# PSW at MBI: Day 2

Arranged to take the shuttle from the hotel this morning since 8:15am was really kicking my butt and giving myself an extra 15 minutes would really be appreciated.  So, where are we?  First let’s describe the process.

A polymer scaffold is set into the bottom of a petri dish and is about 300 microns in thickness with a slight variation.  On top of this 1 ml of tissue culture is pipetted into the dish bringing the fluid level up to about 2 mm in depth.  The tissue culture medium which contains 10% serum which is slightly more viscous than water.  On top this another layer is added with cells.  This is done by once again carefully pipetting this above the first layer.

The nature of the cells is quite special.  There are basically two types.  Stem cells that are coated in antigen that are bound to tiny magnetite beads covered in an antibody and in amongst these stem cells are regular cells without any beads bound to them.  Once loaded, the petri dish is placed on top of a magnet so that the culture experiences a magnetic field.  After about 5 minutes the stem cells and some of the other cells embed themselves in the scaffold.  The trick is to try to get the stem cells to distribute into the scaffold uniformly and simultaneously minimize the amount of contaminant (non-stem) cells.

Some resolution with the magnetic field.  The paper Particle trapping by an external body force in the limit of large Peclet number: applications to magnetic targeting in the blood flow gives a form for the magnetic field of
$$F(z) = \frac{3\mu_0\chi_m I^2 \tilde{R}^4}{4(1+\chi_m)^2} \frac{N (z+d)}{\left((z+d)^2 + \tilde{R}^2\right)^2}$$ where the values of $$d, I$$ and $$\tilde{R}$$ are fit to the given experimental data.

We are beginning by developing a particle trajectory model that determines the path of individual cells that are covered with $$N$$ magnetite beads assuming that there is magnetic, buoyancy and viscous drag effects. That is, the velocity of the particles are given by
$$m\frac{dv}{dt} = -F_0\frac{N (z+d)}{\left((z+d)^2 + \tilde{R}^2\right)^2} – 6\pi\mu R v – \frac{4}{3} \pi R_m^3 (\rho_m-\rho_f)$$
at least in a crude sense if we assume that the cells are neutrally buoyant, $$R_m = 1 \mu$$m and $$R \sim 7.5 \mu$$m are the radii of the magnetite beads and the cell respectively. This is complicated by the fact that $$N$$ has a distribution of values that to can take on with an average of about 4 beads per stem cell.

Parallel to this is a concentration model that evolves in time and space. As cells move through the scaffold at the bottom they are removed with some probability. Whether or not this will work nicely will be seen tomorrow.