## Abstract

Biophysics is rarely mentioned as one of the most useful parts of dental and medical students' curricula. However, with the growing complexity of tools and methods used in diagnostics and therapy, the knowledge of their physical foundations becomes important and helps with choosing the optimal solutions for both, a patient and a doctor. The aim of the proposed activity is to develop students' intuition about simple physical models that help with understanding fundamental properties of temporomandibular joint (TMJ). A simple device, which allows for bite force measurement, is proposed. It is based on beam bending and a strain gauge Wheatstone bridge circuit mounted on two connected arms: the stiff one and the more elastic one. Linear regression is the only mathematical concept needed for understanding the physical background of the proposed activity. During the proposed activity—measuring of bite force for incisors, premolars, and molars—students are confronted with basic concepts, such as lever, torque, electrical circuit, calibration curve. By utilizing a simple idea, instead of a commercially available device, students can understand where the data come from. Proposed system delivers physiologically reasonable results.

## Introduction

Modern medicine draws on the achievements of many disciplines of science and engineering. It is then important that medical professionals—MDs of different specializations, technicians, paramedics, medical physicists, and analysts, to name only a few—can speak each other's language. In the following paper we demonstrate how basic physics can be introduced into dental students' curriculum with the intention to foster their future collaboration with other medical professionals, such as radiologists, orthodontists, or engineers. For this, we provide a description of a system which demonstrates some of the biomechanical fundamentals of biting.

Temporomandibular joint (TMJ) is a beautiful example of a complex biomechanical system. It has been studied for years, but the consensus about the basic operational principles (e.g., how important for its functioning is joint compression under load and to what extent a simple lever models, which assume generation of huge stress on the tissue, can be used) has not been reached [14]. Theoretical models become quite complicated when features such as deformation and distributed loads are taken into account [5]. The discussion is subtle and yet important from both, practical and theoretical points of view. It is therefore important to equip the students with tools that may be used for simplifying what seems complicated and later, after they gain some confidence in their craft, confront them with all the ambiguities, for example, with the between-patients variability. Physical models—levers or wedges—are a natural way of simplifying the description of joints, even as complicated as TMJ, so they seem valuable elements of every biomedical curriculum.

Unfortunately, as most medical students, future dentists have rather little interest in physics, considering it to be of little importance in their future work [6,7]. We argue that physics is important to medical and paramedical students for the reasons provided in these studies [811], that this is not the case. Intuitive understanding of simple mechanics may only be strengthened by introducing some basic concepts of physics. Physics might be interesting for life science students, but requires careful choice of topics, which has to incorporate realistic applications in which physics encourages understanding of some relevant mechanisms [12,13]. Here we present a simple experimental setup to measure the bite force and explore basic biomechanical operating principles of TMJ. We start with a simplified biomechanical model of the joint, then we describe the experimental setup, present the results of measurements with interpretation and summarize with some proposals of problems the students may like to consider.

## Theoretical Framework

The movement of some body parts can be described using simple physical models. One of such models is a lever, i.e., a device for transferring force in such a way as to obtain a greater load by applying relatively low effort. The model lever is a rigid rod, rotated around a fulcrum due to the forces acting on it (Fig. 1(a)).

Fig. 1
Fig. 1

The three fundamental elements of the system, in which we consider the action of a lever, are:

• a fulcrum defining the axis of rotation;

• the effort arm of the applied force, which is the distance between the support point and the point of application of a smaller force that overcomes the lever resistance;

• the resistance arm, which is the distance between the fulcrum and the point of application of the greater resistance force which the lever is to overcome.

Using the anatomical data of a standard human, a mechanical model of the mandible as a third-class lever can be created (Figs. 1(b), 1(c)). In the very simplified, yet accurate version, the condyle plays the role of the fulcrum, the masseter applies force and the bite force is the resistance [1,14]. Depending on the biting conditions, we consider forces acting on the first molar, H, and on the central incisors, W. In the idealized model, the reaction force of the temporal bone on the mandible, R, acts exactly in the fulcrum, so the radius from axis of rotation to point of application of the force is equal to zero. Another simplification is that the weight of the mandible is disregarded. The characteristic distance between the joint and the first molar, L, is on average 6.5 cm for women and 8 cm for men [9]. The masseter has a trailer at a distance of about 0.4 L from the fulcrum (i.e., TMJ), and the incisors are at a distance of 1.2 L.

The main equation needed for understanding the proposed experiment describes the balancing of torques about the joint. For the case of biting with the first molar we get
$0.4L·FM=0·R+L·H$
(1)
$FM=R+H$
(1a)
while for the incisors
$0.4L·FM=0·R+1.2L·W$
(2)
$FM=R+W$
(2a)

where $FM$ is the force exerted by the masseter. Typically reported maximum values of $FM$for adult humans are of the order of 650 N or smaller [2,14,15]. Equations (1) and (2) are torque balances about the fulcrum, whereas Eqs. (1a) and (2a) are force balances in the vertical direction at static equilibrium. Students should be reminded that mechanical equilibrium requires balancing torques and forces. The same muscle force $FM$ can be reacted by bite force $H$ at first molar as well as by bite force $W$ at the incisor. Since at static equilibrium torques around the TMJ must be in balance, for a fixed $FM H>W$ due to different distances of molar and incisor from the fulcrum (Fig. 1(b)). We will come back to this important observation in the last part of the paper.

## Materials and Methods

Figure 2 shows the device used in the experiment (original design, developed as a diploma project, by Michał Lodek). Its core is a set of strain gauges, i.e., sensors whose electrical resistance depends on their strain, which in turn depends on the material the gauge is made of (its modulus), the geometry of the system (i.e., the relative orientation of the interacting sensor and measured object) and the applied force. Since the first two remain constant for a specific device, it allows for indirect measurement of forces by the direct measurement of the change in resistance. It is often introduced at schools as an element of a strain gauge balance made from flexible pencils, in which graphite core is bound as a composite of graphite and elastic polymer [16]. Our setup uses commercially available strain gauges TF-30/350 (active grid length 30 mm, resistance 350 Ω) type K-s (Tenmex, Poland).

Fig. 2
Fig. 2

Measurement device consists of two stainless steel arms connected by a screw, forming a kind of a second-class lever (Fig. 2). Due to different cross-sectional geometry lower arm is stiffer than the upper one. Therefore, applied force mostly deflects the upper arm, where four strain gauges are mounted: two on the bottom and two on the upper side of the arm. The proposed geometry, as well as the material of the device, can be modified—the only requirements are that the lever can be safely placed in the mouth and deformed during biting so that the high enough output electrical signal would be generated.

Since typical changes of resistance of an individual strain gauge rarely exceed 1% it is common to use four sensors, connected to form the so-called Wheatstone bridge (Fig. 3(a)) which significantly improves accuracy of the measurements. The arms ends, shielded by single-use silicate covers, are placed between teeth and the investigated person starts to bite. As a result of the biting the more elastic device arm will deflect, resulting in a change in resistance of the gauges: the resistance of two upper sensors (R1, R4) will increase, while the resistance of two bottom ones (R2, R3) will decrease (Fig. 3(b)).

Fig. 3
Fig. 3

The power supply for the Wheatstone bridge can be DC or AC voltage. When no strain is applied to the lever, the bridge is balanced, i.e., electric potentials VA and VB, shown in Fig. 3, are equal. Under the strain one of the potentials rises up and the other falls down, due to resistance changes of the gauges. This change is usually detected by connecting points A and B with the galvanometer (Fig. 3(c)). Instead of the potential difference (i.e., the output voltage) between those points, galvanometer measures the current flow (i.e., the output current). Very small changes of the resistance provide one advantage: the relation between the applied force and the output current is nearly linear, at least in the range of couple hundred Newtons (around 800 N for our system). This is why, after calibrating the experimental setup with several known weights, the proposed device could be used to determine the bite force.

To keep the design simple and affordable for any school laboratory one has to consider some important parameters of the equipment. For our system, deflection of the lever in the range of a few millimeters gives the resistance change of a single strain gauge of about 0.3%. As mentioned before this value could be altered by changing the thickness of the bar acting as the upper arm, using a different material or repositioning the sensors. It is important though to keep the strain gauges within the limits of their elasticity. With the resistance of the sensor of 350 Ω and the input voltage of 5 V, the obtained output current reaches tens of micro-amps. Therefore, it is possible to replace the galvanometer with a much cheaper multimeter. Applying higher voltage and choosing sensors with lower resistance one can possibly increase the current even further, although in such a case the problem of heat dissipation should be addressed.

## Proposed Activity

Below we present an experiment in the form in which it is performed by medical/dental students. Possible modifications of the experiment, which may be addressed by more physics-oriented students, are presented at the end of this section. Proposed experiment consists of three stages. The first one is calibration of the device. Students are asked to connect the device with voltage power supply and ammeter correctly. We suggest measuring an output current rather than output voltage in order to reduce the possibility of making mistakes by students, i.e., taking input voltage for the output. Then, for a chosen, constant input voltage (UIN), students measure the current at the output of the system (IOUT) when the lever is loaded with known forces—weights of known mass, placed over the upper arm (Figs. 4(a) and 4(b)). Collected data could be used for plotting the electrical current (IOUT) versus force (F) graph in order to obtain so-called calibration curve IOUT(F) (or, in this case, calibration line). Since they are linearly dependent, simple regression can be used to find a linear equation that links both physical quantities (Fig. 5). The general equation for linear regression function is
$Y=aX+b$
(3)
Fig. 4
Fig. 4
Fig. 5
Fig. 5
In presented case Y = IOUT is the measured electrical current (in μA), X =F is the applied force (in N), a is the slope and b is the intercept of a fitted function, hence
$IOUT= aF+b$
(4)

With this equation, one can easily obtain the unknown force by measuring the output electrical current.

In the second stage students use their own mandible to bite the device and measure the electrical current at the output of the circuit (Fig. 4(c)). They protect their teeth with disposable silicon covers. After measuring the output current, it is then straightforward to obtain the value of bite force from Eq. (3), since the parameters a and b are known from the first stage. Students have to decide which teeth they would like to use to bite a device, which translates into the language of mechanics as: choosing the lever arm. At least two different teeth should be measured: one incisor and one molar, with an ideal addition of premolar. The first and the second stages can then be repeated for different values of input voltage.

While biting the paddle with incisors is simple, doing it with molars may be more challenging. Our experience shows that the proposed geometry allows for satisfactorily accurate measurements—that is, for the comparable force exerted by the masseter, the closer the tooth is to the joint, the greater force it exerts on the paddle. To make the measurements even more precise, we propose placing additional silicon patches on both arms of the silicon cover, using nontoxic glue or tape. This will facilitate symmetrical biting of the device with the back teeth. But then similar patches should be used for incisors, to maintain the same measurement conditions. Another possible modification would be shaping the paddle in a more sophisticated way. But we would like to stress that the proposed simple geometry, that we have tested during laboratories, gave physically sound, repeatable results.

One could notice that calibration curve was performed for maximum load of 120 N. However, we have tested the device with load as great as 800 N, for which the dependence between the current and the force was still linear. We have decided to limit the applied load during the regular laboratories due to safety concerns.

The last stage involves comparison of the experimental results with theoretical predictions of the simplified biomechanical model of the TMJ. The main point here is to realize that if the muscle force, $FM$, is the same while biting with molars and incisors, the torque reacted by each tooth must also be the same (cf. Eqs. (1) and (2)). Therefore, the magnitude of the bite force (i.e., the force measured at the site of the teeth) increases when moving toward the fulcrum, i.e., TMJ.

This raises an important issue: how to “control” the muscle force? In reality, it is almost impossible to bite two or more times in a row, making masseter to generate exactly the same force. Additionally, with the increasing number of attempts factors such as fatigue or weariness will surely affect the results. During our labs, each student is told to “apply maximum load with which she or he still feels comfortable”. Quite surprisingly, the results obtained by students are consistent enough to make it clear for them, that the torques reacted by different teeth indeed are the same if the muscle force remains constant. Sample results for one student are shown in Table 1.

Table 1

Sample results obtained for one person, biting with different teeth under input voltages equal to 5 V and 25 V, written as: mean value (standard deviation)

Input voltage UIN = 5V
Incisor teethPremolar teethMolar teeth
IOUT [μA]29.4(1.0)37.5(1.1)45.2(1.2)
F [N]207(11)261(12)313(13)
r [cm]1076
M [Nm]20.7(1.1)18.3(0.8)18.8(0.8)
Input voltage UIN = 5V
Incisor teethPremolar teethMolar teeth
IOUT [μA]29.4(1.0)37.5(1.1)45.2(1.2)
F [N]207(11)261(12)313(13)
r [cm]1076
M [Nm]20.7(1.1)18.3(0.8)18.8(0.8)
Input voltage UIN = 25V
Incisor teethPremolar teethMolar teeth
IOUT [μA]145(8)204(9)232(9)
F [N]206(13)288(15)326(15)
r [cm]1076
M [Nm]20.6(1.3)20.1(1.0)19.6(0.9)
Input voltage UIN = 25V
Incisor teethPremolar teethMolar teeth
IOUT [μA]145(8)204(9)232(9)
F [N]206(13)288(15)326(15)
r [cm]1076
M [Nm]20.6(1.3)20.1(1.0)19.6(0.9)

Results are averaged over five repetitions of biting with each tooth. Distance of a given tooth from the fulcrum, r, is taken as a fixed value.

To find the distance between the TMJ and the tooth, one can use the standardized distances, discussed above and in more depth elsewhere [14]. Another possibility is to make some approximated measurements of tooth-joint distance with a ruler or, more accurately, by carefully palpating the TMJ center and using a compass or a caliper to get the distance. This is the most inaccurate part of the experiment—rather estimation than measurement. But it can be useful in triggering discussion of sources of possible errors and, in general, of the impact of methodology on the final results. This, in fact, may be the most important lesson of any student's laboratory.

Even with such a simple device, one can obtain physiologically reasonable results of the bite force, which differs between incisors, premolars, and molars [2,15]. Applying Equations (1) and (2) and the results for the force exerted on incisors and molars one can obtain the value for the strength generated by the masseter, which is in the physiologically relevant range [14]. Since the setup is a linear electrical circuit, there is consistency between results obtained for 5 V and 25 V. Also, the joint torque (which can be used to find a masseter force) calculated from results obtained for different teeth are consistent within 10% range.

We note that the obtained results, although consistent and convincing, are subject to significant uncertainty due to the simplified assumptions of the model as well as technical issues. For example, it is extremely hard to standardize the force exerted by the masseter for different cases (i.e., when the arms of the device are bitten with different teeth). Also, the silicon covers, used for hygiene and to protect the teeth, impact the precision. However, it is also valuable to let students discuss those and other factors affecting the final results.

## Questions for Students and Possible Extensions of the Experiments

The proposed experiment can be enriched by discussing crucial points with student. Here, we present a few questions which could be posed to students with short descriptions of them.

1. (1)

What does it mean to calibrate a measuring device?

Calibration is done by using a measuring tool to measure a given feature (or, more precise, physical value) of an object, for which the result of such a measurement is known—hence, the object is a reference standard. It is important to realize that the conditions—e.g., the value of voltage—for which the calibration is being done are important and that one cannot expect the same quantitative results under different conditions: a change in voltage will change the current output (see Fig. 5). The idea of calibration is also a good starting point for discussing the variability of the experimental results. Even for the same conditions, the obtained results differ slightly—and that is exactly what should be expected.

2. (2)

Is the bite force distributed equally among all the teeth?

No! The force depends on the distance of each tooth from the fulcrum (joint). This is really simple physics, which in a beautiful way explains, why we use molar teeth for crunching food.

3. (3)

Is it easier to break a molar tooth rather than an incisor while biting a cherry stone?

Due to the forces and the size of an occlusion surface. Bigger force causes bigger pressure when the surface area stays constant. For cherry stone, the pressure is especially big, since its surface at the edge is really small. But bigger force is only part of the explanation. The other one, as significant, is the geometry of the system: a cherry stone is much more stable between molars—with more contact points and different distribution of forces due to geometry of occlusal pits and fissures—than incisors.

4. (4)

While the force exerted by the molars is greater than the one exerted by the incisors, why do we use the latter for bite off, e.g., an apple?

This is caused by the shape of incisors. They look and behave as a wedge—another simple machine. But even smarter answer would be: it is very hard to get the apple between the molars. In fact, one can try biting off a carrot with one's molars—it is not easy, despite larger force (for the same action of the masseter!). One solution is to fix one end of the vegetable between molars while pulling down the other one with hand. For more biology-oriented students it may be interesting to discuss evolutionary history of teeth: were they different in shape from early beginning or rather differentiation was a consequence of changing conditions and specialization?

5. (5)

What are the sources of the uncertainty that impact the accuracy of the results? How to calculate the uncertainty?

Different ammeters measure the current with different accuracy. The uncertainty of analog ammeter is easily obtained by interpreting the scale of the device. In case of digital ammeters, it may be necessary to find this information in the user's manual. The masses of the standards used for calibrating the device can be measured several times (or with different scales) or treated as exactly specified by the manufacturer (this is a good opportunity to discuss the standard—i.e., a measured value's template—is or should be). Also, a way in which the mass is distributed on the loading arm of the device can affect the result—are all weights put exactly on the same part of the device? Do they have the same size and mass distribution? If not, what can be the consequences of the differences?

Another factor worth discussing is the position of the teeth during biting versus the position of the weight during calibration: were they the same? And were they distributed in the same way? And what about the way in which the students measured tooth-joint moment arm—was it accurate? Why? Why not? Do they see a way of improving this part of the experiment? And if they used anatomical data of a standard human—was it a reasonable approximation of a student who really did the measurement?

It is also good to point out to the students that human body is not a perfectly repeatable machine. This point can be strengthened by the observation of the results while biting: it is difficult to bite the device with a constant force for a longer time. Varying force is reflected by the variation in the measured current. Hence, an important decision needs to be made by the students: which value should be included into the final report? The highest? The average over, e.g., 10 measurements, taken every 3 seconds? Each measurement should be repeated multiple times and at least the average value and standard deviation should be calculated to strengthen the final result (see Table 1).

6. (6)

How is the load applied by the weight of a known mass? And by teeth?

In the above discussion, we limited the geometry to the most basic situation, in which the point load is applied vertically down at the end of the lever's arm and assumed that the tool is not deforming while being bitten. However, there are several possible modifications of the system and the analysis of the results:

• a distributed force can be introduced, instead of a single force acting on the lever; the force can then be assumed to be distributed uniformly around the whole surface of the plate at the end of the lever or some other modifications, for example, those resulting from the teeth geometry, can be used, according to the instructor's needs;

• in particular, one may consider force, which is applied not perpendicular, buy at a different angle—in such a case, students should use trigonometry to write the correct versions of Eqs. (1) and (2);

• if distributed forces and deformations of the experimental system (bending of the arm), as well as biological tissues (muscles, teeth, bones in a moving joint), are to be included, one may decide to use the opportunity and introduce more advanced computational methods, like the finite element methods (FEM) and analysis (FEA); there is a substantial literature on using those tools to model human mandible [17,18] and design therapeutic interventions [19,20]; this approach may also be interesting for students of different disciplines, such as zoology [21].

7. (7)

How does the force exerted by the masseter change with its elongation?

In general, it can be assumed that the force exerted by the masseter depends on the elongation of the muscle. This observation—which, for cardiac muscle, is known as the Frank-Starling mechanism—results from overlapping of thick and thin myofilaments in a sarcomere [22,23]. Muscle length-tension curve shows two distinct behaviors of the elongated muscle: the tension grows with growing elongation, reaches a plateau, and then it drops, usually more sharply than it grew. Those features are possible to capture with the proposed set up. However, our device is quite large, and it forces significant mouth opening width—i.e. masseter elongation—even when the screw is removed (see Fig. 2). To demonstrate the full complexity of tension-elongation relation, one needs to construct smaller lever, which will be robust enough to withstand the applied loads and at the same time allow for scanning a wider range of values of the muscle extension.

## Conclusions

This experiment, which relies on a cheap and DIY equipment, provides the opportunity to introduce basic physics for dental students in a setup which they are familiar with, namely, the TMJ. The basic approach, which we have focused on, is by no means the only way for dealing with the system. In the 21st century, students grow up with digital devices. However, oftentimes they do not realize how those devices work—they are just black boxes that transfer user's actions into given results. We strongly believe that every opportunity to show them, how the input is translated into output thanks to more or less sophisticated application of fundamental principles and phenomena, is worth taking. The proposed activity is by no means the ultimate method of doing it in the context of jaw's biomechanics, but it may be a convenient starting point.

## Acknowledgment

B. L. acknowledges funding by the Priority Research Area DigiWorld under the Strategic Programme Excellence Initiative at the Jagiellonian University.

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