# Computational Modelling for Efficient Transdentinal Drug Delivery

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Transport of Therapeutic Compounds through the Dentinal Tissue

^{2}. In addition, the dentinal tubules are connected to each other by branching canalicular systems.

## 3. Materials and Methods

#### 3.1. Code Validation

^{2}) and it was attached with a minimum quantity of aquarium marine silicone (Top sil, Mercola, Athens, Greece) carefully placed on the enamel margins of the dentine disc. After the silicone was set, all of the joints were reinforced externally with sticky wax (Kem-den, Purton, UK) and the whole system was filled with Ringer’s solution (Vioser, Iraklio, Greece) to be checked for leakages. The experimental conduit, which is schematically presented in Figure 2, is a closed system that was constructed at the Dental School of Aristotle University of Thessaloniki.

^{−10}m

^{2}/s [23]) through the dentinal disc to the opposite area (i.e., the observation area) is measured using the non-intrusive experimental technique μ-LIF (μ-Laser Induced Fluorescence). Two concentration groups of the aqueous solution were used, namely G1 for M

_{0}= 0.1 mg/L and G2 for M

_{0}= 0.05 mg/L.

- Prior to the actual measurements, the system is calibrated using suitable solutions. In our case aqueous Rhodamine B solutions with known concentrations, namely, M = 0.05, 0.025 and 0.00 mg/L were employed, while, for each concentration C
_{i}, an image C_{(x,y)}is taken. - To reduce noise, image masking of the acquired images is performed before defining an appropriate Region of Interest (ROI), at which the fluorescence intensity is measured.
- The relationship between the measured fluorescence intensity field I
_{(x,t)}and the concentration field C_{(x,y)}is determined. - A set of 20 images is acquired and the mean image is calculated.
- Each mean image is compared with the previously defined μ-LIF calibration curve.

_{L}) values of Rhodamine B at the observation area of a dentinal disc with porosity ϕ = 10% are compared with the corresponding CFD simulation results.

#### 3.2. Numerical Procedure

^{−10}kg/(m

^{2}·s). Finally, a parametric study is performed that includes the effect of:

- the porosity of the tissue (ϕ),
- the thickness of the tissue (Remaining Dentinal Thickness, RDT),
- the initial concentration (M
_{0}) of the substances to be diffused, - the molecular size of the substances to be diffused, i.e., their Diffusion Coefficient and
- the consumption rate (R) of the diffusate at the pulpal side

^{2}, while the diffusate occupies a volume of 32 mm

^{3}.

^{2}/8. They extend and adapt Equation (1) to the equivalent channel model of the porous body by taking into consideration the effects of (i) the effective tortuous path, (ii) the effective volume and (iii) the concept of effective radius R:

_{e}due to the tortuous path and the length L of the porous body (τ ≡ L

_{e}/L). Comparison of Equation (2) with Darcy’s law gives the following expression for defining the permeability (k) (Equation (3)):

_{min}and the diffusion coefficient D of a protein can be estimated by Equations (4) and (5) [30]:

_{min}is given in nanometers and the molecular weight M

_{w}in Daltons (Da)

^{−16}g·cm

^{2}·s

^{−2}·K

^{−1}is the Boltzmann constant and T the absolute temperature. k is given here in centimeter–gram–second units because D is expressed in centimeter–gram–second, while μ is the solute viscosity in g/(cm·s). In our case, it is assumed to be that of water, as its thermophysical properties are very close to those of the actual dentinal fluid [31]. R

_{s}, called Stokes radius, represents the radius of a smooth sphere that would have the same frictional coefficient f with a protein and is expressed in centimeters in this equation. Assuming that f equals f

_{min}, i.e., the minimal frictional coefficient that a protein of a given mass would obtain if the protein were a smooth sphere of radius R

_{min}, in Equation (5), R

_{s}can be replaced by R

_{min}.

_{s}of the molecules tested is in the range of 2.2–22.0 nm and consequently the corresponding diffusion coefficient, D, in water at 37 °C, is in the range of 1.36–0.14 × 10

^{−10}m

^{2}/s. For example, bone morphogenetic protein (BMP-7) is a bioactive protein used in the dental clinic practice with approximately 50 kDa molecular weight and R

_{s}of 2.2 nm. To conduct a parametric study, three initial mass concentration values for each diffusing substance are employed, i.e., 0.10, 0.05 and 0.01 mg/mL following our previous study [17].

^{®}package (18, ANSYS, Canonsburg, PA, USA) and following the Design of Experiments (DOE) methodology, a set of “computational experiments” was initially designed. The DOE methods allow the designer to extract as much information as possible from a limited number of test cases and makes the method ideal for CFD models that are significantly time-consuming [32]. Eventually, the temporal and spatial concentration at the pulpal side of the disc will be calculated using the computational procedure. A summary of the design variables range is given in Table 1.

^{6}of tetrahedral elements, while the porosity model is used to account for the space occupied by dentinal tubules per unit area in the disc. All simulations were run in transient mode, while the total simulation time for each run varied in the range of 10–25 h, depending on the consumption rate (R) applied. A time-step dependence study was also performed (i.e., time steps in the range of 5–10 s). Zero flux was inposed as boundary conditions at all the walls of the computational domain, while, when a consumption rate is employed, negative flux is imposed at the pulpal side. For the discretization scheme, the High Resolution one was employed and as for the transient scheme, a Second Order Backward Euler scheme was used. A high-performance unit for parallel computing, which runs on a Gentoo Linux distribution, was used.

## 4. Results

_{L}) are presented. The general case, i.e., when a consumption rate is imposed at the pulp, is presented first. For the simulations concerning the effect of the other parameters, no consumption is assumed, i.e., R = 0.

#### 4.1. Effect of the Agent Consumption Rate at the Pulpal Side

_{s}= 2.2 nm and M

_{0}= 0.10 mg/mL is employed at the top of the dentinal disc of ϕ = 10%. As it can be seen, the concentration value at the pulpal side is lower than the initial one regardless of the consumption rate applied. The concentration value may exhibit 100% decrease when the lowest and the highest consumption rates are compared. During the first time steps of the agent application on the disc, the concentration gradient along the disc thickness is high and the concentration at the pulpal side C

_{L}increases until t* = 1 (i.e., dashed line), if the agents are consumed, and, after this critical time, equilibrium cannot be reached. On the contrary, as time passes, the agent is being continuously consumed, which means that the concentration at the pulp will decrease until the agent disappears.

^{−10}kg/(m

^{2}·s), the total estimated time for an agent of R

_{s}= 2.2 nm to be fully consumed is approximately 18 h in a porous disc of ϕ = 10%.

_{L}values. In order to facilitate the evaluation of the effect of the other parameters, a zero consumption rate at the pulp end is assumed for the remaining CFD simulations.

#### 4.2. Effect of Remaining Dentin Thickness (RDT)

_{L}) is shown in Figure 7, when a substance of R

_{s}= 4.4 nm at M

_{0}= 0.10 mg/mL is diffused through a dentinal disc of ϕ = 5%. As it is expected, since the process is considered one-dimensional, it is dominated by the diffusion length. For discs with significant low thickness, the delivery of substances to the pulp is rapid, and a steady state condition can be quickly achieved. For example, when a dentinal disc of 0.5 mm is employed, the concentration at the pulpal side reaches its final value after two hours of application. The time barrier for steady state condition can be easily estimated from the dimensionless time variable t = D/L

^{2}.

#### 4.3. Effect of the Molecular Size

_{L}of the substance with the greatest molecular size to reach steady state conditions (i.e., the concentration gradient to be zero).

#### 4.4. Effect of Dentine Porosity

_{s}= 4.4 nm and M

_{0}= 0.01 mg/mL). From Figure 12, it can be observed that the magnitude of the porosity value does not practically affect the calculated molar flux through the dentinal discs.

#### 4.5. Effect of the Initial Diffusate Concentration

_{L}, are presented in Figure 13. One can observe that approximately 0.5 h is needed for the first molecules of the substance to reach the bottom end of a dentinal disc of 1.5 mm RDT. This is in agreement with the literature and especially with Pashley and Matthews [11] who have reported that diffusion is a low process and, depending on their size, it requires 30–120 min for the molecules to reach the pulp across a 1–2 mm dentin thickness.

_{1}and t

_{2}in Figure 12). This is important for clinical practice of dentistry, since, by this procedure, the behavior of each therapeutic agent can be predicted prior to its application.

#### 4.6. Prediction of the Drug Pulpal Concentration

^{2}·s) and RDT is expressed in m. The outcome of Equation (7) is in very good agreement with the available data from the computational simulations (overall uncertainty is less than ±15%). Figure 14 presents this comparison when an agent of R

_{s}= 2.2 nm is transported through a 2.5 mm thick dentinal disc of ϕ = 10% for M

_{0}= 0.10 (Figure 14a) and 0.05 (Figure 14b) mg/mL, respectively.

_{L}). Equation (7) permits the estimation of the necessary initial concentration of the therapeutic agent to be imposed for an efficient therapy to be achieved, i.e., the drug concentration at the pulp to reach a critical signaling value dictated by the dental clinical practice, when the maximum acceptable time of application and the consumption rate are given. From Figure 14, it can be observed that there is always a time margin during which the agent retains its maximum value at the pulp (i.e., ~2 h in the specific case of Figure 14; grey area). However, since the R values have been arbitrarily chosen, the scientific community that deals with phenomena related with the dental pulp irrigation and dentinal tissue regeneration must quantify the desirable consumption rate of biomolecules/proteins in the vicinity of the pulp.

## 5. Conclusions

- the transdentinal diffusion of drugs is mainly affected by the molecular size and the RDT, as it was expected,
- a porosity change of 5% to 20% results in less than ±15% C
_{L}difference, - a variation of the agent consumption rate at the pulpal side between 0 and 10
^{−10}kg/(m^{2}·s), leads to a 100% C_{L}decrease, while the consumption time is 18–25 h.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

C | Molar concentration, mol/m^{3} |

C_{L} | Molar concentration at the bottom end, mol/m^{3} |

C_{o} | Initial molar concentration, mol/m^{3} |

C_{∞} | Final molar concentration (for t = L^{2}/D), mol/m^{3} |

D | Coefficient of diffusion, m^{2}/s |

j | Concentration flux, mol/m^{2}·s |

L | Length, m |

Μ_{0} | Mass concentration, g/m^{3} |

M_{w} | Molecular weight, g/mol |

R | Agent consumption rate, kg/(m^{2}·s) |

RDT | Remaining dentinal thickness, m |

R_{s} | Stokes radius, m |

r | Radius of conduit, m |

T | Temperature, °C |

t* | Dt/L^{2}, dimensionless |

t | Time, s |

U | Velocity, m/s |

x | Distance, m |

μ | Viscosity, g/cm·s |

ϕ | Porosity, % |

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**Figure 1.**(

**a**) dentinal tubules running perpendicularly from pulpal wall towards dentino-enamel junction (DEJ) scanning electron microscopy (SEM), reproduced with permission from [17] and (

**b**) dentinal disc of a human tooth.

**Figure 2.**Schematic of the experimental conduit. Reproduced with permission from [17].

**Figure 3.**Schematic representation of the μ-LIF experimental setup. Reproduced with permission from [17].

**Figure 4.**Typical comparison of the molar concentration values at the observation area of the μ-LIF technique with the corresponding computational results (ϕ = 10%).

**Figure 6.**Molar concentration at the pulpal side of the dentinal discs as a function of different consumption rates for ϕ = 10% and RDT = 2.5 mm.

**Figure 7.**Effect of the Remaining Dentin Thickness (RDT) on the diffusion characteristics; C

_{L}versus time.

**Figure 10.**Effect of the molecular size on the diffusion characteristics; concentration gradient versus time.

**Figure 11.**Effect of the dentinal disc porosity value on the diffusion characteristics for: (

**a**) RDT = 0.5 mm, R

_{s}= 2.2 nm and (

**b**) RDT = 1.5 mm, R

_{s}= 2.2 nm.

**Figure 12.**Temporal molar flux distribution for three different porous discs (RDT = 1.5 mm, R

_{s}= 4.4 nm and M

_{0}= 0.10 mg/mL).

**Figure 14.**Comparison of computational fluid dynamics (CFD) data concerning the concentration of agents at the bottom end of the porous disc as a function of time with the outcome of Equation (7) for: RDT = 2.5 mm, ϕ = 10%, R

_{s}= 2.2 nm and (

**a**) M

_{0}= 0.10 mg/mL and (

**b**) M

_{0}= 0.05 mg/mL.

Parameter | Lower Bound | Upper Bound |
---|---|---|

Porosity (ϕ), % | 5 | 20 |

Remaining Dentinal Thickness (RDT), mm | 0.5 | 2.5 |

Initial Concentration (M_{0}), mg/mL | 0.01 | 0.10 |

Molecular Size (R_{s}), nm | 2.2 | 22.0 |

Consumption Rate (R), kg/(m^{2}·s) | 0 | 10^{−10} |

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**MDPI and ACS Style**

Passos, A.D.; Tziafas, D.; Mouza, A.A.; Paras, S.V.
Computational Modelling for Efficient Transdentinal Drug Delivery. *Fluids* **2018**, *3*, 4.
https://doi.org/10.3390/fluids3010004

**AMA Style**

Passos AD, Tziafas D, Mouza AA, Paras SV.
Computational Modelling for Efficient Transdentinal Drug Delivery. *Fluids*. 2018; 3(1):4.
https://doi.org/10.3390/fluids3010004

**Chicago/Turabian Style**

Passos, Agathoklis D., Dimitris Tziafas, Aikaterini A. Mouza, and Spiros V. Paras.
2018. "Computational Modelling for Efficient Transdentinal Drug Delivery" *Fluids* 3, no. 1: 4.
https://doi.org/10.3390/fluids3010004