2.2.1. Sociodemographic Data and Anthropometrics

Anthropometric measurements were collected under standardized conditions with patients wearing light clothing and no shoes. Bioelectrical impedance analysis, using a TANITA TBF-410 M over a horizontal and hard surface and to the nearest 0.1 kg, was the selected method to measure body weight and composition. Height was measured to the nearest 0.5 cm using a portable height board (SECA 216). BMI z-score (BMIz) was calculated for each patient. BMIz correlates to growth percentile charts and allows children's relative weight follow-up from childhood up to adolescence. Participants' obesity was defined with BMIz scores greater than 2 SD following World Health Organization (WHO) growth charts [56].


**5. Results**

Results from the Machubeni case study are presented here to test our two hypotheses that: (a) transformative spaces will enable broader levels of participation among actors between and within scales (Section 5.1), and (b) strong social-relational linkages (i.e., network ties) will increase over time across and within scales (Sections 5.2–5.4). Results from our two hypotheses will help identify which key barriers and strategies enable effective collaboration and coordination in transformative spaces (Section 5.5).

#### *5.1. Trends in Participation*

Figure 3A,B shows the trend in the number of actors who attend workshop meetings in Machubeni over time. Our results show that the total number of actors who attended workshop meetings steadily increased between Phase 1 and Phase 2 (Figure 3A,B). There was a 134% increase in the total number of actors who attended workshop meetings between Phase 1 and Phase 2 (Figure 3A). In total, 58 members—28 local community members, 18 government officials, and 12 researchers—were identified as active members in both phases (Figure 3A). Twelve were based at the local municipality, three at the district level, and six at the province level, indicating that the transformative space was composed of different actors across and within scales (Figure 3B).

**Figure 3.** Trends and changes in the number of workshop attendees over time. (**A**) depicts changes in the total number of attendees and (**B**) depicts changes in the total number of attendees per scale. W represents workshop number.

Network members highlighted that the change of workshop venue led to an increase in attendance (coded 87 times). Provision of transport (coded 89 times) for local hub members to attend workshop meetings was cited as one of the reasons that led to an increase in attendance. Based on interview responses, network members cited environmental reasons for participating in the transformative space including:

"*I joined the transformative space because I would like to help transform the Machubeni landscapes into a better sustainable area. I have realised that land degradation has rapidly increased over the past 60 years. Therefore, its has become a necessity to stop land degradation and bush encroachment which is a*ff*ecting our livelihoods.*"

Others cited cross-scale collaboration as a reason for joining the transformative space (coded 71 times): "*I joined the transformative space because I like the idea of working together. Moreover, the workshop allows us to interact and share knowledge with government o*ffi*cials who hardly visit Machubeni.*"

While the transformative space enabled broader levels of participation in Machubeni, there was a slight decrease in the number of active actors in Phase 2 (Figure 3B). Based on workshop attendance data, there was a high turnover of government officials and the youths (Figure 3B). In Phase 2, we observed that district municipality staff and the youths stopped attending the workshop meetings (Figure 3B). High turnover of network members was identified as a major leakage of human capital

(coded 31 times). In terms of youth, the high turnover was due to a disinterest in home gardening and community-based land rehabilitation (coded 24 times).

Further probing revealed that most youth did not aspire to having green jobs. Here, green jobs refers to work related to the conservation of the environment. One youth member suggested that the high turnover was due to the fact that youth anticipated employment opportunities as data enumerators and not as land rehabilitation workers. In some instances, the youths emphasised that the transformative space was not a safe space and their safety was often compromised. As one youth member noted:

"*Whenever we make suggestions during forum meetings, the older participants always interject us. However, we cannot argue with the elders in public because this is against our culture.*"

**Interviewer**: *But the forum is a safe space where all network members are equal?*

**Respondent**: *In principle, it is a safe space, but usually after the workshops, we often get a rebuke from the elders for highlighting divergent points to theirs. This resulted in youths keeping quiet during workshop discussions. "It is better to be safe than sorry."*

Out of the 22 government officials identified in Phase 1, 36% represented the public service, while 64% represented the local government. Inclusively, 9% (n = 2) represented senior management, 72% (n = 16) middle management, and 18% (n = 4) junior management. In Phase 2, there was a 12% decline (n = 2) in the number of middle management actors. Middle management actors interviewed related this 12% turnover rate to being overworked within the district (coded 12 times) and collaboration fatigue (coded 19 times). Figure 4 depicts the changes in the frequency of attendance by government actors over time. Further analysis revealed that the attendance ratio of senior management actors was highly fluid (Figure 4).

**Figure 4.** Changes in the frequency of attendance by government actors over time.

We observed that the frequency of attendance by junior management in 2019 steadily increased from 20% to 80% over two years, and this coincided with the rapid decline of senior management attendance. Based on interview responses, all senior management actors indicated apathy for future collaboration (coded 12 times). As one senior management actor articulated:

... *"When the university finally exits the project in 2022, I do not see any senior government actors from District or Provincial level attending forum meetings because it will be the mandate of the local municipality as an implementer to take over* ... *..Our role at the Provincial level is to help with policy interpretation, but if the local municipality is non-functional there is nothing we can do"* ...

Further probing revealed that there were deep structural tensions between the local municipality and the district municipality. One district municipality official highlighted that the Municipal Structure Act of 1998 states that the role of the district municipality is to provide technical assistance to local municipalities, including sharing of specialised capacity. However, local municipality officials emphasised that the idea of shared priorities is an ideal concept on paper, but it is rare in practice, signifying governance mismatches. Overall, edge density, clustering coefficient, and reciprocity scores steadily increased on average over the two phases, despite a decline in the number of active organisations within the polycentric governance network (Figure 5). We observed that the reciprocity score of the inter-organisational trust network increased steeply from 0.33 to 0.85 (Figure 5). This might signify an increase in collaborative cohesion among different actors within the polycentric governance network.

**Figure 5.** Network metrics evolution. (**A**) depicts changes in edge density; (**B**) changes in average clustering; (**C**) changes in reciprocity; and (**D**) changes in average betweenness. CG (Common Goals network), IS (Information-Sharing network) and IT (Inter-organisational trust network).

Furthermore, 90% of the respondents reported that the transformative space demonstrated strong cohesion (e.g., "*The transformative space discussions have enabled us to form new bonds and build strong relationships with other villages and government actors. Furthermore, I have realised that by working together as di*ff*erent villages, government departments and researchers, we can arrest land degradation because we all bring di*ff*erent strengths. For example, Rhodes University is glueing us together, the government*

*departments are providing technical support, and the community is providing the energy to rehabilitate the land*"). The social facilitation process enabled attitudinal changes: network members generated a "together-we-can" mentality. We observed an increase in edge density, as well as clustering and reciprocity within the network (Figure 5). The increase of these metrics scores over time suggests that there is an increase in cooperation and knowledge sharing among di fferent semi-autonomous institutions within the polycentric governance network. While there is an increase in network density, clustering coe fficient, and reciprocity, the average betweenness steeply declined over time (Figure 5, supplementary: Tables S2–S5).

Most importantly, we observed that the overall number of strong ties increased on average, while the number of weak ties decreased over time (Figure 6). The changes in the proportion of direct ties observed within respective governance networks in Machubeni ranged widely, with the common goals network having an increase of 6%, information-sharing network 25%, and inter-organisational trust network 33%. Furthermore, the proportion of strong ties steadily increased, with the common goals network having an increase of 66%, information-sharing network 29%, and inter-organisational trust network 17%. In terms of multi-actor ties, strong linkages were prevalent between local–local hubs, while weak vertical ties were prevalent between the local hubs and the local municipality (Figure 6). Overall, the proportion of strong ties between the university and the government departments remained stable over time across three networks (Figure 6).

**Figure 6.** Distribution of multi-actor and multi-level ties across three networks. ( **A**) depicts the distribution of ties across three governance networks, i.e., IT (inter-organisational trust network), IS (information-sharing network), and CG (Common goals network) between Phase 1 and Phase 2. (**B**), ( **C**) and ( **D**) depict the distribution of strong versus weak ties across three governance networks (common goals, information sharing, and inter-organisational trust networks). LC (Local hubs), LM (Local municipality), GV (Government), RU (Rhodes University).

#### *5.2. Evolution and Description of the Common Goals Network over Time*

Our analysis revealed that the distribution of strong ties increased among actors and across scales over time (Figure 6). Semi-structured interviews revealed four processes that contributed to the increase of strong ties among local–local hubs over time. Network members highlighted that enhanced social facilitation (coded 43 times), personal relationships (coded 21 times), behavioural change (coded 13 times), and personal learning (coded nine times) were related to the increase of strong ties within the network over time. Network members elucidated that social facilitation enabled members to develop shared goals, thereby enhancing relational and collective agency. Most importantly, ties among local–local hubs were mostly strong with like-minded organisations such as CA, LR, and RG closely connected in both phases (Figure 7) because of personal connections, with one hub leader suggesting that "*over the past two years we have enjoyed working together as local organisations, and we have become stronger together.*"

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**Figure 7.** Overview of the common goals network. Arrows represent directed, weighted ties and circle size represents degree centrality. Rhodes University (RU), Department of Economic Development, Environmental Affairs, and Tourism (DEDEAT), Department of Rural Development and Agrarian Reform (DRDAR), Chris Hani District Municipality (CHDM), Emalahleni Local Municipality (ELM), Traditional Leaders Council (TLC), Masibambisane multi-actor forum executive (MMF exe), GEF5 Community liaison officers (CLO), Conservation Agriculture Hub (CA), Land Rehabilitation Hub (LR), Rangeland Management Hub (RG), Training and Knowledge Hub (TK), M&E Hub (M&E), Gender Equity Hub (GE), and Youth Hub (YH).

Our data reveal that the local municipality was the least connected organisation across and within scales in both phases (Figure 7). Further probing revealed that municipal officials were hesitant in aligning their goals with the rest of the network because of limited resources for service delivery (coded 13 times) and a lack of clearly outlined rules of engagement between local municipalities and civil society (coded eight times). An ex-municipal employee elucidated that most municipal extension officers who attend the forum meetings were mere 'pen-pushers' with little or no authority to make decisions that enable institutional transformations.

#### *5.3. Evolution and Description of the Information-Sharing Network over Time*

Our analysis revealed that the distribution of strong ties increased between actors and across scales over time (Figure 6). In Phase 1, we observed that Rhodes University was the most central and dominating actor (Figure 8). However, in Phase 2, five organisations were identified as central in the information-sharing network. Based on interview responses, network members linked this to an increase of cross-scale coordination meetings, i.e., Intergovernmental Relations (IGR). Both the Department of Economic Development, Environmental Affairs, and Tourism and Department of Rural Development and Agrarian Reform were seen as central government agencies that shared technical information within the transformative space. However, network members reported weak ties between local organisations and the local municipality in both phases (Figure 6).

**Figure 8.** Overview of the information-sharing network. Arrows represent directed, weighted ties and circle size represents degree centrality. Rhodes University (RU), Department of Economic Development, Environmental Affairs, and Tourism (DEDEAT), Department of Rural Development and Agrarian Reform (DRDAR), Chris Hani District Municipality (CHDM), Emalahleni Local Municipality (ELM), Traditional Leaders Council (TLC), Masibambisane multi-actor forum executive (MMF exe), GEF5 Community liaison officers (CLO), Conservation Agriculture Hub (CA), Land Rehabilitation Hub (LR), Rangeland Management Hub (RG), Training and Knowledge Hub (TK), M&E Hub (M&E), Gender Equity Hub (GE), and Youth Hub (YH).

Document analysis of grey literature from the Emalahleni municipality revealed that the local municipality did not have adequate scientific information to share with local communities concerning land degradation and rangelands. Further probing revealed that a lack of resource planners at the local municipality level, such as soil scientists, pasture scientists, and natural scientists, created communication barriers. Network members interviewed elucidated that political instability (coded nine times), high-level leadership instability (coded four times), deep structural tensions (coded eight times), and discontinuous participation by senior municipality officials (coded 48 times) from the local municipality impeded effective communication within the transformative space across scales. Despite most organisations and hubs having some informational ties in both phases, there were no informational ties reported for the district municipality in Phase 2. Two potential explanations for this emerged from interviews with district municipality officials, who highlighted that the participatory process was time-consuming considering that the transformative space focused on five villages within the district. Another reason might be insufficient budgets from the district municipality to fulfil demands of the transformative space, thereby resulting in self-exclusion, e.g., "*the district municipality is financially constrained, and we support six local municipalities, it will be di*ffi*cult to commit to knowledge sharing activities.*"

#### *5.4. Evolution and Description of the Inter-Organisational Trust Network over Time*

Our analysis revealed that the distribution of strong ties increased between actors and across scales over time (Figure 6). Figure 9 depicts the overview of the Inter-organisational trust network over time. Although there was an increase in strong ties between local–local hubs, mistrust between the Traditional Leaders Council and the MMF executive was becoming more apparent. The MMF executive was accused of "capturing" the project and directing all project benefits to their cronies (coded 34 times). As one of the headmen stated:

"*We are the custodians of this community, MMF executive is distributing lucerne seedlings without our blessings. What criteria did they use to distribute the seeds? As the headman, I am the father of the five villages, and I will never let anyone divide my community. Therefore, the selection criteria should include all local semi-autonomous hub leaders.*"

**Figure 9.** Overview of the Inter-organisational trust network. Arrows represent directed, weighted ties, and circle size represents degree centrality. Rhodes University (RU), Department of Economic Development, Environmental Affairs, and Tourism (DEDEAT), Department of Rural Development and Agrarian Reform (DRDAR), Chris Hani District Municipality (CHDM), Emalahleni Local Municipality (ELM), Traditional Leaders Council (TLC), Masibambisane multi-actor forum executive (MMF exe), GEF5 Community liaison officers (CLO), Conservation Agriculture Hub (CA), Land Rehabilitation Hub (LR), Rangeland Management Hub (RG), Training and Knowledge Hub (TK), M&E Hub (M&E), Gender Equity Hub (GE), and Youth Hub (YH).

However, the MMF responded by highlighting that:

"*when making decisions, we include the two headmen and their seven sub-headmen. However, when a headman misses an important meeting, it is the responsibility of the sub-headman to inform their headman.*"

Further probing revealed that some of the tensions emanated from unclear communication channels among the multiple semi-autonomous institutions at the local level. Other network members highlighted that the diminishing legitimacy of traditional leaders, due to multiple independent institutions, created tensions within the transformative space (coded eight times). Further analysis revealed that there were more strong ties among different hubs and the MMF executive than with the Traditional Leaders Council.


*5.12. Statistical Analysis*

Data are presented as means ± standard error (SEM) and analyzed by Graph Pad Prism, version 6.0 for Windows (Graph Pad Software, San Diego, CA, USA). One-way ANOVA test and multiple comparisons by Tukey HSD were used for comparisons of one variable in more than two groups. For comparisons of two or more variables, twoway ANOVA followed by Tukey HSD were used. For all tests, the values *p* < 0.05 were considered significant. The assays were conducted in duplicate and repeated at least twice in independent days.

**Supplementary Materials:** The following are available online at https://www.mdpi.com/article/ 10.3390/toxins13120906/s1, Figure S1. Differentiation of THP-1 macrophages. (A) Human premonocytes of the THP-1 lineage were cultured in 75 cm3 bottles containing supplemented RPMI medium. The cells were kept in an incubator at 37 ◦C and 5% CO2, at a concentration of 2–8 × 105 cells/mL. (B) Macrophages were differentiated from THP-1 pre-monocytes with 100 ng/mL PMA for 3 days, followed by a 4-day rest period in the absence of PMA. 200× magnification. (C) CD11b expression in differentiated THP-1 macrophages. Adherent THP-1 macrophages differentiated with PMA and pre-monocytes in suspension were analyzed for CD11b expression by flow cytometry. Result expressed as mean of duplicates ± SEM and analyzed by two-way ANOVA followed by Tukey's post-test. (\*) Significant difference in relation to the respective controls (*p* < 0.0001). Figure S2. Cell cytotoxicity. Differentiated THP-1 macrophages was treated with 0.5–15 μg/well of BaV or with 0.5 and 1 μg/well Kn-Ba for 24 h, 48 h, and 72 h, and cell viability was evaluated by the quantification of lactate dehydrogenase (LDH) enzyme in cell-free supernatants. Results expressed as mean of duplicates ± SEM and analyzed by one-way ANOVA followed by Tukey's post-test. (\*) Significant difference in relation to the respective control. *p* < 0.01 (\*\*), and *p* < 0.0001 (\*\*\*\*).

**Author Contributions:** Conceptualization, Â.A.A.M. and W.D.-d.-S.; methodology, Â.A.A.M., F.C.V.P. and W.D.-d.-S.); validation, Â.A.A.M., F.C.V.P. and W.D.-d.-S.; formal analysis, Â.A.A.M. and W.D.-d.-S.; investigation, Â.A.A.M. and F.C.M.; resources, W.D.-d.-S. and F.C.V.P.; data curation, Â.A.A.M., F.C.M., F.C.V.P. and W.D.-d.-S.; writing—original draft preparation, Â.A.A.M., F.C.V.P. and W.D.-d.-S.; writing—review and editing, Â.A.A.M., F.R.G., K.S.G., F.C.V.P. and W.D.-d.-S.; supervision, W.D.-d.-S. and F.C.V.P.; project administration, W.D.-d.-S.; funding acquisition, W.D.-d.-S. and F.C.V.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Coordination for the Improvement of Higher-Level Education Personnel—CAPES (Â.A.A.M.: fellowship Ph.D. PROEX; W.D.S.: 23038.000814/2011–83); São Paulo Research Foundation (FAPESP 2013/07467-1 and Fapesp 2019/20832-7) and the National Council for Scientific and Technological Development (*PROÁFRICA*, CNPq: 490048/2005-6).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Acknowledgments:** The authors thank Denise Tambourgi for the Immunochemistry Laboratory structure and to Martin Wesley for English reviewing in the article.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


*Article* **Study on Shear Velocity Profile Inversion Using an Improved High Frequency Constrained Algorithm**

**Qing Ye 1, Huafeng Sun 2,\*, Zhiqiang Jin 1 and Bing Wang 1,\***

**Abstract:** The formation shear-wave (S-wave)'s velocity information around a borehole is of great importance in evaluating borehole stability, reflecting fluid invasion, and selecting perforation positions. Dipole acoustic logging is an effective method for determining a formation S-wave's velocity radial profile around the borehole. Currently, the formation S-wave's radial-profile inversion methods are mainly based on the impacts of radial velocity changes of formations outside the borehole on the dispersion characteristics of dipole waveforms, without considering the impacts of an acoustic tool on the dispersion curves in the inversion methods. Accordingly, the inversion accuracy is greatly impacted in practical data-processing applications. In this paper, a novel inversion algorithm, which introduces equivalent-tool theory into the shear-velocity radial profile constrained-inversion method, is proposed to obtain the S-wave's slowness radial profile. Based on the equivalent-tool theory, the acoustic tool can be modeled using two parameters, radius and elastic modulus. The tool's impact on the dipole waveform's dispersion is eliminated first by using the equivalent-tool theory. Then, the corrected dispersion curve is used to carry out the constrained inversion processing. The results of this processing on the simulation data and the real logging data show the validity of the proposed algorithm.

**Keywords:** oil and gas exploration; logging tool; velocity radial profile; constrained inversion method; equivalent-tool theory

#### **1. Introduction**

The rock properties of formations around a borehole are of great significance in the exploration and development of oil and gas. The velocity information of the formations around the borehole, which is one of the most important rock properties, can be addressed with an acoustic logging method. However, abnormal in-situ stress, mud invasion, and borehole damage would result in radial changes in formation velocity around the borehole, thereby bringing inaccurate interpretation results [1]. Based on a dipole waveform obtained by multipole array acoustic logging data, a formation's shear-velocity radial profile can be inverted for evaluating the borehole stability, estimating in-situ stress, and reflecting mud invasion [2–7]. Consequently, it is crucial to study the inversion method of the formation's S-wave velocity radial profile around the borehole.

When the formation's S-wave velocity changes with radial depth, the measured dipole's S-wave dispersion characteristics will also change accordingly. The existing inversion methods regarding the formation's S-wave velocity radial profile are mainly based on the above relationship. Generally, there are two inversion methods. One is Sinha's method, which combines a perturbation method and the Backus–Gilbert (B-G) theory to invert the formation's S-wave velocity radial profile [8]. This is a compromise method between radial distance and velocity change, resulting in an average radial velocity profile that is closest to the actual situation [9,10]. Subsequently, this inversion method is successfully applied

**Citation:** Ye, Q.; Sun, H.; Jin, Z.; Wang, B. Study on Shear Velocity Profile Inversion Using an Improved High Frequency Constrained Algorithm. *Energies* **2023**, *16*, 59. https://doi.org/10.3390/en16010059

Academic Editors: Xingguang Xu, Kun Xie and Yang Yang

Received: 29 November 2022 Revised: 14 December 2022 Accepted: 19 December 2022 Published: 21 December 2022

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

in well completions, stress parameters, and velocity estimations of Transversely Isotropic with a Vertical axis of symmetry (TIV) media [11–13]. For example, some researchers have improved this method by using a parametric velocity-characterization equation, and this method was successfully implemented in many cased wells [14,15]. However, when the signal-to-noise ratio is low, there may be deviations between the inversion results and a real formation model, failing to reach a high resolution and high precision. Furthermore, the convergence of this algorithm is seriously affected by the selecting of the initial values. The other inversion method is the constrained inversion method using a high-frequency band, which obtains the conclusion that the radial shear profile is mainly affected by the formation's shear velocity and with regards to the radial variation formation as a formation described by two parameters [16]. This method assumes that the radial change of the formation's S-wave velocity is mainly controlled by two parameters, which are the velocity change and the thickness change of a transition zone (TZ), and an inversion objective function is constructed based on these two parameters. This method constrains a high-frequency band and increases the contribution of a high-frequency component in the objective function, which can effectively improve the accuracy of inversion the results, especially in the formation around the borehole. This constrained inversion method is then applied in a logging-while-drilling quadrupole waveform processing to obtain the formation radial velocity, and to assess the rock-brittleness and fracability of the near-borehole formation [4,5]. In real data processing, the acoustic tool in the borehole has a great impact on the dispersion characteristics of the dipole waveform. Meanwhile, the existing two inversion methods of the formation's S-wave velocity radial profile do not consider the impacts of the tool on the dispersion, which affects the processing effect of the real data. In this case, an equivalent-tool theory was proposed to eliminate impacts of the tool on the flexural wave dispersion in the dipole acoustic logging, and it has been effectively applied in filed data processing and formation information-extraction [17,18].

In this paper, based on the equivalent-tool theory, an improved high-frequency constrained inversion method is proposed to invert the formation's S-wave velocity radial profile around the borehole. The impacts of the tool on the dipole dispersion characteristics are eliminated by equivalently treating the acoustic tool as an elastic rod characterized by two parameters, which are the radius and the equivalent elastic modulus. The improved high-frequency constrained inversion method can eliminate the impacts of the tool and obtain a high-precision formation S-wave velocity radial profile. By processing and comparing the theoretical simulation data and the measured data, the reliability of the improved inversion method is validated.

#### **2. High-Frequency Constrained Inversion Method of Formation Shear-Velocity Radial Profile**

#### *2.1. Basic Method Principle*

Tang and Patterson found that a dispersion curve of a radially continuously changing formation is very similar to that of a single-layer altered formation [16]. According to this characteristic of the dispersion curve, the formation with a radial change of S-wave velocity is regarded as a double-layer medium formation with a thickness around the borehole and a velocity difference from the undisturbed formation. In cases of constantly changing values, multiple theoretical models were established, and then the dispersion curve of each theoretical model is compared with the dispersion curve of the real formation. When the comparison results are closest, it can be considered that the theoretical model in this case is the real formation, and an objective function can be constructed:

$$E(\mathcal{A}r, \mathcal{A}V) = \sum\_{\Omega} \left[ V\_m(\omega, \Delta r, \Delta V) - V\_d(\omega) \right]^2 \tag{1}$$

where *Vm* is a dispersion curve calculated by the theoretical model established using Δ*r* and Δ*V*; *Vd* is a dispersion curve of a real radially changing formation; Ω is a frequency range; and *ω* is the circular frequency. When the value of the objective function (1) is the smallest, the velocity variation Δ*V* and the TZ thickness Δ*r* can be used to represent the velocity profile of the real radially changing formation.

The objective function (1) can be used to invert a relatively accurate velocity profile, but there is a large error for the model's calculation of a near-borehole formation. In order to obtain more accurate results, Tang and Patterson delved into this study, and found that a phase velocity of a low-frequency part of the dispersion curve in a radially changing model is close to the shear-velocity of the undisturbed zone [16]. When the S-wave velocity of a homogeneous model is equal to the near-borehole S-wave velocity of an inhomogeneous formation model, the high-frequency parts of the dispersion curves of the two formation models are very close. This shows that the low-frequency part of the flexural wave is influenced by the undisturbed zone, and the high-frequency part is influenced by the formation near the borehole. This is because the bending wave is a dispersive wave, leading to a longer wavelength and a further detection range, whereas a shorter wavelength travels along the shallow formation area. From this, Tang and Patterson proposed the high-frequency constrained inversion method to reduce the near-bore error of the S-wave's slowness profile by constraining the high-frequency band [16]. The objective function (1) is modified to:

$$E(\Delta r, \Delta V) = \sum\_{\Omega} \left[ V\_{\mathfrak{m}}(\omega, \Delta r, \Delta V) - V\_d(\omega) \right]^2 + \lambda \sum\_{\Omega'} \left[ V\_{\mathfrak{m}}(\omega, \Delta r, \Delta V) - V\_{\mathfrak{h}}(\omega, V\_1) \right]^2 \tag{2}$$

where *Vh* is the dispersion curve calculated by establishing a uniform formation model; *V*1 is the wellbore formation shear-velocity; Ω' is the high-frequency range for constraint, usually (9~10 kHz); and *λ* is used to adjust the relative contribution of the high-frequency term.

The radial change of a formation's S-wave velocity is usually monotonically increasing or monotonically decreasing. According to this phenomenon, the formation's S-wave velocity radial-change model is established [4]:

$$V(r) = V\_0 - \left(\frac{\overline{\Delta V}}{1 - e^{-1}}\right) \exp\left(-\frac{r - r\_0}{\Delta r}\right), (r \ge r\_0) \tag{3}$$

where *V*0 is the S-wave velocity of the undisturbed formation and *r*0 is the borehole's radius. The shear-velocity profile can be obtained by substituting the result obtained from Equation (2) into Equation (3).

#### *2.2. Theoretical Model Validation*

A radial layered formation was modeled by the numerical simulation method, and the high-frequency constrained inversion method was used for the inversion calculation. The relevant parameters of this formation model are shown in Table 1. In the model, the number of sampling points for each receiver was 400, the spacing was 0.1524 m, the time-sampling interval was 36 μs, and the center frequency was 3 kHz. The velocity alteration schematic is shown in Figure 1.

**Figure 1.** Structure diagram of numerical simulation model.

For the model in Table 1, a real-axis-integration method was used to simulate its waveform [19], as shown in Figure 2:

**Figure 2.** Waveforms of the numerical simulation model.

The dispersion curve was calculated using a weighted spectral semblance algorithm [19], and the results are as shown in Figure 3a. For the dispersion curve without a frequency range of 8~9 kHz, an arctangent function fitting should be applied to obtain the high frequency range (shown in Figure 3b).

**Figure 3.** (**a**) Dispersion analysis by the weighted spectral semblance method and (**b**) the fitting results.

According to the model parameters (Table 1), the velocity changes of the TZs during the inversion process were set to: 1000 m/s: 2 m/s: 1200 m/s, and the thickness changes of the TZs were set to: 0 m: 0.005 m: 0.3 m. Then, it was necessary to construct 101 × 61 theoretical models, calculate the dispersion curve of each theoretical model one by one, and substitute them into Equation (2). The full inversion frequency range was 3~9 kHz, and the high-frequency constraint range was 8~9 kHz. A two-dimensional image of the correlation coefficients of the objective function was obtained, as shown in Figure 4:

**Figure 4.** Two-dimensional correlation image of objective function in high-frequency constrained inversion.

The ordinate of Figure 4 is the equivalent TZ thickness Δ*r*, the abscissa is the velocity change Δ*V*, and the color represents the size of the objective function value. When the objective function value is the smallest, the corresponding Δ*V* and Δ*r* are 146 m/s and 0.125 m, respectively. By putting these two parameters into Equation (3), the shear-velocity radial profile can be obtained, as shown in Figure 5, and its relative error is shown in Figure 6.

**Figure 5.** Inversion result of high-frequency constrained inversion method.

**Figure 6.** Radial distribution of relative error between inversion result and real formation velocity.

From Figure 6, due to the high-frequency constraints, the inversion method had a relatively high accuracy, and the overall error was basically below 4%, and especially in the near-borehole formation a high inversion-accuracy could be obtained.

#### **3. Application of Equivalent-Tool Theory in Constrained Inversion**

#### *3.1. Equivalent-Tool Theory*

When using real logging data to invert the formation's S-wave velocity radial profile, the impacts of the acoustic logging tool cannot be ignored. Due to a complex structure, the equivalent-tool theory can be introduced to simulate the impact of the tool [17]. Assuming that the logging tool is a cylinder with a radius of *a*, no matter how complex the structure of the tool is, as long as the wavelength of the bending wave is greater than the radius of the tool, the tool can be equivalent to an elastic rod with an elastic modulus of *MT*. One parameter can used to simulate the impacts of multiple elastic parameters of the tool. Considering that there is a logging tool in center of the borehole, the displacement (*u*) and pressure (*p*) in the annular fluid between the tool and the borehole caused by the acoustic wave can be expressed as follows:

$$\mu = \left\{ A\_{\text{il}} \left[ \frac{n}{r} I\_{\text{n}}(fr) + f I\_{\text{n}+1}(fr) \right] + B\_{\text{n}} \left[ \frac{n}{r} K\_{\text{n}}(fr) - f K\_{\text{n}+1}(fr) \right] \right\} \cos[n(\theta - \phi)] \tag{4}$$

$$p = \left\{ A\_{\text{ll}} \left[ \rho\_f \omega^2 I\_{\text{n}}(fr) \right] + B\_{\text{n}} [\rho\_f \omega^2 K\_{\text{n}}(fr)] \right\} \cos[n(\theta - \phi)] \tag{5}$$

where *An* is the amplitude coefficients of the incident wave; *Bn* is the amplitude coefficients of the outgoing wave; n is the sound-source order, *n* = zero for monopole, *n* = one for dipole, and *n* = two for quadrupole; *In* and *Kn* are the first-kind and second-kind of nth-order variant Bessel functions, respectively; f is the radial wave number of the fluid; *k* is the axial wavenumber of the mode wave; *Vf* and *ρf* are the velocity and density of the fluid, respectively; *ω* is the circular frequency; and *θ* − *ϕ* is the azimuth of the alignment of the multipoles to a reference azimuth *ϕ* [19].

The boundary conditions at the fluid–solid interfaces (fluid in the borehole and the logging tool's surface, fluid in the borehole and borehole wall) are fully matched by the acoustic admittance (i.e., *u/p*). At longer wavelengths, the logging tool can be equivalent to the elastic rod with only one modulus. Norris studied the case of the solid elastic rod in the center of the borehole, and used the following acoustic admittance matching for the Stoneley wave between the elastic rod and the borehole wall, and the specific expression is [20]:

$$\left(\frac{u}{p}\right)\_{r=a} = -\frac{a}{M\_T} \tag{6}$$

Assuming that this result is still applicable to multipole waves, and extending Equation (6) to the frequency domain, it is only necessary to calculate the acoustic admittance of the elastic waves according to Equations (4) and (5) and then match it with the acoustic admittance of the logging tool at the tool surface. This gives the relationship between the unknown coefficients *An* and *Bn*:

$$E\_{tool} = \frac{B\_n}{A\_n} = -\frac{\left(\frac{M\_T}{a}\right)\left[\frac{n}{a}I\_n(fa) + fI\_{n+1}(fa)\right] + \rho\_f\omega^2I\_n(fa)}{\left(\frac{M\_T}{a}\right)\left[\frac{n}{a}K\_n(fa) - fK\_{n+1}(fa)\right] + \rho\_f\omega^2K\_n(fa)}\tag{7}$$

where *Etool* represents the elastic parameter related to the tool, and it can be added to the dispersion equation for the simulation of the impacts of the tool on the dispersion curve. The specific method is: add the terms related to the equivalent tool to two matrix elements *M*11, *M*21 related to the sound field in the borehole in the dispersion equation, which is:

$$\begin{array}{c} \begin{array}{c} M\_{11} : \frac{n}{R} I\_{n}(fR) + fI\_{n+1}(fR) \\ \frac{n}{R} I\_{n}(fR) + fI\_{n+1}(fR) \end{array} \\ \begin{array}{c} \begin{array}{c} \frac{n}{R} I\_{n}(fR) + fI\_{n+1}(fR) \end{array} \Big{[} + E\_{\text{total}}\left[\frac{n}{R} K\_{n}(fR) - fK\_{n+1}(fR)\right] \\ M\_{21} : I\_{\text{n}}(fR) \to I\_{\text{n}}(fR) + E\_{\text{total}}K\_{\text{n}}(fR) \end{array} \end{array} \tag{8}$$

where the symbol "→" in the above formula means to convert the expression on the left to the expression on the right. From this, the multipole dispersion equation when the logging tool is located in the center of the borehole is obtained:

$$D\left(k\_{\prime}\omega\_{\prime}V\_{p\prime}V\_{s\prime}\rho\_{\prime}V\_{f\prime}\rho\_{f\prime}R\_{\prime}M\_{\mathrm{T}},a\right)=0\tag{9}$$

where *Vp*, *Vs*, *ρ* are compressional velocity, shear velocity, and the density of the formation, respectively. The Newton–Raphson method is normally used to solve the dispersion equation [19]; thus, the multipole dispersion curve with the impact of the tool is obtained.

$$V\_{\rm pl} = \omega / k \tag{10}$$

In order to validate the feasibility of the equivalent-tool theory, an annular heavy-mud column is used to simplify and simulate the wireline dipole acoustic logging tool, and a formation model with the tool in the borehole is established; its parameters are shown in Table 2.

According to the parameters in Table 2, the real-axis integration method is used to simulate and calculate the corresponding waveforms and, respectively, calculate the reference dispersion curve without considering the impacts of the tool, the reference dispersion curve calculated by the equivalent-tool theory, and the dispersion curve calculated by simulating the waveform with the impacts of the tool. The results are shown in Figure 7.

**Figure 7.** Dispersion curve without tool impacts (blue); reference dispersion curve calculated by the equivalent-tool theory (black); dispersion curves of the waveform affected by the tool (red).

In the calculation, the equivalent radius a of the tool is equal to the outer diameter of the heavy-mud column (0.045 m), and the value of the equivalent modulus of the tool *MT* is 32 GPa. It can be seen from Figure 7 that in the frequency range of 2–9 kHz, the calculation results using the equivalent-tool theory are consistent with the theoretical curve, which validates the applicability of the equivalent-tool theory. Furthermore, in the frequency band higher than the cutoff frequency, logging tools make the flexural wave dispersion curve shift to low frequencies, which indicates that the impact of the tool must be considered when we use the dipole dispersion curve to carry out any inversion processing.

#### *3.2. Application of Equivalent-Tool Theory in Constrained Inversion Method*

The parameters of the annular heavy-mud column in Table 2 are used to simulate the formation model with the impacts of the tool. The high-frequency constraint method is used for the inversion calculation. When calculating the dispersion curve of the theoretical model, the equivalent theory is used to calculate the dispersion curve with the impacts of the tool, and a two-dimensional image of the objective function is obtained by substituting it into Equation (2).

Figure 8a,b represents the two-dimensional images of the objective function using the equivalent-tool theory and without using the equivalent-tool theory, respectively. When the value of the objective function is the smallest, the velocity change and equivalent thickness using the equivalent-tool theory are 146 m/s and 0.125 m, respectively, and the velocity change and equivalent thickness without using the equivalent-tool theory are 171 m/s and 0.145 m, respectively. The impacts of the tool make the velocity change and the equivalent thickness larger. By bringing these two results into Equation (3), the obtained dipole formation S-wave velocity radial profile is as follows:

**Figure 8.** Two-dimensional image of objective function: (**a**) using equivalent-tool theory and (**b**) without using equivalent-tool theory.

From Figures 9 and 10, it can be seen that the relative error of the inversion results without considering the tool is larger. This is because the tool's existence makes the dispersion curve shift to the low-frequency of the real formation, the velocity change, and the equivalent thickness increase when the objective function value is the smallest, resulting in a generally smaller radial S-wave velocity and a larger relative error. Therefore, the impact of the tool can be eliminated by using the equivalent-tool theory, and the result is more in line with true values.

**Figure 9.** Inversion results using equivalent-tool theory (red line) and without using equivalent-tool theory (blue line).

**Figure 10.** Relative error between inversion results and real values using equivalent-tool theory (red line) and without using equivalent-tool theory (blue line).

#### **4. Real Data Processing**

Based on the previous theoretical analysis, we can identify that the existence of the acoustic tool affects the dipole dispersion characteristic dramatically. The impacts of the tool should be considered carefully when performing the formation S-wave velocity radial profile inversion. For the real data processing, due to unknown tool parameters, the tool would be different for different borehole-logging measurements, and thus constant parameters cannot be used as equivalent to tool impacts. The following procedure is adopted in real data processing:

3. Step 3, implement the formation's S-wave velocity radial profile inversion algorithm corrected by the equivalent-tool theory in other intervals. The tool equivalent parameters *MT* and *a* in this step use the value from Step 2.

The formation's S-wave velocity radial profile is processed for the real data of an oilfield in Western China, as shown in Figure 11.

**Figure 11.** Comparison of formation shear-velocity profile radial with several logging results.

The depth of the borehole is X200–X260 m. In Figure 11, Trace 1 is the well diameter curve (black line) and the natural gamma curve (red line), Trace 2 is the time difference between the formation's P and S-waves in this interval, and Trace 3 gives the orthogonal dipole anisotropy analysis and fast and slow S-wave waveforms. Trace 4 is the formation's S-wave velocity radial profile calculated by the high-frequency constrained inversion method. In Trace 4, the shade of color represents the difference with the undisturbed formation velocity. Trace 5 is the formation's S-wave velocity profiles with the improved inversion method. Trace 6 gives the deep and shallow resistivity logging curves. From Figure 11, it can be seen that there are obvious differences between the two velocity profiles in the interval marked by the red frame. In this marked interval, there is no change in the borehole diameter, the anisotropy is not obvious, the GR value has no offset, there is no obvious difference between the deep and shallow resistivity curves, and thus the radial S-wave velocity should not change. Therefore, the result of Trace 5 is more in line with the real formation. At the interval between X245 m and X255 m, the radial S-wave velocity on Trace 5 has obvious changes, while the depth and shallow resistivity curves on Trace 6 are significantly different, the GR value on Trace 1 is small, and the borehole diameter (Trace 1) and anisotropy (Trace 3) have no obvious changes. This may be caused by mud invasion. The radial velocity changes are not obvious in other intervals, but the radial velocity change exists in the whole interval on Trace 4, which is obviously not in line with the actual situation.

#### **5. Conclusions**

In this paper, the basic theory of the high-frequency constrained inversion method for the formation's S-wave velocity radial profile is studied, and examples of a forward and an inversion simulation are given to illustrate that this method has a large error in the case of considering the impacts of the acoustic logging tool. Then, the equivalent-tool theory is introduced to eliminate the impacts of the tool through theoretical and real data studies. Finally, an improved high-frequency constraint inversion method using the equivalent-tool theory is proposed and used to process the real logging data. The following conclusions are drawn:

**Author Contributions:** Conceptualization, Q.Y. and B.W.; methodology, Q.Y.; software, H.S.; validation, Z.J.; writing—original draft preparation, Q.Y.; writing—review and editing, H.S.; visualization, Z.J.; supervision, B.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China, grant number 42274141.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

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**5. Numerical Example**

In this section, an example is used to present the application of the proposed approach. Consider an AMS example illustrated in Figure 15a. Its Petri net model is given in [6,7,15,22,45,46]. The system consists of four machines M1–M4 for processing parts; two robots R1 and R2 for loading and unloading parts. Each machine (robot) can process (hold) one part at a time. There are two input buffers I1 and I2 and two output buffers O1 and O2. Two raw part types, A and B, are considered to be processed in the system. Figure 15b shows the operation sequences of the two raw part types. The S3PR net of this AMS example is illustrated in Figure 16. It comprises 19 places and 14 transitions. The places can be defined as the following set partitions: *PA* = {*p*2, *p*3, ... , *p*12}, *PR* = {*p*13, *p*14, ... , *p*18}, and *P0* = {*p*1, *p*19}. The S3PR net contains 282 reachable markings.

**Figure 15.** (**a**) An AMS example and (**b**) production sequence.

**Figure 16.** S3PR net (*No*, *Mo*) of the AMS illustrated in Figure 13a.

Suppose that the first configuration of the system involves removing old machine. In this case, an old machine M1 is removed from the system (*No*, *Mo*). To model the removed machine by using the synthesis procedure of Algorithm 1, we construct a configuration as a rewriting rule R= {*rr1*} with *rr1* = {*L1*, *R1,* ϕ*1*, •ϕ*1*, ϕ*1* •}, where *L1* and *R1* are illustrated in Figures 17a and 17b, respectively. In addition, we have ξ*1: N1* → *No*, ϕ*1*= ({*p2*, *p3*, *p4*, *p5*, *p13*, *p16*, *p17*},{*t1*, *t2*, *t3*, *t4*, *t5*, *t6*}), •ϕ*1*= ({*L1.t1*},{*R1.t1*}), and ϕ*1* • = ({*L1.t6*},{*R1.t6*}). The second configuration includes adding new product. If a new product (part C) is assigned to a system, which indicates that a new operation sequence is assigned and the system requires an adjustment to its Petri net model structure. To model the addition of new product by using the synthesis procedure of Algorithm 1, we construct a configuration as a rewriting rule R = {*rr2*} with *rr2* = {*L2*, *R1,* ϕ*2*, •ϕ*2*, ϕ*2* •}, where *L2* and *R2* are illustrated in Figures 18a and 18b, respectively. Moreover, we have ξ*2: N2* → *N1*, ϕ*2*= ({*p15*, *p17*, *p20*, *p21*, *p22*, *p23*},{*t15*, *t16*, *t17*, *t18*}), •ϕ*2*= ({*L2.p15*, *L2.p17*},{*R2.t15*}), and ϕ*2* • = ({*L2.p15*, *L2.p17*},{*R2.t18*}).

The third system configuration involves rework. In this scenario, a part can be inspected after all operations have been completed. By using the proposed Algorithm 1, the production operations of the reworked part can be exactly and easily modeled by considering rework operations as alternative sequences. Suppose that an inspection machine M5 is added to a system and that part A is processed in M1 and M3. Then, part A is moved to an M5 by Robot 2 to check if there are defects in part A. If part A performs properly, then it will leave the system by Robot 2. Otherwise, if part A has defects, rework is needed, and part A is moved to M3 by Robot 2. To model the rework operation by using the synthesis procedure of Algorithm 1, we construct a configuration as a rewriting rule R = {*rr3*} with *rr3* = {*L3*, *R3,* ϕ*3*, •ϕ*3*, ϕ*3* •}, where *L3* and *R3* are illustrated in Figures 19a and 19b, respectively, ξ*3: N3* → *N2*, ϕ*3*= ({*p6*, *p7*, *p14*, *p18*, *p24*, *p25*, *p26*, *p27*},{*t6*, *t7*, *t8*, *t19*, *t20*, *t21*, *t22*}), •ϕ*3*= ({*L3.t6*},{*R3.t6*}), and ϕ*3* • = ({*L3.t8*},{*R3.t21*}). The Specifications of S3PR net illustrated in Figure 16 under changeable control specifications are shown in Table 2. In addition, the required monitors using Algorithm 2 of the system illustrated in Figure 16 under changeable control specifications are shown in Table 3.

**Figure 17.** A reconfigured S3PR net by removing a machine. (**a**) Left hand side net *L*. (**b**) Right hand side net *R*.

**Figure 18.** A reconfigured S3PR net by adding a product. (**a**) Left hand side net *L*. (**b**) Right hand side net *R*.

**Figure 19.** A reconfigured S3PR net by rework. (**a**) Left hand side net *L*. (**b**) Right hand side net *R*.

The controlled net after adding above changeable control specifications is illustrated in Figure 20. The place and arcs of the controller are illustrated with blue lines.

**Figure 20.** Controlled reconfigurable S3PR net after adding changeable control specifications.

#### **6. Conclusions**

This paper develops a novel two-step solution for quick and accurate reconfiguration of supervisory controllers for deadlock control in RMSs with dynamic changes. In the first step, the net rewriting system is used to design a reconfigurable PN model under dynamic configurations. The obtained model guarantees boundedness behavioral property but may not guarantee the other properties of a Petri net model (i.e., liveness and reversibility). The second step proposes an automatic deadlock prevention policy for reconfigurable Petri net using the siphon control method based on a place invariant to solve the deadlock problem with dynamic structure changes in RMSs and guarantee the liveness and reversibility properties for the system. The proposed method is validated using the GPenSIM tool and compared with existing methods in the literature to highlight its ability of adapting to RMS configuration changes.

The major advantages of the developed approach are as follows: (1) It does not need to compute reachability graphs as illustrated in Algorithm 2, Section 3, and has low-overhead computation as proved in Theorems 10 and 11, Section 4.4. (2) It can automatically and dynamically modify the structure of a Petri net model without affecting its behavioral properties, i.e., liveness, boundedness, and reversibility as illustrated in Algorithm 2, Section 3. (3) It allows rapid reconfigurability and reusability of the controller during reconfiguration as shown in Algorithm 2, Section 3. (4) It can easily handle any dynamical changes in RMSs compared with the studies in Badouel et al. [39], Llorens and Oliver [34], Wu and Zhou [25], and Kaid et al. [7] as shown in Algorithm 2, Section 3. (5) The GPenSIM code is developed for designing, simulation, validation, and performance analysis of deadlock problems with dynamic structure changes in RMSs and the correctness of the proposed

approach is proven and compared with the studies in Ezpeleta et al. [43], Li and Zhou [44], and Kaid et al. [6] as shown in Section 4.5. (6) Based on Theorems 10 and 11, the computational complexity of the proposed approach has polynomial time complexity. Therefore, it has low computational complexity and can be applicable to other types of complex systems such as mass customization manufacturing, lean productivity, agile manufacturing, and flexible manufacturing systems. (7) It can consider systems with sequential and complex resource requirements, meaning that a set of system resources can be used and shared to process each component according to sequential processes that depend on the step-by-step discrete execution and multiple processes that depend on the execution at the same time as shown in numerical example.

The limitation of the developed approach is that the obtained models lack an appropriate conversion approach from the PN model into control languages for application. Thus, our future research will examine the developed approach to have an automatic method to examine the applicability of the obtained models for real world manufacturing systems.

**Author Contributions:** Conceptualization, H.K. and A.A.-A.; software, H.K. and R.D.; resources, H.K., A.A.-A. and R.D.; formal analysis, H.K. and A.A.-A.; investigation, H.K., A.A.-A. and Z.L.; validation, H.K., A.A.-A., Z.L. and R.D.; writing—original draft preparation, H.K., A.A.-A. and Z.L.; writing—review and editing, H.K., A.A.-A., Z.L. and R.D.; visualization, H.K., A.A.-A. and Z.L.; supervision, A.A.-A., and Z.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by King Saud University through Researchers Supporting Project Number (RSP-2020/62).

**Acknowledgments:** The authors would like to thank King Saud University for funding and supporting this research through Researchers Supporting Project Number (RSP-2020/62).

**Conflicts of Interest:** The authors declare no conflict of interest.


**3. Use of comet assay in buccal cells**

The comet assay in buccal cells has been used to evaluate DNA damage induced by different materials such as mouthrinses [50], metals released from orthodontic appliances [51–59], ionizing radiation [60], as well as assessment of DNA damage, and its modulation by life-style, dietary, genetic and healthy factors [61–74], occupational exposure [66–69,75–82], and environmental exposure [83–86]. Different procedures have been used in collecting and processing the samples that are presented and discussed in Rojas et al. [33]. The **Table 2** represents classification according to the type of population study based on exposure and lifestyle factors with the results of comet assay.

#### **3.1. Mouthrinses and metal released from orthodontic appliances**

The genotoxic properties of mouthrinses and metals from orthodontic appliances are essential for determining the biological safety of those materials in patients. Current in vivo human studies are aimed at representing the real condition of the oral cavity by sampling buccal cells, which are directly exposed to the appliances [51,52].

Eren et al. [50] evaluated the stability of buccal epithelial cells for SCGE assay after the use of chlorhexidine digluconate (CHX), a mouthrinse used by dentists as disinfecting agent for operation sites washing and for disinfection of root canals. A statistical increase was observed in the DNA damage after the CHX application. Considering orthodontic appliances, the first in vivo study was performed by Faccioni et al. [51], who conducted the alkaline comet assay in orthodontic patients. They reported genotoxic damage and found positive correlations between the concentrations of released cobalt and nickel and the number of comets as well as correlations between Co levels and comet tails. However, Westphalen et al. [52] did not find genetic damage after the placement of the orthodontic appliances.

According to Fernández-Miñano et al. [53], genotoxicity induced in buccal cells could be related to the composition of orthodontic appliances. Orthodontic apparatus made with titanium was not genotoxic for oral mucosa cells, whereas the stainless steel alloy and nickelfree alloy induced DNA damage in buccal mucosa cells. In contrast, Hafez et al. [54] observed that stainless steel brackets with stainless steel archwires produce the least damage, whereas titanium brackets with nickel–titanium archwires produced the highest amount of genotoxic‐ ity, assessed with the comet assay. Baričević et al. [55] assessed subjects with Co–Cr–Mo alloy and Ni–Cr alloy showed significantly higher comet assay parameters when compared with controls. Gonçalves et al. [59] showed the genotoxic effects of Hyrax auxiliary orthodontic appliances containing silver-soldered joints.

On the other hand, Hafez et al. [54] reported damage to the DNA in mucosa cells at 3 months of orthodontic treatment but not at 6 months. Thus, the difference in exposure period of prosthodontic and orthodontic appliances in oral cavity might explain discrepancies observed between results obtained by Faccioni et al. [51], and those of Westphalen et al. [52] and Baričević et al. [55].

Visalli et al. [56] found that both amalgams and resin-based composite fillings can induce genotoxic damage in human oral mucosa cells. They also report that lifestyle variables, including alcohol intake and smoking habits, did not affect the genotoxic response and did not act as confounding factors. Martín-Cameán et al. [57] observed induction of genotoxicity in buccal cells of subjects with orthodontic appliances and orthodontic appliances with micro‐ screws when compared with controls. In addition they found that damage was higher in women.

#### **3.2. Radiation**

Only one work that analyses and compares the DNA damage and repair following radiation challenge in buccal cells and lymphocytes using SCGE assay was found. The results suggested that baseline DNA damage in oral epithelial cells is greater than that in lymphocytes [60].

#### **3.3. Life style, dietary, genetic and healthy factors**

As mentioned above in the first work of this type, Rojas et al. [14] found a significantly increased tail length in a smoker group compared with a non-smoker group. Differences between genders either in the smoker or non-smoker group were not observed and were neither related to age or number of cigarettes smoked. Waterpipe smoking (a type of tobacco smoking) and its condensate have been examined for the genotoxic effects on buccal cells. The tail moment in buccal cells of smokers was found to be 186 ± 26, which is 371.9% higher than the tail moment in buccal cells of non-smokers. The other comet parameters such as tail length, % tail DNA, and fragmented DNA were 456 ± 71, 97.0 ± 19, and 32.0 ± 3.3, respectively, in buccal cells of smokers, whereas in control group (non-smokers), the values of tail length, % tail DNA, and fragmented DNA were extremely low [72].

Oral habits have also been associated with DNA damage. Khanna et al. [70] reported a case of a tobacco chewer in which the percentage of damaged cells was significantly higher than in the control. Also the effect of gutkha (a preparation of crushed areca nut, tobacco, catechu, paraffin wax, slaked lime, and sweet or savory flavorings) and pan masala (an herb, nut, and seed mixture that is commonly served in the Middle East countries) chewing along with and without smoking was studied in buccal epithelial cells using single-cell gel electrophoresis [71]. The increase in the mean comet tail length was observed as follows: non users < smokers < pan masala chewers < gutkha chewers < pan masala + smoking < gutkha + smoking. Like Rojas et al. [14], they conclude that these bioassay and biomarker are easier and safe methods to detect DNA damage among humans.

Assessment of DNA damage and its modulation by dietary and genetic factors in smokers using the comet assay has also been developed [87]. Pal et al. [62,63] analyzed the influence of regular black tea consumption on tobacco-associated DNA damage and human papilloma virus (HPV) prevalence in human oral mucosa. The increase in DNA damage was significantly associated with increase in age and tenure of tobacco habit. Reduced DNA damage was found to be significantly associated with increase in tea intake. In case of oral cancer patients, comparatively high frequency of DNA damage was observed. The frequency of DNA damage and HPV infection was comparatively high in oral cancer patients than in the normal subjects. These studies indicated a chemopreventive role of black tea against reducing DNA damage risk of buccal cells due to tobacco exposure. Authors concluded that buccal cells could be used as cytological markers for detection of risk and risk reduction in normal population. Since, as mentioned above, more than 90% of human cancers arise from epithelial cells, it has been postulated that experiments with these cells may have particular relevance for the detection of cancer preventive effects [47].

On the other side, several polymorphisms in DNA repair genes have been reported to be associated with cancer risk [88]. The repair of DNA damage has a key role in protecting the genome from the insults of genotoxic agents. Tobacco-related compounds cause a variety of DNA damage, and DNA repair capacity plays an important role in agent-induced damage genotoxic. Several polymorphisms in genes that participate in different DNA repair pathways, such as XRCC1 399, hOGG1 326 [65], GSTP1 [66], CYP2E1 [67], CYP1A2 [68], and CYP1A1 [69], have been evaluated for their effects on different biomarkers [89], including comet tail length in buccal cells.

DNA damage effects of the used substances were confirmed in mechanical workshops workers, but with no confirmation of the influence of GSTP1 [66] or CYP1A1 [69] gene polymorphism on DNA damage, considering the comet assay performed on buccal cells. Conversely, workers with the wild genotype for CYP2E1 showed statistically significant higher comet tail length at the occupational exposure, while the mutated genotype did not have influence on this biomarker [67]. With CYP1A2 gene, the results showed that DNA damage in cells of workers carrying the mutated genotype was higher than workers carrying the wild genotype [68].

Sellappa et al. [65] found significant differences in the comet scores between smokeless tobacco users and control subjects when XRCC1399 and hOGG1326 polymorphisms and the frequen‐ cies of genetic damage among tobacco chewers were studied.

These findings provided evidence for the view that polymorphisms in DNA repair genes may modify individual susceptibility to genotoxic agents and justify additional studies to investi‐ gate their potential role in development of genetic damage.


*4.6. Response Function*

The response function was derived and integrated following four steps: (1) fit beta-function for all treatments, (2) run FSP model to obtain perceived BPFD of internodes for each treatment, (3) derive response function across all treatments for internode length in dependence of simulated perceived BPFD and (4) integrate the derived response function into the model for all treatments.

Based on the experimental observations, the parameters for the beta-function were derived for each light treatment. Based on these parameters, the FSP model of soybean dynamically simulated the plant architecture over time under each of the six light treatments.

At this stage, the growth of the internodes stopped at a final length according to the measurements. During the simulations, BPFD perceived by the internodes was recorded for each hour. The simulated perceived BPFD was used to fit a response function to the internode length observed under the different BPFD treatments:

$$\text{Internode length} = \text{L}\_{\text{min}} \left( \mathbf{1} + \left( BPFD\_{\text{min}} - BPFD\_{\text{per}} \right) \mathbf{a} \right) \text{ BPFD}\_{\text{min}} - BPFD\_{\text{per}} > \mathbf{0} \tag{5}$$

where *Lmin* is the minimum possible length of the internode, *BPFDmin* is the lowest BPFD giving *Lmin*, *BPFDper* is the average perceived BPFD of the internode during the first four days of growth and *a* is the slope of the response to *BPFDper*. The first four days of growth were used due to a very rapid growth inhibiting effect of blue light [59] and according to the results of Kahlen and Stützel [46] who found four successive days starting one week before reaching maximum growth rate to be particularly sensitive to changes in PPFD.

The found BPFD response function was integrated in the FSP model to determine *Lmax* of the beta-function and the final internode length. They were hereby simulated in dependence of the perceived BPFD during the simulations and at this stage no longer determined by experimental measurements. Internode two until nine elongated according to the integrated response function, with *Lmin*, and *tm* set for each internode according to the found parameter values for the treatment

B310 (baseline scenario with shortest internodes). Because the first internode (hypocotyl) grew before thinning and beginning of experimental measurements, its length was set to grow to the final average length of all treatments.


**2. BAV syndrome pathology and pathophysiology**

The bicuspid aortic valve morphology has been characterized by raphe number and the position of cusps and raphes [8]. This morphology includes complete or partial leaflet fusion. The most common morphology in BAV patients is fusion of left and right coronary cusps. The next most common morphology in BAV patients is fusion of right coronary cusp with noncoronary cusp [8, 9]. The morphology of the bicuspid aortic valve has been described by various classification systems, most notably the Sievers classification [8–10]. Sievers Type 0 BAV has an incidence of 7% in the original Sievers and Schmidtke series and a BAV valve geometry of no raphe and 2 valve cusps [8, 11]. Sievers Type 1 BAV morphology is the most common and has an incidence of 88% in the original series with a BAV valve geometry of a single raphe and 2 valve cusps [8, 11]. Sievers Type 2 BAV has an incidence of 5% in the original series and is the rarest morphology of the three types in this classification with a BAV valve geometry of 2 raphe and 2 valve cusps [8, 11].

Patients with BAV are known to have high rates of various valvular pathologies in adult life [7], particularly aortic stenosis (especially in males) [12], aortic regurgitation [13], and infective endocarditis involving the aortic valve [14]. BAV is three to four times more frequent in men than in women. Approximately, 50–75% of patients with BAV will require aortic valve replacement procedure during their lifetime and up to 25% may require an aortic procedure often concurrent with their valve replacement [15].

BAV syndrome pathophysiology is associated with dilation of the aorta [2]. BAV may be associated with an ascending aortic aneurysm phenotype in around 26–50% patients or a root phenotype in up to 20–32% of patients [2, 3, 16]. These distinct phenotypes have been stratified into three categories. Type 1 is dilation of ascending *Bicuspid Aortic Valve: Current Therapeutic Strategies DOI: http://dx.doi.org/10.5772/intechopen.113315*

aorta along its convexity and can involve root dilation. Type 2 is arch dilation and ascending aortic dilation with sparing of the root. Type 3 involves an isolated root phenotype and has been associated with a genetic causality [16–18]. BAV associated with aortic dilation increases the risk of dissection as the aorta dilates further. In BAV patients, aortic dissection has been observed to occur almost 5–10 times more commonly than trileaflet aortic valve population [19].

Aortic dilatation and aneurysm formation in BAV patients has been attributed to two different mechanisms: hemodynamic stress and the inherent aortic wall tissue abnormality. Hemodynamically, shear stress on the aortic wall due to blood flowing through a stenotic valve has been hypothesized to cause early dilatation of the aorta [20]. Abnormal flow patterns have been noted due to the configuration of the BAV even in absence of stenosis [21].

Dysregulation of the aortic wall can also contribute to aneurysmal dilation. Human and animal studies have identified that extracellular matrix dysregulation along with dysfunctional signaling pathways can contribute to hemodynamic effects observed in calcific aortic valve disease and regurgitation [22]. A recent study demonstrated newborns with BAV having aortopathy and dilated aorta even in the presence of relatively normal velocities across the valve [23] and this suggests inherent aortic tissue dysregulation as a factor which may impact the population of patients with BAV.

#### **3. Embryology and genetics of BAV and associated aortic aneurysm**

The aortic valve develops from endocardial cushions within the maturing heart tube and begins to form around the fifth week of embryonic development. In normal cardiac development, these cushions later divide into three distinct aortic valve leaflets. In patients with BAV disease, the cushions either fail to divide or fuse leading to the characteristic bicuspid morphology. It has been postulated that multifactorial variables including genetic/embryologic factors impact the formation of the bicuspid aortic valve.

There is a recognized genetic component to bicuspid aortic valve disease. It may occur sporadically or as an autosomal dominant disorder with variable penetrance [24]. And, it has been noted that some family members of BAV patients may present with isolated BAV, some with associated aortic (ascending/root) aneurysms and some may be carriers with no manifest disease. The spatial and anatomical sequences in development of congenital heart disease (CHD) continues to be defined. Outflow tract defects specifically those involving the aortic valve have been difficult to categorize as they appear to be multifactorial in origin with many signaling and transcriptional gene anomalies possible for the outcome. Autosomal dominant inheritance pattern has been described for BAV specifically involving the NOTCH1 gene pathway however, this is not exclusive [25].

The genetic mechanism for a majority of BAV cases remains unknown. Although these cases may seem 'sporadic' there is still a 10% increase in risk of having BAV in siblings and offspring based on epidemiological studies [26]. There is a similar rate of incidence of aortic aneurysms in family members with or without BAV demonstrating some role of shared environmental or genetic causes [27]. High number of genetic variants, associated with structural variation and mixed inheritance patterns of the disease have complicated the discovery of the BAV-associated genes [28]. Chromosomal mutations involving 9q have been linked to BAV disease [29]. Mutations at various loci on chromosomes 5, 15, and 18 are linked to familial BAV and aortic syndromes [24].

The occurrence of aortic aneurysm and coarctation in patients with BAV underlines a possible common genetic pathway for these disease entities. Microscopic examination of BAV associated aneurysm tissue has demonstrated non-inflammatory medial degeneration also known as cystic medial necrosis [30]. Dysregulation of the canonical (Smad2/ Smad3) TGF-B signaling genes has been implicated to be a possible common defect for BAV and aneurysm formation. TGF-B signaling plays a role in cell migration and valvulogenesis that are pivotal in proper valve formation and functioning [30]. Similarly, Loeys-Dietz syndrome (LDS) is caused by mutations in genes encoding for TGF-B receptors. BAV along with thoracic aortic aneurysms are commonly found in patients with LDS.

With these multifactorial variables that impact BAV syndrome presenting as familial clusters and variable penetrance, screening is recommended in first degree relatives of patients with BAV disease [31]. An echocardiogram is commonly utilized to monitor the bicuspid aortic valvular pathophysiology. A CTA or MRA may also be utilized for monitoring the bicuspid aortic valve aortic phenotype.

### **4. Genetic syndromes with BAV**

BAV is associated with several complex valvuloaortopathies and specific syndromes. Approximately 30–50% of patients with coarctation of the aorta have BAV [32–34].

Turner Syndrome is one of the most common genetic syndromes involving patients having a BAV phenotype [34]. The syndrome is associated with X chromosome monosomy in females. Approximately 30% of females with Turners can have a BAV [35]. The frequency of aortic aneurysms associated with BAV is also known to be higher in this group [36]. This observation along with the male predominance of BAV has led to the hypothesis that the X chromosome reduction maybe related to BAV incidence [37]. Generally short statured females with coarctation of the aorta should raise suspicion and lead to surveillance for BAV disease [38].

Loeys-Dietz Syndrome (LDS) is the second most common syndrome associated with BAV with approximately 10% of these patients manifesting the BAV phenotype [39]. TGF-B pathway gene mutations are known to be associated with the LDS. These mutations are common in non-syndromic thoracic aortic aneurysmal disease as well demonstrating a possible common pathway. Compared to non-syndromic patients, the LDS patients tend to present earlier in their life usually with symptomatic aortic regurgitation due to accelerated aortic dilation. Increased arterial tortuosity in major blood vessels and male sex have been determined to be associated with a higher risk of dissection in these patients [40].

Velocardiofacial Syndrome (DiGeorge Syndrome) is caused by deletions in gene 22; this syndrome involves cleft palate, immune deficiency, hypoparathyroidism, ventricular septal defect (VSD), and conotruncal defects of the heart (truncus arteriosus and tetralogy of Fallot). BAV and aneurysmal disease is more prevalent in this set of patients compared to the non-syndromic population [41]. The syndrome itself is a combination of genetic defects that are found in BAV disease demonstrating the multigenetic components that are involved in the BAV phenotype.

### **5. Surveillance of the aorta in BAV disease**

BAV syndrome is a heterogenous disease presenting with variable aortic and valvular pathology over a spectrum of age groups. Asymptomatic BAV may be an *Bicuspid Aortic Valve: Current Therapeutic Strategies DOI: http://dx.doi.org/10.5772/intechopen.113315*

incidental finding on imaging [32, 33, 38]. Patients with BAV syndrome require individualized treatment according to the degree of involvement of the aortic and valvular apparatus with patient comorbidities and age considerations. BAV patients should therefore undergo routine, periodic surveillance, to delineate the optimal timing of therapeutic intervention.

Surveillance for BAV syndrome patients is performed based on the pathophysiology of the aortic valve phenotype or aortic phenotype. Surveillance requires serial echocardiography for valvulopathy. The growth rate of the aorta in BAV patients can be 0.2–2.3 mm/year [16, 42, 43], and serial CTA or MRI should be performed to monitor the growth rate. For patients with ascending aortic and root dimensions within normal limits, imaging can be done every 3–5 years [35]. For dimensions ranging from 40 to 49 mm, imaging should be performed annually. For BAV patients with an aorta measuring 50–54 mm or with family history of aortic dissection or rapid growth of the aorta, imaging should be performed every 6–12 months [44].

#### **5.1 Family screening**

Current guidelines suggest family screening with echocardiography for all firstdegree relatives with BAV probands. Relatives found to have BAV should have complete evaluation and CTA or MRA imaging [45]. When multiple signs of a disorder are present, genetic testing should be conducted for BAV patients especially those in their early years of life. Other high-risk features that should lead to genetic testing are family history of dissection or sudden death, congenital heart lesions, or other aneurysmal disease. Once identified, genetic counseling plays an important role in the holistic care for BAV patients. Due to the variable expression of causative genes, parents of BAV patients may not have a bicuspid aortic valve. Lifetime follow-up and aortic surveillance is also important as the timing of incidence of valve or aortic disease may be different amongst different family members.


**3.6. Practical significance of the new (Ea) approach in tribo- and mechanochemistry**

56 Tribology in Engineering

equation.

When (eo cos

inhibitor effect.

range of value of Ea.

can be shown as follows:

therefore, it is concluded that:

in angle .

Relation between RTa and RTs is as follows:

RTa) – reaction inhibition, when (eo cos

Concluding the general dependence (1):

Ea/ RTs = Ea/ RTa + ln (eo cos

than ambient energy (RTs > RTa) – catalytic effect, for (eo cos

to the molecules of reactants and this energy should be introduced into Arrhenius

exp (B - Ea/ RTs) / exp (B - Ea/ RTa) = 1 / (eo cos

In case the same value of reaction rate constant for reaction without and with catalyst is compared using Arrhenius equation, there should be noticed difference between activation energy for reaction with catalyst (Eac) and without catalyst (Ea). The hypothesis based on i

Ea = 40 kJ/mol = 240 x 1022 eV/mol = 4 eV / molecule Empirically determined energy emitted by solid surface is in the range 3 to 7eV and it is the

The results of these calculations are in line with hypothesis based on i model saying that catalytic effect is due to energy emission from catalysts surface in the form of electrons / photons stream, additional energy of which makes possible to reach the same reaction rate in lower ambient temperature or increase the reaction rate in the same ambient temperature. This hypothesis is described by C in Eqn. 8: C is the quotient of reaction rate constant described by Arrhenius equation and the stream of energy emitted by the surface of catalyst

i = {[f ( b ) – f ( a )] / [ ( b ) - (a ) ]} / [f' ( b ) / ' ( b ) ] (1)

i = (L – L0) A [exp (B – Ea/RTa)] / e0cos() (19)

{[f ( b ) – f ( a )] / [ ( b ) - (a ) ]} / [f' ( b ) / ' ( b ) ] =

Function f(y) represents the stream of energy introduced into the system, function (y) is connected with catalytic / tribocatalytic reaction critical rate, which is represent by the ratio

= (L – L0) A [exp (B – Ea/RTa)] / e0cos() (20)

model is that this difference is equal to energy emitted by solids surface in angle .

k = A exp (B - Ea/ RTs) (16)

) > 1 the real energy near catalyst surface is less than ambient energy (RTs <

) < 1 the real energy near catalyst surface is higher

) (17)

) (18)

) = 1 there is no catalytic nor

Mechanochemistry is the coupling of the mechanical and the chemical phenomena on a molecular scale and includes mechanical breakage, chemical behaviour of mechanicallystressed solids (e.g., stress-corrosion cracking), tribology, polymer degradation under shear, cavitation-related phenomena (e.g., sonochemistry and sonoluminescence), shockwave chemistry and physics, and even the burgeoning field of molecular machines. Mechanochemistry can be seen as an interface between chemistry and mechanical engineering. A smart method was proposed recently, in order to measure the energy involved during mechanical transformations. Displacement reactions between a metal oxide and a more reactive metal can be induced by ball milling. In some cases the reaction progresses gradually and a metal/metal-oxide nanocomposite is formed. Ball milling may also initiate a self propagating combustive reaction. The information available about these processes is reviewed. It is argued that the gradual or combustive nature of the reaction depends on thermodynamic parameters, the microstructure of the reaction mixture, and the way they develop during the milling process.

Baláž, et al. [15] investigated the mechanochemical treatment of solids which lead to a positive influence on the solid – liquid kinetics. They used Arrhenius equation for activation energy analysis. The breaking of bonds in the crystalline lattice of solids brings about a decrease (ΔE\*) in the activation energy and an increase in the rate of leaching

$$(\Delta \mathbf{E}^\*) = \mathbf{E} \cdot \mathbf{E}^\* \tag{21}$$

$$\mathbf{k}^\* = \mathbf{k} \exp\left(\Delta \mathbf{E}^\*/\mathbf{R}\mathbf{T}\right) \tag{22}$$

where E is the apparent activation energy of the non-disordered solid, E\* is the apparent activation energy of the disordered solid, k, R and T stand for the rate constant of leaching for the non-disordered solid, (the pre-exponential factor) gas constant and reaction temperature, respectively; k\* is the rate constant of leaching for the disordered solid. If E > E\*, then exp (ΔE\*/RT) > 1 and thus it follows from Eqn. (18) that k\* > k, i.e., the rate of leaching of a disordered solid is greater than that of an ordered mineral.

Thermodynamic methods are essentially macroscopic by origin and nature. They appear in the analysis of macroscopic engineering systems. They have been reliably validated in numerous macroscopic experiments and observations. Most probably there can be found areas that permit analysis of mechanochemical systems by means of relatively simple thermodynamic methods. From the purely thermodynamic point of view, the central problem of mechanochemistry is the exchange of energy between the (long-range) elastic energy and the (short-range) energy accumulated in individual bonds.

A New Attempt to Better Understand Arrehnius Equation and Its Activation Energy 59

These kinetic equations used in heterogeneous catalytic reactions description concern no one parameter characterizing catalyst. Consequently the effect of catalyst action can be explained only by the decreasing of activation energy value – the only one calculated parameter in kinetic equations. It is the reason that all theories of catalysts action try explain

The i model, particularly equation (19) describes catalytic and tribocatalytic reactions by

all kinds of energy introduced into the reaction system, including mechanical energy - L

properties of catalyst, explained by energy emitted from its surface to reaction space –

Resulted from the i model concept of the mechanism of heterogeneous catalysis and tribocatalysis, shown above, is confirmed partly by tribochemistry and mechanochemistry. This mechanism should be directly confirmed, particularly by materials engineering, which

Most recent work [19] emphasizes that while the detailed mechanisms by which different mechanochemical phenomena arise are not always well understood, mechanical forces are capable of effecting novel reactivity. Additionally, it strengthens that using force, one can effectively shepherd a chemical reaction down specific reaction pathways, for instance *by selectively lowering the energy of a transition state.* At this point it is of note, that the field of polymer mechanochemistry, has also the potential to change this paradigm by

In recent years the mechanochemistry field approach has found a renaissance, and different techniques have been applied to activate chemical reaction [20-22] and thereby to lower their

i. The i model put forward in this paper attempts to correlate mechanical work performed on a solid with its catalytic activity. This model was worked out on the basis

ii. The analysis of basic dependences resulted from i model lead to the conclusion that the mechanism of catalysis related to tribological processes can be adapted to

iii. The reactants molecules energy resulted from ambient temperature of reaction mixture is enhanced near solid surface by additional energy – emitted electrons / photons. Due

the influence of catalyst surface structure on the electrons / photons emission

revolutionizing the way chemists think about controlling chemical reactions [19].

of tribological tests results and was dedicated to tribochemistry.

heterogeneous catalysis including mechanochemical reactions.

to the additional energy the reaction can reach a critical rate.

the mechanism of activation energy decreasing.

the role of support in catalysts activity,

the possibility of energy storage by different materials

in eqn. (19)

e0 cos .

should explain:

activation energy.

**4. Conclusions**

dependence concerning parts, which quantitatively characterizes:

There is no clear theory which could be adapted to mechanochemistry, however the most recent approach [16] should be mentioned here. The i model applied for tribochemical applications can also be adapted to mechanochemistry. This model can be helpful in general dependences formulation, related to kinetics of mechanochemical reactions and to mechanical forces used for reactions activation. The theory based on i model assumes that mechanical energy introduced into solid body – reagent or catalyst, is accumulated in it and then emitted as low energy electrons or photons of energy equal or higher than activation energy of the reaction. The general Eqn. (19) can be used to determine quantitatively relationship between mechanical stress (L), the possibility of solid body to accumulate and then emit energy (e0 cos ) and kinetics of mechanochemical reaction.

Mechanochemistry, especially results of investigations shown above can be explained by hypothesis based on i model. On the other hand the positive effect of mechanical stress on catalyst efficiency confirms this hypothesis. However the reason of this effect can be mechanically (eg. during milling) produced changes of catalysts surface.

Rodriguez et al. [17] found out the influence of ultrasound radiation on catalysts effectiveness. They tested a new advanced method for dechlorination of 1,2,3-, 1,2,4-, and 1,3,5-trichlorobenzenes in organic solvent catalysed by palladium on carbon support and solid hydrazine hydrochloride yields benzene in short reaction times. The catalyst system can be efficiently reused for several cycles. Ultrasound radiation of the heterogeneous catalyst reaction increases remarkably the rate of dechlorination. Moreover, Rodriguez found that there is optimum energy of ultrasound radiation which results maximum catalysts efficiency. This effect is not seen when ultrasound radiation act liquid reactants. Rodriguez results confirm thesis that energy, in this case of ultrasound radiation is useful in reaction rate increasing when solid body – particles of catalyst are present in reaction mixture. This energy is cumulated by catalyst and emitted to the space near catalyst surface, what is the reason of reaction rate increasing. This effect can not be explained by the changes of the structure of catalysts surface, like in other mechanochemical effects (eg. during milling) so the only one probably mechanism is emission of cumulated in solid catalyst energy to reaction space.

The concept of mechanochemistry to modify molecular reactivity has a rich history for a long time. For instance Kauzmann an Eyring as early as 1940 [18] suggested that the mechanical perturbation of diatomic molecules could alter the reaction coordinates combined with their homolytic dissociation.

The chemical kinetic quantitatively describes homogenous reactions, where the rate of reaction depends only on heat introduced to reaction system. Kinetic equations concern reagents concentration, and according to Arrhenius equation: temperature of reaction mixture as well as activation energy.

These kinetic equations used in heterogeneous catalytic reactions description concern no one parameter characterizing catalyst. Consequently the effect of catalyst action can be explained only by the decreasing of activation energy value – the only one calculated parameter in kinetic equations. It is the reason that all theories of catalysts action try explain the mechanism of activation energy decreasing.

The i model, particularly equation (19) describes catalytic and tribocatalytic reactions by dependence concerning parts, which quantitatively characterizes:

Resulted from the i model concept of the mechanism of heterogeneous catalysis and tribocatalysis, shown above, is confirmed partly by tribochemistry and mechanochemistry. This mechanism should be directly confirmed, particularly by materials engineering, which should explain:

Most recent work [19] emphasizes that while the detailed mechanisms by which different mechanochemical phenomena arise are not always well understood, mechanical forces are capable of effecting novel reactivity. Additionally, it strengthens that using force, one can effectively shepherd a chemical reaction down specific reaction pathways, for instance *by selectively lowering the energy of a transition state.* At this point it is of note, that the field of polymer mechanochemistry, has also the potential to change this paradigm by revolutionizing the way chemists think about controlling chemical reactions [19].

In recent years the mechanochemistry field approach has found a renaissance, and different techniques have been applied to activate chemical reaction [20-22] and thereby to lower their activation energy.