**Commentary**

## Commentary on an Article by Kwon TK, et al.: “FirstOrder Mathematical Correlation between Damping and Resonance Frequency Evaluating the Bone-Implant Interface”

In-Sung Luke Yeo^{*}

Department of Prosthodontics and Dental Research Institute, Seoul National University School of Dentistry, Korea

** ^{*}Corresponding author: **In-Sung Luke Yeo, Department of
Prosthodontics and Dental Research
Institute, Seoul National University
School of Dentistry, 101, Daehak-ro,
Jongro-gu, Seoul 03080, Korea

**Published**: 05 Jan 2017

**Cite this article as**: In-Sung Luke Yeo. Commentary on an
Article by Kwon TK, et al.: “First-Order
Mathematical Correlation between
Damping and Resonance Frequency
Evaluating the Bone-Implant Interface”.
J Dent Oral Biol. 2017; 2(2): 1025.

## Abstract

A recent study on implantology reported a statistical linear correlation between two measures
at the bone-implant interface: Periotest evaluation of the damping effect at the interface and
resonance frequency analysis. The publication formulated a linear function of the dependent
variable, the Periotest value (PTV), to the independent variable, the implant stability quotient
value, by homogenizing the density of bone blocks and thoroughly controlling the engaged depth
of implants inserted into these blocks. A correlation such as this can be mathematically explained
using a second-order differential equation to describe a mechanical vibration model in physics;
free vibration with damping. Although several clinical studies have found no significant correlation
between the two values, it is important to investigate the reasons for the discrepancy between theory
and experimental results in order to understand the nature of the bone around an implant.

**Keywords: Bone-implant interface; Periotest; Resonance frequency analysis; Correlation**

## Commentary

In a recent publication, “First-order mathematical correlation between damping and resonance
frequency evaluating the bone-implant interface,” Kwon et al. [1] found a statistical linear
correlation between the Periotest and implant stability quotient values. The Periotest device (Gulden
Messtechnik, Bensheim, Germany) evaluates the quality of the bone-implant interface using the
damping effect at the interface. When the implant firmly contacts the bone, the damping effect
becomes smaller, which reduces the Periotest value (PTV) (the overall PTV range is from −8 to 50)
[2,3]. Resonance frequency analysis can detect the frequency of the peak output amplitude responsive
to input signals, which varies depending on the quality of the interface between the implant and
the bone [3,4]. This frequency is used to derive the implant stability quotient (ISQ) value, which is
higher (within an overall range from 0 to 100) for more stable bone-implant interfaces. Although
the authors formulated a linear equation showing a statistical correlation between the PTV and the
ISQ value [1], this finding is in conflict with many studies that have found little or no significant
correlation even though the two values would intuitively seem to be associated [5].

It appears that a mathematical relation can be derived between the resonance frequency and
damping effect at the bone-implant interface. The primary devices used for resonance frequency
analysis (Osstell, Integration Diagnostics AB, Sävedalen, Sweden) and damping effect estimation
both use reflective outputs based on vibrational inputs applied to the bone-implant interface. When
an implant is placed in the bone, three interface layers must be considered: the implant, bone, and
an in-between layer filled with blood, blood clot, fibrous tissue, or other tissues (Figure 1).When
the relative size of the in-between layer grows, the implant becomes unstable, resulting in a change
in resonance frequency and a larger damping effect. Thus, the reflective response to the vibrational
inputs at the interface can be interpreted using amass-spring-damper model of mechanical vibration,
which is a well-known model in physics (Figure 2).

The second-order differential equation for this model is mx cx kx + + = 0 , where m is the mass
of the system, c is the damping constant, k is the spring constant, x is displacement, and x and x
are the second- and first-order time derivatives of x, respectively. The mass, damping, and spring
constants are all positive. When the bone-implant interface is under damped, a situation that is
clinically common (the critically damped and over damped cases
both indicate failed implants that can be considered to be mobile),
the general solution for this equation is
( ) 2 2
2 4 4
x t cos cos sin sin
2 2
c t m mk c mk c Ae B t C t m m
− − − = +
where A, B, and C are constants.
From this solution, a mathematical relation is found between the
resonance frequency, Rf
, and the damping constant, c:
2 4 2 1 , , : 4 2 f
mk c c D r r D constant m km
R π
− − = = =

Using 2 1 R D r f = − , a linear mathematical relation can be
theoretically found using logarithms such as 1 2 ln (1 ) (constant). 2
R r E f = − +
Assuming that ln Rf
is y, which is associated with the ISQ value, and
that ln(1-r2
) is x, which is associated with the PTV, the equation can
be expressed as y = ax | b, where a and b are constants. That is, a firstorder
function is obtained. The authors demonstrated a statistical
proof of this using bone models with precisely controlled thickness
and density [1]. Another study supported such a statistical correlation
between PTVs and ISQ values and produced a similar linear function
[3].

As the bone has a heterogeneous density distribution, and bone
quality differs for each individual, it is difficult to find the linear
PTV-ISQ correlation from real-world selected sample groups.
Nevertheless, is it logical to conclude that the damping effect and
resonance frequency are insignificantly correlated in reality, as
other studies have stated, despite the fact that the two phenomena
are theoretically related through a differential equation based on
a mechanical vibration model? [6-9]. It would be more rational to
attempt to determine the following: how the real bone resorption
pattern and bony contour differ from their respective theoretical
counterparts; why the PTV-ISQ correlation deviates from the
theoretically expected value according to bone quality; and what is the
nature of osseointegration at the bone-implant interface. Addressing
these questions will lead to the development of a more elaborately
designed resonance frequency analyzer and damping effect estimator
that can predict the crestal bone geometry around an implant and
differentiate the simple initial contact of implant surface to bone from
the bone integrated into the implant surface.

**Figure 1 **

**Figure 1**

Schematic of the bone-implant interface. Direct contact of implant
surface with the bone clinically equates to a stable implant. Empty space
scan occur between the bone and the implant; these are often filled with
blood (red area) or other tissues (pink area).

**Figure 2 **

**Figure 2**

Schematic of an implant inserted into bone. The red area on the
left represents a gap between the implant and bone, which results in a larger
damping effect when force is applied to the implant. The clinical state of the
implant inserted into the bone is considered to be physically equivalent to
free vibration with damping, as shown on the right. m: mass of the system; k:
spring constant; c: damping constant.

## Acknowledgment

This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean Government (MSIP) (No. NRF-2016R1A2B4014330).

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