Murray's law was derived for animal vascular networks by Cecil Murray [23]. It predicts how the blood vessels should change in diameter across branch points to maximize hydraulic conductance for a given investment in vascular volume and a particular branching architecture. The derivation for a single bifurcating blood vessel (Figure 4.1) is useful for pointing out the underlying assumptions as highlighted in italics. An initial assumption is that the volume flow rate is conserved from mother to daughter branch ranks — no fluid is lost in transit. The total volume (V) assuming cylindrical geometry of the vessels is:

where l is length, r is inner radius of vessel lumen, F is the number of daughter branches per mother (F = 2), and subscripts 0 and 1 designate mother and daughter ranks, respectively. We refer to this ratio as the "conduit furcation number." The total flow resistance (R, pressure drop per volume flow rate), assuming laminar flow through cylindrical tubes, is given by the Hagen-Poiseuille equation. Resistances are used rather than the reciprocal conductance because they are additive in series:

where ^ is the dynamic viscosity of the xylem sap.

Comparing Equations 4.1 and 4.2 demonstrates the basic conflict between minimizing flow resistance (and thus the energy consumed by the heart) while also minimizing vascular volume and the energy required to maintain the blood. Low flow resistance requires large radii (R °c 1/r4; Equation 4.2), but large radii make for larger and more expensive volume (V °c r2l, Equation 4.1). If the individual branch lengths and total network volume are held constant, the relative change in conduit radii (r0 and r1) across branch ranks that minimizes the flow resistance per fixed volume can be determined.

Setting the volume equation to equal zero (0 = V - C, where C is a constant), Lagrange's theorem can be used to solve for the values of r0 and rj that minimize the resistance:

where the Lagrange multiplier, X, is a nonzero constant. From Equations 4.1 and 4.2, the partial derivatives are:

dR/dr0 = -32^l0/:rcr05 |
(4.4a) |

dV/dr0 = 2l0nr0 |
(4.4b) |

dR/dr1 = -32^l1/Fnr15 |
(4.4c) |

dV/dr1 = 2Fl1nr1 |
(4.4d) |

Substituting Equations 4.4a and 4.4b into Equation 4.3a and solving for X yields:

Inserting Equations 4.4c and 4.4d in Equation 4.3b and solving for X yields:

Substituting Equation 4.5 for X into Equation 4.6, canceling terms, and rearranging, gives Murray's law:

or more generally,

which states that the hydraulic conductance of the fixed-volume network is maximized when the radius cubed of the mother vessel equals the sum of the radii cubed of the two daughter vessels. The shape of the conductivity optimum is shown for a network of three bifurcating branch ranks in Figure 4.2A.

The law does not require that all vessels of a branch rank be equal in radius as in our simple example; this can be seen by adjusting the equations accordingly and following the same derivation. The law also does not depend on F when conduit size is expressed as the conservation of 2r3 across branch ranks (Equation 4.7b). Note that Murray's law is independent of the lengths of the branches, which cancel out in the derivation even if they are unequal within a branch level (e.g., at Equations 4.6 and 4.7). The law is equivalent to solving for the maximum hydraulic conductivity (volume flow rate per pressure gradient, and length-independent) for a fixed cumulative cross-sectional area (volume per unit length) summed across each branch rank. Measurements have largely supported Murray's law in animals, at least outside of the network of leaky capillaries and beyond the influence of pulsing pressures at the exit from the heart [24-26].

As the derivation shows, Murray's law does not solve for the optimal size or individual branch lengths — only the optimal tapering of the conduit diameter across a given branching topography. The law is less ambitious than the "West et al. model," which attempts to solve for both the optimum tapering and the relative lengths (l0, l1, etc.) of branches at each level. The limitations of the West et al. approach [20,21] recommend a more limited analysis based on Murray's law.

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