A nanoparticle has been injected into a small artery and you are riding it. You have no engine and no rudder. What you have is a magnet outside the body, a few piconewtons of pull, and sixteen forks between here and the tissue. At every fork the fluid has already decided where most of it is going, and the only question that matters is which side of the dividing streamline you were on when you arrived.
The tree is built from Murray's law and nothing else: each fork splits one vessel into two whose radii cube to the parent's. Everything below follows from that, from Poiseuille flow and from the Pries fit for how thick blood is at this diameter. No number here was chosen to look plausible.
Apparent viscosity relative to plasma, against tube diameter, at your haematocrit. The dot is the vessel you are in. The minimum near seven microns is the whole effect: at the diameter of a capillary, blood flows almost as easily as water.
When an artery divides, the two daughters are not half the parent. Their radii cube to
the parent's: r₀³ = r₁³ + r₂³. For a symmetric
fork that makes each daughter 2-1/3, about 0.794, of the parent, and
the whole tree becomes a single power law. Murray derived it in 1926 by minimizing the sum
of two costs: the power burned pushing blood through a narrow pipe, and the metabolic cost
of keeping the blood in a wide one.
The consequence is the interesting part. Flow splits in half at each fork while area
falls by 2-2/3, so the mean velocity falls by the same 0.794 as the
radius. Wall shear stress is 4μv/r, and if v and r fall
together it does not move at all. Murray's law is what a vascular tree looks like when every
vessel in it is holding its endothelium at the same shear.
Watch the instrument panel as you descend. Sixteen generations take the radius from six
hundred microns to fifteen, and the velocity from a hundred millimeters a second to two and
a half, and the wall shear stress drifts only from about 26 down to 14 dyn/cm². It does
not hold perfectly, and the reason it does not is the next fold: μ is not a
constant.
Put blood in a viscometer and it is about three and a bit times as thick as plasma. Push the same blood down a tube seven microns across and it behaves as though it were only 1.25 times as thick. Fahraeus and Lindqvist found this in 1931, and the reason is that red cells migrate away from the wall and leave a thin sleeve of nearly pure plasma behind. The tube lubricates itself. Below about four microns it reverses sharply, because now a red cell has to fold itself to get through at all.
This page uses the in vitro fit published by Pries, Neuhaus and Gaehtgens in 1992, which gives relative apparent viscosity as a function of tube diameter and discharge haematocrit. Two independent checks that it is the right fit and not a lookalike: it puts the minimum at 6.9 microns with a relative viscosity of 1.25, and it settles to about 3.19 for large vessels, both of which are the published values. The curve above is that function, evaluated live at whatever haematocrit you set.
It matters here for a reason beyond arithmetic. The plasma sleeve at the wall is where a drug carrier has to end up if it is ever going to touch the endothelium, and it is the same sleeve that makes the blood easy to push.
Fluid does not mix as it approaches a fork. There is a surface in the parent vessel, the dividing streamline, and everything on one side of it goes left while everything on the other side goes right. Its position is set purely by how the flow splits: if the upper daughter takes 30 percent of the flow, the dividing streamline sits wherever 30 percent of the parent's flux is above it.
Because the velocity profile is parabolic, that is not 30 percent of the way across. The
middle of the vessel carries far more than its share, so the streamline for a minority
branch sits well out toward the wall. The simulator solves
∫v(y)dy over the parabola for the exact position every time, and draws it.
This is not a game mechanic borrowed from somewhere. It is why the branch that takes less flow also receives blood with fewer red cells in it, since the cells are concentrated in the fast middle: plasma skimming, and the Zweifach-Fung effect that follows from it. Your nanoparticle obeys the same rule, and steering it means being on the correct side of a line before the fork arrives, not after.
Magnetic drug targeting is a real technique: load the carrier with iron oxide, put a strong magnet against the skin, and pull the particles out of the flow and into the tissue you care about. Your steering control here is that magnet, and the physics of it is unforgiving in a way worth feeling.
Whatever force the magnet applies, the particle is in Stokes drag, so it does not
accelerate at all: it immediately moves sideways at v = F/6πμa and no
faster. A piconewton on a hundred nanometer carrier in plasma buys 442 microns a second
across the vessel, against a hundred millimeters a second along it. Over one segment of the
artery you start in that is thirteen percent of the radius, so you are carried past the
first few forks more or less whatever you do.
Sixteen forks later the same piconewton moves you clean across the vessel several times over, because the vessel is forty times narrower and the blood is forty times slower while the magnet has not changed at all. Steering becomes easy at almost exactly the point where it stops mattering, since every branch that decided which tissue you are heading for is by then behind you. The panel prints the lateral reach per segment, so you can watch that crossover happen as you descend.
Making the carrier bigger helps more than it looks. Force follows volume and drag follows radius, so ten times the radius is a thousand times the force against ten times the drag: a hundredfold gain in sideways speed. It also lodges in the first vessel narrower than it is, which in this tree happens well above the capillaries. That trade is the whole design problem in magnetic targeting and you can run into both ends of it here. What the page does not model is the constraint that actually limits the technique: the field gradient from a magnet outside the body falls off steeply with depth, so the force available is really a function of how far under the skin the target is.
The vessel is a straight rigid cylinder with fully developed flow, seen in two dimensions. Real arteries are curved, elastic, taper between branches, and never let the profile fully develop before the next fork. Curvature alone produces secondary flows this model has none of.
Flow is treated as steady at the local mean. Pulsatility is computed and reported, and the Womersley number tells you honestly how much it should matter, but the trajectory does not oscillate with the heartbeat. At the sizes here that is defensible; in the aorta it would not be.
Red cells are drawn as advected discs to show the profile and the cell-free layer. They do not deform, do not interact, and do not push your particle around. Real margination, which is how a drug carrier reaches the wall at all, comes precisely from those collisions. The cell-free layer here is set to two microns, inside the measured range of roughly one to three, rather than derived.
Adhesion in the delivery mode is modeled as a binding rate that rises with contact area and falls with wall shear stress. That shape is right and the constant in front of it is chosen, not measured. Treat the delivery score as a game, and the vessel panel as the instrument.
The particle is a hard sphere with no coating, no protein corona, no immune system removing it, and no clearance. A real injected nanoparticle is opsonized within minutes and most of the dose ends up in the liver. Nothing here models the reason drug targeting is hard.