*2.1. Study Design and Samples*

This study was approved by the ethical committee of the Ecole Nationale Vétérinaire d'Alfort, Paris, France, (reference number: 02343.03) and it is reported according to the ARRIVE (Animal Research Reporting of in Vivo Experiments) guidelines [27]. Ten female Finnish Dorset crossbred sheep (average; years old and 54 kg) were used and housed together for one week before surgery.  The present implant surfaces, investigated by a broad variety of preclinical and clinical research [28–30], presents the following surface characteristics according to Wennerberg and Albrektsson with the parameters Sa (the arithmetic

average height deviation from mean plane) and Sdr (the developed surface ratio): 1.1 μm and 0.9 μm (Sa); 37% and 34% (Sdr) for TiUnite® and turned surface, respectively [31].

### *2.2. Surgical Procedures*

All surgeries were done under general anesthesia with ketamine (Imalgene 1000®, Merial, Villeurbanne, France), and diazepam (Valium, Roche, Boulogne-Billancourt, France) for injection, and with 2.5% isoflurane (Forane®/Forene®, Drägerverk AG, Lubeck, Germany) for inhalation. The surgical room had a controlled temperature set at 19 ◦C.

In each sheep, the metatarsal bone of one leg was shaved and disinfected with 40% ethanol and 0.5% chlorhexidine. After skin incision, the bone was exposed with periosteal elevator, and 4 implants were installed in the ventral metatarsal bone plate, consisting of approximately 4 mm of cortical bone, along the longitudinal direction.

Based on the drilling protocol and implant surface topography, four experimental groups were designed as showed in Table 1.

**Table 1.** Description of the experimental groups.

Each metatarsal bone received one implant per group. The sequence of the implantation for each metatarsal bone, according to the experimental group (A, B, C, D) was randomized by a computer-generated method, Microsoft® excel (Version 15.30). The sequential implant osteotomies were free-hand prepared at approximately 20 mm inter-implant distance.

The surgical instrumentation sequence was performed following two different drilling protocols. In all groups, a 2.0 mm drill, 2.8 mm step drill, and 3.0 mm drill were used. Hereafter, the osteotomy was finalized based on the group (Figure 1a):

All drilling procedures were performed at 1200 rpm under abundant saline irrigation using a total of two new drill sets. Implant installation was carried out at 25 rpm without saline irrigation. Both the drilling procedure and implant installation were performed using a SA-310 W&H Elcomed implant unit (W&H, Burmoos, Austria). The insertion torque value (ITV) was recorded using the specific function and saved to a USB memory. Implants where ITV exceeded 80 Ncm during installation were manually installed by a manual torque wrench. Based on the ITV, three classes were distinguished: ITV ≤ 45 Ncm; 45 < ITV < 80 Ncm; ITV ≥ 80 Ncm. The flap was closed using a resorbable suture (Vicryl™, Ethicon®, Sommerville, NJ, USA) for the inner layer and non-absorbable suture for the external layer (Ethicon™, Ethicon®, Sommerville, NJ, USA). Morphine (0.1 mg/kg) was given intravenously every second hour during the operation and subcutaneously for the three first postoperative days.

**Figure 1.** (**a**) Representation of the osteotomy preparation: the undersized drilling protocol on the left and non-undersized drilling protocol on the right. The osteotomy dimensions are depicted in the figures. The thermocouple site was prepared 2 mm deep into the cortical bone and 1 mm from the implant surface. (**b**) Overview of the surgical field with the thermocouple in position before implant installation (**left**) and during implant installation (**right**). (**c**) Representation of the regions of interest for histomorphometrical parameters. Bone-to-implant Contact (BIC) was calculated only for the cortical portion. Bone Area Fraction Occupancy (BAFO) was calculated in the blue part. Bone Area 1.5 (BA1.5) was calculated in the yellow part. (**d**) The Finite Element Model (FEM) was composed of three cylindrical elements: Cortical bone, bone marrow, and implant.

#### *2.3. Temperature Measurement*

Intraosseous temperature was measured using the type-K thermocouple (Omega Engineering Limited, Manchester, UK) with a 0.5 mm tip, coupled to a HH12C handheld thermometer (Omega Engineering Limited, Manchester, UK). This dual-channel meter device is able to record the lower and maximum temperature with 2.5 measurements per second and a resolution of 0.1 ◦C. The tip of the thermocouple was inserted in a prepared drilled hole in the proximity of the implant osteotomy. More in details, the thermocouple site was prepared in a predetermined position in a proximal or distal site at 1 mm from the implant surface (Figure 1a) with the aid of a metal surgical template. To prepare the site, which had a diameter of 0.5 mm and a depth of 2 mm, a lance drill (Precision Drill, Nobel Biocare AB, Göteborg, Sweden) was used. To ensure the reproducibility of the hole depth, the drill was inserted until the marked line met the upper border of the template. Once the osteotomy was prepared, the thermocouple was inserted and kept the position until the thermometer showed a stable value (basal temperature). The installation of the implants was carried out approximately 1 min after the finalization of the drilling procedure. During the implant installation, the temperature was measured, and the maximum value was recorded as the maximum value (Figure 1b). The difference between the maximum and basal temperature for each site was calculated as the temperature change.

#### *2.4. Preparation of Samples*

After a healing period of five weeks, all animals were euthanized with an intravenous injection of a combination of 4000 mg embutramide, 538.4 mg mebezonium, and 87.8 mg tetracaine (T61, Intervet International, Unterschleißheim, Germany) and metatarsal bone blocks containing the implants were retrieved. Blocks were fixed in 4% formalin for seven days before dehydration by ascending concentration of ethanol, and embedded in light-curing methylmethacrylate (Technovit 7200 VLC, Heraeus Kulzer, Wehrheim, Germany) for undecalcified ground section procedures. The embedded bone samples with implant were sectioned parallel to the long axis at the center position of the implant using a diamond saw cutting machine (EXAKT 300, EXAKT Advanced Technologies GmbH, Norderstedt, Germany), and grinded and polished until a final section thickness of 30 μm (EXAKT 400CS, EXAKT Advanced Technologies GmbH, Norderstedt, Germany). The non-decalcified sections were stained in toluidine blue and pyronin G, and photographed using light microscopy with a digital imaging system (NanoZoomer S210, HAMAMATSU, Shizuoka, Japan).

#### *2.5. Histomorphometric Analysis*

Quantitative histomorphometry was performed considering the cortical layer only. All measurements were carried out with an image analysis software (Image J v.1.43u, National Institute of Health). The following variables were calculated:

#### *2.6. Finite Element Model*

In order to have a deeper understanding of the thermal behavior at the peri-implant bone, a Finite Element Model (FEM) was generated (Figure 1d). The model was composed of three cylindrical elements: Cortical bone (dimensions: 5 mm × 4 mm), bone marrow (dimensions: 5 mm × 6 mm), and implant (dimensions: 3.75 mm × 7 mm) designed by the CAD software (Solidworks Simulation 2011, Dassault Systèmes Solidworks, Waltham, MA, USA). Two different calorific values were set to the upper and bottom surface of the fixture related to the drilling hole with different diameters.

Steady conduction analysis was conducted by voxel-based Finite Element Analysis (FEA) software (VOXELCON2015, Quint, Fuchu, Japan). The thermal conductivity of cortical bone, bone marrow, and implant were set to 0.6 W/mK, 0.3 W/mK, and 20 W/mK, respectively [32,33]. The air temperature around those models was set to 19 ◦C.

The temperature change in the location of the thermocouple (2 mm depth from cortical bone surface and 1 mm away from the fixture surface) was approximated to 8 ◦C and 4 ◦C for the undersized drilling protocol groups and non-undersized drilling protocol groups, respectively. Until converging the calorific value, the steady conduction analysis was repeatedly conducted.

#### *2.7. Statistical Methods*

Categorical variables were reported as relative frequency, while continuous variables were reported as mean ± standard deviation after checking the normality of the distribution. Wilcoxon rank-sum test was used to test the influence of the drilling protocol and implant surface on temperature change.

Multiple regression models were used to evaluate the effect of the drilling protocol, implant surface, temperature change on the histomorphometric parameters (BIC, BAFO, BA1.5). *p*-values < 0.05 were considered statistically significant. The R statistical software package was used for the statistical evaluation and modelling (available at www.r-project.org/).

### **3. Results**

#### *3.1. ITV, General Healing and Temperature*

All animals survived during surgical procedures and the experimental period. No implant was lost during the healing period. Signs of minor periosteal reaction were noted in three metatarsal bone samples. Such a response did not undermine the peri-implant bone healing and was limited to the periosteal area, which was not considered in this study. All implants were included in the statistical analysis. The relative distribution of the ITV class among the groups is displayed in Figure 2a.

**Figure 2.** (**a**) Cumulative percentage of insertion torque value (ITV) classes divided per group. Note that all implants installed for group A had a ITV ≥ 80 Ncm. (**b**) Temperature change represented in box-plots.

The values for basal and maximum temperature are indicated in Table 2. Temperature change values are shown in Figure 2b. The temperature change was affected by the drilling protocol (*p* < 0.001), while it was not influenced by the surface topography (*p* = 0.879).

**Table 2.** Basal temperature and maximum temperature for the different groups are shown in ◦C. SD: Standard deviation; Max: maximum value; Min: minimum value.


**2. Space reactor research and development programs**

### **2.1 Rover/NERVA**

Nuclear thermal propulsion systems were studied extensively during the 1950s and 1960s, but they were considered to be too heavy and expensive for deployment. At the time, chemical rockets and solar power were more economical for near-Earth operations that were the focus of the world's space agencies. However, recent interest in deep space exploration, and especially interest in sending astronauts to Mars, has reinvigorated NTP research for several reasons. First, the longer thrust duration than chemical rockets that can be produced with an NTP system could cut the travel time to Mars by 20–25%. The reduced travel time is important because it would reduce the amount of dose that astronauts would receive from cosmic radiation during the voyage. Second, the higher thrust would extend the available launch window for missions to Mars. Conventional chemical rocket engines can only be used to reach Mars during a 30-day window that opens every 26 months due to the relative positions of the planets, while nuclear propulsion systems can provide enough thrust to leave Earth's orbit and reach Mars during more points during the orbital profiles of the two planets. Finally, NTP systems could extend the amount of time during which a mission to Mars could be aborted and still allow astronauts to safely return to Earth. Chemical rocket fuel needed for the return trip from Mars to Earth would likely need to be sent to Mars in advance of a crewed mission, so astronauts would have to reach Mars in order to

## *Nuclear Thermal Propulsion Reactor Materials DOI: http://dx.doi.org/10.5772/intechopen.91016*

return to Earth after their initial supply of fuel is consumed, if chemical rockets were used for the mission. An NTP system, on the other hand, would be able to turn around and return to Earth before reaching Mars, if the mission had to be aborted.

Several successful research and development programs focused on space reactor fission power technologies have been established over the past 60 years. The earliest, and most extensive of these efforts, were the Rover and Nuclear Engine for Rocket Vehicle Application (NERVA) programs that were sponsored by the US Atomic Energy Commission (AEC) between 1958 and 1971 [5]. A total of 13 research reactors and 6 nuclear engines were built and tested under the Rover/ NERVA programs at the AEC's Nevada Test Site (NTS) Nuclear Rocket Development Station (NRDS) and other facilities located across the country [6]. The Rover reactor development and testing efforts were led by the Los Alamos Scientific Laboratory (LASL), and the NERVA reactors were designed and built by Westinghouse Electric Corporation (Astronuclear) and Aerojet-General Corporation following a 1961 design competition. The Kiwi (1955–1964), Phoebus (1964–1969), and Peewee (1969–1972) series of reactors were developed and tested under Rover to demonstrate the basics of nuclear rocket technology and to study characteristics of high-temperature nuclear fuels and long-life fuel elements. The NERVA NRX and XE engines were also built between 1964 and 1969 and tested at NRDS to study the complexities of nuclear engine start-up, full-power operation, and shutdown. A list of the best performance parameters achieved during the Rover/NERVA programs is presented in **Table 1** [7].

Early work in the Rover/NERVA program was performed with uranium carbide (UC or UC2) and uranium dioxide (UO2) fuel particles embedded into graphite. The effects of fission product interaction with the graphite matrix quickly led to the use of pyrolytic graphite-coated UO2 particles in the Rover/NERVA fuel designs. The pyrolytic carbon contained fission products produced during reactor operations before the fission products could cause dislocations in the graphite structural materials. The pyrolytic carbon layer also protected the UO2 particles from oxidation during fabrication and handling of the fuel.

A major drawback associated with using graphite in space reactors is that graphite converts to methane (CH4) under hydrogen exposure, and the conversion causes the graphite to corrode. During the Rover/NERVA programs, the graphite matrix used in the reactor fuel was protected from corrosion through application of a high-temperature niobium carbide (NbC) coating. However, the use of the NbC coating leads to issues associated with "mid-band corrosion," which was higher corrosion rates in the center third of the reactor fuel elements where power density

#### **Table 1.**

*Maximum performance results achieved during the Rover/NERVA programs.*

was high. The corrosion was found to be caused by cracking of the NbC coatings used in the fuel's hydrogen coolant channels due to a mismatch between the thermal expansion coefficient of the NbC coating and the thermal expansion coefficient of the fuel matrix. The NbC coatings were eventually replaced with zirconium carbide (ZrC) coatings because of zirconium carbide's superior resistance to fission product diffusion at high temperatures.

The search for solutions to the corrosion problem also led to the development of {(U, Zr) C, graphite} composite fuel. The coefficient of thermal expansion for the composite fuel was 6–6.5 μm/mK (versus 3 μm/mK for fuel made from UO2 dispersed in graphite), which matched the NbC coefficient of 7.1 μm/mK fairly well [7].

Carbide material systems have several favorable features applicable to NTP reactors including:

However, the complex fuel designs used in the early Rover/NERVA tests were challenged by the mechanical loads, thermal stresses, and high radiation fields found in NTP reactors. The intense operational conditions contributed to the formation of stress fractures in the fuel coatings and surfaces and the cracking encouraged increased hydrogen penetration into the fuel that produced fuel degradation.

The Small Nuclear Rocket Engine (SNRE) was the last engine design studied by Los Alamos National Laboratory (LANL) under the Rover/NERVA program. The SNRE was a nominal 16,000 lbf thrust engine that was originally intended for short run-time, unmanned missions, and the SNRE stage design was constrained to fit within the payload volume of the planned space shuttle. The reactor's core design used hexagonal fuel elements and hexagonal structural support elements (i.e., tie tubes), and the number of elements could be varied to support different thrust requirements. Higher thrust designs for SNRE meet or exceed performance characteristics identified in the DRA 5.0 study, so SNRE would be suitable for use in human missions to Mars.


**3. Cavity nucleation**

#### **3.1. Thermodynamic considerations**

The mechanisms for cavitation nucleation have been a puzzle for a long time as explained in the introduction. However part of the explanation came from studies on copper. Pure copper can show extensive cavitation during creep [39], but the number of particles present is so low that they cannot explain the large number of cavities. Lim suggested that it was the substructure of the dislocations that could nucleate the cavities [40]. He also presented a model that can be used to demonstrate whether a nucleation mechanism is thermodynamically feasible or not. He assumed that pile ups of grain boundary dislocations generate the necessary high stresses for the nucleation. Since these high stresses are stationary as a result of the creep process, it avoids the problem of fast stress relaxation in many models. Lim's model is fairly complex and details in the model will not be given here. When a cavity is formed the free energy Δ*G* is changed in a number of ways that are represented by the terms in the following equation [19]

$$
\Delta G = -r^3 F\_{\text{v}} \sigma\_{\text{approx}} + r^2 F\_{\text{s}} \gamma\_{\text{s}} - r^2 F\_{\text{GB}} \gamma\_{\text{GB}} - \{\Delta G\_1 + \Delta G\_2 + \Delta G\_3\}.\tag{7}
$$

γs and γGB are the surface and grain boundary energies per unit area. *F*v = 2π/3 (2–3cos α + cos3 α), *F*s = 4π(1−cos α), *F*b = πsin2 α and *F*v ′ = 1.5 *F*v , where α is half the tip angle of the cavity. The first term in Eq. (7) is the work done by the applied stress. The second and third terms represent the modification in the surface and grain boundary energies. The fourth term is the decrease in the strain energy. Δ*G*1 is the change in the line energy of the grain boundary dislocations (GBD). Δ*G*2 is the interaction energy between the remaining and the consumed GBD. The strain energy Δ*G*3 is the reduction of the strain energy of GBDs outside the cavity. Full details can be found in [19, 40].

Lim's model has been applied to copper and austenitic stainless. As long as energy is gained when a cavity is formed, that is, Δ*G* in Eq. (7) is negative, cavitation is possible. From Eq. (7) Δ*G* is reduced when the applied stress σappl is raised, that is, cavitation becomes more likely which is natural. On the other hand when σappl is reduced cavitation is more difficult. There is minimum stress where cavitation is no longer possible because Δ*G* becomes positive. This minimum stress is plotted as a function of temperature for copper in **Figure 5**.

These minimum stresses are compared with design stresses during creep in copper. It is clear that the stresses required for nucleation are well below the stresses that typically appear in the material. This demonstrates that nucleation based on the substructure is a viable process.

From a technical point of view it is well established that the creep ductility of oxygen free pure copper Cu-OF can be very much lower than for phosphorus alloyed copper Cu-OFP. As a consequence the latter material should be used in creep exposed components [28, 41]. It is evident from **Figure 5** that much lower stresses are needed in Cu-OF than in Cu-OFP, which makes the cavitation in the former material much more abundant. This is believed to be the main reason for the low creep ductility of Cu-OF.

**Figure 5.** Minimum stress to form cavities at cell boundaries versus temperature for oxygen free pure Cu-OF and phosphorus alloyed copper Cu-OFP. For comparison the stress that gives creep rupture after one year (10,000 h) is included. After Ref. [19].

It has also been verified that the minimum nucleation stresses are below typical design stresses for the common stainless steels 304H (18Cr10Ni), 316 (17Cr12Ni2Mo), 321 (18Cr12NiTi) and 347 (18Cr12NiNb). For example this is illustrated for 347 (18Cr12NiNb) in **Figure 6**. The design stresses are 10,000 h rupture data. The minimum cavitation stress lies in the interval 35–50 MPa in the interval from 500°C to 750°C. The minimum cavitation stresses are again below the design stresses. The temperature dependence of Lim's model is probably not fully correct. In general it is thought that the amount of cavitation will increase with temperature. However the temperature dependence of the minimum cavitation stress is weaker than that of the design stress, which suggests the opposite behaviour.

#### **3.2. Strain dependence**

**3. Cavity nucleation**

26 Study of Grain Boundary Character

equation [19]

γs

cos3

**3.1. Thermodynamic considerations**

α), *F*s = 4π(1−cos α), *F*b = πsin2

the decrease in the strain energy. Δ*G*1

Full details can be found in [19, 40].

main reason for the low creep ductility of Cu-OF.

dislocations (GBD). Δ*G*2

viable process.

GBD. The strain energy Δ*G*3

The mechanisms for cavitation nucleation have been a puzzle for a long time as explained in the introduction. However part of the explanation came from studies on copper. Pure copper can show extensive cavitation during creep [39], but the number of particles present is so low that they cannot explain the large number of cavities. Lim suggested that it was the substructure of the dislocations that could nucleate the cavities [40]. He also presented a model that can be used to demonstrate whether a nucleation mechanism is thermodynamically feasible or not. He assumed that pile ups of grain boundary dislocations generate the necessary high stresses for the nucleation. Since these high stresses are stationary as a result of the creep process, it avoids the problem of fast stress relaxation in many models. Lim's model is fairly complex and details in the model will not be given here. When a cavity is formed the free energy Δ*G* is changed in a number of ways that are represented by the terms in the following

*ΔG* = −*r* 3 *F*v σappl + *r* 2 *F*s γs − *r* 2 FGB γGB − (*Δ G*1 + *Δ G*2 + *Δ G*3). (7)

α and *F*v

minimum stress is plotted as a function of temperature for copper in **Figure 5**.

and γGB are the surface and grain boundary energies per unit area. *F*v = 2π/3 (2–3cos α +

, where α is half the tip angle of the cavity.

is the change in the line energy of the grain boundary

is the interaction energy between the remaining and the consumed

is the reduction of the strain energy of GBDs outside the cavity.

′ = 1.5 *F*v

The first term in Eq. (7) is the work done by the applied stress. The second and third terms represent the modification in the surface and grain boundary energies. The fourth term is

Lim's model has been applied to copper and austenitic stainless. As long as energy is gained when a cavity is formed, that is, Δ*G* in Eq. (7) is negative, cavitation is possible. From Eq. (7) Δ*G* is reduced when the applied stress σappl is raised, that is, cavitation becomes more likely which is natural. On the other hand when σappl is reduced cavitation is more difficult. There is minimum stress where cavitation is no longer possible because Δ*G* becomes positive. This

These minimum stresses are compared with design stresses during creep in copper. It is clear that the stresses required for nucleation are well below the stresses that typically appear in the material. This demonstrates that nucleation based on the substructure is a

From a technical point of view it is well established that the creep ductility of oxygen free pure copper Cu-OF can be very much lower than for phosphorus alloyed copper Cu-OFP. As a consequence the latter material should be used in creep exposed components [28, 41]. It is evident from **Figure 5** that much lower stresses are needed in Cu-OF than in Cu-OFP, which makes the cavitation in the former material much more abundant. This is believed to be the Experimentally it has been found many times that the number of cavities is proportional to the creep strain, cf. Eq. (2). To explain this strain dependence, Sandstrom and Wu introduced the *double ledge model* [43]. They considered a sliding grain boundary with dislocation substructures on both sides of the boundary that moved along with the grains. The substructures consist of subgrains that contain fairly few dislocations in their interior but with well-developed subgrain walls. The positions where the subgrain walls meet at the grain boundary are referred to as subgrain corners. Nucleation was assumed to take place when a subboundary on one side of the boundary hits a subgrain corner on the other side. The nucleation rate can be expressed as

$$\frac{dn}{dt} = \frac{u\_{\rm cas}}{d\_{\rm sub}} \frac{1}{d\_{\rm sub}^2} \,' \tag{8}$$

**Figure 6.** Minimum cavitation stress versus temperature for TP347H austenitic stainless steel. 10,000 h rupture data from ECCC [42] for TP347 are shown for comparison.

where *d*sub is the subgrain diameter. The last factor takes into account that one nucleus can be formed in each subgrain on the boundary. The subgrain size is directly related to the applied stress [19]

$$d\_{\rm sub} = K\_{\rm sub} \, \text{GB/} \sigma\_{\rm app}. \tag{9}$$

The constant *K*sub is about 20 for austenitic stainless steels and about 11 for copper. The same model can be applied to particles in the grain boundary that are known to contribute to the nucleation. In the model the subgrain corners are replaced by the particles in the grain boundaries with an interparticle distance of λ. Taking both subgrain corners and particles into account, the resulting expression for the nucleation rate is [44]

$$\frac{dn}{dt} = \frac{0.9 \text{ C}\_s}{d\_{sub}} (\frac{1}{d\_{sub}^2} + \frac{1}{\lambda^2}) \text{ } \dot{\varepsilon} = B\dot{\varepsilon}. \tag{10}$$

In Eq. (10), Eq. (1) has been used. The factor 0.9 in Eq. (10) takes into account the averaging of different orientations [44].

The model in Eq. (10) is compared with experimental data for austenitic stainless steels in **Figure 7**. For three of the experimental data sets TP347 at 550°C and 650°C and TP304 at 727°C, the model gives quite an acceptable description. For TP304XX at 750°C the deviation between model and experiment is larger.

**Figure 7.** Modelling and experimental number of cavities per unit grain boundary area as a function of creep strain [44]. Experimental data from Hong and Nam [45] for TP304 steel, Laha et al. [46] for three different types of austenitic stainless steels and Needham and Gladman [27] for TP347 steel.

#### **3.3. Particle size**

where *d*sub is the subgrain diameter. The last factor takes into account that one nucleus can be formed in each subgrain on the boundary. The subgrain size is directly related to the applied

**Figure 6.** Minimum cavitation stress versus temperature for TP347H austenitic stainless steel. 10,000 h rupture data from

*d*sub = *K*sub GB/*σ*appl. (9)

The constant *K*sub is about 20 for austenitic stainless steels and about 11 for copper. The same model can be applied to particles in the grain boundary that are known to contribute to the nucleation. In the model the subgrain corners are replaced by the particles in the grain boundaries with an interparticle distance of λ. Taking both subgrain corners and particles into

In Eq. (10), Eq. (1) has been used. The factor 0.9 in Eq. (10) takes into account the averaging of

The model in Eq. (10) is compared with experimental data for austenitic stainless steels in **Figure 7**. For three of the experimental data sets TP347 at 550°C and 650°C and TP304 at 727°C, the model gives quite an acceptable description. For TP304XX at 750°C the deviation

*λ*2 ) *ε*˙ = *Bε*˙. (10)

account, the resulting expression for the nucleation rate is [44]

*dt* = 0.9 *C* \_\_\_\_\_*s d*sub ( \_\_\_1 *d*sub 2 + \_\_1

\_\_\_ *dn*

ECCC [42] for TP347 are shown for comparison.

28 Study of Grain Boundary Character

between model and experiment is larger.

different orientations [44].

stress [19]

It has been proposed that a critical particle radius exists for nucleation [45, 46]. The radius must exceed a minimum value in order for nucleation to take place. Harris developed a model that related the critical particle size to the GBS velocity [47, 48]. His basic assumption was that particles are not able to stop GBS if the diffusion is fast enough. According to Harris this critical particle radius also represented the minimum radius that could nucleate cavities. Harris gave the following relation between the GBS velocity *u* . GBS and the critical particle radius *r*c

$$\dot{\mu}\_{\rm cons} = \frac{\delta \, D\_{\rm cs}}{r\_e^{\prime} \ln \frac{\lambda}{2r\_e}} \left( \exp \frac{2}{k\_b} \frac{\gamma\_\* \, \Omega}{T r\_e} - 1 \right) , \tag{11}$$

**Figure 8.** Comparison of experimental and modelling GBS velocity as a function of particle radius [24]. Experimental data for 304, 321 and 347 austenitic stainless steels from Refs. [45, 46, 49, 50]. Modelling results from Eq. (11).

where λ is the interpaticle spacing, γs the surface energy, δ the grain boundary width, *D*GB the grain boundary self-diffusion coefficient, Ω the atomic volume, *k*B Boltzmann's constant and *T* the absolute temperature. The application of Eq. (11) is illustrated in **Figure 8** for different types of austenitic stainless steels [24].

In **Figure 8** the particle parameters are taken from the experimental references. The minimum particles that nucleated cavities in the experiments are chosen for the critical particle radius. The experimental data clearly support Harris' model.

From the particle size distributions [24] the number of nucleated cavities can be estimated if the critical particle size is known. The computed number of nuclei is compared with the observed ones for austenitic stainless steels in **Figure 9.**


*Edited by Maria Laura Garcia*

*and Victor Romanowski*

Viruses are small infectious agents that can replicate only inside the living cells of susceptible organisms. The understanding of the molecular events underlying the infectious process has been of central interest to improve strategies aimed at combating viral diseases of medical, veterinary and agricultural importance. Some of the viruses cause dreadful diseases, while others are also of interest as tools for gene transduction and expression and in non-poluting insect pest management strategies. The contributions in this book provide the reader with a perspective on the wide spectrum of virus-host systems. They are organized in sections based on the major topics covered: viral genomes organization, regulation of replication and gene expression, genome diversity and evolution, virus-host interactions, including clinically relevant features. The chapters also cover a wide range of technical approaches, including high throughput methods to assess genome variation or stability. This book should appeal to all those interested in fundamental and applied aspects of virology.

ISBN 978-953-51-0098-0

Viral Genomes - Molecular Structure, Diversity, Gene Expression Mechanisms

and Host-Virus Interactions

Photo by Gio\_tto / iStock


*Article* **A Relation-Theoretic Matkowski-Type Theorem in Symmetric Spaces**

**Based Ali 1, Mohammad Imdad 1 and Salvatore Sessa 2,\***

**Abstract:** In this paper, we present a fixed-point theorem in R-complete regular symmetric spaces endowed with a locally *T*-transitive binary relation R using comparison functions that generalizes several relevant existing results. In addition, we adopt an example to substantiate the genuineness of our newly proved result. Finally, as an application of our main result, we establish the existence and uniqueness of a solution of a periodic boundary value problem.

**Keywords:** fixed points; symmetric spaces; binary relations; *T*-transitivity; regular spaces

**MSC:** 47H10; 54H25

## **1. Introduction**

In 1922, one of the most pivotal results in analysis was proved by Banach [1] in his doctoral thesis, which asserts that every contraction mapping on a complete metric space admits a unique fixed point. This principle continues to inspire generations of researchers in metric fixed-point theory. Thus far, this classical result has been generalized and improved in various ways, and by now, there exists an extensive literature on and around this premier result. Over the last several decades, there have been many interesting generalizations of this classical result in various directions.

There exist several extensions of the Banach contraction principle to various spaces obtained by lightening the underlying involved metric conditions.  In 1976, Cicchese [2] established the first ever fixed-point theorem in the framework of symmetric spaces. The idea of such spaces was coined by Wilson [3] by relaxing the triangle inequality from metric conditions. By now, there exists a considerable literature on fixed-point theory in symmetric spaces. For work of this kind, one can be referred to [4–11].

On the other hand, there have been various generalizations that were obtained by varying the class of contractions (e.g., see [12–14]). In 2004, Ran and Reurings [15] obtained a very useful generalization of the Banach fixed-point theorem in a partially ordered metric space by taking a relatively weaker contraction condition that is required to hold only on those elements that were comparable in the underlying ordering. In doing so, they were essentially motivated by Turinici [16]. This result was further generalized by Nieto and Rodríguez-López in [17,18] in 2005 and 2007, respectively. Subsequently, in 2015, Alam and Imdad [19] furnished a natural extension of the Banach contraction principle in a complete metric space endowed with a binary relation that generalizes all of the above-mentioned results [15,17,18].

The existing literature contains several results on nonlinear contractions, which were initiated by Browder [13] and were followed by similar works by Boyd and Wong [14]

**Citation:** Ali, B.; Imdad, M.; Sessa, S. A Relation-Theoretic Matkowski-Type Theorem in Symmetric Spaces. *Axioms* **2021**, *10*, 50. https://doi.org/10.3390/ axioms10020050

Academic Editor: Wei-Shih Du

Received: 1 March 2021 Accepted: 25 March 2021 Published: 2 April 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 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/).

1 Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India<EMAIL_ADDRESS>(B.A<EMAIL_ADDRESS>(M.I.)

and Matkowski [20]. In 2014, Bessenyei and Páles [21] extended Matkowski's result in symmetric spaces, which required an additional regularity condition.

The intent of this paper is to prove a relation-theoretic version of a theorem due to Bessenyei and Páles [21]. In doing so, we are essentially motivated by [15,17–19].

#### **2. Preliminaries**

In this section, we recall some definitions, propositions, and lemmas that will be utilized in our subsequent discussions. The following are taken from Wilson's paper [3] on symmetric space. Throughout the paper, R, R+, N, N0, and Q denote the sets of reals, nonnegative reals, natural numbers, whole numbers, and rational numbers respectively.

**Definition 1.** *Let X be a nonempty set and let d* : *X* × *X* → R+ *be a mapping satisfying the following axioms: for each a*, *b* ∈ *X,*

*Then, d is called symmetric on X and the pair* (*X*, *d*) *is a symmetric space.*

In such spaces, the notions of convergent and Cauchy sequences are as usual.

The space is said to be complete if every Cauchy sequence converges. For an open ball centered at *p* with radius *r*, the notation *B*(*p*,*r*) is used. The diameter of *B*(*p*,*r*) is the supremum of distances taken over the pairs of points of the ball. The topology of such spaces is the topology induced by the open balls.

Because of the unavailability of the triangle inequality, the following problems are obvious:

**Definition 2.** *Consider a symmetric space* (*X*, *d*)*. A function ψ* : R2 + → R+ *is a triangle function [21] for d if the following hold:*

*(i) ψ*(*u*, *v*) = *ψ*(*v*, *u*) ∀*u*, *v* ∈ R+*;*

*(ii) ψ is monotone increasing in both of its arguments;*

It has been shown in [21] that every symmetric space (*X*, *d*) admits a unique triangle function Φ*d*, which has the property that if *ψ* is any other triangle function for *d*, then Φ*d* ≤ *ψ*. Such a triangle function Φ*d* is called the basic triangle function.

**Definition 3.** *A symmetric space* (*X*, *d*) *is called a regular space if the basic triangle function with respect to the symmetric d is continuous at* (0, 0)*.*

Throughout this paper, we shall restrict our attention to regular spaces only. The utility of such spaces is enlightened by the next important result.

**Lemma 1** ([21])**.** *The topology of a regular symmetric space is Hausdorff. A convergent sequence in a regular symmetric space has a unique limit and it has the Cauchy property. Moreover, a symmetric space* (*X*, *d*) *is regular if and only if*

lim *r*→0 sup *p*∈*X* diam *B*(*p*,*r*) = 0.

**Definition 4.** *A monotone increasing function ϕ* : R+ → R+ *is called a comparison function if* lim*n*→∞ *ϕn*(*t*) = 0 *for each t* ∈ R+*. A mapping T* : *X* → *X is called a ϕ-contraction if*

> *d T*(*x*), *T*(*y*) ≤ *ϕ d*(*x*, *y*) *for each x*, *y* ∈ *X*.

These concepts are due to Matkowski [20]. The following is the main result of [21].

**Theorem 1.** *If* (*X*, *d*) *is a complete regular symmetric space and ϕ is a comparison function, then every ϕ-contraction on X has a unique fixed point.*

#### **3. Relation-Theoretic Notions and Related Results**

**Definition 5** ([22])**.** *Let X be a nonempty set. A subset* R *of X* × *X is called a binary relation on X. For x*, *y* ∈ *X when* (*x*, *y*) ∈ R*, we say that x is related to y, or in other words, x relates to y under* R*. Sometimes, we write x*R*y instead of* (*x*, *y*) ∈ R*. If* (*x*, *y*) ∈ R/ *, we say x is not related to y.*

**Definition 6.** *Let* R *be a binary relation on a nonempty set X and x*, *y* ∈ *X. We say that x and y are* R*-comparative if either* (*x*, *y*) ∈ R *or* (*y*, *x*) ∈ R*. When x and y are* R*-comparative, we write it as* [*x*, *y*] ∈ R*.*

**Proposition 1** ([19])**.** *If* (*X*, *d*) *is a symmetric space,* R *is a binary relation on X, T is a selfmapping on X, and ϕ* : R+ → R+ *is a comparison function, then the following conditions are equivalent:*

*(i) d*(*Tx*, *Ty*) ≤ *ϕ d*(*x*, *y*) ∀(*x*, *y*) ∈ R*;*

*(ii) d*(*Tx*, *Ty*) ≤ *ϕ d*(*x*, *y*) ∀[*x*, *y*] ∈ R*.*

*The proof is simple and follows from symmetry of d.*

**Definition 7.** *A binary relation* R *defined on a nonempty set X is called*

**Definition 8** ([19])**.** *Let X be a nonempty set endowed with a binary relation* R*. A sequence xn* ⊂ *X is called* R*-preserving if* (*xn*, *xn*+1) ∈R∀*n* ∈ N*.*

**Definition 9** ([19])**.** *Let X be a nonempty set and let T be a self-mapping on X. A binary relation* R *on X is called T-closed if, for any x*, *y* ∈ *X,*

$$(\mathfrak{x}, \mathfrak{y}) \in \mathcal{R} \Rightarrow (T\mathfrak{x}, T\mathfrak{y}) \in \mathcal{R} \Box$$

**Definition 10** ([23])**.** *Let X be a nonempty set and let T be a self-mapping on X.  The restriction of* R *to E, denoted as* R|*E, is the set* R ∩ *E*2*. Indeed,* R|*E is a relation on E induced by* R*.*

**Definition 12** ([23])**.** *A binary relation* R *on a nonempty set X is called locally transitive if, for each* R*-preserving sequence xn* ⊂ *X with range E* = {*xn*}*n*∈N0 *, the binary relation* R|*E is transitive.*

**Definition 13** ([23])**.** *Let X be a nonempty set and let T be a self-mapping on X. A binary relation* R *on X is called locally T-transitive if, for each (effectively)* R*-preserving sequence xn* ⊂ *T*(*X*) *with range E* = {*xn*}*n*∈N0 *, the binary relation* R|*E is transitive.*

**Definition 14** ([24])**.** *Let X be a nonempty set and let* R *be a binary relation on X. For x*, *y* ∈ *X, a path of length k (where k is a natural number) in* R *from x to y is a finite sequence* {*x*0, *x*1, *x*2,..., *xk*} ⊂ *X satisfying the following conditions:*

**Definition 15** ([23])**.** *Let X be a nonempty set and let* R *be a binary relation on X. A subset E of X is called* R*-connected if, for each pair x*, *y* ∈ *X, there exists a path (in* R*) from x to y.*

**Definition 16** ([23])**.** *Let X be a nonempty set and let* R *be a binary relation on X. A subset E of X is called* R*S-connected if, for each pair x*, *y* ∈ *X, there is a finite sequence* {*x*0, *x*1, *x*2, ... , *xk*} ⊂ *X satisfying the following conditions:*

Now, we define the analogue of the notion of *d*-self-closedness in metric space due to [23] in the framework of symmetric spaces.

**Definition 17.** *Let* (*X*, *d*) *be a symmetric space. A binary relation* R *defined on X is called d-self-closed if, for any* R*-preserving sequence xn converging to x, there exists a subsequence xnk of xn with* (*xnk* , *x*) ∈ R*.*

We will use the following notations in this paper:

#### **4. Main Result**

In an attempt to prove a relation-theoretic version of Matkowski's theorem [20] in symmetric spaces, we prove the following.

**Theorem 2.** *Let* (*X*, *d*) *be a regular symmetric space,* R *a binary relation on X, and T a selfmapping on X. Suppose that the following conditions hold:*

$$d(Tx, Ty) \le \wp(d(\mathfrak{x}, y)) \,\,\forall (\mathfrak{x}, y) \in \mathcal{R}.$$

*Then, T has a fixed point.*

*Moreover, if*

*(f) F(T) is* R*S-connected, then T has a unique fixed point.*

**Proof.** As *X*(*T*, R) is nonempty, let *x*0 be such that (*x*0, *Tx*0) ∈ R. If *Tx*0 = *x*0, then we are done. Suppose that *Tx*0 = *x*0. Since (*x*0, *Tx*0) ∈ R and R is *T*-closed, we obtain by induction that

$$(T^{\mathfrak{u}}\mathfrak{x}\_{0\prime}T^{\mathfrak{u}+1}\mathfrak{x}\_{0}) \in \mathscr{R} \,\,\forall \mathfrak{u} \in \mathbb{N}.$$

Construct the sequence *xn* of Picard iterates with initial point *x*0, i.e., *xn* = *Tn*(*x*0). So, (*xn*, *xn*+1) ∈R∀*n* ∈ N0, i.e., the sequence is R-preserving. As R is locally *T*-transitive, we have (*xn*, *xm*) ∈R∀*m* > *n*. Observe that the sequence *d*(*xn*, *xn*+*k*) tends to zero for all fixed *k* ∈ N;

$$\begin{array}{rcl}d(\mathbf{x}\_{n},\mathbf{x}\_{n+k})&=&d(T\mathbf{x}\_{n-1},T\mathbf{x}\_{n+k-1})\\&\leq&\varrho\left(d(\mathbf{x}\_{n-1},\mathbf{x}\_{n+k-1})\right)\\&\leq&\varrho^{2}\left(d(\mathbf{x}\_{n-2},\mathbf{x}\_{n+k-2})\right)\\&\vdots\\&\leq&\varrho^{n}d(\mathbf{x}\_{0},\mathbf{x}\_{k})\rightarrow 0 \text{ as } n\rightarrow\infty.\end{array}$$

Now, we are going to prove that {*xn*} is a Cauchy sequence. Let > 0 be any positive number. As (*X*, *d*) is regular, the basic triangle function Φ*d* is continuous at (0,0). So, there exists a neighborhood *U* of the origin such that Φ*d*(*u*, *v*) < ∀(*u*, *v*) ∈ *U*. In other words, ∃*δ* > 0 such that Φ*d*(*u*, *v*) < ∀*u*, *v* : 0 ≤ *u*, *v* ≤ *δ*. We take *δ* < . As *ϕ* is a comparison function, *ϕn*(*t*) → 0 ∀*t* > 0; so there exists *N* ∈ N such that *ϕN*() < *δ*. Set *S* = *TN*. We can see that

$$d(\mathcal{S} \mathfrak{x}, \mathcal{S}y) = d(T^N \mathfrak{x}, T^N y) \le \mathfrak{q}^N d(\mathfrak{x}, y) \text{ when } (\mathfrak{x}, \mathfrak{y}) \in \mathcal{R}.$$

Define *nk* : *d*(*xn*, *TkSxn*) < *δ* ∀*n* ≥ *nk* and set *M*= max{*n*0, *n*1,..., *nN*}. If *V* = {*xM*, *xM*+1, *xM*+2, ... .*xM*+*k*, ... } then for any *y* ∈ *B*(*xM*, ) ∩ *V*, *y* = *xM*, we

have

$$d(T^k \mathbf{S} \mathbf{x}\_M, T^k \mathbf{S} \mathbf{y}) = d(ST^k \mathbf{x}\_M, ST^k \mathbf{y}) \le \varrho^N d(T^k \mathbf{x}\_M, T^k \mathbf{y}) \text{ as } (T^k \mathbf{x}\_M, T^k \mathbf{y}) \in \mathcal{R}$$

$$\le \varrho^N \varrho^k d(\mathbf{x}\_M, \mathbf{y}) < \varrho^N d(\mathbf{x}\_M, \mathbf{y}) < \varrho^N(\varepsilon) < \delta.$$

So,

$$\begin{array}{rcl}d(T^k S y, \mathbf{x}\_M) & \leq & \Phi\_d(d(T^k S y, T^k S \mathbf{x}\_M), d(T^k S \mathbf{x}\_M, \mathbf{x}\_M)) \\ & \leq & \Phi\_d(\delta, \delta) \; \forall k = 0, 1, 2, \dots, N; \\ \Longrightarrow \; d(T^k S y, \mathbf{x}\_M) & < & \epsilon, \forall k = 0, 1, 2, \dots, N.\end{array}$$

and for *y* = *xM*, *d*(*TkSxM*, *xM*) < *δ* < , ∀*k* = 0, 1, 2, . . . , *N*.

Thus we see that *TkS* maps *V* ∩ *B*(*xM*, ) into itself. In particular, each iteration of *S* maps *V* ∩ *B*(*xM*, ) into itself. Now, if *n* > *M* is any arbitrarily given natural number, i.e., *n* = *Nk* + *p* where *k* ∈ N0 and 0 ≤ *p* < *N*, then

$$T^n \mathcal{S} = T^{\mathbb{N}k + p} \mathcal{S} = T^p \mathcal{S}^{k+1} \mathcal{S}$$

and hence,

$$\begin{aligned} \left(T^{p}S\left(V\cap B(\mathbf{x}\_{M\prime}\varepsilon)\right)\right) &=& T^{p}S^{k+1}\left(V\cap B(\mathbf{x}\_{M\prime}\varepsilon)\right) \\ &=& T^{p}S\left(S^{k}\left(V\cap B(\mathbf{x}\_{M\prime}\varepsilon)\right)\right) \\ &\subset& T^{p}S\left(V\cap B(\mathbf{x}\_{M\prime}\varepsilon)\right) \\ &\subset& V\cap B(\mathbf{x}\_{M\prime}\varepsilon);\text{ as } 0\leq p$$

Therefore, *TnS*(*xM*) ∈ *B*(*xM*, ) ∀*n* > *M*, i.e., *xM*+*N*+*k* ∈ *B*(*xM*, ) ∀*k* ∈ N. As the space is regular, diam(*xM*, ) → 0 when → 0, and from this, we conclude that the sequence {*xn*} is a Cauchy sequence. The completeness of the space (*X*, *d*) gives some element *x* ∈ *X* such that *xn* → *x*.

Now, if *T* is R-continuous, then *T*(*xn*) → *T*(*x*), i.e., *xn*+1 → *T*(*x*). As the space is regular, we conclude that *T*(*x*) = *x*, as the limit is unique in regular spaces. So, the limit of the sequence constructed above is a fixed point.

If R is *d*-self-closed, then there is a subsequence *xnk* of *xn* such that [*xnk* , *x*] ∈ R ∀*k* ∈ N0. So,

$$d(\mathbf{x}, T\mathbf{x}) \le \Phi\_d(d(\mathbf{x}, \mathbf{x}\_{n\_k+1}), d(\mathbf{x}\_{n\_k+1}, T\mathbf{x})) \le \Phi\_d[d(\mathbf{x}\_{n\_k+1}, \mathbf{x}), \boldsymbol{\varrho}(d(\mathbf{x}\_{n\_k}, \mathbf{x}))].$$

Now, for > 0, there exists *δ* > 0 such that Φ*d*(*u*, *v*) < ∀*u*, *v* : 0 ≤ *u*, *v* ≤ *δ*, and for *δ* > 0, there exists *K* ∈ N such that *d*(*xn*, *x*) ≤ *δ* ∀*n* ≥ *K*. Therefore, if we take *nk* ≥ *K*, we have

$$d(\mathfrak{x}, T\mathfrak{x}) \le \Phi\_d(\delta, \mathfrak{q}(\delta)) \le \Phi\_d(\delta, \delta) < \epsilon.$$

So, *T*(*x*) = *x* i.e., *x* is a fixed point.

To show that *T* has a unique fixed point, let *y* be any other fixed point of *T*. Now, *F*(*T*) is R*S*-connected and *x*, *y* ∈ *F*(*T*); so, there is a finite sequence of elements {*z*0, *z*1, *z*2, ... , *zk*} ⊂ *X* satisfying the following conditions:

(i) *z*0 = *x*, *zk* = *y*;

(ii) [*zi*, *zi*+1] ∈ R for each *i* (0 ≤ *i* ≤ *k* − 1).

Now, as *T* is a *ϕ*-contraction on R, *d*(*Tzi*, *Tzi*+1) ≤ *ϕ d*(*zi*, *zi*+1) . Using induction, we get *d*(*Tnzi*, *Tnzi*+1) ≤ *ϕnd*(*zi*, *zi*+1). We already have, for > 0, ∃*δ* > 0 such that Φ*d*(*u*, *v*) < ∀*u*, *v* : 0 ≤ *u*, *v* < *δ*.

Let *δ*1 = *δ*, define *δi* (2 ≤ *i* ≤ *k* − 1) : Φ*d*(*u*, *v*) < *δi*−1 ∀*u*, *v* : 0 ≤ *u*, *v* < *δi*, and set *α*=min{*δ*1, *δ*2,..., *δk*−1}.

In addition, set *M*=max{*N*1, *N*2, ... , *Nk*−1}, where *Ni* : *d*(*Tnzi*, *Tnzi*+1) ≤ *ϕnd*(*zi*, *zi*+1) < *α* ∀*n* ≥ *Ni*. Hence, for *n* ≥ *M*, we have

$$\begin{array}{rcl}d(T^{n}z\_{k-1},T^{n}y) &< \mathsf{a} & \leq \delta\_{k-1} \\ d(T^{n}z\_{k-2},T^{n}y) & \leq & \Phi\_{d}[d(T^{n}z\_{k-2},T^{n}z\_{k-1}),d(T^{n}z\_{k-1},T^{n}y)] \\ & \leq & \Phi\_{d}(d,\delta\_{k-1}) \leq \Phi\_{d}(\delta\_{k-1},\delta\_{k-1}) < \delta\_{k-2} \\ d(T^{n}z\_{k-3},T^{n}y) & \leq & \Phi\_{d}[d(T^{n}z\_{k-3},T^{n}z\_{k-2}),d(T^{n}z\_{k-2},T^{n}y)] \\ & \leq & \Phi\_{d}(a,\delta\_{k-2}) \leq \Phi\_{d}(\delta\_{k-2},\delta\_{k-2}) < \delta\_{k-3} \\ & \vdots \\ d(T^{n}z\_{1},T^{n}y) & \leq & \Phi\_{d}(d(T^{n}z\_{1},T^{n}z\_{2}),d(T^{n}z\_{2},T^{n}y)) \\ & \leq & \Phi\_{d}(a,\delta\_{2}) \leq \Phi\_{d}(\delta\_{2},\delta\_{2}) < \delta\_{1} \\ d(T^{n}x,T^{n}y) & \leq & \Phi\_{d}(d(T^{n}x,T^{n}z\_{1}),d(T^{n}z\_{1},T^{n}y)) \\ & \leq & \Phi\_{d}(a,\delta\_{1}) \leq \Phi\_{d}(\delta\_{1},\delta\_{1}) < \epsilon. \end{array}$$

Therefore, *d*(*Tnx*, *Tny*) = *d*(*x*, *y*) = 0, i.e., *x* = *y*. Hence, the fixed point of *T* is unique.

Now, we consider some special cases, where our result deduces some well-known results from the existing literature.

(4) The result of Neito and Rodríguez-López becomes a corollary of our result because of the same reasons as the earlier one. Notice that the *d*-self-closedness property is a generalization of the ICU (increasing-convergence upper bound) property.

Finally, we produce an illustrative example to substantiate the utility of our result, which does not satisfy the hypotheses of the existing results [1,15,17–19,21,23], but satisfies the hypotheses of our result, and hence has a fixed point.

 Consider the binary relation*

$$\mathcal{R} = \{(x, y) \in \mathbb{R}^2 : x \ge y \ge 0, x \in \mathbb{Q}\}.$$

*We define a mapping T* : *X* → *X as follows:*

*T*(*x*) = 2*x*, *if x* ≤ 0, *x* 3 , *if x* > 0.

*We see that the self-mapping T on X is not a ϕ-contraction on the whole space X for any comparison function ϕ. So, the result of Bessenyei and Páles [21] does not apply here. However, when we consider the elements x*, *y such that* (*x*, *y*) ∈ R*, then T is a ϕ-contraction on* R *for ϕ*(*t*) = *t* 2 *, and all the other hypotheses of our result hold.*

*In addition, we see that the fixed-point results of [1,15,17–19,23] do not apply here, as the space is not a metric space.*

#### **5. Application to Ordinary Differential Equations**

In this section, we study the existence and uniqueness of a first-order periodic boundary value problem as an application of our main fixed-point theorem.

Consider the first-order periodic boundary value problem

$$\begin{aligned} \mathbf{x}'(t) &= \mathbf{g}\left(t, \mathbf{x}(t)\right), t \in [0, \lambda] \\ \mathbf{x}(0) &= \mathbf{x}(\lambda), \end{aligned} \tag{1}$$

where *λ* > 0 and *g* : [0, *λ*] × R → R is a continuous function.

We consider the space *X* = *C*[0, *λ*] of all continuous functions on [0, *λ*] under the symmetric given by

$$d(\mathfrak{x}, y) = \sup\_{t \in [0, \lambda]} \left( \mathfrak{x}(t) - y(t) \right)^2.$$

We define a relation R on *X* as

$$\mathfrak{x}\mathfrak{R}y \iff \mathfrak{x}(t) \le y(t) \,\,\forall t \in [0, \lambda].$$

Now, we give the following definition, which will be useful in the subsequent theorem.

**Definition 18.** *A function z is said to be a lower solution of* (1) *if*

$$\begin{aligned} z'(t) &\le g\left(t, z(t)\right) \text{ for } t \in [0, \lambda],\\ z(0) &\le z(\lambda). \end{aligned}$$

**Theorem 3.** *Consider problem* (1) *with g* : *I* × R → R*, a continuous function, and suppose that there exists some k* > 0 *such that for s*1,*s*2 *in* R *with s*1 ≥ *s*2*,*

$$0 \le \mathfrak{g}(t, \mathfrak{s}\_1) + k\mathfrak{s}\_1 - \left[\mathfrak{g}(t, \mathfrak{s}\_2) + k\mathfrak{s}\_2\right] \le k\sqrt{\mathfrak{g}(\mathfrak{s}\_1 - \mathfrak{s}\_2)^2},$$

*where ϕ is a comparison function. Then, the existence of a lower solution for* (1) *guarantees the existence of a unique solution of* (1)*.*

**Proof.** Problem (1) can be rewritten as

$$\begin{aligned} \mathbf{x}'(t) + k\mathbf{x}(t) &= \mathbf{g}\left(t, \mathbf{x}(t)\right) + k\mathbf{x}(t), t \in [0, \lambda], \\ \mathbf{x}(0) &= \mathbf{x}(\lambda). \end{aligned}$$

This problem is equivalent to the integral equation

$$\mathbf{x}(t) = \int\_0^\lambda \mathbf{G}(\mathbf{t}, \mathbf{s}) [\mathbf{g}(\mathbf{s}, \mathbf{x}(\mathbf{s})) + k\mathbf{x}(\mathbf{s})] d\mathbf{s} \,\mathrm{s}$$

where

$$G(t,s) = \begin{cases} \frac{\epsilon^{k(\lambda+s-t)}}{\epsilon^{k\lambda-1}}, 0 \le s < t \le \lambda, \\\frac{\epsilon^{k(s-t)}}{\epsilon^{k\lambda}-1}, 0 \le t < s \le \lambda. \end{cases}$$

Consider the self-mapping *T* on *X* defined as

$$(T\mathbf{x})(t) = \int\_0^\lambda \mathbf{G}(t, \mathbf{s}) [\mathbf{g}\left(\mathbf{s}, \mathbf{x}(\mathbf{s})\right) + k\mathbf{x}(\mathbf{s})] d\mathbf{s}.$$

Here, it is apparent that a fixed point of *T* is, in fact, a solution of the above problem (1). Now, we will show that the hypotheses in Theorem 2 are satisfied. To prove that the relation R is *T*-closed, take *x*, *y* ∈ *X* such that *x*R*y*, i.e.,

$$\text{The corresponding } \mathcal{C} \text{-algebras are } \mathcal{C} \text{-algebras, and } \mathcal{C} \text{-algebras, respectively.}$$

*x*(*t*) ≤ *y*(*t*) ∀*t* ∈ [0, *λ*].

As *y*(*t*) ≥ *x*(*t*), from the hypothesis, we obtain

$$
\lg\left(t, x(t)\right) + kx(t) \le \lg\left(t, y(t)\right) + ky(t) \; \forall t \in [0, \lambda].
$$

As *G*(*t*,*s*) > 0 ∀*t*,*s* ∈ [0, *λ*], we have

$$\begin{aligned} (Tx)(t) &= \int\_0^\lambda \mathcal{G}(t,s)[\mathcal{g}(s,x(s)) + kx(s)]ds \\ &\leq \int\_0^\lambda \mathcal{G}(t,s)[\mathcal{g}(s,y(s)) + ky(s)]ds \\ &= (Ty)(t). \end{aligned}$$

Hence, R is *T*-closed. In addition, for *x*R*y*, we have

$$\begin{split} \sqrt{d(\mathcal{X}x,Ty)} &= \sup\_{t \in [0,\lambda]} |(\mathcal{X}x)(t) - (\mathcal{Y}y)(t)| \\ &\leq \sup\_{t \in [0,\lambda]} \int\_0^\lambda G(t,s) |g(s,\mathbf{x}(s)) + k\mathbf{x}(s) - g(s,y(s)) - ky(s)| ds \\ &\leq \sup\_{t \in [0,\lambda]} \int\_0^\lambda G(t,s) \sqrt{q(y(s) - \mathbf{x}(s))^2} ds \\ &\leq \sqrt{\gcd(x,y)} \sup\_{t \in [0,\lambda]} \int\_0^\lambda G(t,s) ds \\ &= \sqrt{\gcd(x,y)} \sup\_{t \in [0,\lambda]} \frac{1}{\epsilon^{k\lambda} - 1} \left(\frac{1}{k} \epsilon^{k(\lambda + s - t)}\right)\_0^t + \frac{1}{k} \epsilon^{k(s-t)}\Big|\_t^\lambda \\ &= \sqrt{\gcd(x,y)} \sup\_{t \in [0,\lambda]} \frac{1}{\epsilon^{k\lambda} - 1} (\epsilon^{k\lambda} - 1) \\ &= \sqrt{\gcd(x,y)}. \end{split}$$

Thus, we have

$$d(Tx, Ty) \le \gcd(x, y).$$

Hence, the required contraction condition (2) holds. Now, as there is some lower solution, say *x*0 ∈ *X*, we have

$$\mathbf{x}\_0'(t) \le \mathbf{g}\left(t, \mathbf{x}\_0(t)\right),$$

which can be rewritten as

$$
\lambda x\_0'(t) + kx\_0(t) \le \lg(t, x\_0(t)) + kx\_0(t) \text{ for } t \in [0, \lambda].
$$

Multiplying both the sides by *ekt*, we obtain

$$\left(\mathbf{x}\_{0}(t)e^{kt}\right)' \le \left[\mathbf{g}\left(t, \mathbf{x}\_{0}(t)\right) + k\mathbf{x}\_{0}(t)\right]e^{kt} \text{ for } t \in \left[0, \lambda\right]\_{\ast}.$$

and thus, we get

$$\mathbf{x}\_{0}(t)e^{kt} \le \mathbf{x}\_{0}(0) + \int\_{0}^{t} [\mathbf{g}\left(\mathbf{s}, \mathbf{x}\_{0}(s)\right) + k\mathbf{x}\_{0}(s)]e^{ks}ds \text{ for } t \in [0, \lambda],\tag{2}$$

which implies that

$$\mathbf{x}\_0(0)e^{k\lambda} \le \mathbf{x}\_0(\lambda)e^{k\lambda} \le \mathbf{x}\_0(0) + \int\_0^\lambda [\mathbf{g}\left(\mathbf{s}, \mathbf{x}\_0(s)\right) + k\mathbf{x}\_0(s)]e^{k\mathbf{s}}ds$$

thereby yielding

$$\mathbf{x}\_0(0) \le \int\_0^\lambda \frac{\mathbf{c}^{ks}}{\mathbf{c}^{k\lambda} - 1} [\mathbf{g}\left(\mathbf{s}, \mathbf{x}\_0(s)\right) + \mathbf{x}\_0(s)] ds.$$

Using the above inequality (2), we get

*x*0(*t*)*e kt* ≤ *t* 0 [*g s*, *x*0(*s*) + *x*0(*s*)]*e ksds* + *λ* 0 *eks ekλ* − 1 [*g s*, *x*0(*s*) + *x*0(*s*)]*ds* = *t* 0 [*g s*, *x*0(*s*) + *x*0(*s*)] *ek*(*s*+*λ*) *ekλ* − 1 *ds* + - 0 *t* [*g s*, *x*0(*s*) + *x*0(*s*)] *eks ekλ* − 1 *ds* + *λ* 0 *eks ekλ* − 1 [*g s*, *x*0(*s*) + *x*0(*s*)]*ds* = *t* 0 [*g s*, *x*0(*s*) + *x*0(*s*)] *ek*(*s*+*λ*) *ekλ* − 1 *ds* + *λ t* [*g s*, *x*0(*s*) + *x*0(*s*)] *eks ekλ* − 1 *ds*.

Hence,

$$\mathbf{x}\_{0}(t) \le \int\_{0}^{t} \frac{\mathbf{e}^{k(s+\lambda-t)}}{\mathbf{e}^{k\lambda} - 1} [\mathbf{g}\left(\mathbf{s}, \mathbf{x}\_{0}(s)\right) + \mathbf{x}\_{0}(s)] ds + \int\_{t}^{\lambda} \frac{\mathbf{e}^{k(s-t)}}{\mathbf{e}^{k\lambda} - 1} [\mathbf{g}\left(\mathbf{s}, \mathbf{x}\_{0}(s)\right) + \mathbf{x}\_{0}(s)] ds, \quad t \in [t\_{0}, t\_{c}]$$

i.e.,

$$\mathbf{x}\_{0}(t) \le \int\_{0}^{\lambda} \mathbf{G}(t, \mathbf{s}) [\mathbf{g}\left(\mathbf{s}, \mathbf{x}\_{0}(\mathbf{s})\right) + \mathbf{x}\_{0}(\mathbf{s})] d\mathbf{s} = (T\mathbf{x}\_{0})(t).$$

Thus, the existence of some element *x*0 ∈ *X* such that *x*0R*Tx*0 is ensured.

To show that R is *d*-self-closed, let *xn* be an R-preserving Cauchy sequence converging to *x* ∈ *X*. As *xn* is R-preserving, we have

$$\mathbf{x}\_0(t) \le \mathbf{x}\_1(t) \le \mathbf{x}\_2(t) \le \dots \le \mathbf{x}\_n(t) \le \mathbf{x}\_{n+1}(t) \le \dots \le \mathbf{x}(t) \,\forall t \in [0, \lambda]\_\prime$$

thereby yielding *xn*R*x* ∀*n* ∈ N. Therefore, R is *d*-self-closed.

The remaining hypotheses of Theorem 2 also hold and are easy to check. Hence, *T* possesses a fixed point in *X*.

**Author Contributions:** The authors B.A., M.I. and S.S. contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

**Funding:** This research received no external funding.

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** All the authors are grateful to the anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper.

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

#### **References**


**Abstract**

The well-being of the fetus depends on the efficiency of its circulatory system and the proper maternal-fetal exchange. Hemodynamic changes can occur due to disturbance of fetal and maternal homeostasis, malformations, pregnancy pathology, and medications. Preterm labor directly affects maternal-fetal haemodynamics, both due to uterine contractions and medications used to inhibit it. Research on maternal-fetal haemodynamics in preterm labor is currently focused mainly on the safety of the used tocolytics. In this chapter, we will discuss the basic principles of fetal haemodynamics, ultrasound methods of maternal-fetal circulation assessment, and the influence of preterm labor on maternal-fetal haemodynamics, with particular emphasis on medications used in threatening and progressive preterm labor.

**Keywords:** preterm labor, maternal-fetal haemodynamics, doppler


*3.8. Thermal Analysis*

*3.8. Thermal Analysis*  Thermal analysis is an essential tool used for the characterization of metal complexes where, when heating a compound, its weight loss increases. In addition, DTG differential thermogravimetric analysis (DTGA) is used to study the thermal stability of compounds and the weight loss at different temperatures. TGA confirmed the successful chelation of CFX with different metal ions. Thermogravimetric analysis (TGA) and differential (DTGA) analysis for Mg (II), Cu (II), Fe (III), Zn (II) and Se (IV) CFX complexes were carried out in a temperature range of 30–800 °C under an N2 atmosphere. The first decomposition endothermic step occurred in a temperature range 30–130 °C and corre-Thermal analysis is an essential tool used for the characterization of metal complexes where, when heating a compound, its weight loss increases. In addition, DTG differential thermogravimetric analysis (DTGA) is used to study the thermal stability of compounds and the weight loss at different temperatures. TGA confirmed the successful chelation of CFX with different metal ions. Thermogravimetric analysis (TGA) and differential (DTGA) analysis for Mg (II), Cu (II), Fe (III), Zn (II) and Se (IV) CFX complexes were carried out in a temperature range of 30–800 ◦C under an N2 atmosphere. The first decomposition endothermic step occurred in a temperature range 30–130 ◦C and corresponded to a loss of molecules of crystalline water. The second cracking step was carried out between 130–380 ◦C owing

sponded to a loss of molecules of crystalline water. The second cracking step was carried

to a loss of ceftriaxone ligand. The third stage of decomposition occurred in temperature ranges of 380–530 ◦C and 530–620 ◦C, representing the evolution of NH3, CO2 and HNCO. The final residual products for all ceftriaxone metal complexes are metal oxides.

#### *3.9. Ceftriaxone Metal Complexes Alleviate Hepatic Injury in Male Rats*

Excessive CFX exposure for 30 consecutive days afforded a significant increment in serum activities of ALT and AST in male rats. However, in contrast, the supplementation of CFX complexes induced significant improvements in liver enzymes, and the best ameliorations were recorded in CFX/Mg, CFX/Zn and CFX/Se, respectively, as compared to the CFX treated group alone, as shown in Table 5.

### *3.10. Ceftriaxone Metal Complexes Alleviate Oxidative Injury in the Hepatic Tissues and Structural Alterations of Male Rats Exposed to Ceftriaxone*

Reactive oxygen species levels manifested a marked elevation in the hepatic tissues of CFX-exposed male rats. Supplementation of CFX metal complexes reduced the excessive hepatic generation of free radicals, as recorded in the CFX-treated group. Hepatic MDA levels were significantly elevated in the CFX-treated group only. These changes were mainly inverted in the group to which CFX metal complexes were administered, especially CFX/Zn and CFX/Mg, respectively.

On the contrary, male rats exposed to CFX exhibited a high decline in hepatic antioxidant enzyme (GSH, SOD and CAT) activities. GSH, SOD activity and CAT declined markedly as compared with the control group. Oral supplementation of CFX metal complexes elevated the hepatic antioxidant enzymes as compared to CFX alone, especially for CFX/Mg, CFX/Zn and CFX/se, CFX/Cu and CFX/Fe, in the same order as shown in Table 6.

**Table 6.** Antioxidant enzyme activities in groups treated with either ceftriaxone or ceftriaxone metal complexes.

Values are means ± standard error. Mean values with different letters in the same row show significance at *p* ≤ 0.05, where the highest mean value has the symbol a and b−g : Decreases in value were assigned alphabetically, with similar letters implying partial or complete non-significance and different letters implying significance. CFX = group supplemented with ceftriaxone; CFX/Mg = group supplemented with ceftriaxone/Mg complex; CFX/Zn = group supplemented with ceftriaxone/Zn complex; CFX/Se = group supplemented with ceftriaxone/Se complex; CFX/Cu = group supplemented with ceftriaxone/Cu complex.

Histological sections of hepatic tissues of different treated groups as shown in (Figure 8) which showed marked hepatic alterations and some atrophy in CFX treated group only, while there was marked improvement recorded in hepatic tissues of other treated groups with either CFX/Mg, CFX/Zn, CFX/Se, CFX/Cu and CFX/Fe, Meanwhile the marked improvement was recorded in CFX/Mg, CFX/Zn and CFX/Cu treated groups respectively.

**Figure 8.** Photomicrographs showing histological sections of the hepatic tissues. (**A**) Control group showing normal hepatic structure and normal central vein (CV) (H&EX400). (**B**) CFX-treated group showing severe toxicity in the form of the hypertrophy of hepatocytes (orange arrow) with the appearance of binucleated hepatocytes and increased eosinophilia, granular cytoplasm and vesic-**Figure 8.** Photomicrographs showing histological sections of the hepatic tissues. (**A**) Control group showing normal hepatic structure and normal central vein (CV) (H&EX400). (**B**) CFX-treated group showing severe toxicity in the form of the hypertrophy of hepatocytes (orange arrow) with the appearance

ular nuclei (\*\*\*); the central vein is dilated and filled with hemorrhage and necrotic tissue (black arrow), with focal necrosis in some hepatocytes with increased eosinophilia and nuclear disap-

of binucleated hepatocytes and increased eosinophilia, granular cytoplasm and vesicular nuclei (\*\*\*); the central vein is dilated and filled with hemorrhage and necrotic tissue (black arrow), with focal necrosis in some hepatocytes with increased eosinophilia and nuclear disappearance and the accumulation of a few mononuclear inflammatory cells in blood sinusoids (H&EX400). (**C**) CFX/Mg-treated group showing an almost normal hepatic structure (green arrow) with a normal sized central vein (CV) (H&EX400). (**D**) CFX/Zn-treated group showing very mild toxicity in the form of the hypertrophy of hepatocytes with granular eosinophilic cytoplasm and vesicular nuclei and the appearance of some binucleated cells (orange arrow), with a mild congested central vein which contains (CV) mild brown particles of bilirubin and single hepatocyte necrosis (H&EX400).  (**F**) CFX/Cu-treated group showing an almost normal hepatic structure with ballooning cytoplasm (blue arrow) and a central vein (CV) filled with some red blood cells (H&EX400). (**G**) CFX/Fe-treated group showing mild toxicity in the form of the hypertrophy of hepatocytes with the. appearance of binucleated hepatocytes and increased eosinophilia, granular cytoplasm and vesicular nuclei (red arrow); the central vein is normal in diameter with mild congestion (CV) and ballooning degeneration in some hepatocytes (\*\*\*), with increased eosinophilia around the central vein indicating the beginning of necrosis (H&EX400). pearance and the accumulation of a few mononuclear inflammatory cells in blood sinusoids (H&EX400). (**C**) CFX/Mg-treated group showing an almost normal hepatic structure (green arrow) with a normal sized central vein (CV) (H&EX400). (**D**) CFX/Zn-treated group showing very mild toxicity in the form of the hypertrophy of hepatocytes with granular eosinophilic cytoplasm and vesicular nuclei and the appearance of some binucleated cells (orange arrow), with a mild congested central vein which contains (CV) mild brown particles of bilirubin and single hepatocyte necrosis (H&EX400).  (**F**) CFX/Cu-treated group showing an almost normal hepatic structure with ballooning cytoplasm (blue arrow) and a central vein (CV) filled with some red blood cells (H&EX400). (**G**) CFX/Fe-treated group showing mild toxicity in the form of the hypertrophy of hepatocytes with the. appearance of binucleated hepatocytes and increased eosinophilia, granular cytoplasm and vesicular nuclei (red arrow); the central vein is normal in diameter with mild congestion (CV) and ballooning degeneration in some hepatocytes (\*\*\*), with increased eosinophilia around the central vein indicating the beginning of necrosis (H&EX400). *3.11. Antibacterial Activity Evaluation*

#### *3.11. Antibacterial Activity Evaluation* (*Bacillus subtilis*, *Streptococcus pneumoniae* and *Staphylococcus aureus*) and Gram-negative

Biological evaluations were performed for CFX complexes against Gram-positive (*Bacillus subtilis*, *Streptococcus pneumoniae* and *Staphylococcus aureus*) and Gram-negative (*Escherichia coli* and *Pseudomonas aeruginosa*) bacteria. Results from the agar disc diffusion tests for the antimicrobial activities are presented in Table 6 and demonstrated in Figure 9. The diameters of the zone of inhibition (in mm) of the standard drug Amikacin (Aminoglycoside) (C22H43N5O13) against Gram-positive bacteria *B. subtilis* and *S.aureus* and Gram-negative bacteria *E. coli* and *P. aeruginosa* were found to be 36, 30, 31 and 35 mm, respectively. Under identical conditions, Table 7 and Figure 9 show that all complexes were found to be efficient, with a high antimicrobial activity that exceeded CFX itself. (*Escherichia coli* and *Pseudomonas aeruginosa*) bacteria. Results from the agar disc diffusion tests for the antimicrobial activities are presented in Table 6 and demonstrated in Figure 9. The diameters of the zone of inhibition (in mm) of the standard drug Amikacin (Aminoglycoside) (C22H43N5O13) against Gram-positive bacteria *B. subtilis* and *S.aureus* and Gram-negative bacteria *E. coli* and *P. aeruginosa* were found to be 36, 30, 31 and 35 mm, respectively. Under identical conditions, Table 7 and Figure 9 show that all complexes were found to be efficient, with a high antimicrobial activity that exceeded CFX itself.

Biological evaluations were performed for CFX complexes against Gram-positive

**Figure 9.** Antibacterial activity of CFX and Zn (II) and Se (IV) metal complexes**. Figure 9.** Antibacterial activity of CFX and Zn (II) and Se (IV) metal complexes.

**Table 7.** Inhibition zone diameter (mm/mg sample) of CFX and CFX metal complexes.

Means within the same column (mean ± SE) carrying different letters are significant at *p* ≤ 0.05 using Duncan's multiple range test, where the highest mean value has the symbol a and b−e those decreasing in value are assigned alphabetically.

#### **4. Discussion**

A serious problem now facing humanity is oxidative damage and its dangerous consequences for human health and hepatic vitality. The world now needs powerful antibiotics with potent antioxidant activities and fewer side effects for the liver tissues to reinforce the immune system and fight against resistant microbes and microorganism. Thus, the current study was designed to synthesize novel metal complexes of ceftriaxone (CFX) with Mg (II), Fe (III), Cu (II), Zn (II) and Se (IV) to investigate the potency of the hepatoprotective effects and antioxidant capacities of CFX complexes and determine if they succeed in the elevation of antioxidant capacities to investigate novel compounds with high antioxidant capacities.

The oxidative injury of biomolecules is the major concern in the pathogenesis of a large number of diseases such as cancer, degenerative diseases, metabolic diseases and even dangerous instances of SARS-CoV-2. Thus, it is very important to investigate the role of potent novel metal complexes to prevent oxidative stress and the treatment of side effects and diseases induced by such oxidative stress.

Liver injury caused by drugs ranges from mild biochemical abnormalities to chronic liver failure. The majority of adverse liver reactions occur in most antibiotic treatments [23].

Some antibiotics are considered to be a common cause of liver injury. The hepatotoxicity that occurs is usually associated with hepatic impairment [40]. CFX is a broad-spectrum antibiotic with potent activity against Gram-positive and Gram-negative bacteria [41].

The hepatotoxicity caused by CFX appears after an average of 9–11 days [42,43]. Previous studies have reported high aspartate aminotransferase (ALT) and alanine aminotransferase (AST) activities with the administration of CFX [44,45], and this concept coincides with the current findings that confirmed that CFX causes hepatic damage as a result of elevations in some biochemical parameters such as AST, ALT and low-density lipoprotein (LDL) as well as a decline in high-density lipoprotein (HDL) concentrations [46–48].

Based on the previous background, researchers have demonstrated that CFX is widely used as a third-generation cephalosporin antibiotic that has a broad spectrum of bactericidal activity. However, previous evidence has indicated that CFX carries a risk of the elevation of liver enzymes, with liver injury as an adverse effect [48].

In agreement with previous findings [23] that reported that the oxidative stress mediated by oxygen free radicals (ROS) has been implicated as a common link between chronic liver damage and hepatic fibrosis, the current study demonstrated that the administration of CFX resulted in a significant decline in hepatic GSH. Conversely, a significant increment in the level of hepatic MDA (a marker of lipid peroxidation) was shown. The increase in MDA level was more pronounced in rats in CFX-treated groups.

The novel finding of our current study is the improvement effects of CFX metal complexes, especially in groups treated with CFX with Mg, Zn, Se and Cu complexes, respectively, as they improved hepatic function serum levels and ameliorated the hepatic structure greatly.

Our explanation depends on the functional activity of the used metals, as previous researchers demonstrated the importance of Se metal in enhancing the glutathione levels of hepatic tissues, leading to enhancements of the whole antioxidant defense system [49–52] and a decline in the excessive generation of reactive oxygen species.

Metal ions are required to keep the human body healthy because several critical biological functions in humans depend upon their presence, and their absence or scarcity may lead to diseases. Magnesium (Mg) belongs to the main group of elements and is required mainly for fat and carbohydrate metabolism. Meanwhile, zinc (Zn) belongs to the transition metal group of elements. Zn is a vital element due to its strong binding to proteins. Zn also is exceptionally stable with respect to oxidation and reduction, and thus it does not participate in redox reactions. Additionally, Zn2+ shows a strong preference in enzymes for a tetrahedral coordination over an octahedral one [53], and all these previous aspects confirm the findings of the current study and the great improvement activities of CFX metal complexes, especially with Zn2+, in the alleviation of the hepatic side effects induced by CFX alone.

A deficiency of Fe leads to anemia, as it is known that Fe is a part of hemoglobin; besides this, a deficiency of Cu leads to heart diseases and anemia. The importance of all of these metals and especially in conjugation with CFX has been proved chemically in alleviating oxidative stress and providing high hepatic protection. The Gram-negative strains are exposed to various stress conditions during pathogenesis, of which the stress of acids serves as a major defense mechanism in the host [54]. Such environments are encountered, and this supports the main hypothesis of the study and proved the hepatoprotective and antioxidant capacities of two synthesized CFX complexes with either Zn (II) or Se (IV); in addition, the two complexes showed antibacterial activity besides their abilities to reduce free radicals' production.

#### **5. Conclusions**

The current study aimed to synthesize five ceftriaxone complexes by the reaction of ceftriaxone sodium salt with (Mg2+, Zn2+, Fe3+, Cu2+ and Se4+) ions. The structures of the CFX complexes have been explained using microanalytical, molar conductance, IR, 1HNMR, UV–Vis, magnetic, SEM, TEM and X-ray diffraction analyses. Mg (II), Cu (II), Zn (II), Fe (III) and Se (VI) complexes form six-coordinate systems with a distorted octahedral geometry. The obtained results clarified that ceftriaxone metal complexes, especially (CFX/Mg, CFX/Zn and CFX/Se), greatly ameliorated hepatic enzyme functions and enhanced the antioxidant activities of hepatic tissues while reducing the excessive triggering of excessive reactive oxygen species (ROS) and the protection of hepatic structures as compared with ceftriaxone-treated groups alone. For antibacterial activity, the CFX/Zn complex was highly effective against *Streptococcus pneumoniae*, while CFX/Se was highly effective against *Staphylococcus aureus* and *Escherichia coli*. These results are very promising in providing protection for the hepatic tissues and reducing damaging effects and the severe oxidative stress induced by antibiotics on liver tissues, especially during the COVID-19 pandemic; in addition, it brings the recent research that correlated the damage of hepatic functions and severe instances of SARS-CoV-2 on the health of the body up to date.

**Author Contributions:** Conceptualization, S.M.E.-M., S.H.Q., F.S.A. and R.Z.H.; methodology, S.M.E.-M., S.H.Q. and R.Z.H.; validation, S.M.E.-M., S.H.Q., F.S.A. and R.Z.H.; formal analysis, S.M.E.-M., S.H.Q., F.S.A. and R.Z.H.; investigation, S.M.E.-M., S.H.Q., F.S.A. and R.Z.H.; resources, S.M.E.-M., S.H.Q., F.S.A. and R.Z.H.; data curation, S.M.E.-M. and R.Z.H.; writing—original draft preparation, S.M.E.-M., S.H.Q., F.S.A. and R.Z.H.; writing—review and editing, S.M.E.-M., S.H.Q., F.S.A. and R.Z.H.; supervision, S.M.E.-M., F.S.A., S.H.Q. and R.Z.H.; project administration, S.M.E.-M. and R.Z.H.; funding acquisition, S.M.E.-M., S.H.Q., F.S.A. and R.Z.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work received no external funding.

**Institutional Review Board Statement:** The animal study was conducted in accordance with the Declaration of Helsinki, and approved by Ethics Committee of Taif University.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** All the data are available inside the text.

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


**8. Rabies in camelids**

#### **8.1 Epidemiology**

Camelids are susceptible to rabies. However, most of the publications described clinical rabies in dromedary camels (*Camelus dromedarius*), with few reports in Bactrian camels (*Camelus bactrianus*) [8, 46] and the New World camelids [67]. From 2006 to 2013 in Oman, foxes reported the highest positivity rate (70.1%), camels accounted for the second higher positivity (59.7%) for rabies diagnosis [63]. Camels were the second most likely livestock after cattle to test positive for rabies in Mongolia from 1970 to 2005 [45]. Camels were the fourth (4.2%) mainly rabies tested positive animals during 2004–2018 in China [46]. In China, rabies infection in camels was reported [8]. According to the confirmed rabies cases in Saudi Arabia during 2010–2017 [15], Dogs and dromedary camels were the most affected species. In Iran, a review on zoonotic diseases published articles [11] revealed that camels are one of the essential sources as well as carriers of infection for humans, livestock, and wildlife in Iran and worldwide. Rabies is highly endemic in Iran; it is circulated easily in wildlife and livestock. The authors reviewed reported camel rabies cases in Iran, 3 cases during 1996–2006; an outbreak of camel rabies was reported for the first time in 2008 in central Iran. A rabid wolf attacked 8 camels; another camel rabies in the east was reported in 2012 [11]. In Jordan, Rabies in 8 camels was described [12]. In India, a report of clinical signs of rabies in she-camel was described [4].

Reports of rabies in dromedary camels from Morocco, Mauritania, Sudan, Yemen, Saudi Arabia, UAE, Niger, Jordan, India, Israel, and Iran were reviewed by Abbas and Omer [19]. Ali et al. (2004) reviewed dromedary camel rabies in Sudan; the first confirmed rabies cases were in 1926, then laboratory-confirmed cases continued to be reported, 17 cases from 1927 to 1939, 21 from 1940 to 1970. Other cases were reported in different parts of the country, 21 in the north and 16 in the Western States. Camel population in Sudan is about 4.8 million; the reported rabies cases are very few compared to the population, which is mainly due to under-reporting. Other cases in camels were reported as well in Sudan [20].

In a review about camel diseases, many rabies cases have been reported in Mauritania, Saudi Arabia, Iran, and Pakistan; infection is usually due to rabid dog bites [68]. In Niger, an outbreak of rabies in the camel herd due to feral dog bites was reported [69].

Rabies outbreak in camels in Iran was described [70]. An outbreak of dog rabies in the camel herd was reported in Sudan in 1998, and it resulted in the death of 19 camels [71]. Antibodies against rabies were detected in non-vaccinated camels imported for slaughtering in Nigeria, which may indicate subclinical rabies infection [72]. Very few suspected rabies cases in camels were reported in Morocco during 1991–2015 [27].

#### **8.2 Clinical signs**

Camels with furious rabies form show restlessness, anxiety, salivation, and attacking and biting form followed by terminal paralysis, lateral recumbency (**Figure 6**) and a characteristic flexion of four limbs, reviewed by Abbas and Omer [19]. Mohammadpour et al. (2020) reviewed a report stating that a furious form of rabies is seen in most of cases in camels.

Rabies clinical signs observed in Bactrian camels were reduced appetite, excessive activity and agitation, cessation of rumination, lip twitching, hypersalivation, tachypnea and howling, paralysis [8]. Some reports described the occurrence of

**Figure 6.** *A case of rabies in a dromedary camel showing lateral recumbency and excessive salivation.*

rabies dumb form more frequently. Most rabies cases (67%) in camels in Oman were of the dumb form; observed clinical signs were restlessness, salivation, head and neck rotation in all directions, paralysis, recumbency, and death [73]. Clinical signs seen in rabid camels were hyperesthesia, profuse salivation, anorexia, and paralysis [12]. In India, clinical signs of rabies in the camel described the appearance of hyperexcitability, bellowing, aimless running, salivation, convulsions, swaying of the hindquarters and recumbency, biting tendency to the owner and wooden objects [4]. Noticed rabies clinical signs in camels were unusual behavior including aggression, pica, ptyalism, and terminal paralysis [69]. During an outbreak of rabies in camels, reported clinical signs were high sensitivity, ferocity, biting faces of other camels, bloat, restlessness, limb paralysis, and yawning [70]. In a rabies outbreak, clinical signs noticed in most of the affected camels were restlessness, irritability with very harsh and loud sounds. Later excitement became noticeable. There were then rubbing incoordination (staggering gait), tenesmus, abnormal sexual behavior: (she-camels mounting each other), raising of tails, and slight salivation, terminated by paralysis of the hindquarter, recumbency, and death of 19 camels [71]. In Khartoum, two camel rabies cases in 1996, 1997 showed off food, salivation, nervous signs followed by biting fence and their forelimbs and abdomen, then recumbency, hind limbs paralysis, and death [74]. An almost similar picture was previously reported by Afzal et al. [75], which were hyper-excitability, salivation, attacking inanimate objects, biting of its forelimbs, sternal recumbency, paralysis of hind legs, and death.

### **8.3 Transmission**

Transmission of rabies to camels is mainly through dog bites, except in some countries where wild animals like foxes and wolves are involved. In China, the main rabies transmitters are dogs; meanwhile, in border areas, the wild fox is [46]; camels were reported to be rabies-infected by dogs and wild foxes [8]. In Iran, a rabid wolf was reported to transmit the infection to camels [11]. In Oman, the main animals involved in rabies transmission are foxes [63]. In another study in Oman, it was noticed that most rabid camels were bitten by foxes, which confirmed the major role of foxes in rabies transmission [13]. In Saudi Arabia, the majority (70%) of camel rabies cases were due to wild dog bite, while wild foxes accounted for about 17% of cases [73]. Almost in all reported camel rabies cases in Sudan, dogs were the source of infection [18, 62, 71, 74]. The same finding was reported in other African as well as Asian countries, Niger [69], Mauritania, Saudi Arabia, Iran, and Pakistan [68], China [8] Saudi Arabia [15].