# In-Vivo Analysis and Model-Based Prediction of Tensides’ Influence on Drug Absorption

^{*}

## Abstract

**:**

## 1. Introduction

## 2. In-Vivo Experiment―Material and Method

## 3. Model-Based Experiment

#### 3.1. Structure of the Model

_{S}is a saturated concentration at the close vicinity around the particle, C is the concentration of the bulk solution, h is a thickness of the saturated layer around the particle, A is a surface area of the particle, D is a diffusion coefficient. As the mass is equal to the product of the volume and concentration, Equation (1) is a first-order differential equation indicating that the liberation obeys the first-order dynamics.

_{1}and m

_{2}are drug amounts and V

_{1}, V

_{2}are volumes of distribution. The constant parameters in (3) are commonly lumped into the membrane permeability G = DA/d, which is experimentally determinable. The Equation (3) is illustrated in Figure 4.

#### 3.2. General Properties of Compartment Model

_{0}and the output y(t) (samples of concentrations). In other words, the relation between the input (dose administered) and the samples measured at the system output (drug concentrations) defines an input/output (I/O) model, which says nothing about the internal structure of the model. One such I/O model is known as a transfer function. What is worse, the same transfer function corresponds to infinitely many structures (so-called realizations). This fact can cause significant difficulties if one wants to identify values of the model’s parameters―the rate constants of the drug disposition. Therefore, besides the I/O model, a closer look is thrown on the state-space model, also the “internal model”.

#### 3.3. State-Space (Internal) Model

_{0}) is an initial state of the system, i.e., a vector of initial values of the state variables

^{T}is the transposition of c, hence c

^{T}is a row vector

_{0}from a nonnegative initial state vector x(t

_{0}) must remain consistently nonnegative. It can be shown that this happens if and only if the vectors b and c are component-wise nonnegative and the matrix A is the so-called Metzler matrix, i.e., a non-zero matrix with nonnegative off-diagonal entries [13,14,15]. Let us note that the requirement of the system positivity significantly hardens the synthesis of possible control algorithms, e.g., the dosing regimes.

**Remark**

**1.**

_{1}(t) × k

_{a}). However, in the case of the model shown in Figure 5, it means that the rate of absorption does not depend on the drug amount x

_{2}(t) in the blood compartment. In the cybernetic parlance, one would say that the rate of absorption “lacks the feedback information” about the current drug amount in the acceptor compartment. As a consequence, the rate of absorption is not influenced by the drug amount x

_{2}(t). Clearly, in reality, such a feedback always exists. No wonder that in some cases the authors approximate the rate of absorption by the exponential K × exp (−k

_{a}t), which in general is not quite correct, especially for the special drug forms like the transdermal patches, chewing gums, etc.

_{0}is a system input―an instantaneously administered dose

_{e1}, k

_{e2}are rate constants of the drug elimination

_{1}(t) is a time course of the drug amount in the peripheral compartment―the GIT

_{2}(t) is a time course of the drug amount in the central compartment—the blood

_{1}, x

_{2}is not included in these samples. The reason is simple: The state variables x

_{1}, x

_{2}are in general artificial quantities, which may not have a realistic interpretation. Their meanings are defined by the model’s structure which was chosen by the designer. Hence, to resolve the problem of identification of the matrix A one should make a record of an input–output model (I/O model). We decided to use the I/O model in the form of the transfer function G(s), which is the ubiquitous concept in the realm of system analysis. The G(s) is defined by the ratio of the image y(s) of the output y(t) to the image u(s) of the input u(t). Omitting the detailed explanations, we only declare that the symbol “s” is a certain complex variable. The transfer function G(s) of the system (4) can be computed in accordance with Expression (7).

_{0}, a

_{0}, a

_{1}) and (k

_{a}, k

_{e1}, k

_{e2}) is that while the former were identified (!) from the in-vivo samples, the latter were calculated (!) from already known values of the coefficients b

_{0}, a

_{0}, and a

_{1}). Comparing the homothetic coefficients in the numerator and denominator in (10) the following relations will be obtained:

_{12}, k

_{21}and k

_{e}take the form (12).

_{e1}, k

_{e2}, k

_{a}, for already known values of the parameters b

_{0}, a

_{0}, a

_{1}, the compartment model (6) would be parametrically identifiable. However, looking at the second expression in (12), it is clear that there exist two possible solutions for k

_{e2}. Hence, the compartment model (6) is not uniquely identifiable.

**Remark**

**2.**

_{0}in Figure 5 corresponds to the instantaneous administration of 50 mg of sulfathiazole. Let us recall that the computer implementation of the dose u may be either in the form of the function u(t) = 50 δ(t) mg, where δ(t) means the Dirac unit function [15], or in the form of the initial condition x

_{1}(0) = 50 mg though the latter alternative may seem to be more natural (because the suspension Though the latter alternative may seem to be more natural (because the suspension was instantly inserted into the stomach), this paper prefers the first one. A reason is that the drug must first transit from the stomach into the intestine where it is dissolved, and only then it can be absorbed. In other words, the drug amount in the GIT―x

_{1}(t) in (6) cannot instantly jump to the initial value X

_{1}(0) = 50 mg. Using the initial condition in the role of the input can be fully accepted in the case of an instantaneous intravenous administration but not in the case discussed here. As to the Dirac function δ(t) is defined as an infinitely short and infinitely high impulse with unit surface area, the input u(t) = 50 δ(t) was approximated by a very thin and very high rectangle of the surface area equal to 50 units. In particular, the following dimensions of this rectangle were used: width × height = 0.083333 × 600 m = 50 h·mg.

_{a}+ k

_{e1}) exclusively influences the dynamic behaviour of the 1st compartment, while the dynamics of the 2nd compartment is exclusively influenced by k

_{e2}. The lager is the sum (k

_{a}+ k

_{e1}) the more quickly the 1st compartment responds to the input u(t).

_{0}, a

_{0}, a

_{1}were identified from in-vivo measurements and the following results were obtained: b

_{0}= 0.030272, a

_{0}= 0.230137, a

_{1}= 1.018397. With parameters b

_{0}, a

_{0}, a

_{1}identified, the rate constants k

_{a}, k

_{e1}, k

_{e2}were calculated in accordance with (12) and two different sets of solutions were obtained:

_{a}= 0.030272 h

^{−1}k

_{e1}= 0.649656 h

^{−1}k

_{e2}= 0.338477 h

^{−1}

_{a}= 0.030272 h

^{−1}k

_{e1}= 0.308204 h

^{−1}k

_{e2}= 0.679928 h

^{−1}

_{a}are the same in both sets, values of k

_{e1}and k

_{e2}are different. The existence of two different sets of solutions means that model (6) is parametrically un-identifiable. Nevertheless, this fact is not relevant for the purposes of this paper, as information about the rate of absorption is exclusively carried by the absorption rate constant k

_{a}, which was uniquely identified. Keeping in mind that the major route of the drug elimination is renal excretion, in case of need a passable way to determine which of these two sets of solutions is correct can lead through evaluation of the drug amount excreted into the urine. In that case, it would be enough to evaluate the half-time (t

_{1/2}) and then use the known relation k

_{e1}= 0.693/t

_{1/2}.

#### 3.4. Prediction of the Tenside’s Influence on the Rate of Absorption

_{0}= 0.045815, a

_{0}= 0.244092, a

_{1}= 1.247769

_{a}= 0.045815 k

_{e1}= 0.959041 k

_{e2}= 0.242913

_{a}= 0.045815 k

_{e1}= 0.197097 k

_{e2}= 1.004856

_{a}is a measure of the strength of the tenside’s influence. As follows from (13) and (15), the 0.001% solution of tenside increased absorption rate constant k

_{a}from 0.030272 h

^{−1}to 0.045815 h

^{−1}, that is by 51.34%.

_{1}(t), without and with the tenside are shown in Figure 8. The dashed curve corresponds to parameters (k

_{a}= 0.030272 h

^{−1}k

_{e1}= 0.649656 h

^{−1}k

_{e2}= 0.338477 h

^{−1}) while the full one corresponds to the increased absorption rate constant k

_{a}, namely (k

_{a}= 0.045815 h

^{−1}, k

_{e1}= 0.959041 h

^{−1}, k

_{e2}= 0.242913 h

^{−1}).

_{1}(t) (dashed curve) at a particular time instant, say t = 2 h was cca 28 mg, but after the addition of the tenside (full curve), it decreased to cca 25 mg. The decrease is due to faster outflow from the 1st compartment caused by the increased k

_{a}.

_{1}(t) represents the drug amount in the peripheral compartment (GIT) and for that reason, it cannot be measured in-vivo. It can be only predicted by the compartment model (6). The same goes for the predicted rate of absorption given by Expression (16).

_{a}x

_{1}(t) [mg·h

^{−1}]

_{1}(t) without tenside.

_{1}(0) = 50 mg. The deviation is natural and indicates that the drug amount cannot jump instantly to its initial value.

#### 3.4.1. Cumulative Eliminated Amounts in Time

_{e1}and k

_{e2}were identified is irrelevant w.r.t. purposes of the paper. This is demonstrated in Figure 10 showing cumulative drug amounts which left the 1st compartment in the time t, (given by the integral ${{\displaystyle \int}}_{0}^{t}\left[{k}_{{e}_{1}}+{k}_{a}\text{}\right]{x}_{1}\left(t\right)dt)$for two different parameter sets. Let us consider the parameter sets (15). As shown in Figure 10, both cumulative amounts converge to the administrated dose M

_{0}= 50 mg, though their shapes are slightly different.

_{a}+ k

_{e1}) is larger than in the lower triplet.

#### 3.4.2. Total Amount of Absorbed Drug and Absolute Bioavailability

_{1}(t) for the drug with and without added tenside are shown in Figure 8. Due to the added tenside, the absorption rate constant k

_{a}increased from 0.030272 h

^{−1}to 0.045815 h

^{−1}, i.e., by 51.34447% and the total absorbed amount (18) increased virtually two-fold, namely from 4.384808 mg to 8.919179 mg.

_{0}, which is nothing else than the absolute bio-availability defined by (19), increased virtually two-fold, namely from 8.76964% to 17.83835%. Hence, the presence of surfactant significantly increases the bioavailability of sulfathiazole from the suspension. Similarly, it was found that the addition of 0.001% solution of the tenside increased the rate constant k

_{a}by cca 51.34%

## 4. Discussion and Conclusions

_{a}t) and then identify parameters K and k

_{a}. However, who knows whether the absorption follows the exponential or any other function? This is especially true for special dosage forms, like chewing gums, dragées, dermal patches and the like. On other hand, the presented system-based analysis can directly predict both the rate of absorption and the fraction of the dose which enters the blood circulation.

_{24}H

_{44}O

_{12}) at the temperature of the experiment, i.e., 38 °C. It is an easy task to calculate the molecular weight of the tenside, namely 524.6 g/mole. Keeping this in mind, the tenside concentration 0.001% corresponds to CMC =1/ 524.6 = 1.91 × 10

^{−3}mole/l. Note that we neither measured the CMC exactly nor found it in the literature.

_{a}k

_{e1}, and k

_{e2}cannot be unambiguously determined from already identified values of the parameters b

_{0}, a

_{0}, a

_{1}. The same goes for the model shown in the appendix―see Figure A1. In spite of this, the absorption rate constant k

_{a}was unambiguously determined. In this view, the ambiguous identification of values of k

_{e1}and k

_{e2}does not matter. In relation to that, it would be reasonable to note the following: As the major route of the drug elimination is the renal excretion, a passable way to specify which of these two sets of model’s parameters is correct may lead through evaluation of the drug amount in the urine. In particular, it would be enough to evaluate the half-time t

_{1/}

_{2}of the renal excretion and then use the known relation k

_{e1}= 0.693/t

_{1/2.}However, this would be beyond the scope of the paper.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## Appendix A

_{a}, still another parsimonious compartment model (Figure A1) was analysed.

_{12}= 0.045817, k

_{21}= 0.943229, k

_{e}= 0.258784)

_{12}= 0.045817, k

_{21}= 0.258784, k

_{e}= 0.943229)

_{12}, (which is in essence the absorption rate constant k

_{a}) obtained before and after additions of the tenside are exactly the same for both models (6) and (A1). Specifically, k

_{a}= 0.030272 h

^{−1}before addition and k

_{a}= 0.045817 after addition of the tenside. These results also confirm that absence of the full parametric identifiability may not preclude the correctness of the results.

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**Figure 1.**Drug amounts M and m obtained from the in-vivo experiment with (dashed curve) and without (full curve) of the added tenside, respectively.

**Figure 7.**The in-vivo measured drug amounts x

_{2}(t) without the added tenside (full curve) and the model predicted amounts with the tenside (dashed curve).

**Figure 8.**Predicted drug amounts x

_{1}(t) in the GIT without (dashed curve) and with the tenside (full curve). Below is the difference (x

_{1without}–x

_{1with}).

**Figure 10.**Cumulative amounts of the eliminated drug from the 1st compartment for 1st triplet in (15) (

**above**) and the 2nd triplet in (

**below**).

T [h] | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|

c [mg/mL] | 0 | 0.0715 | 0.0855 | 0.0780 | 0.0735 | 0.0490 | 0.0535 |

m [mg] | 0 | 0.914 | 1.093 | 0.997 | 0.952 | 0.626 | 0.684 |

M [mg] | 0 | 1.262 | 1.403 | 1.280 | 1.222 | 0.803 | 0.878 |

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**MDPI and ACS Style**

Vitková, Z.; Tárník, M.; Pavlovičová, J.; Murgaš, J.; Babinec, A.; Vitko, A.
In-Vivo Analysis and Model-Based Prediction of Tensides’ Influence on Drug Absorption. *Molecules* **2021**, *26*, 5602.
https://doi.org/10.3390/molecules26185602

**AMA Style**

Vitková Z, Tárník M, Pavlovičová J, Murgaš J, Babinec A, Vitko A.
In-Vivo Analysis and Model-Based Prediction of Tensides’ Influence on Drug Absorption. *Molecules*. 2021; 26(18):5602.
https://doi.org/10.3390/molecules26185602

**Chicago/Turabian Style**

Vitková, Zuzana, Marián Tárník, Jarmila Pavlovičová, Ján Murgaš, Andrej Babinec, and Anton Vitko.
2021. "In-Vivo Analysis and Model-Based Prediction of Tensides’ Influence on Drug Absorption" *Molecules* 26, no. 18: 5602.
https://doi.org/10.3390/molecules26185602