# Dynamic Correlations and Disorder in the Masticatory Musculature Network

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Objectives and Aims

## 3. Materials and Methods

^{m}: frequency, F (m = 1); stiffness, S, (m = 2); decrement, D (m = 3); mechanical stress relaxation time, R (m = 4); and creep, C (m = 5).

_{max}/Δl, where M is the mass of the testing end of the myometer, a

_{max}is the maximal acceleration of oscillation, and Δl is the deformation depth of the muscle mass [34]. The reliability/precision of the measurement is 3.9%. The decrement directly measures the dissipation of the oscillation when a tissue recovers the shape after being deformed and is related to muscle elasticity. It is given by D = ln (a

_{1}/a

_{3}), where a

_{1}and a

_{3}are the first two positive amplitudes of the accelerogram. The viscoelastic properties are measured by the relaxation time, R, and the creep, C (known as number of Deborah). The relaxation time is the time taken by the muscle to restore its initial shape after external force is removed [35]. It is measured as the time interval between the maximum displacement of the tissue and the return to its initial shape. The reliability of the R measurement is 1.5%. Furthermore, the gradual elongation of a tissue over time when placed under a constant tensile stress is the last extracted quantity, i.e., creep, C, also known as the Deborah number; it is measured as the ratio of the relaxation time, R, to deformation time. The device has been used in the multiscan mode, where one measurement corresponded to the mean of 3 mechanical taps [21,36].

## 4. Results

#### 4.1. Measurement Maps A^{m}

^{m}

_{lp}, shown as colour maps in Figure 3a. Each pixel (l,p) corresponds to the measure of the parameter A

^{m}on a specific link, l, of a specific patient p. The patients have been sorted by age, while links are numbered counterclockwise in Figure 2. The link axis has been grouped by horizontal lines in four different cycles indicated by I, II, III, and IV. The first cycle, I, groups links from l = 1…4 and shows higher values for frequency (m = 1) and stiffness (m = 2) in comparison with all the remaining links. Correspondingly, it assumes smaller values for relaxation time (m = 4) and creep (m = 5).

^{m}. The results (Figure 3b) show a bimodal distribution for F, S, R, and C (A

^{m}with m = 1, 2, 4, and 5, respectively), which is well modelled by a mixture of two Gaussians. The first Gaussian, given by the lower PDF values in F and S, corresponds with F and S values measured on the II, III, and IV cycles, while the second Gaussian corresponds to higher values of F and S measured on the first cycle, I. The opposite behaviour is found for the viscoelastic R and C parameters. The proportion and the mean of the two Gaussian components in the F, S, R, and C measurements are reported in Table 1.

^{m}matrix on all 52 patients for each link, l, giving <A

^{m}

_{l}>

_{p}

_{=(1,...Np)}= <F>

_{P}, <S>

_{P}, <D>

_{P}, <R>

_{P}, and <C>

_{P}shown in the five plots of Figure 3c. The larger frequency and stiffness for the first cycle, alongside the larger viscoelastic properties of creep and relaxation time of the II, III, and IV cycles, are well depicted. At the same time, <D>

_{P}assumes competing slightly different values in the four cycles. These competing values produce a more complex and dynamic landscape of muscle elasticity in comparison with muscle tone, stiffness, and viscoelasticity.

#### 4.2. Cycle Dynamics: Exponential Growth and Decaying of A^{m}_{cy} Measurements

^{m}in links belonging to the different cycles suggest studying the time evolution of the four cycles in our measurements, as shown in the plot of the cycle average A

^{m}

_{cy}as a function of the age of patients (Figure 3). We observe a stretched exponential growth in A

^{1}

_{cy}= F

_{cy}and A

^{2}

_{y}= S

_{cy}, while the viscoelastic A

^{4}

_{cy}= R

_{cy}and A

^{5}

_{cy}= C

_{cy}show a stretched exponential decay in the same range. We have modelled our data by using the following stretched exponential equations:

^{m}

_{cy}is a constant, τ

^{m}

_{cy}is a characteristic time, and γ

^{m}

_{cy}is the stretching exponent relative to the measurement m on the cycle cy. Similarly, the stretched exponential decay of viscoelastic quantities is given by

^{m}

_{cy}is the baseline value and k

^{m}

_{cy}is a constant value. All the parameters, including B

^{m}

_{cy}, k

^{m}

_{cy}, τ

^{m}

_{cy}, and γ

^{m}

_{cy}for each cycle, cy, and each MyotonPRO measured quantity, m, are tabulated and shown in Figure 4. The k

^{m}

_{cy}is larger for the first cycle, indicating that the final value reached with late age is always larger in the first cycle for F

_{cy}(t) and S

_{cy}(t), while it is smaller in the same cycle for the decaying viscoelastic R

_{cy}(t) and C

_{cy}(t).

^{3}

_{cy}= D

_{cy}(t) evolution. Here, the decrement grows in the first cycle, while it decays in the other cycles. The characteristic time, τ

^{m}

_{cy}, is around 20 ± 1 years for all cycles in frequency (m = 1) and biomechanical muscle evolution (m = 2, 3). In the decay of viscoelastic evolution, the characteristic time, τ

^{m}

_{cy}, assumes lower values of around 13 ± 1 years. The stretching exponent (γ

^{m}

_{cy}) is 1 for all A

^{m}

_{cy}in the II, III and IV cycles, while it is 3 for the first cycle (cy = 1). Thus, the growth of tone and stiffness, as well as the decay of viscoelastic relaxation time and creep, are faster in the first cycle. In this way, we have well characterised model lines describing the physiological evolution of our system that can serve as references for measurements in non-healthy patients. This will allow for the assessment of disorder degrees at different ages as new diagnostic tools.

#### 4.3. Correlation Matrix Analysis of MyotonPRO Measurements

^{m}

_{lp}matrices. We first calculated the Pearson correlation coefficients between A

^{m}maps, given by

^{m}and A

^{m′}. It ranges from −1 to 1, where −1 indicates a perfect negative correlation, meaning as one variable increases, the other decreases; a value of 0 indicates no correlation, meaning the variables do not move together; and a value of 1 indicates a perfect positive correlation, meaning as one variable increases, the other also increases. The results are shown in Table 2. Strong positive correlations occur between frequency and stiffness (C

^{23}= C

^{32}= 0.96) and between relaxation time and creep (C

^{45}= C

^{54}= 0.99). Consequently, frequency and stiffness have high negative correlations with both relaxation time (C

^{14}= C

^{41}= −0.93, C

^{24}= C

^{42}= −0.95) and creep (C

^{15}= C

^{51}= −0.91, C

^{25}= C

^{52}= −0.93).

^{31}= C

^{13}= 0.23) and stiffness (C

^{32}= C

^{23}= 0.25) as well as higher negative correlations between decrement and both viscoelastic relaxation time (C

^{34}= C

^{43}= −028) and creep (C

^{35}= C

^{53}= −0.19).

#### 4.3.1. Pairwise Patient–Patient and Link–Link Correlations of MyotonPRO Measurements A^{m}

^{m}(m = 1, 2, 4, 5,). As a result, we obtain patient–patient cross-correlation matrices, c

^{m}

^{,m′}

_{p}

_{,p′}:

^{m}

^{,m}

_{p}

_{,p′}is a symmetric matrix made by the pairwise linear correlation coefficient between the same measured parameter, m, for different patients, p and p′. Its main diagonal, where p = p′, is thus composed of Np elements equal to 1 since it represents the correlation of a measurement of a parameter for a patient with the same parameter measured in the same patient. As one gets away far from this diagonal, following the dashed arrows in the symmetric maps of Figure 4a, the mean of the diagonal elements is expected to change for dynamic systems. Thus, to describe analytically how each symmetric map, c

^{m}

^{,m}

_{p}

_{,p′}, changes, we have calculated the 1D autocorrelation function, ACF, by the mean values of the diagonals of each c

^{m}

^{,m}

_{p}

_{,p′}matrix:

^{m}

^{,m}

_{p}

_{,p′}map are shown by different symbols in the upper panel of Figure 5b. The specific decay of the ACF is used to quantify the specific dynamic of the process. Also, in this case, we have modelled the ACF’s decay by using the stretched exponential function:

_{0}is related to a characteristic age for the correlation decay of the measured parameter. The stretching exponent, β, is a shape parameter characterizing the degree of deviation from an exponential function and the fastness of the decay, b is the baseline, and (a − b) is defined as the contrast and indicates the strength of the decay. These parameters, for all measurements, m, are reported in Table 3, while the best fitted curves are shown by the thick lines in Figure 5b. We note how decrement correlation decay is stronger since its contrast, a − b, is larger; at the same time, it has a stretching exponent near to 1, lower than 1.8, found for the decay of the other parameters. Thus, the decrement correlation decay is stronger and slower. Finally, we observe that the characteristic age for the correlation decrement decay is 50 years, which is lower than the characteristic age of the other parameters’ correlation decay. Indeed, patient–patient correlations decay at 60 years in frequency and stiffness, while decay between these correlations occurs at 70 years in viscoelastic creep and relaxation time.

^{m}

^{,m′}

_{l}

_{,l′}, calculated for all healthy patients, p, in each pair of links, l, l′, as follows:

^{m}

^{,m′}

_{l}

_{,l′}maps shown in Figure 5c give, in this case, the pairwise linear cross-correlation coefficients between link l in A

^{m}

_{lp}matrix and link l′ in A

^{m’}

_{l’p}. We observe that this link–link correlation decay is quite different from the decay in the patient–patient correlation map, c

^{m}

^{,m′}

_{p}

_{,p′}, described in Figure 5b. In this case, the 1D autocorrelation function is given by

^{1,3}

_{l}

_{,{1,..,4}}, c

^{2,3}

_{l}

_{, {1,..,4}}, c

^{4,3}

_{l}

_{, {5,..,20}}, and c

^{5,3}

_{l}

_{, {5,..,20}}; on the opposite side, c

^{1,3}

_{l}

_{, {5,..,20}}, c

^{2,3}

_{l}

_{, {5,..,20}}, c

^{4,3}

_{l}

_{, {1,..,4}}, and c

^{5,3}

_{l}

_{, {1,..,4}}are positive. The decrement symmetric matrix presents negative correlations c

^{3,3}

_{{1,..,4},{5,..,20}}and c

^{3,3}

_{{5,..,20},{1,..,4}}. Thus, also from the above correlation analysis, a peculiar role of cycle I in the muscular network is apparent.

#### 4.3.2. Correlations between Network Cycles of MyotonPRO Measurements

^{m}, for each patient in the four cycles I, II, III, and IV (cy = 1, 2, 3, 4) and then we have extracted i) the Pearson coefficient C

^{m}

_{cy}

_{,cy′}between A

^{m}

_{cy}and A

^{m}

_{cy′}matrices and ii) the Pearson coefficient C

^{cy}

_{m}

_{,m′}between A

^{m}

_{cy}and A

^{m′}

_{cy}matrices:

^{m}

_{cy,cy}

_{′}= corr

^{2}(A

^{m}

_{cy}, A

^{m}

_{cy}

_{′})

^{cy}

_{m,m}

_{′}= corr

^{2}(A

^{m}

_{cy}, A

^{m′}

_{cy})

^{m}

_{cy}

_{,cy′}tell us how a measurement of a cycle is correlated with another cycle in the same measurement A

^{m}. Coefficients C

^{cy}

_{m m}

_{′}describe how a measurement, m, of a cycle is correlated with another measurement, m′, for the same cycle. The results are shown by clustergrams in Figure 6. A clustergram is composed of a heat map of the correlation matrix (C

^{m}

_{cy}

_{,cy′}and C

^{cy}

_{m}

_{,m′}, where the rows and columns are sorted in the order suggested by the hierarchical clustering). This allows for the grouping of various subsets of the cycles that are highly correlated within the subset, as highlighted by a dendogram.

^{m}

_{cy}

_{,cy′}, as shown in Figure 6a, has a positive result, indicating positive correlations between A

^{m}measurements in all cy cycles, except for the decrement (m = 3), where the first cycle is negatively correlated with the other cycles. Stronger positive correlations occur between the lateral cycles in creep, where c

^{m}

_{24}= c

^{m}

_{42}>0.86. C

^{cy}

_{m m}

_{′}are shown in Figure 6b. Equal dendrograms with similar heat maps are found for cycles II, III, and IV. Here, C

^{cy}

_{m m}

_{′}is positive between F, S, and D (m,m′ = 1, 2, 3) and between R and C (m,m′ = 4, 5), while it is negative between R and C and F, S, and D. This clustering changes in cycle I, where negative correlations occur between F and D.

## 5. Discussion

^{m}, (visualised as a colour map) for each measured parameter, m = F, S, D, R, C, where a specific element A

^{m}

_{lp}provides the measured parameter of the link, l, in the patient, p. In this way, statistical physics has been used to characterise the functional state of the system and to describe its evolution with age. All measured maps have shown a bimodal distribution due to the different values measured for the first cycle, I, with respect to the other (II, III, and IV) cycles, except for the decrement, which assumes nearly competing values on all links, giving rise to a stable distribution typical of nonlinear and metastable phenomena. Indeed, the decrement (inversely proportional to the elasticity) seems to be a critical parameter of the masticatory musculature, while the first cycle might play a prominent role in its functionality. The time evolution analysis of our data clustered in the four cycles shows a clear stretched exponential growth of F and S for all cycles, but the first cycle shows a faster growth. Similarly, we find a decay of C and R for all cycles, with a faster decay in the first cycle. The decrement behaviour is quite different, confirming the critical nature of elasticity in the functionality of the masticatory system. Indeed, it decays in the first cycle, while, at the same time, it increases for the other (II, III, and IV) cycles at the same rate.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Muscle taken into consideration for measurement with the use of MyotonPRO. The coloured points show the 20 points where measurements have been taken by MyotonPRO on each subject. The legend describes the five pairs (right (RT) and left (LT)) of tested muscles, divided by colour, and their corresponding bony insertions, (www.visiblebody.com) as reported in the following list: M. temporal RT (interacts with) frontal (white); m. temporal RT (interacts with) mandible (white); m. masseter RT (interacts with) temporal (yellow); m. masseter RT (interacts with) mandible (yellow); m. mylohyoid RT (interacts with) mandible (green); m. mylohyoid RT (interacts with) hyoid bone (green); m. platysma RT (interacts with) mandible (black); m. platysma RT (interacts with) clavicle RT (black); m. sternocleidomastoid RT (interacts with) clavicle RT (red); m. sternocleidomastoid RT (interacts with) temporal (red); m. temporal LT (interacts with) frontal (white); m. temporal LT (interacts with) mandible (white); m. masseter LT (interacts with) temporal (yellow); m. masseter LT (interacts with) mandible (yellow); m. mylohyoid LT (interacts with) mandible (green); m. mylohyoid LT (interacts with) hyoid bone (green); m. platysma LT (interacts with) mandible (black); m. platysma LT (interacts with) clavicle (black); m. sternocleidomastoid LT (interacts with) clavicle (red); m. sternocleidomastoid (interacts with) temporal (red).

**Figure 2.**Temporomandibular muscle network. The nodes and links are indicated by full circles and tick lines, respectively. The four cycles are indicated by I, II, III, and IV. The list of nodes and links is also reported. The thickness of the links refers to the measured values of the frequency averaged across all patients for each link (see Figure 3c, top panel). LT (

**left**); RT (

**right**).

**Figure 3.**Maps of link measurements. (

**a**) Measurements of MyotonPRO in each of the 20 links for each of the 52 healthy patients The horizontal lines delimit the links belonging to cycles I, II, III, and IV. (

**b**) Probability distributions of measurement maps showing the bimodal distributions of frequency, stiffness, relaxation, and creep. The bimodal distributions have been modelled as a mixture of two Gaussians whose mean and proportion values of the two components are reported in Table 1. The bimodal distribution is not able to fit decrements in data, modelled by a stable distribution (red line). (

**c**) Averaged values of A

^{m}

_{lp}on all 52 patients, giving <A

^{m}

_{l}>

_{P}= <F>

_{P}, <S>

_{P}, <D>

_{P}, <R>

_{P}, and <C>

_{P}(full circles). Each link is represented by a full circle whose colour is the same as the corresponding link in the graph of Figure 2. The standard deviations of <A

^{m}

_{l}>

_{P}for all links, l, are plotted by error bars and represented by shaded areas. PDF (Probability density function); Hz (Hertz); N/m (N/meter); ms (millisecond).

**Figure 4.**Growing, decaying, and competing cycles in muscle networks. Measurements of MyotonPRO in each cycle as a function of the age of the 52 healthy patients are shown. Values of fit parameters extracted by Equations (1) and (2) are also tabulated. We note that, after 25 years, the parameters’ evolution shows a saturation-like behaviour after τ values. Hz (Hertz); N/m (N/meter); ms (millisecond).

**Figure 5.**Patient–patient and link–link map correlations in masticatory muscle network. (

**a**) Correlation maps, c

^{m}

^{,m′}

_{p}

_{,p′}, where each pixel (p,p′) is calculated by Equation (4). (

**b**) Autocorrelation function, ACF(m,p), calculated by Equation (5), for each measurement correlation map, c

^{m}

^{,m}

_{p}

_{,p′}. The ACF frequency, stiffness, relaxation time, and creep are represented by black squares, circles, diamonds and triangles, respectively; the ACF of decrement corresponds to the red circles. The continuous lines in the ACF panel are the best fitted curve obtained by modelling data with the decaying stretched exponential of Equation (6). (

**c**) Correlation maps, c

^{m}

^{,m′}

_{l}

_{,l′}, where each pixel (l,l’) is calculated by Equation (7). The zones delimited by the black thick lines represent cycles I, II, III, and IV. (

**d**) Autocorrelation function, ACF(m,l), calculated by Equations (4) and (5), respectively, for the symmetric correlation maps, c

^{m}

^{,m}

_{l}

_{,l′}. ACF (AutoCorrelation Function).

**Figure 6.**Heat map with dendrogram of coefficients (

**a**) C

^{m}

_{cy}

_{,cy′}, calculated by Equation (8) and (

**b**) C

^{cy}

_{m}

_{,m’}calculated by Equation (9). The Pearson coefficients are reported also reported. F (frequency); S (stiffness); D (decrement); C (creep); R (relaxation).

**Table 1.**Proportion and mean of the two Gaussian components used to fit the probability density function (PDF) of A

^{m}with m = 1, 2, 4, 5. F (frequency); S (stiffness); R (relaxation); C (creep).

Cycles | F (%) | S (%) | R (%) | C (%) | <F> | <S> | <R> | <C> |
---|---|---|---|---|---|---|---|---|

I | 0.32 | 0.27 | 0.20 | 0.20 | 25.50 | 636.0 | 7.17 | 0.49 |

II–IV | 0.68 | 0.73 | 0.80 | 0.80 | 15.35 | 284.4 | 18.90 | 1.16 |

**Table 2.**Correlation coefficients, C

^{m}

^{,m′}, between A

^{m}maps calculated by Equation (1). The green cells highlight the different lower values of the C

^{3,m′}coefficients involving decrement.

Frequency | Stiffness | Decrement | Relaxation | Creep | |
---|---|---|---|---|---|

Frequency | 1 | 0.96 | 0.23 | −0.93 | −0.91 |

Stiffness | 0.96 | 1 | 0.25 | −0.95 | −0.93 |

Decrement | 0.23 | 0.25 | 1 | −0.28 | −0.19 |

Relaxation | −0.93 | −0.95 | −0.28 | 1 | 0.99 |

Creep | −0.91 | −0.93 | −0.19 | 0.99 | 1 |

M | Measurement | Contrast | β | p_{0} | Age (y) |
---|---|---|---|---|---|

1 | Frequency | 0.028 | 1.85 | 42 | 60 |

2 | Stiffness | 0.028 | 1.85 | 43 | 60 |

3 | Decrement | 0.265 | 1.05 | 33 | 50 |

4 | Relaxation | 0.050 | 1.85 | 48 | 70 |

5 | Creep | 0.040 | 1.85 | 48 | 70 |

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## Share and Cite

**MDPI and ACS Style**

Campi, G.; Ricci, A.; Costa, N.; Genovesi, F.; Branca, J.J.V.; Paternostro, F.; Della Posta, D.
Dynamic Correlations and Disorder in the Masticatory Musculature Network. *Life* **2023**, *13*, 2107.
https://doi.org/10.3390/life13112107

**AMA Style**

Campi G, Ricci A, Costa N, Genovesi F, Branca JJV, Paternostro F, Della Posta D.
Dynamic Correlations and Disorder in the Masticatory Musculature Network. *Life*. 2023; 13(11):2107.
https://doi.org/10.3390/life13112107

**Chicago/Turabian Style**

Campi, Gaetano, Alessandro Ricci, Nicola Costa, Federico Genovesi, Jacopo Junio Valerio Branca, Ferdinando Paternostro, and Daniele Della Posta.
2023. "Dynamic Correlations and Disorder in the Masticatory Musculature Network" *Life* 13, no. 11: 2107.
https://doi.org/10.3390/life13112107