# Thermal Load and Heat Transfer in Dental Titanium Implants: An Ex Vivo-Based Exact Analytical/Numerical Solution to the ‘Heat Equation’

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## Abstract

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## 1. Introduction

**1**) firstly, in the other studies, authors tend to solve the heat equation with a source term that accounts for the temperature excess T-Ts, with (T) being the temperature at a given point and (Ts) being the temperature from the surroundings. Herein, we find it more natural to integrate the heat equation without the additional source term and opted to take into account the thermal load into the imposed boundary conditions; and (2) secondly, in other works, the investigation is often based on a typical or traditional numerical analysis (approach), and, to the best of our knowledge, an exact analytical solution is still missing. Obtaining an exact analytic expression for the solution is always challenging and desirable, since the physics are more transparent, while at the same time, being accessible to everyone. In particular, the interested reader may use the analytic expression either to check/reproduce the results shown in articles or to study other aspects of the solution not considered in publications.

**A**) We propose to solve the standard heat equation, modifying the imposed boundary conditions without any additional source term, and on the other hand, (

**B**) we fill a gap in the literature via obtaining an exact analytical solution of a somewhat simplified problem, which nevertheless, encapsulates the physics and reproduces the results already found in previous works via numerical analyses. In addition, for the first time, we introduce and involve herein, the intrinsic time, a “proper” time that characterizes the geometry of the dental implant fixture and overall system, and we show how the interplay between that time and the exposure time influences temperature changes, and subsequent implant survival.

## 2. Thermal Load and Heat Transfer

#### 2.1. Formulation of the Physical Problem

_{surface}(t) and T(t, y = 0) = T

_{bone}(t), where the functions T

_{bone}(t), T

_{surface}(t), and f(y) are given functions depending on the physics of the problem at hand. This problem is well posed, and it has a unique solution [31]. For example, in the simplest case, in which

_{0}is a constant temperature (e.g., room temperature or body’s natural temperature) the unique solution must be T(t, y) = T

_{0}, as it clearly satisfies the heat equation and all conditions.

_{1}= 60 degrees, and then it decreases monotonically until it eventually reaches the body temperature T

_{2}= 37 degrees.

_{1}down to T

_{2}. In particular, in the discussion to follow, we shall consider an exponential function as follows:

_{0}is the thermal stress exposure time that shows how fast the temperature of the oral cavity drops to the body’s natural temperature (Figure 2). In fact, if we take t

_{0}= 0.2 s, which corresponds to a duration of approximately 1 s, or t

_{0}= 2 s, which corresponds to a duration of 10 s, our thermal loading resembles the ones commonly considered in previous works [21,24,26,27,28,29,30]. Therefore, in what follows, we assume an initial condition T(t = 0, y) = T

_{2}, a boundary condition at the bone T(t, y = 0) = T

_{2}, and a boundary condition at the surface T(t, y = L) = T

_{oralcavity}(t) given in (8), and we define ΔT = T

_{1}− T

_{2}= 23 degrees.

#### 2.2. Exact Analytical Solution of the Initial/Boundary Value Problem

_{n}that are given by:

_{n}/L, while if we plug the ansatz into the partial differential equation, we obtain an ordinary differential equation for Cn(t) as follows:

_{2}as t $\to \infty $.

_{2}. As we will see shortly, the temperature changes versus time at a certain location, e.g., close to the surface or close to the bone, depend on the interplay between the exposure time t

_{0}and the intrinsic time of the implant τ = ${L}^{2}$/(α ${\pi}^{2}$), which, for a given length L, is low for good thermal conductors and high for poor conductors.

## 3. Main Features of the Solution

_{0}; and (iii) location point along the axis of the implant. Our approach is theoretical, with no experimental validation feasible or at disposal. Nonetheless, the figures below clearly show that all key features observed in the earlier related studies are reproduced.

_{0}of the thermal load, and not on characteristics such as age, sex, etc., of the patients. In practice, however, the behavior of the solution depends on the interplay between t

_{0}and the intrinsic time of the implant T (please see discussion below).

_{0}; and (ii) on the other hand, the intrinsic time τ of the geometry of the dental implant. The length of the dental implant varies from 7 mm to 20 mm [24]. Here, we fix the length at L = 1.3 cm as in [25]. Moreover, for a given geometry and a certain thermal load, temperature changes depend on the location along the implant. Following the standard notation, we introduce the location points B1 at y = 3 L/4 (superficial), B2 at y = L/2 (middle), and B3 at y = L/4 (deep) [26]. Finally, we consider two different types of implants, type A with α = 2 × ${10}^{-6}$($\begin{array}{c}{\mathrm{m}}^{2}\end{array}$/s) (moderate conductor), and type B with α = 5 × ${10}^{-6}$($\begin{array}{c}{\mathrm{m}}^{2}\end{array}$/s) (good conductor), with values comparable to those employed in [25,26]; consequently, our approach and findings may be directly compared to the results obtained therein.

_{0}) and for a given implant material (i.e., known thermal diffusivity α), the temperature depends on two independent variables, namely, one one hand, on the time t, and on the other hand, on the location point y along the axis of the implant. Therefore, one may plot T versus t for a certain point y, or plot T versus y at a given instant of time. This is shown in Figure 3 and Figure 4 below.

_{0}= 2 s. At every instant of time, the temperature at the end points remains the same due to the imposed boundary conditions, while at a certain location, i.e., fixed y point, the temperature decreases with time.

_{0}= 2 s. As we go deeper, the maximum temperature reached decreases and the time needed to reach it increases. The highest temperature is observed at point B1 due to its proximity to the thermal load.

_{0}= 2 s. The good conductor (type B in brown) reaches the highest temperature fast, while the moderate conductor (type A in orange) reaches a lower highest temperature later, due to the fact the heat is transferred slower in the case of implant A. Our results shown in Figure 3 and Figure 4 have been also observed in [25].

_{0}for implants A and B, respectively. In particular for implant A, with an intrinsic time of 8.6 s, we have considered t

_{0}= 2, 5, 8, 11, and 14 s, from bottom to top, while for implant B, with an intrinsic time of 3.4 s, we have considered t

_{0}= 1, 2, 3, 5, and 6 s, from bottom to top. For both implants, the curve in the middle corresponds to the case where the exposure time is very close to the intrinsic time of the implant. We see that the highest temperature observed increases with the exposure time both for implant A and implant B. Note that when the exposure time approaches the intrinsic time of the implant, the highest temperature reached is approximately 41 degrees, just below the critical temperature of 42 degrees, irrespectively of the material chosen.

## 4. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

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**Figure 1.**Oro-Dental Implantology, illustrating the main parts of a dental implant system and the implant–bone interface.

**Figure 2.**Thermal load for two different exposure times, t

_{0}= 0.2 s (black color) and t

_{0}= 2 s (blue color). The first one drops faster.

**Figure 3.**Temperature distribution (in degrees °C) versus location at four different instants of time (from top to bottom 9 s, 12 s, 15 s, and 18 s) for implant A and for t

_{0}= 2 s.

**Figure 4.**Temperature changes (in degrees °C) versus time (in sec) for implant A and for t

_{0}= 2 s at three different locations, namely B1 (y = 3 L/4) in red, B2 (y = L/2) in orange, and B3 (y = L/4) in blue.

**Figure 5.**Temperature changes versus time at point B2 for implants A (orange) and B (brown) and for t

_{0}= 2 s.

**Figure 6.**Temperature changes versus time at point B2 for implant A (=8.6 s) and for t

_{0}= 2 s, 5 s, 8 s, 11 s, and 14 s from bottom to top.

**Figure 7.**Temperature changes versus time at point B2 for implant B (=3.4 s) and for t

_{0}= 1 s, 2 s, 3 s, 5 s, and 6 s from bottom to top.

**Figure 8.**Experimental Set-up; developed in-House (at the BioMAT’X R&D&I Laboratory—Haidar Lab, CiiB, Faculties of Medicine and Dentistry, Universidad de los Andes, Santiago de Chile) as an ex-vivo heat distribution model employing human patient-grade titanium dental implants placed into porcine ribs (without coolant) and thermal changes monitored/recorded (quantified) using a CorDEX TP3R ToughPix DigiTherm Digital Thermal Camera (an ongoing investigation).

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**MDPI and ACS Style**

Panotopoulos, G.P.; Haidar, Z.S.
Thermal Load and Heat Transfer in Dental Titanium Implants: An Ex Vivo-Based Exact Analytical/Numerical Solution to the ‘Heat Equation’. *Dent. J.* **2022**, *10*, 43.
https://doi.org/10.3390/dj10030043

**AMA Style**

Panotopoulos GP, Haidar ZS.
Thermal Load and Heat Transfer in Dental Titanium Implants: An Ex Vivo-Based Exact Analytical/Numerical Solution to the ‘Heat Equation’. *Dentistry Journal*. 2022; 10(3):43.
https://doi.org/10.3390/dj10030043

**Chicago/Turabian Style**

Panotopoulos, Grigorios P., and Ziyad S. Haidar.
2022. "Thermal Load and Heat Transfer in Dental Titanium Implants: An Ex Vivo-Based Exact Analytical/Numerical Solution to the ‘Heat Equation’" *Dentistry Journal* 10, no. 3: 43.
https://doi.org/10.3390/dj10030043