Murray's law

As stated above, the underlying assumption of Murray's law is that the power (energy per time) for transport and upkeep is minimal in a natural transport system. Hence we seek to minimize ${\displaystyle P=P_{t}+P_{m}}$, where ${\displaystyle P_{t}}$ is the power required for transport and ${\displaystyle P_{m}}$ the power required to maintain the transport medium (e.g. blood).

We first minimize the power of transport and upkeep in a single channel of the system, i.e. ignore bifurcations. Together with the assumption of mass-conservation this will yield the law. Let ${\displaystyle Q}$ be the laminar flow rate in this channel, which is assumed to be fixed. The power for transport in a laminar flow is ${\displaystyle P_{t}=Q\Delta p}$, where ${\displaystyle \Delta p}$ is the pressure difference between the entry and exit of a tube of radius ${\displaystyle r}$ and length ${\displaystyle l}$. The Hagen–Poiseuille equation for laminar flow states that ${\textstyle Q={\frac {\pi \,r^{4}}{8\,\mu \,l}}\Delta p}$ and hence ${\textstyle \Delta p={\frac {8\,\mu \,l}{\pi \,r^{4}}}Q}$ , where ${\displaystyle \mu }$ is the dynamic viscosity of the fluid. Substituting this into the equation for ${\displaystyle P_{t}}$, we get

$${\displaystyle P_{t}={\frac {8\,\mu \,l}{\pi \,r^{4}}}Q^{2}.}$$

We furthermore make the assumption that the power necessary for maintenance is proportional to the volume of the cylinder ${\displaystyle V=\pi \,l\,r^{2}}$ :

$${\displaystyle P_{m}=\lambda \,V=\lambda \,\pi \,l\,r^{2}}$$

where ${\displaystyle \lambda >0}$ is a constant number (metabolic factor). This yields

$${\displaystyle P=P_{t}+P_{m}={\frac {8\,\mu \,l}{\pi \,r^{4}}}Q^{2}+\lambda \,\pi \,l\,r^{2}.}$$

Since we seek the radius with minimal power, we calculate the stationary point with respect to r

$${\displaystyle {\operatorname {d} \over \operatorname {d} \!r}P=0\;\Leftrightarrow \;-{\frac {32\,\mu \,l}{\pi \,r^{5}}}Q^{2}+2\,\lambda \,\pi \,l\,r=0\;\Leftrightarrow \;Q^{2}={\frac {\pi ^{2}}{16}}{\frac {\lambda }{\mu }}r^{6}\;\Leftarrow \;Q={\frac {\pi }{4}}{\sqrt {\frac {\lambda }{\mu }}}r^{3}.}$$

With this relationship between the flow rate ${\displaystyle Q}$ and radius ${\displaystyle r}$ on an arbitrary channel, we return to the bifurcation: Since mass is conserved, the flow rate of the parent branch ${\displaystyle Q}$ is the sum of the flow rates in the children branches ${\displaystyle Q_{1},\ldots ,Q_{n}}$. Since in each of these branches the power is minimized, the relationship above holds and hence, as claimed:

$${\displaystyle Q=\sum _{i=1}^{N}Q_{i}\;\Rightarrow \;{\frac {\pi }{4}}{\sqrt {\frac {\lambda }{\mu }}}\,r^{3}=\sum _{i=1}^{N}{\frac {\pi }{4}}{\sqrt {\frac {\lambda }{\mu }}}\,r_{i}^{3}\;\Rightarrow \;r^{3}=\sum _{i=1}^{N}r_{i}^{3}.}$$

The power for transport spent by diffusion is given by

${\displaystyle P_{t}=Q\,\Delta C=\left(\pi \,D\,r^{2}{\frac {\Delta C}{l}}\right)\Delta C}$

where the flow rate is given by Fick's law, whose ${\displaystyle D}$ is the diffusivity constant and ${\displaystyle \Delta C}$ is the difference of concentration between the ends of the cylinder. Similarly to the case of laminar flow, the minimisation of the objective function results in

${\displaystyle {\frac {\Delta C}{l}}={\sqrt {\frac {\lambda }{D}}}\Rightarrow Q=\pi \,D\,r^{2}{\sqrt {\frac {\lambda }{D}}}\Leftrightarrow Q=k\,r^{2}}$

Hence,

${\displaystyle Q_{p}=\sum _{i=1}^{N}Q_{i}\Rightarrow k\,r_{p}^{2}=\sum _{i=1}^{N}k\,r_{i}^{2}\Leftrightarrow r_{p}^{2}=\sum _{i=1}^{N}r_{i}^{2}}$