The Incorporation of Fractal Kinetics in the PK Modeling of Chemotherapeutic Drugs with Nonlinear Concentration-Time Profiles

Abstract

Paclitaxel is a well-known chemotherapeutic drug which has been successfully used in the treatment of various cancers. Clinical studies confirm that the concentration-time profile of this very important drug is nonlinear. In order to capture this non-linearity, various multi-compartmental models with both saturable distribution and elimination (usually under the assumption that various PK processes are governed by Michaelis–Menten kinetics) have been developed and published in the literature. These models have been successful (to some degree), even though it has been observed that complex biological systems are often heterogeneous media displaying fractal geometry. Furthermore, the assumption of classical Michaelis–Menten kinetics could be a shortcoming when attempting to capture the nonlinear behavior of many drugs. In this paper, we propose a two compartmental model, incorporating steady state, fractal Michaelis–Menten Kinetics. The model is an extension of an existing model that uses classical Michaelis–Menten kinetics. However, comparison of both models suggests that our model is better able to capture the nonlinear behavior of paclitaxel.

Appendices

Appendix 1: Genetic Algorithm

To find an optimal or best solution, optimization techniques have been used in different areas of science and engineering. These techniques are considering the following factors [36]:

• An objective function: the function we want to minimize or maximize, for example, we want to minimize the cost and maximize the profit in manufacturing.

• A set of unknown variables: the variables by which the objective function can be effected.

• A set of constraints: which can be used to include certain conditions while evaluating values of the unknown parameters.

So, an optimization technique is a technique which can be used to find the variables that maximize or minimize the objective function while satisfying the constraints. Genetic algorithm (GA) is a heuristic search algorithm to optimize a problem, inspired by Darwin evolution theory, which uses random search process. The basics of GA can be stated as follows:

• It starts by generating a random population of n chromosome which can be think of the solution of the problem.

• Calculate f(x), the fitness function of x (chromosome) in the population.

• Creates new population by repeating the following processes:

  • Two different parent chromosomes are selected from the population which we can think as Selection by comparing with evolution process (when the fitness is better, then the chance to be selected is bigger).

  • Perform a Crossover probability. The offspring (children) will be exact same copy of the parents if there is no crossover (but this does not mean that the new generation is same) and the offspring will be made from parts of parents chromosome, then crossover probability performed.

  • How often the parts of the chromosome mutated is measured by the Mutation probability and offspring can be taken as is after the crossover if there is no mutation.

  • Place the new offspring in the new population which is known as Accepting the new population.

• The new generated population is replaced for the next run of the algorithm.

• By testing the constraint conditions it will stop and return the best solution of the current population.

• And it will go to the step 2 to continue the loop as long as the tolerance is achieved.

Appendix 2: Akaike Information Criterion (AIC)

In statistical study, we are engaging ourselves to estimate the effects for a given variable using certain parameters. While doing this, we certainly may include parameters for which we might lose the physical information we are trying to predict via a mathematical model. And we may also over fit the available data. Form the Occam’s razor philosophical principle we know that if we can describe something with a simple model or with also a more complicated model then we should choose the simple one, which indicates to rely on the simpler model than to the complicated model. Now when we are fitting some observation how do we know that we are including less or more parameters? In 1973, Akaike H., a Japanese Statistician came with a theory which will compare with the models and that comparison will give the idea which model we should choose among the models are available to choose. By doing so, it restricts us from under or over fit the model. Since this is a comparison with the available models we have, it will not tell us that this is the best model rather give us the information that this a better one among the models we have. In statistics this means that a model with less parameters is preferable than a model with more parameters. Again a model with too less parameters will be biased and a model with too many parameters will have low precision [37].

The theory is from the information field theory and named as Akaike information criterion or AIC. Good model would be the one which will minimize the loss of information. Akaike in 1973 proposed an information criterion as follows:

$$\displaystyle \begin{aligned} \mbox{AIC}=-2(\mbox{log-likely hood})+\mbox{2}{\mathit{K}} \end{aligned}$$

where K is the number of estimated parameters used in the model. For a given sets of data log-likelihood can be calculated and from there one can tell about the model, is that it is over or under fit or not (smaller value means worse fit) [37]. If the models are based on conventional least squares regression, then the assumption is that the error obeys Gaussian distribution. And we can compute the AIC formula as follows:

$$\displaystyle \begin{aligned} {\mathrm{AIC}}=N_{obs}+ln({\mathrm{WRSS}})+2N_{par}; \end{aligned}$$

where N _obs is the number of observed data point, N _par is the number of model parameters, WRSS is the weighted residual sum squares [38]. WRSS can be calculated from the following relation:

$$\displaystyle \begin{aligned} {\mathrm{WRSS}}=\Sigma_{i=1}^{n}\frac{(C_i-\hat{C_i})^2}{\hat{C_i}^2} \end{aligned}$$

where $\hat {C_i}$ is the predicted value and C _i is the true value. A lower AIC value indicates a better fit. Idea about the weighted factor is, if at high concentration, i.e., at the beginning of the time plasma profile, data are showing more accuracy than the tail, then the data can be weighted with WRSS ∼ 1 [38]. On the other hand, if the tail end of the profile showing more accuracy, then the data might be weighted with $1/\hat {C^2}$ [38]. Following this idea in our model we have used, the weighted factor for WRSS is $1/\hat {C_i}$.