{\mathrm{BR}}^i\cdot {\mathrm{BR}}^{-\frac{i}{3}}}{{\left({\mathrm{BR}}^i\cdot {\mathrm{BR}}^{\frac{-2i}{2+\varepsilon }}\right)}^2}\right)=\frac{K_{\mathrm{s}}\cdotp {L}_{\mathrm{s}}}{{\left({D}_{\mathrm{s}}^2\right)}^2}\left(1+\sum \limits_{i=1}^{N_{\mathrm{total}}}\frac{{\mathrm{BR}}^{\frac{2i}{3}}}{{\mathrm{BR}}^{\frac{2 i\varepsilon}{2+\varepsilon }}}\right)\\ {}& =\frac{K_{\mathrm{s}}\cdotp {L}_{\mathrm{s}}}{{\left({D}_{\mathrm{s}}^2\right)}^2}\left(1+\sum \limits_{i=1}^{N_{\mathrm{total}}}{\mathrm{BR}}^{\left(\frac{2}{3}-\frac{2\varepsilon }{2+\varepsilon}\right)\cdotp i}\right)\end{array}} $$

(7.48)

Equation (7.48) relates the crown resistance to the branching ratio of vascular tree. From Taylor expansion, it is known that $\frac{1}{1-x}=1+x+{x}^2+{x}^3+{x}^4+\cdots$ for −1 < x < 1. When 0 ≤ ε < 1, the last term $\left(1+{\sum}_{i=1}^{N_{\mathrm{total}}}{\mathrm{BR}}^{\left(\frac{2}{3}-\frac{2\varepsilon }{2+\varepsilon}\right)\cdotp i}\right)$ can be written as ${\mathrm{BR}}^{\left(\frac{2}{3}-\frac{2\varepsilon }{2+\varepsilon}\right)\cdot {N}_{\mathrm{total}}}\cdot \left(1+{\sum}_{i=1}^{N_{\mathrm{total}}}{\left(\frac{1}{{\mathrm{BR}}^{\left(\frac{2}{3}-\frac{2\varepsilon }{2+\varepsilon}\right)}}\right)}^i\right)$ with $-1<\left(\frac{1}{{\mathrm{BR}}^{\left(\frac{2}{3}-\frac{2\varepsilon }{2+\varepsilon}\right)}}\right)<1$. With Taylor expansion, Eq. (7.48) can be written as:

$$ {\displaystyle \begin{array}{l}{R}_{\mathrm{c}}=\frac{K_{\mathrm{s}}\cdotp {L}_{\mathrm{s}}}{{\left({D}_{\mathrm{s}}^2\right)}^2}\cdot {\mathrm{BR}}^{\left(\frac{2}{3}-\frac{2\varepsilon }{2+\varepsilon}\right)\cdotp {N}_{\mathrm{total}}}\cdot \left(1+\sum \limits_{i=1}^{N_{\mathrm{total}}}{\left(\frac{1}{{\mathrm{BR}}^{\left(\frac{2}{3}-\frac{2\varepsilon }{2+\varepsilon}\right)}}\right)}^i\right)\\ {}=\frac{K_{\mathrm{s}}\cdotp {L}_{\mathrm{s}}}{{\left({D}_{\mathrm{s}}^2\right)}^2}\cdot {\mathrm{BR}}^{\left(\frac{2}{3}-\frac{2\varepsilon }{2+\varepsilon}\right)\cdotp {N}_{\mathrm{total}}}\cdot \frac{1}{1-\frac{1}{{\mathrm{BR}}^{\left(\frac{2}{3}-\frac{2\varepsilon }{2+\varepsilon}\right)}}}\\ {}=\frac{K_{\mathrm{s}}\cdotp {L}_{\mathrm{s}}}{{\left({D}_{\mathrm{s}}^2\right)}^2}\cdot {\mathrm{BR}}^{\frac{2}{3}\cdot {N}_{\mathrm{total}}}\cdot {\mathrm{BR}}^{\left(-\frac{2\varepsilon }{2+\varepsilon}\right)\cdotp {N}_{\mathrm{total}}}\cdot \frac{1}{1-\left(\frac{1}{{\mathrm{BR}}^{\left(\frac{2}{3}-\frac{2\varepsilon }{2+\varepsilon}\right)}}\right)}\end{array}} $$

(7.49)

The crown length is defined as the sum of the lengths of each vessel in the crown, such that ${L}_{\mathrm{c}}={L}_{\mathrm{s}}+{\sum}_{i=1}^{N_{\mathrm{total}}}{N}_i{L}_i$. Based on a similar Taylor expansion of Eq. (7.49), the crown length can be written as:

$$ {L}_{\mathrm{c}}={L}_{\mathrm{s}}\cdot \left(1+\sum \limits_{i=1}^{N_{\mathrm{total}}}{\left({\mathrm{BR}}^{\frac{2}{3}}\right)}^i\right)={L}_{\mathrm{s}}\cdotp {\mathrm{BR}}^{\frac{2}{3}{N}_{\mathrm{total}}}\cdot \frac{1}{1-\frac{1}{{\mathrm{BR}}^{\frac{2}{3}}}} $$

(7.50)

From Eqs. (7.49) and (7.50), one obtains the following equation:

$$ {\displaystyle \begin{array}{l}{R}_{\mathrm{c}}=\frac{K_{\mathrm{s}}\cdotp {L}_{\mathrm{s}}}{{\left({D}_{\mathrm{s}}^2\right)}^2}\cdot {\mathrm{BR}}^{\frac{2}{3}\cdot {N}_{\mathrm{total}}}\cdotp \frac{1}{1-\left(\frac{1}{{\mathrm{BR}}^{\frac{2}{3}}}\right)}\cdotp \left({\mathrm{BR}}^{\left(-\frac{2\varepsilon }{2+\varepsilon}\right)\cdotp {N}_{\mathrm{total}}}.\frac{\left(1-\frac{1}{{\mathrm{BR}}^{\frac{2}{3}}}\right)}{1-\left(\frac{1}{{\mathrm{BR}}^{\left(\frac{2}{3}-\frac{2\varepsilon }{2+\varepsilon}\right)}}\right)}\right)\\ {}=\frac{K_{\mathrm{s}}\cdotp {L}_{\mathrm{c}}}{D_{\mathrm{s}}^4}\cdot \left({\mathrm{BR}}^{\left(-\frac{2\varepsilon }{2+\varepsilon}\right)\cdotp {N}_{\mathrm{total}}}.\frac{{\mathrm{BR}}^{\frac{2}{3}}-1}{{\mathrm{BR}}^{\frac{2}{3}}-{\mathrm{BR}}^{\left(\frac{2\varepsilon }{2+\varepsilon}\right)}}\right)\end{array}} $$

(7.51)

When ε = 1, this expression reduces to:

$$ {R}_{\mathrm{c}}=\frac{K_{\mathrm{s}}\cdotp {L}_{\mathrm{c}}}{D_{\mathrm{s}}^4}\cdot \left({\mathrm{BR}}^{-\frac{2}{3}{N}_{\mathrm{total}}}\cdotp \frac{{\mathrm{BR}}^{\frac{2}{3}}-1}{{\mathrm{BR}}^{\frac{2}{3}}}\cdotp \left({N}_{\mathrm{total}}+1\right)\right) $$

(7.52)

When ε > 1, the following holds:

$$ {R}_{\mathrm{c}}=\frac{K_{\mathrm{s}}\cdotp {L}_{\mathrm{c}}}{D_{\mathrm{s}}^4}\cdot \left({\mathrm{BR}}^{-\frac{2}{3}{N}_{\mathrm{total}}}\cdot \frac{{\mathrm{BR}}^{\frac{2}{3}}-1}{{\mathrm{BR}}^{\frac{2}{3}}-{\mathrm{BR}}^{\left(\frac{4}{3}-\frac{2\varepsilon }{2+\varepsilon}\right)}}\right) $$

(7.53)

If one defines ${K}_{\varepsilon }=\left({\mathrm{BR}}^{\left(-\frac{2\varepsilon }{2+\varepsilon}\right)\cdotp {N}_{\mathrm{total}}}\cdot \frac{{\mathrm{BR}}^{\frac{2}{3}}-1}{{\mathrm{BR}}^{\frac{2}{3}}-{\mathrm{BR}}^{\left(\frac{2\varepsilon }{2+\varepsilon}\right)}}\right)$, $\left({\mathrm{BR}}^{-\frac{2}{3}{N}_{\mathrm{total}}}\cdot \frac{{\mathrm{BR}}^{\frac{2}{3}}-1}{{\mathrm{BR}}^{\frac{2}{3}}}\cdot \left({N}_{\mathrm{total}}+1\right)\right)$, $\left({\mathrm{BR}}^{-\frac{2}{3}{N}_{\mathrm{total}}}\cdot \frac{{\mathrm{BR}}^{\frac{2}{3}}-1}{{\mathrm{BR}}^{\frac{2}{3}}-{\mathrm{BR}}^{\left(\frac{4}{3}-\frac{2\varepsilon }{2+\varepsilon}\right)}}\right)$ for 0 ≤ ε < 1, ε = 1, and ε > 1, respectively, Eqs. (7.51–7.53) can be written as:

$$ {R}_{\mathrm{c}}={K}_{\mathrm{s}}{K}_{\varepsilon}\frac{L_{\mathrm{c}}}{D_{\mathrm{s}}^4} $$

(7.54)

Finally, if K_c = K_sK_ε, Eq. (7.54) can be written as:

$$ {R}_{\mathrm{c}}={K}_{\mathrm{c}}\frac{L_{\mathrm{c}}}{D_{\mathrm{s}}^4} $$

(7.55)

where K_c depends on the branching ratio, diameter ratio, total number of tree generations, and blood viscosity.

It should be noted that although K_ε is a constant for a given crown, it does vary over the crowns due to N_total. Although modest variation of K_c is found for most vasculatures (factor of 5) except for the pulmonary tree (factor of 10), this is negligible variation given the range of variables is very large (8 orders of magnitude on both x-axis and y-axis). In such a broad range, the relatively small variation of K_c can be neglected as verified by the present validation.

Table 7.3 Parameters A₁ in Eq. (7.5b) and (K_s/K_c)_ML in Eq. (7.6) of this chapter with correlation coefficient calculated from the Marquardt-Levenberg algorithm, respectively, for various species

Appendix 6: Scaling Laws of Mass

Table 7.4 A least squares fit as well as a scaling model fit (Y = Y₀m^b) of the morphometry-mass data of swine hearts

Appendix 7: Validation of Volume Scaling Law

A cost function is proposed for a crown, F_c, consistent with formulation in Appendix 1:

$$ {F}_{\mathrm{c}}={Q}_{\mathrm{s}}\cdotp \Delta {P}_{\mathrm{c}}+{K}_{\mathrm{m}}{V}_{\mathrm{c}}={Q}_{\mathrm{s}}^2\cdot {R}_{\mathrm{c}}+{K}_{\mathrm{m}}{K}_{\mathrm{v}}{D}_{\mathrm{s}}^{2/3}{L}_{\mathrm{c}} $$

(7.56)

where Q_s and ΔP_c = Q_s ⋅ R_c are the flow rate through the stem and the pressure drop in the distal crown, respectively, and K_m is a metabolic constant of blood in a crown. Since ${R}_{\mathrm{c}}={K}_{\mathrm{c}}\frac{L_{\mathrm{c}}}{D_{\mathrm{s}}^4}$ (Appendix 5), the cost function of a crown tree in Eq. (7.56) can be written as:

$$ {F}_{\mathrm{c}}={Q}_{\mathrm{s}}^2\cdotp {R}_{\mathrm{c}}+{K}_{\mathrm{m}}{K}_{\mathrm{v}}{D}_{\mathrm{s}}^{2/3}{L}_{\mathrm{c}}={K}_{\mathrm{c}}{Q}_{\mathrm{s}}^2\frac{L_{\mathrm{c}}}{D_{\mathrm{s}}^4}+{K}_{\mathrm{m}}{K}_{\mathrm{v}}{D}_{\mathrm{s}}^{2/3}{L}_{\mathrm{c}} $$

(7.57)

Equation (7.57) can be normalized by the metabolic power requirements of the entire tree of interest, ${K}_{\mathrm{m}}{V}_{\mathrm{m}\mathrm{ax}}={K}_{\mathrm{m}}{K}_{\mathrm{v}}{D}_{\mathrm{m}\mathrm{ax}}^{2/3}{L}_{\mathrm{m}\mathrm{ax}}$, to obtain:

$$ {\displaystyle \begin{array}{c}{f}_{\mathrm{c}}=\frac{F_{\mathrm{c}}}{K_{\mathrm{m}}{K}_{\mathrm{v}}{D}_{\mathrm{m}\mathrm{ax}}^{2/3}{L}_{\mathrm{m}\mathrm{ax}}}\\ {}=\frac{Q_{\mathrm{m}\mathrm{ax}}^2{R}_{\mathrm{m}\mathrm{ax}}}{K_{\mathrm{m}}{K}_{\mathrm{v}}{D}_{\mathrm{m}\mathrm{ax}}^{2/3}{L}_{\mathrm{m}\mathrm{ax}}}{\left(\frac{Q_{\mathrm{s}}}{Q_{\mathrm{m}\mathrm{ax}}}\right)}^2\cdotp \frac{\left({L}_{\mathrm{c}}/{L}_{\mathrm{m}\mathrm{ax}}\right)}{{\left({D}_{\mathrm{s}}/{D}_{\mathrm{m}\mathrm{ax}}\right)}^4}+{\left(\frac{D_{\mathrm{s}}}{D_{\mathrm{m}\mathrm{ax}}}\right)}^{2/3}\left(\frac{L_{\mathrm{c}}}{L_{\mathrm{m}\mathrm{ax}}}\right)\end{array}} $$

(7.58)

where f_c is the nondimensional cost function. Stem flow and crown length are linearly related as:

$$ {Q}_{\mathrm{s}}={K}_{\mathrm{Q}}{L}_{\mathrm{c}}\Rightarrow \frac{Q_{\mathrm{s}}}{Q_{\mathrm{max}}}=\frac{L_{\mathrm{c}}}{L_{\mathrm{max}}} $$

(7.59)

where K_Q is a flow-crown length constant. When Eq. (7.59) is applied to Eq. (7.58), the dimensionless cost function can be written as:

$$ {f}_{\mathrm{c}}=\frac{Q_{\mathrm{m}\mathrm{ax}}^2{R}_{\mathrm{m}\mathrm{ax}}}{K_{\mathrm{m}}{K}_{\mathrm{v}}{D}_{\mathrm{m}\mathrm{ax}}^{2/3}{L}_{\mathrm{m}\mathrm{ax}}}\cdotp \frac{{\left({L}_{\mathrm{c}}/{L}_{\mathrm{m}\mathrm{ax}}\right)}^3}{{\left({D}_{\mathrm{s}}/{D}_{\mathrm{m}\mathrm{ax}}\right)}^4}+{\left(\frac{D_{\mathrm{s}}}{D_{\mathrm{m}\mathrm{ax}}}\right)}^{2/3}\left(\frac{L_{\mathrm{c}}}{L_{\mathrm{m}\mathrm{ax}}}\right) $$

(7.60)

Similar to Murray’s law (Murray, 1926), the cost function can be minimized with respect to diameter at a fixed L_c/L_max to obtain the following:

$$ {\displaystyle \begin{array}{ll}\frac{\partial {f}_{\mathrm{c}}}{\partial \left(\frac{D_{\mathrm{s}}}{D_{\mathrm{m}\mathrm{ax}}}\right)}& =0\Rightarrow \frac{\left(-4\right){Q}_{\mathrm{m}\mathrm{ax}}^2{R}_{\mathrm{m}\mathrm{ax}}}{K_{\mathrm{m}}{K}_{\mathrm{v}}{D}_{\mathrm{m}\mathrm{ax}}^{2/3}{L}_{\mathrm{m}\mathrm{ax}}}\cdot \frac{{\left({L}_{\mathrm{c}}/{L}_{\mathrm{m}\mathrm{ax}}\right)}^3}{{\left({D}_{\mathrm{s}}/{D}_{\mathrm{m}\mathrm{ax}}\right)}^5}\\ {}& =-\left(\frac{2}{3}\right){\left(\frac{D_{\mathrm{s}}}{D_{\mathrm{m}\mathrm{ax}}}\right)}^{\frac{2}{3}-1}\left(\frac{L_{\mathrm{c}}}{L_{\mathrm{m}\mathrm{ax}}}\right)\\ {}& \Rightarrow \frac{6{Q}_{\mathrm{m}\mathrm{ax}}^2{R}_{\mathrm{m}\mathrm{ax}}}{K_{\mathrm{m}}{K}_{\mathrm{v}}{D}_{\mathrm{m}\mathrm{ax}}^{2/3}{L}_{\mathrm{m}\mathrm{ax}}}\cdot {\left(\frac{L_{\mathrm{c}}}{L_{\mathrm{m}\mathrm{ax}}}\right)}^2={\left(\frac{D_{\mathrm{s}}}{D_{\mathrm{m}\mathrm{ax}}}\right)}^{4+\frac{2}{3}}\end{array}} $$

(7.61)

Equation (7.61) applies to any stem-crown unit. When L_c = L_max and D_s = D_max in Eq. (7.61), the following holds:

$$ \frac{6{Q}_{\mathrm{m}\mathrm{ax}}^2{R}_{\mathrm{m}\mathrm{ax}}}{K_{\mathrm{m}}{K}_{\mathrm{v}}{D}_{\mathrm{m}\mathrm{ax}}^{2/3}{L}_{\mathrm{m}\mathrm{ax}}}=1\Rightarrow \frac{Q_{\mathrm{m}\mathrm{ax}}^2{R}_{\mathrm{m}\mathrm{ax}}}{K_{\mathrm{m}}{K}_{\mathrm{v}}{D}_{\mathrm{m}\mathrm{ax}}^{2/3}{L}_{\mathrm{m}\mathrm{ax}}}=\frac{1}{6} $$

(7.62)

Therefore, Eq. (7.62) can be written as:

$$ \left(\frac{D_{\mathrm{s}}}{D_{\mathrm{max}}}\right)={\left(\frac{L_{\mathrm{c}}}{L_{\mathrm{max}}}\right)}^{\frac{3}{7}} $$

(7.63)

From Eqs. (7.24b) to (7.63), one desired relation is obtained as:

$$ \left(\frac{V_{\mathrm{c}}}{V_{\mathrm{max}}}\right)={\left(\frac{L_{\mathrm{c}}}{L_{\mathrm{max}}}\right)}^{1\frac{2}{7}} $$

(7.64)

From Eqs. (7.59) and (7.63), another desired relation is obtained:

$$ \left(\frac{Q_{\mathrm{s}}}{Q_{\mathrm{max}}}\right)={\left(\frac{D_{\mathrm{s}}}{D_{\mathrm{max}}}\right)}^{2\frac{1}{3}} $$

(7.65)

where Q_max is the flow rate through the most proximal stem. From Eqs. (7.63) to (7.64), the following relation holds:

$$ \left(\frac{V_{\mathrm{c}}}{V_{\mathrm{max}}}\right)={\left(\frac{D_{\mathrm{s}}}{D_{\mathrm{max}}}\right)}^3 $$

(7.66)

Equations (7.63–7.66) are the structure–function scaling relations in the vascular tree, based on the “Minimum Energy Hypothesis.” Equations (7.63), (7.64), and (7.66) represent the diameter–length, volume–length, and volume–diameter relations, respectively, and Eq. (7.65) represents the general Murray’s law in the entire tree. Tables 7.5, 7.6, 7.7 provide validations of the above relations based on vascular morphometric data of various organs and species.

Table 7.5 Scaling law can be represented in the form Y = A ⋅ X^B, where X and Y are defined such that A and B should have theoretical values of unity

Table 7.6 Parameters B (obtained from least squares fits) and A (obtained from nonlinear regression with B constrained to one) in various organs similar to Table 7.5

Table 7.7 The parameter A obtained from nonlinear regression in various organs when (diameter–length, volume–length, flow–diameter, and volume–diameter relations, respectively) are represented by Y = A ⋅ X^B

Appendix 8: Summary of Horton’s Law for Various Vascular Trees

Reproduced from Kassab (2000) with permission.

Table 7.8 A summary of length, diameter, and branching ratios (R_L, R_D, and R_N, respectively) for various vascular trees

Appendix 9: Fractal-Based Derivation of Volume–Diameter Scaling Law

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) (Huo & Kassab, 2012). The volume of a crown, V_c, can be written as:

$$ {V}_{\mathrm{c}}={V}_{\mathrm{s}}+\sum \limits_{i=1}^{N_l}{n}_i{V}_i;\kern1em {V}_i=\frac{\pi }{4}{D}_i^2{L}_i;\kern1em i=1,\dots {N}_i $$

(7.67)

where

$$ {V}_{\mathrm{s}}=\frac{\pi }{4}{D}_{\mathrm{s}}^2{L}_{\mathrm{s}} $$

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 (7.67) can be written as:

$$ {V}_{\mathrm{c}}=\frac{\pi }{4}{D}_{\mathrm{s}}^2{L}_{\mathrm{s}}\left(1+\sum \limits_{i=1}^{N_1}{n}_i{\left(\frac{D_i}{D_{\mathrm{s}}}\right)}^2\left(\frac{L_i}{L_{\mathrm{s}}}\right)\right) $$

(7.68)

Appendix 5 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 $\mathrm{DR}={\mathrm{BR}}^{\frac{-1}{2.64\pm 0.64}}$ and $\mathrm{LR}={\mathrm{BR}}^{\frac{-1}{2.55\pm 0.49}}$. Parameters ε and γ equal 0.64 ± 0.64 and 0.45 ± 0.49, respectively, which are significantly different from zero (P ≪ 0.05).

From n_i = BRⁱ, ${D}_i={\mathrm{BR}}^{-\frac{i}{2+\varepsilon }}{D}_{\mathrm{s}}$, ${L}_i={\mathrm{BR}}^{-\frac{i}{3-\gamma }}{L}_{\mathrm{s}}$, and Eq. (7.68), the following equation can be obtained:

$$ {\displaystyle \begin{array}{c}{V}_{\mathrm{c}}=\frac{\pi }{4}{D}_{\mathrm{s}}^2{L}_{\mathrm{s}}\left(1+\sum \limits_{i=1}^{N_{\mathrm{i}}}{\mathrm{BR}}^i{\left({\mathrm{BR}}^{-\frac{i}{2+\varepsilon }}\right)}^2{\mathrm{BR}}^{-\frac{i}{3-\gamma }}\right)\\ {}=\frac{\pi }{4}{D}_{\mathrm{s}}^3\frac{L_{\mathrm{s}}}{D_{\mathrm{s}}}\left(1+\sum \limits_{i=1}^{N_{\mathrm{i}}}{\mathrm{BR}}^{i\left(\frac{2-\gamma }{3-\gamma }-\frac{2}{2+\varepsilon}\right)}\right)\end{array}} $$

(7.69)

Equation (7.69) relates the crown volume to the branching ratio of vascular tree. Since $\frac{2-\gamma }{3-\gamma }-\frac{2}{2+\varepsilon }<0$ for most vascular trees (mean of ≈ −0.15 for vascular trees of various organs), the last term in Eq. (7.69) has $0<{\mathrm{BR}}^{i\left(\frac{2-\gamma }{3-\gamma }-\frac{2}{2+\varepsilon}\right)}<1$. This implies a decrease of total blood volume of vessels in each level from the stem (level zero) to the terminal (level N_i) as supported by experimental data (Kassab, Rider, Tang, & Fung, 1993). Equation (7.69) is written as:

$$ {V}_{\mathrm{c}}=\frac{\pi }{4}{D}_{\mathrm{s}}^3\frac{L_{\mathrm{s}}}{D_{\mathrm{s}}}\frac{1-{\mathrm{BR}}^{\left(\frac{2-\gamma }{3-\gamma }-\frac{2}{2+\varepsilon}\right)\left({N}_i+1\right)}}{1-{\mathrm{BR}}^{\left(\frac{2-\gamma }{3-\gamma }-\frac{2}{2+\varepsilon}\right)}}\kern1em \frac{2-\gamma }{3-\gamma }-\frac{2}{2+\varepsilon }<0 $$

(7.70)

Since ${D}_i={\mathrm{BR}}^{-\frac{i}{2+\varepsilon }}{D}_{\mathrm{s}}$ and ${L}_i={\mathrm{BR}}^{-\frac{i}{3-\gamma }}{L}_{\mathrm{s}}$, Eq. (7.70) can be written as:

$$ {\displaystyle \begin{array}{ll}{V}_{\mathrm{c}}& =\frac{\pi }{4}{D}_{\mathrm{s}}^3\frac{{\left({L}_{\mathrm{s}}\right)}_{N_i}}{{\left({D}_{\mathrm{s}}\right)}_{N_i}}{\mathrm{BR}}^{\left(\frac{1}{3-\gamma }-\frac{1}{2+\varepsilon}\right){N}_i}\frac{1-{\mathrm{BR}}^{\left(\frac{2-\gamma }{3-\gamma }-\frac{2}{2+\varepsilon}\right)\left({N}_i+1\right)}}{1-{\mathrm{BR}}^{\left(\frac{2-\gamma }{3-\gamma }-\frac{2}{2+\varepsilon}\right)}}\\ {}& =\frac{\pi }{4}{D}_{\mathrm{s}}^3\frac{{\left({L}_{\mathrm{s}}\right)}_{N_i}}{{\left({D}_{\mathrm{s}}\right)}_{N_i}}\frac{{\mathrm{BR}}^{\left(\frac{1}{3-\gamma }-\frac{1}{2+\varepsilon}\right)\left({N}_i+1\right)}-{\mathrm{BR}}^{\left(\frac{\varepsilon -1}{2+\varepsilon}\right)\left({N}_i+1\right)}}{{\mathrm{BR}}^{\left(\frac{1}{3-\gamma }-\frac{1}{2+\varepsilon}\right)}-{\mathrm{BR}}^{\left(\frac{\varepsilon -1}{2+\varepsilon}\right)}}\end{array}} $$

(7.71)

where ${\left({L}_{\mathrm{s}}\right)}_{N_i}$ and ${\left({D}_{\mathrm{s}}\right)}_{N_i}$ are the length and diameter of a pre-capillary vessel segment (i.e., the smallest arterioles or venules). Given that $\frac{1}{3-\gamma }-\frac{1}{2+\varepsilon }=\frac{1}{2.55\pm 0.49}-\frac{1}{2.64\pm 0.64}\approx 0.01$, $\frac{\varepsilon -1}{2+\varepsilon}\approx -0.14$, 0 ≤ N_i ≤ 16, and 2 ≤ BR ≤ 4, there is $1\le \frac{{\mathrm{BR}}^{\left(\frac{1}{3-\gamma }-\frac{1}{2+\varepsilon}\right)\left({N}_i+1\right)}-{\mathrm{BR}}^{\left(\frac{\varepsilon -1}{2+\varepsilon}\right)\left({N}_i+1\right)}}{{\mathrm{BR}}^{\left(\frac{1}{3-\gamma }-\frac{1}{2+\varepsilon}\right)}-{\mathrm{BR}}^{\left(\frac{\varepsilon -1}{2+\varepsilon}\right)}}<10$ for different crowns in various vascular trees (negligible variation given the range of variables is very large, ten decades).

To be exhaustive, consider two alternate scenarios are considered: (1) it may be that $\frac{2-\gamma }{3-\gamma }-\frac{2}{2+\varepsilon }=0$ (unchanged total blood volume of vessels in each level) or (2) $\frac{2-\gamma }{3-\gamma }-\frac{2}{2+\varepsilon }>0$ (increase of total blood volume of vessels in each level from the stem to the terminal), which lead to:

$$ {V}_{\mathrm{c}}=\frac{\pi }{4}{D}_{\mathrm{s}}^3\frac{{\left({L}_{\mathrm{s}}\right)}_{N_i}}{{\left({D}_{\mathrm{s}}\right)}_{N_i}}\left({N}_i+1\right){\mathrm{BR}}^{\left(\frac{1}{3-\gamma }-\frac{1}{2+\varepsilon}\right){N}_i}\kern1em \mathrm{for}\kern1em \frac{2-\gamma }{3-\gamma }-\frac{2}{2+\varepsilon }=0 $$

(7.72)

$$ {V}_{\mathrm{c}}=\frac{\pi }{4}{D}_{\mathrm{s}}^3\frac{{\left({L}_{\mathrm{s}}\right)}_{N_i}}{{\left({D}_{\mathrm{s}}\right)}_{N_i}}\frac{{\mathrm{BR}}^{\left(\frac{1}{3-\gamma }-\frac{1}{2+\varepsilon}\right)\left({N}_i+1\right)}-{\mathrm{BR}}^{\left(\frac{\varepsilon -1}{2+\varepsilon}\right)\left({N}_i+1\right)}}{{\mathrm{BR}}^{\left(\frac{1}{3-\gamma }-\frac{1}{2+\varepsilon}\right)}-{\mathrm{BR}}^{\left(\frac{\varepsilon -1}{2+\varepsilon}\right)}}\kern1em \mathrm{for}\kern1em \frac{2-\gamma }{3-\gamma }-\frac{2}{2+\varepsilon }>0 $$

(7.73)

For a symmetric tree, the ratio of vessel length to diameter, ${\left({L}_{\mathrm{s}}\right)}_{N_i}/{\left({D}_{\mathrm{s}}\right)}_{N_i}$, is constant in pre-capillary vessels. If ${K}_{\varepsilon }=\frac{{\mathrm{BR}}^{\left(\frac{1}{3-\gamma }-\frac{1}{2+\varepsilon}\right)\left({N}_i+1\right)}-{\mathrm{BR}}^{\left(\frac{\varepsilon -1}{2+\varepsilon}\right)\left({N}_i+1\right)}}{{\mathrm{BR}}^{\left(\frac{1}{3-\gamma }-\frac{1}{2+\varepsilon}\right)}-{\mathrm{BR}}^{\left(\frac{\varepsilon -1}{2+\varepsilon}\right)}}$ for $\frac{2-\gamma }{3-\gamma }-\frac{2}{2+\varepsilon }<0$ and >0 and ${K}_{\varepsilon }=\left({N}_i+1\right){\mathrm{BR}}^{\left(\frac{1}{3-\gamma }-\frac{1}{2+\varepsilon}\right){N}_i}$ for $\frac{2-\gamma }{3-\gamma }-\frac{2}{2+\varepsilon }=0$, Eqs. (7.71–7.73) can be written as

$$ {V}_{\mathrm{c}}=\frac{\pi }{4}{K}_{\varepsilon }{D}_{\mathrm{s}}^3={K}_{\mathrm{d}}{D}_{\mathrm{s}}^3;\kern1em {K}_{\mathrm{d}}=\frac{\pi }{4}{K}_{\varepsilon } $$

(7.74)

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, Eq. (7.74) is written as:

$$ {\left({V}_{\mathrm{c}}\right)}_{\mathrm{max}}={K}_{\mathrm{d}}{\left({D}_{\mathrm{s}}\right)}_{\mathrm{max}}^3 $$

(7.75)

where ${K}_{\mathrm{d}}={\left({V}_{\mathrm{c}}\right)}_{\mathrm{max}}/{\left({D}_{\mathrm{s}}\right)}_{\mathrm{max}}^3$ 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 Eqs. (7.74) to (7.75), the desired relation can be obtained:

$$ \left[\frac{V_{\mathrm{c}}}{{\left({V}_{\mathrm{c}}\right)}_{\mathrm{max}}}\right]={\left[\frac{D_{\mathrm{s}}}{{\left({D}_{\mathrm{s}}\right)}_{\mathrm{max}}}\right]}^3 $$

(7.76)

Equations (7.74) and (7.76) are the volume–diameter scaling relation in a vascular tree.

Appendix 10: Fractal-Based Derivation of Flow-Length Scaling Law

The crown length, L_c, can be written as:

$$ {L}_{\mathrm{c}}={L}_{\mathrm{s}}+\sum \limits_{i=1}^{N_i}{n}_i{L}_i;\kern1em i=1,\dots {N}_i $$

(7.77)

where N_i is the same as defined in Appendix 9 (Huo & Kassab, 2012). From n_i = BRⁱ, ${L}_i={\mathrm{BR}}^{-\frac{i}{3-\gamma }}{L}_{\mathrm{s}}$, and Eq. (7.77), the following equation is obtained:

$$ {L}_{\mathrm{c}}={L}_{\mathrm{s}}+\sum \limits_{i=1}^{N_i}{n}_{\mathrm{i}}{L}_i={L}_{\mathrm{s}}\left(1+\sum \limits_{i=1}^{N_i}{\mathrm{BR}}^i\cdot {\mathrm{BR}}^{-\frac{i}{3-\gamma }}\right)={L}_{\mathrm{s}}\cdot \frac{{\mathrm{BR}}^{\frac{2-\gamma }{3-\gamma}\left({N}_i+1\right)}-1}{{\mathrm{BR}}^{\frac{2-\gamma }{3-\gamma }}-1} $$

(7.78)

Since ${L}_{\mathrm{s}}={\left({L}_{\mathrm{s}}\right)}_{N_i}\cdot {\mathrm{BR}}^{\frac{N_i}{3-\gamma }}$, where ${\left({L}_{\mathrm{s}}\right)}_{N_i}$ is the length of pre-capillary vessel segment, the crown length can be expressed as:

$$ {L}_{\mathrm{c}}={\left({L}_{\mathrm{s}}\right)}_{N_i}\cdot {\mathrm{BR}}^{\frac{N_i}{3-\gamma }}\cdot \frac{{\mathrm{BR}}^{\frac{2-\gamma }{3-\gamma}\left({N}_i+1\right)}-1}{{\mathrm{BR}}^{\frac{2-\gamma }{3-\gamma }}-1} $$

(7.79)

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, ${\left({Q}_{\mathrm{s}}\right)}_{N_i}$, as:

$$ {Q}_{\mathrm{s}}={\left({Q}_{\mathrm{s}}\right)}_{N_i}\cdot {\mathrm{BR}}^{N_i} $$

(7.80)

Equation (7.80) divided by Eq. (7.79) results in the following expression:

$$ \frac{Q_{\mathrm{s}}}{L_{\mathrm{c}}}=\frac{{\left({Q}_{\mathrm{s}}\right)}_{N_i}}{{\left({L}_{\mathrm{s}}\right)}_{N_i}}\cdot \frac{{\mathrm{BR}}^{\frac{2-\gamma }{3-\gamma }}-1}{{\mathrm{BR}}^{\frac{2-\gamma }{3-\gamma }}-{\mathrm{BR}}^{-\frac{2-\gamma }{3-\gamma}\cdot {N}_i}} $$

(7.81)

For a symmetric tree, the ratio of flow rate to vessel length, ${\left({Q}_{\mathrm{s}}\right)}_{N_i}/{\left({L}_{\mathrm{s}}\right)}_{N_i}$, is constant in the pre-capillary vessel segment. If ${K}_{\mathrm{Q}}=\frac{{\left({Q}_{\mathrm{s}}\right)}_{N_i}}{{\left({L}_{\mathrm{s}}\right)}_{N_i}}\cdot \frac{{\mathrm{BR}}^{\frac{2-\gamma }{3-\gamma }}-1}{{\mathrm{BR}}^{\frac{2-\gamma }{3-\gamma }}-{\mathrm{BR}}^{-\frac{2-\gamma }{3-\gamma}\cdot {N}_i}}$ (Unit: mL/cm), Eq. (7.81) can be written as:

$$ {Q}_{\mathrm{s}}={K}_{\mathrm{Q}}\cdot {L}_{\mathrm{c}} $$

(7.82)

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, Eq. (7.82) can be written as:

$$ {\left({Q}_{\mathrm{s}}\right)}_{\mathrm{max}}={K}_{\mathrm{Q}}\cdot {\left({L}_{\mathrm{c}}\right)}_{\mathrm{max}} $$

(7.83)

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 Eqs. (7.82) to (7.83), one obtains:

$$ \left[\frac{Q_{\mathrm{s}}}{{\left({Q}_{\mathrm{s}}\right)}_{\mathrm{max}}}\right]=\left[\frac{L_{\mathrm{c}}}{{\left({L}_{\mathrm{c}}\right)}_{\mathrm{max}}}\right] $$

(7.84)

Equations (7.82) and (7.84) are the flow-length relation in a vascular tree.

Appendix 11: Scaling Laws of Flow Rate with Number of Capillaries

The formulation invokes the law of conservation of mass which requires the flow at the inlet of the tree or crown (Q^s stem flow) to be equal to the sum of the flows at the first capillary segment, Q^c, namely:

$$ {Q}^s=\sum \limits_{i=1}^N{Q}_i^c $$

(7.85)

where N is the number of capillaries perfused by a given stem. Since a relation between flow and diameter has been previously determined as Q = K_QDD^δ, Eq. (7.85) becomes:

$$ {Q}^s=\sum \limits_{i=1}^N{\left({D}^c\right)}_i^{\delta } $$

(7.86)

If the diameters of the first segment of capillaries are assumed to be approximately uniform and given by ${\overline{D}}_{\mathrm{c}}$, Eq. (7.86) reduces to:

$$ {Q}^s\cong {kN}_{\mathrm{c}} $$

(7.87)

where $k={K}_{\mathrm{QD}}{\overline{D}}_{\mathrm{c}}^{\delta }$ is approximately constant. Hence, the inlet flow is proportional to the total number of capillary vessels. If the flow and capillarity are normalized with respect to an entire tree, we obtain the following:

$$ \frac{Q_{\mathrm{s}}}{Q_{\mathrm{s},\max }}={\left(\frac{N_{\mathrm{c}}}{N_{\mathrm{c},\max }}\right)}^{\lambda } $$

(7.88)

where Q_s,max and N_c,max are the inlet flow and the total number of capillaries in a vascular system, respectively. The hypothesis that λ is equal to 1 is verified and hence the form of Eq. (7.88) is equivalent to that of Eq. (7.83) for various vascular trees.

Appendix 12

Table 7.9 The validation of scaling relations in the entire stem-crown system of various organs and species using the least squares fits

Appendix 13: Relationship Between Crown Length, Volume and Capillary Numbers

The crown length is the cumulative length of blood vessels of all branching levels within the network, namely:

$$ {L}_{\mathrm{c}}=\sum {n}_i{L}_i $$

(7.89)

Based on the average branching ratio (n_i + 1/n_i = Br , i = 0, …, m; where n and i are number of vessels and branching level respectively) and the average length ratio (L_i + 1/L_i = Br^γ, where L is the average length of vessel in each branching level and γ is an empirical parameter, the average length in each branching level is written as:

$$ {L}_i={\left({\mathrm{Br}}^{\gamma}\right)}^{m-i}{L}_{\mathrm{cp}} $$

(7.90)

where m and L_cp are the maximum branching level and the length of vessels corresponding to capillaries. A combination of Eqs. (7.89) and (7.90) results in:

$$ {L}_{\mathrm{c}}=\left(\sum \limits_{i=0}^m{\mathrm{Br}}^{\left(i-m\right)\left(1-\gamma \right)}\right){L}_{\mathrm{c}\mathrm{ap}}{N}_{\mathrm{c}} $$

(7.91)

where N_c is the number of capillaries in the stem-crown system. Since the geometric series converges to a constant value, the above equation can be written as:

$$ {L}_{\mathrm{c}}={K}_{\mathrm{LN}}{N}_{\mathrm{c}} $$

(7.92)

Similarly, crown volume is the cumulative volume of blood vessels of all branching levels within the network, namely:

$$ {V}_{\mathrm{c}}=\sum {n}_i\left(\frac{\pi }{4}\ast {L}_i{D}_i^2\right) $$

(7.93)

Based on the assumption that vessel length and diameter scales at each branching level (L_i + 1/L_i = Br^γ and D_i + 1/D_i = Br^α), and using branching ratio and number of capillaries (N_c = Br^m, where m is maximum number of branching levels), a relationship between crown volume and number of capillary is derived as follows:

$$ {V}_{\mathrm{c}}=\frac{\pi }{4}\ast {V}_{\mathrm{c}\mathrm{ap}}\sum \limits_{i=0}^m{N}_{\mathrm{c}}^{\left(i+\left(m-i\right)\left(2\alpha +\gamma \right)\right)/m} $$

Hence, we hypothesize a power law relationship between crown volume and capillary number in as follows:

$$ {V}_{\mathrm{c}}={K}_{\mathrm{VN}}{\left({N}_{\mathrm{c}}\right)}^{\lambda } $$

(7.94)

where K_LN is an approximately constant value.

Appendix 14: Relation Between Transit Times and Crown Length and Volume

Based on the assumption that blood particles travel with the mean velocity of bulk flow and that the total number of blood particles passing through a vessel segment is proportional to the time-averaged flow rate in the segment, the mean transit time (MTT) in the vascular network (T_c) can be written as:

$$ {T}_{\mathrm{c}}=\sum \limits_{i=1}^N{\mathrm{FF}}_i\ast {T}_{\mathrm{sg},i} $$

(7.95)

where FF is the flow fraction (ratio of segment flow to stem flow) and ∗ T_sg is the average transit time in a specific segment where i = 1, 2, …, n; and n is the total number of segments in the entire network. It is well known that transit time can be determined by the ratio of blood volume and blood flow (Meier & Zierler, 1954). An elementary derivation by replacing the definition of the flow fraction (FF_i = Q_i/Q_max), transit time in a segment (T_i = V_i/Q_i) and Eq. (7.95) results in:

$$ {T}_{\mathrm{c}}\ast {N}_{\mathrm{c}}={K}_{\mathrm{TN}}{V}_{\mathrm{c}} $$

(7.96)

where K_TN is a proportionality constant (unit of time/volume). A combination of Eqs. (7.92) (Appendix 13) and (7.96) relates mean transit time to crown volume and length as follows:

$$ {T}_{\mathrm{c}}\ast {L}_{\mathrm{c}}={K}_{\mathrm{TL}}{V}_{\mathrm{c}} $$

(7.97)

where the parameter K_TL is a proportionality constant in unit of time/area.