The Mathematics of a System of Pools

A system of N pools, each containing Q material at time t, with a constant fraction K, being transported from each pool into another, or just eliminated from a pool, is diagrammed in the top part of the diagram.

Below the diagram is the general solution for the Qs when the system starts with some Q being positive, and no more material is added: a sum of exponential terms, each with a negative time constant. One reason semi-log plots are generally used (log Q versus time) is that you get a straight line in the case that there is only one term, or at least there are straight segments if one term predominates over a given time range.

If, for one of the pools, all of the As in the solution are positive, the curve of Q versus time for that pool will be concave upward when the log of Q is plotted versus time. A familiar example is drug concentration in the blood after IV injection.

If, for one of the pools, the sum of the As is zero (at least one A must thus be negative), Q will start at zero at time zero, then rise to a peak, then decline. A familiar example is drug concentration in the blood after oral administration of a drug.

Specific solutions for one, two, and three pool systems can be obtained using simple algebra and a little logic (assuming you don't make careless algebraic errors).

However, the theory (and associated numerical methods and software) of simultaneous linear equations allows you to reduce the energy spent on algebra and focus on the problem. To get a flavor for this approach we switch to vector and matrix notation, one of the tools of this field of mathematics.

We use the n dimensional vector q to represent amounts in n pools,
and the n by n matrix K to represent flow rates from pool to pool,
to write the single differential equation for the system, equation 1.

The general exponential form for the solutions is equation 2. Substitution into the differential equation gives equation 3.

If q is to be nontrivial, the determinant of the matrix in equation 3 must be equal to zero. Thus the n roots of the polynomial obtained from expansion of equation 4 give us the n values for lambda.

Substituting back into equation 1 gives equation 5, and the n independent solutions are the u's. In "vector speak" the lambda's are the eigenvalues and the u's the eigenvectors.

The explicit solutions for the q's, equation 6, are linear combinations of the u's, with the c's that define these combinations determined by the n values of the pools at t=0.

It is typically impossible to obtain analytic expressions for the q's when the flows are nonlinear functions of the concentrations. However, the differential equations are usually easily solved numerically, and the oldest desktop computer can handle significant problems in reasonable times. For many scientists, this ability constitutes the most revolutionary advance enabled by the personal computer.