Saturable Fractal Pharmacokinetics and Its Applications - Mathematical Methods and Models in Biomedicine - page 344

Fig. 4 The vascular network
divided into ( a )
homogeneous, well-perfused
conditions, and ( b )
heterogeneous, fractal
conditions in the deep tissues.
Figure reproduced after
Macheras [ 42 ]

300

200

100

20 40 60 80 100

Time (min)

Fig. 5 The early time behavior of the concentration-time data for four different dogs [ 58 ].
Concentration-time data were obtained for the cardiac drug mibefradil in four dogs. The dogs
received a dose of 1 mg/kg of mibefradil infused over 10 min

using mibefradil data [ 58 ]. The results for four dogs are shown in Fig. 5 . The spectral
dimension for the dog liver estimated using the model was between 1.778 and 1.914,
which compares favorably with the measured value of 2.0 by Javanaud [ 29 ].
The value and standard deviation of the power law tail exponent

were calculated
from the concentration-time curves using linear regression analysis of the log-
transformed data.
The various models were fit to the data using a parameter optimization method
based on a simulated annealing (SA) algorithm implemented in C++. The SA
algorithm [ 31 ] minimizes an objective function through an efficient exploration of
the parameter space. All downhill moves are accepted and selective uphill moves
are allowed according to the Metropolis algorithm and an effective temperature. At
the start of the annealing process, the temperature is relatively high compared to
the standard deviation of the objective function, and the probability of accepting
an uphill move is high. Hence, the random walk is able to explore a wide area
of parameter space without getting trapped in local minima. As the temperature is
decreased, the algorithm is able to focus on the most promising areas and locate the
global minimum.

γ