This study focuses on the dynamics of drug concentration in the blood. In general, the concentration level of a drug in the blood is evaluated by the mean of an ordinary and first-order differential equation. More precisely, it is solved through an initial value problem. We proposed a new modeling technique for studying drug concentration in blood dynamics. This technique is based on two fractional derivatives, namely, Caputo and Caputo-Fabrizio derivatives. We first provided comprehensive and detailed proof of the existence of at least one solution to the problem; we later proved the uniqueness of the existing solution. The proof was written using the Caputo-Fabrizio fractional derivative and some fixed-point techniques. Stability via the Ulam-Hyers (UH) technique was also investigated. The application of the proposed model on two real data sets revealed that the Caputo derivative was more suitable in this study. Indeed, for the first data set, the model based on the Caputo derivative yielded a Mean Squared Error (MSE) of 0.03095 with a corresponding best value of fractional order of derivative of 1.00360. Caputo-Fabrizio-based-derivative appeared to be the second-best method for the problem, with an MSE of 0.04324 for a corresponding best fractional derivative order of 0.43532. For the second experiment, Caputo derivative-based model still performed the best as it yielded an MSE of 0.04066, whereas the classical and the Caputo-Fabrizio methods were tied with the same MSE of 0.07299. Another interesting finding was that the MSE yielded by the Caputo-Fabrizio fractional derivative coincided with the MSE obtained from the classical approach.

Drugs are often inoculated in human blood during the treatment of a disease. One interesting fact is that the concentration of the drug decreases over time once it is inoculated. The initial concentration of a drug and many other parameters are used by medical practitioners to determine the frequency at which a specific drug should be administrated to reach the goal of curing a disease.

The clinical development of a new drug involves many phases. During phase 1 of drug development, the experimenter routinely evaluates time elapse until a new regimen reaches its steady state. The steady state here refers to the state at which the concentration of a drug in the blood does not change consistently anymore over time. In practice, patients have to take several shots of the drug in order to recover. However, each time a dose of the drug is administrated, part of it spreads all over the body and exists through excretions. The remaining part will accumulate to form sediment. The steady-state is reached when the concentration of the new dose coincides with the concentration of sediment. To identify the time from the first inoculation to the steady-state, the patient’s blood is sampled at regular time span intervals and the drug concentration is measured from the sample [

Many research works were produced in line with the evaluation of the steady-state of a drug in blood plasma. An overview of the selected method was proposed by Maganti et al. [

Besides the few but consistent methods of evaluating the steady-state during phase I study are many other approaches that we believe can be used successfully. We think of Fractional Calculus (FC) in this regard. Indeed, over the past decades, many researchers have successfully applied FC to solve real-life problems. One can name a few of these successes just to mention. For instance, Reference [

Almost eighty years ago, more precisely in the year 1940, the mathematician Ulam asked about the stability of functional equations [

In this work, we kept a similar approach to earlier discussed examples. Indeed, we introduced modeling of the drug’s concentration blood using two fractional derivatives, namely the Caputo and the Caputo-Fabrizio fractional derivatives. Moreover, some qualitative analyses of the aforementioned models were obtained.

The remaining work is organized as follows. A review of FC definitions is provided in

FC has a strong foundation built upon definitions. Nowadays, they exit many fractional derivatives, each of them having its specific advantages and down points. This section focuses on some FC concepts required for the rest of the paper.

This function was introduced by the author of the same name in 1903 (see [

Another important derivative that is among the basis of FC is the Caputo fractional derivative. It is an integral form-based derivative defined as follows.

It is even possible to derive another useful fractional derivative that is used in this work. Indeed, if the function

The Caputo-Fabrizio fractional integral exists and is also useful in solving FDE. The said integral is defined as follows:

We end the shortlist of fractional derivatives and integrals of this section with Riemann-Liouville definitions.

So far in this section, three fractional derivatives and two fractional integrals have been introduced. To end this section, we will state without proving some important Lemmas. These lemmas are used in sequel in the process of proving the existence of a solution to the problem and the uniqueness of the existing solution to the core problem which is also defined in the coming sections.

Let us consider a Banach space

Then it follows that:

∃

Let

This section provides an overview of pharmacokinetics theories and the classical approach to building the dynamic model. A common approach to overcoming disease is to provide the sick person appropriate drug that will target harmful germs. It is therefore very substantial to inoculate the appropriate dose of the drug at suitable time intervals. Phase I of the clinical study of any drug focus on determining these parameters. Pharmacokinetics is defined as the branch of medicine that studies the dynamic (kinetics) of the drug when it is inoculated in a living body. In phase I of a clinical study, blood samples are taken at a regular time interval to evaluate the drug concentration. This is a purely experimental approach. There also exists a theoretical approach that consists of using a mathematical model and some historical data to develop a predictive model. A model that would predict drug concentration levels in the blood over time. In general, interest among researchers in computational biology has increased significantly over the past decades. Juinn et al. have dedicated research [

It was mentioned in an earlier section that drug concentration decreases over time. Hence, a suitable mathematical model to describe such a process can be an exponential decay model. The following differential equation describes the decrease observed over time.

This study aims to investigate the performance of fractional differential equations in modeling while comparing them with classical differential equations used for solving the same problem. In the next sections, Caputo’s derivative and Caputo-Fabrizio’s derivative are used to build a fractional counterpart of the differential equation

In order the simplify notation and for conformity purpose, let a kernel function

The following Lemmas introduce more generalized formulae of the solutions to the fractional drug concentration models with the initial condition, as defined by

There exists a solution to

Remark: If the real number

Using Lemma 3.1 the solution to

Considering the initial value problem given by

Let an operator

i)

ii)

where,

Moreover, assume that the following inequality holds for the defined constant:

Based on above setting, one is sure of the existence of at least one solution to the problem

Let us introduce two operators,

Given two elements

In the forthcoming step of the proof, we will show that the operator

Moreover,

Beside all the above properties, one can prove compactness of

For

The quantity defined on the right-hand side of the inequality

On the other hand, the following inequality is derived from the quantity or norm of

Using

The second step of this proof is to show that the operator

The relation given by

On can see from

Having proven the existence and the uniqueness of a solution to the problem defined by

Based on Lemma 3.2, the analytical solution for the drug concentration model via Caputo-Fabrizio sense

All the necessary theoretical work on the proposed approach was done in previous sections. In the next section stability of the model is studied.

In this section, we discuss the UH and generalized UH (GUH) stability of the considered problem.

By Remark 5.3, the solution of

Hence, we will make the following remark.

Now we are ready to announce the results of the stabilization results for UH and GUH.

Since

Thus, by our assumptions, Remark 5.5, along with

As

From Lemma 5.4, we have

Set

Hence,

Let us consider the following CF-problem:

For any

Therefore, i) and ii) hold. Also,

Let

From Theorem 4.1, we known (s1) with

Applying

Moreover, we have

For

Hence, problem

In this section, Caputo and Caputo-Fabrizio fractional derivatives are used to solve FDE for drug concentration modeling in blood. The next figure represents the so-called ADME process. It shows that drugs can be administrated through an oral or intravenous route. In this section, our experiments are carried out on the intravenous administration. Bold and red arrows are meant to highlight the paths considered in the study. The figure also shows where blood can be sampled to measure drug concentration at any time. It is important to mention that the oral route administration is not studied here because it falls apart in a completely different category of differential equations.

The following two experiments were designed with respect to

David Bourne (see [

Parameters | |||
---|---|---|---|

Caputo derivative | 0.34590 | 1.00360 | 0.03095 |

Classical method | 0.34785 | - | 0.04324 |

Caputo-Fabrizio derivative | 0.68678 | 0.43532 | 0.04324 |

Note: Best parameters were obtained through optimization routine, in particular error minimization. The non-linear optimization routine,

^{th} hour on the x-axis but zoom out the page containing the figure.

In this study, a newly developed drug was injected (Intravenous route) into a 50 kg female patient at a proportion of 20 mg/kg. Blood samples were taken on a regular basis and, the plasma was analyzed to determine the current drug concentration in the blood. The entire process generated a data set that was fitted by the mean of FDE based on Caputo and Caputo-Fabrizio derivative, defined by

Parameters | |||
---|---|---|---|

Caputo derivative | 0.49620 | 1.11080 | 0.04066 |

Classical method | 0.53355 | - | 0.07299 |

Caputo-Fabrizio derivative | 0.49974 | 0.52179 | 0.07299 |

The goal of this work is achieved. We have proven that fractional differential equations could be efficiently used for drug concentration modeling. In this regard, we have proven the existence and uniqueness of the solution based on some fixed point techniques. We have also provided a stability analysis in the Ulam-Hyers sense of the CF-type FDE model. Moreover, based on the results from the two experiments, it is observed that the Caputo derivative performs the best. On the other hand, the Caputo-Fabrizio and classical methods have the same

In this study, every fractional order of derivative that leads to the lowest

In future work, we aim to build a comprehensive framework for choosing the right fractional derivative for a given problem.

In practice, the outcome of this study can increase the accuracy of a new model’s parameters calibration. Indeed, if a new drug is developed, and if its concentration level in the blood (during phase IV) decreases over time following an exponential decay trend; then an FDE based on the Caputo derivative would be recommended for estimating the best parameters. The statement

The authors express their thanks to the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia, for supporting this work through the Annual Funding Track by [Project No. AN000273].