**1. Introduction**

Studies performed in a variety of animal models including mice and rhesus monkeys have found that prolonged exposure of developing animals to most inhalational and intravenous anesthetic agents including sevoflurane, isoflurane, ketamine, propofol and anesthetic combinations induce neuroapoptosis [1–3]. The period of rapid synaptogenesis has been linked to the time of greatest vulnerability in developing brains to anesthetic-induced neurotoxicity [4–6]. The period of rapid brain growth in mice peaks at roughly 7 days after birth while this time frame in humans ranges from late in pregnancy until the 2nd or 3rd year of life [7, 8]. The neuroapoptosis observed in the growing brains of the young animals was also found to be coupled to long-term memory and learning deficits. For example, Shen and colleagues noted significant deficiencies in both memory and spatial learning in postnatal day 3 Sprague–Dawley rats following exposure to 1% sevoflurane. Their studies relied on the Morris water maze test to assess spatial learning/memory in the animals. Additionally, they found that these effects were both exposure number and dose dependent. They also observed that 7-week-old (adult) rats were insensitive to sevoflurane exposure with both the control and sevoflurane exposed animals displaying comparable results in the Water maze test [3]. The results of this study confirm that vulnerability to anesthetics is confined to an early period in development.

While many anesthetic agents are used clinically in the pediatric population, we choose to study the effects of the widely used anesthetic ketamine on our model of hESC-derived neurons. Ketamine is an N-methyl-D-aspartate-receptor antagonist and provides pain relief and sedation in children undergoing a variety of procedures. Additionally, ketamine is a drug that is often abused and abuse of ketamine during pregnancy is of concern for the developing brain of the fetus [19]. Despite the many studies implicating the negative effects of anesthetic agents on developing brains, the mechanisms behind the toxicity remain largely unknown. Possible roles of neuroinflammation, reactive oxygen species production, epigenetic changes, and calcium signaling in the mechanism of anesthetic-induced neurotoxicity have all been suggested [20–23]. However, considerable work remains to fully elucidate the mechanisms behind the neuronal toxicity. In this study, we investigated the effect of ketamine on the cell viability, intracellular calcium level, mitochondrial signaling, and microRNA profile using

Ketamine Induces Neuroapoptosis in Stem Cell–Derived Developing Human Neurons Possibly…

http://dx.doi.org/10.5772/intechopen.72939

423

Proper maintenance of intracellular calcium levels is critical for nearly every cellular process. In neurons, calcium is involved in the regulation of electrical activity, cell growth, metabolic activity, and many other processes [24]. Several studies have suggested that aberrant intracellular calcium level in neurons plays a critical role in the neuronal degeneration observed in many different neurological disorders [25]. The dysregulated intracellular calcium might induce the neurotoxicity through mitochondrial signaling. Mitochondria are highly dynamic organelles that undergo continuous cycles of fusion and fission in order to maintain cellular homeostasis. Mitochondrial fusion and fission result in a change in mitochondrial shape: either elongated, tubular, interconnected mitochondrial networks, or fragmented and discontinuous mitochondria, respectively. Unbalanced fission-fusion can lead to various pathological processes including neurodegeneration [26–29]. Shifting the balance towards fission has been associated with neuronal death in age-related neurodegenerative disease and brain injury [30, 31]. Inhibition of mitochondrial fission attenuated glutamate-induced neuronal death [31], translating to an increase in cell survival in the presence of oxidative stress [32]. Recent studies from Dr. Jevtovic's groups have shown that mitochondrial fission plays an important role in anesthetic neurotoxicity. The increased mitochondrial fission was found in neonatal rat brains after a sedative dose of midazolam followed by combined nitrous oxide

MicroRNAs are small, non-coding RNAs that function as negative regulators of gene expression. MicroRNAs are transcribed in hairpin structures in the nucleus by RNA polymerase II and the pri-miRNA produced is processed by the enzyme Drosha to cleave off a single hairpin loop. This forms the pre-miRNA which is exported to the cytoplasm. Once in the cytoplasm, the pre-miRNA is further processed to remove the hairpin loop and forms the mature microRNA strand. The mature microRNA strand then incorporates into an RNA-induced silencing complex where it can act to promote silencing of its target genes [34, 35]. Over 1000 microRNAs have been discovered in humans and they have been shown to play an important role in nearly every cellular process. Dysregulation of microRNAs has also been linked to several diseases including certain cancers [36]. The role of microRNAs in anesthetic-induced

the hESC-derived developing human neuron model.

and isoflurane anesthesia for 6 h [33].

neurotoxicity is just beginning to be studied.

The many animal studies have raised concerns regarding the applicability of the models to humans. Of particular concern are the many rodent studies in which hemodynamic properties were not properly controlled. While extensive studies in rhesus macaques, a non-human primate model have been performed [9–14], a suitable human model in the study of anesthetic-induced neurotoxicity is currently lacking and exposure of young children to anesthetics for study purposes would be unethical. Several human epidemiologic studies have been performed and many have suggested a link between anesthetic exposure in young children and learning and behavioral abnormalities when compared to their unexposed counterparts [15–17]. However, the human epidemiologic studies face many confounding variables that are often difficult to properly account for, leaving the results of these studies uncertain. While many of these studies are still ongoing, it is imperative to develop a proper human model to assess the effects of anesthetics in young children. Our study aimed to identify an appropriate human model to assess the effects of anesthetics on the developing human brain and also to dissect out the mechanisms behind the toxicity.

Human embryonic stem cells (hESCs) are derived from the inner cell mass of a human blastocyst. In 1998 James Thompson and colleagues at the University of Wisconsin-Madison developed a technique to isolate and culture hESCs *in vitro* [18]. The formative work of this group opened up the possibility for mechanistic-based studies using a human cell line, effectively eliminating the potential concerns regarding the relevancy of studies using animal models to humans. hESCs are immature, divide indefinitely and are capable of generating cells from all three germ layers which makes these cells an ideal model of early human neurons once differentiated. Using hESC-derived neurons has provided us with a reasonable human model by which to study anesthetic-induced developmental neurotoxicity.

While many anesthetic agents are used clinically in the pediatric population, we choose to study the effects of the widely used anesthetic ketamine on our model of hESC-derived neurons. Ketamine is an N-methyl-D-aspartate-receptor antagonist and provides pain relief and sedation in children undergoing a variety of procedures. Additionally, ketamine is a drug that is often abused and abuse of ketamine during pregnancy is of concern for the developing brain of the fetus [19]. Despite the many studies implicating the negative effects of anesthetic agents on developing brains, the mechanisms behind the toxicity remain largely unknown. Possible roles of neuroinflammation, reactive oxygen species production, epigenetic changes, and calcium signaling in the mechanism of anesthetic-induced neurotoxicity have all been suggested [20–23]. However, considerable work remains to fully elucidate the mechanisms behind the neuronal toxicity. In this study, we investigated the effect of ketamine on the cell viability, intracellular calcium level, mitochondrial signaling, and microRNA profile using the hESC-derived developing human neuron model.

**1. Introduction**

422 Mitochondrial Diseases

in development.

dissect out the mechanisms behind the toxicity.

which to study anesthetic-induced developmental neurotoxicity.

Studies performed in a variety of animal models including mice and rhesus monkeys have found that prolonged exposure of developing animals to most inhalational and intravenous anesthetic agents including sevoflurane, isoflurane, ketamine, propofol and anesthetic combinations induce neuroapoptosis [1–3]. The period of rapid synaptogenesis has been linked to the time of greatest vulnerability in developing brains to anesthetic-induced neurotoxicity [4–6]. The period of rapid brain growth in mice peaks at roughly 7 days after birth while this time frame in humans ranges from late in pregnancy until the 2nd or 3rd year of life [7, 8]. The neuroapoptosis observed in the growing brains of the young animals was also found to be coupled to long-term memory and learning deficits. For example, Shen and colleagues noted significant deficiencies in both memory and spatial learning in postnatal day 3 Sprague–Dawley rats following exposure to 1% sevoflurane. Their studies relied on the Morris water maze test to assess spatial learning/memory in the animals. Additionally, they found that these effects were both exposure number and dose dependent. They also observed that 7-week-old (adult) rats were insensitive to sevoflurane exposure with both the control and sevoflurane exposed animals displaying comparable results in the Water maze test [3]. The results of this study confirm that vulnerability to anesthetics is confined to an early period

The many animal studies have raised concerns regarding the applicability of the models to humans. Of particular concern are the many rodent studies in which hemodynamic properties were not properly controlled. While extensive studies in rhesus macaques, a non-human primate model have been performed [9–14], a suitable human model in the study of anesthetic-induced neurotoxicity is currently lacking and exposure of young children to anesthetics for study purposes would be unethical. Several human epidemiologic studies have been performed and many have suggested a link between anesthetic exposure in young children and learning and behavioral abnormalities when compared to their unexposed counterparts [15–17]. However, the human epidemiologic studies face many confounding variables that are often difficult to properly account for, leaving the results of these studies uncertain. While many of these studies are still ongoing, it is imperative to develop a proper human model to assess the effects of anesthetics in young children. Our study aimed to identify an appropriate human model to assess the effects of anesthetics on the developing human brain and also to

Human embryonic stem cells (hESCs) are derived from the inner cell mass of a human blastocyst. In 1998 James Thompson and colleagues at the University of Wisconsin-Madison developed a technique to isolate and culture hESCs *in vitro* [18]. The formative work of this group opened up the possibility for mechanistic-based studies using a human cell line, effectively eliminating the potential concerns regarding the relevancy of studies using animal models to humans. hESCs are immature, divide indefinitely and are capable of generating cells from all three germ layers which makes these cells an ideal model of early human neurons once differentiated. Using hESC-derived neurons has provided us with a reasonable human model by Proper maintenance of intracellular calcium levels is critical for nearly every cellular process. In neurons, calcium is involved in the regulation of electrical activity, cell growth, metabolic activity, and many other processes [24]. Several studies have suggested that aberrant intracellular calcium level in neurons plays a critical role in the neuronal degeneration observed in many different neurological disorders [25]. The dysregulated intracellular calcium might induce the neurotoxicity through mitochondrial signaling. Mitochondria are highly dynamic organelles that undergo continuous cycles of fusion and fission in order to maintain cellular homeostasis. Mitochondrial fusion and fission result in a change in mitochondrial shape: either elongated, tubular, interconnected mitochondrial networks, or fragmented and discontinuous mitochondria, respectively. Unbalanced fission-fusion can lead to various pathological processes including neurodegeneration [26–29]. Shifting the balance towards fission has been associated with neuronal death in age-related neurodegenerative disease and brain injury [30, 31]. Inhibition of mitochondrial fission attenuated glutamate-induced neuronal death [31], translating to an increase in cell survival in the presence of oxidative stress [32]. Recent studies from Dr. Jevtovic's groups have shown that mitochondrial fission plays an important role in anesthetic neurotoxicity. The increased mitochondrial fission was found in neonatal rat brains after a sedative dose of midazolam followed by combined nitrous oxide and isoflurane anesthesia for 6 h [33].

MicroRNAs are small, non-coding RNAs that function as negative regulators of gene expression. MicroRNAs are transcribed in hairpin structures in the nucleus by RNA polymerase II and the pri-miRNA produced is processed by the enzyme Drosha to cleave off a single hairpin loop. This forms the pre-miRNA which is exported to the cytoplasm. Once in the cytoplasm, the pre-miRNA is further processed to remove the hairpin loop and forms the mature microRNA strand. The mature microRNA strand then incorporates into an RNA-induced silencing complex where it can act to promote silencing of its target genes [34, 35]. Over 1000 microRNAs have been discovered in humans and they have been shown to play an important role in nearly every cellular process. Dysregulation of microRNAs has also been linked to several diseases including certain cancers [36]. The role of microRNAs in anesthetic-induced neurotoxicity is just beginning to be studied.

This chapter will outline: (1) the use of hESCs as a model system to study anesthetic-induced neurotoxicity in the developing human brain, (2) the protocols involved in dissecting the mechanisms behind anesthetic-induced developmental neurotoxicity. We have focused on the role of deregulated intracellular calcium, mitochondrial signaling (e.g., mitochondrial membrane potential, cytochrome c release from mitochondria into cytosol, and mitochondrial fission), and microR-NAs, and (3) a brief discussion on the possible role of intracellular calcium levels, mitochondrial signaling, and altered microRNA profiles in ketamine-induced developmental neurotoxicity.

**2.2. Neuronal characterization by immunofluorescence staining**

ketamine or neurobasal media as a vehicle-control in a 5% CO2

**2.4. Assessment of cell death by TUNEL staining**

Melville, NY).

incubator.

in a field.

**2.5. Calcium imaging**

**2.3. Ketamine exposure**

hESC-derived neurons cultured on matrigel-coated, glass coverslips for 2 weeks were fixed for 30 min in 1% paraformaldehyde at room temperature. The cells were then washed with phosphate-buffered saline (PBS). Next, the cells were incubated for 15 min in 0.5% Triton X-100 (Sigma-Aldrich) in PBS. The cells were washed once again with PBS and blocked for 20 min with 10% donkey serum at room temperature. The cells were then incubated with the primary antibodies [microtubule-associated protein 2 (MAP2), β-tubulin III or doublecortin (Abcam, Cambridge, MA)] for 1 h in a humidified, 37°C incubator. The cells were washed with PBS and incubated for 45 min with Alexa Fluor 488 or 594 donkey anti-mouse or rabbit immunoglobulin G (Invitrogen) secondary antibodies at 37°C. The cell nuclei were stained with Hoechst 33342 (Invitrogen). The coverslips were then mounted onto glass slides and imaged using a laser-scanning confocal microscope (Nikon Eclipse TE2000-U, Nikon Inc.,

Ketamine Induces Neuroapoptosis in Stem Cell–Derived Developing Human Neurons Possibly…

While the brain concentration of ketamine in humans during the induction and maintenance of anesthesia is not well understood, reports have found that the peak blood levels of ketamine could be as high as 103 μM. The levels required to maintain anesthesia are typically in the range of 10–20 μM. hESC-derived neurons were exposed to 6 h of 20 μM

conditions only for the microRNA studies. For all remaining studies, the cells were exposed to 24 h of 20 or 100 μM ketamine or neurobasal media as the vehicle control in the same

Apoptosis of the hESC-derived neurons was assessed using a cell death detection kit (Roche Applied Bio Sciences, Indianapolis, IN) based on terminal deoxynucleotidyl transferasemediated deoxyuridine triphosphate in situ nick end labeling (TUNEL) following directions provided by the manufacturer. This kit identifies single or double-stranded DNA breaks by labeling the free 3′-OH termini with altered nucleotides in a reaction with terminal deoxynucleotidyl transferase (TdT). Cells cultured on glass coverslips were exposed to ketamine or control conditions, rinsed and fixed with 1% paraformaldehyde. DNA fragmentation was analyzed using TdT, which incorporates into sites of DNA breaks. The nuclei were stained with Hoechst 33342 and the cells were imaged using the confocal microscope. Apoptosis/ necrosis was quantified by assessing the ratio of TUNEL-positive nuclei to total cell nuclei

Neurons were plated on matrigel-coated glass coverslips for the calcium imaging studies. Intracellular calcium was assayed using Fluo-4AM (Thermo Fisher Scientific). Neurons were

incubator with normoxic

http://dx.doi.org/10.5772/intechopen.72939

425


**1. Introduction**

The growth in total shoulder arthroplasty over the past 20 years has been exponential. From 1993 to 2007, primary procedures increased 369% [1]. Revision shoulder arthroplasty increased 431% during the same period. During the timing of this explosion, numerous innovations have occurred, most importantly the development of the reverse shoulder arthroplasty (RSA). Currently, RSA accounts for over 70% of the shoulder arthroplasties performed in the U.S. With the expansion of shoulder arthroplasty, and specifically RSA, numerous other innovative designs have been developed in recent years to address more and more complicated pathology to hopefully reduce the increasing revision burden and improve outcomes.

Innovations currently available with anatomic total shoulder arthroplasty include shorter stem designs and augmented inlay glenoid components. Regarding RSA, metal augmentation, including custom augments on both the glenoid and humeral side have expanded indications in cases of bone loss. In the setting of revision arthroplasty, humeral options include convertible stems and newer tools to improve humeral implant removal. New strategies for treatment and surgical techniques have been developed for recalcitrant shoulder instability, acromial fractures, and infections after RSA. Finally, computer planning, navigation, PSI, and augmented reality are imaging options now available that have redefined preoperative planning and indications as well intraoperative component placement.

#### **2. New Perspectives and Innovations in Anatomic Shoulder Arthroplasty**

*2.1. Humeral Component Innovations*


*Article* **Structure-Based Design and Synthesis of a New Phenylboronic-Modified Affinity Medium for Metalloprotease Purification**

**Shangyong Li 1,†, Linna Wang 1,†, Ximing Xu 2,†, Shengxiang Lin 3,\*, Yuejun Wang 1, Jianhua Hao 1,4 and Mi Sun 1,4,\***

Academic Editors: Vassilios Roussis and Antonio Trincone Received: 17 November 2016; Accepted: 21 December 2016; Published: 27 December 2016

**Abstract:** Metalloproteases are emerging as useful agents in the treatment of many diseases including arthritis, cancer, cardiovascular diseases, and fibrosis. Studies that could shed light on the metalloprotease pharmaceutical applications require the pure enzyme. Here, we reported the structure-based design and synthesis of the affinity medium for the efficient purification of metalloprotease using the 4-aminophenylboronic acid (4-APBA) as affinity ligand, which was coupled with Sepharose 6B via cyanuric chloride as spacer. The molecular docking analysis showed that the boron atom was interacting with the hydroxyl group of Ser176 residue, whereas the hydroxyl group of the boronic moiety is oriented toward Leu175 and His177 residues. In addition to the covalent bond between the boron atom and hydroxyl group of Ser176, the spacer between boronic acid derivatives and medium beads contributes to the formation of an enzyme-medium complex. With this synthesized medium, we developed and optimized a one-step purification procedure and applied it for the affinity purification of metalloproteases from three commercial enzyme products. The native metalloproteases were purified to high homogeneity with more than 95% purity. The novel purification method developed in this work provides new opportunities for scientific, industrial and pharmaceutical projects.

**Keywords:** metalloprotease; adsorption analysis; molecular docking; affinity purification; aminophenylboronic acid


**Author details**

Dilawar Hassan1 , Sidra Amin1,3, Amber Rehana Solangi1 \* and Saima Q. Memon2

1 National Centre of Excellence in Analytical Chemistry, University of Sindh, Jamshoro, Pakistan

2 Dr. M. A Kazi Institute of Chemistry, University of Sindh, Jamshoro, Pakistan

3 Department of Chemistry, Shaheed Benazir Bhutto University, Sindh, Pakistan

\*Address all correspondence to<EMAIL_ADDRESS>
© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


**8. Association of OPRM1 functional alleles with immune cell function**

Sharad et al*.* [93] used a genetic approach to correlate a functional *OPRM1* gene polymorphism with known action of opiates on immunity and undertook a prospective study to understand the relationship of the 118G variation with the amount of exogenous opiates consumed and correlated the immunosuppressive effects of exogenous opiates with the *OPRM1* allele type. They studied the immune status of opiate users by measuring serum Ig (IgG and IgA) levels, in association with specific *OPRM1* genotype, and confirmed that the mean circulating levels of Ig were significantly lower in opiate users when compared with levels in cohort controls. Among opiate-dependent subjects, individuals with AA genotype were found to have the lowest levels of circulating immunoglobulins, both IgG and IgA (p = 0.0001), while the AG genotype carrying individuals had a higher level of both immunoglobulins. The homozygous GG genotype was in between the AA and AG genotypes. Alternatively, in opiate naïve subjects, the AA individuals showed the highest titers of

**93**

molecular exploration.

*Immune Cell Activation: Stimulation, Costimulation, and Regulation of Cellular Activation*

circulating IgG, and the GG individuals showed the lowest with AG having intermediate values [94]. The immunosuppressive effects documented in opiate naïve individuals can be attributed to altered regulation of PKA and pERK1/2 due to the levels of endogenous opioid. In addition to the absence of G genotype in the immortalized cell lines and based on the cell culture data showing cell cycle arrest observed in the present study (in A549 and HepG2 cell lines), we hypothesize that coactivation

of μOR in presence of 118G allele leads to a suppression/arrest of cell division. To test this hypothesis, another pilot study was carried out in which healthy opiate naïve volunteers were enrolled and the cell count for circulating lymphocyte subsets was studied as a measure of immune competence. Genotypic association studies showed a correlation between the immune cell numbers. Total lymphocyte count showed a significant lowering in cell numbers in 118G-allele-bearing individuals when compared to 118A-bearing individuals. However, cell numbers in all individuals remained within the documented normal range of 500–4000 cell/ml. The GG allele individuals showed significantly lower cell count, averaging 490, which differed markedly from cell numbers observed in AA-bearing individuals with mean numbers of 1976 cells/ml, (p = 0.008) and a correlation factor, r2

0.79 between the genotype and average cell numbers. Our data show a significant lowering in all immunocytes, namely leucocyte populations (CD45+ve cells), B lymphocytes (CD10+ve cells), T lymphocytes (CD3, CD4, and CD8+ve cells), NK cells (CD56+ve cells), activated monocytes (CD 11b+ve cells), and mesenchymal progenitors in GG-bearing individuals when compared to AA-allele-bearing individuals but not always in comparison with those bearing the AG allele. This baseline lowering of cell numbers in GG-bearing individuals supports the hypothesis that GG genotype suppresses cell division, and since mounting of a successful immune response and/or overstimulation of immune system, as in case of patients with autoimmune disorders, depends on activation of both innate and adaptive immune responses, the 118G-bearing individuals would be prone to immune suppression due to lack of amplification by selective cell division, a critical step in elimination of

In conclusion, there is a significant correlation between the circulating number of lymphocytes TH and Tc, B cells and NK cells, and the μOR allele present, and this difference can be further increased in the presence of exogenous opioids either during clinical treatment or substance dependence, as the 118G allele affects the process of cell division arresting cells at the S or M phase of the cell cycle, or by modulating the action of cell division–linked secretion of stimulating cytokines/chemokine known to induce clone-specific and cell type–specific proliferation, because of the propensity of opioid receptor to heterodimerize and to selectively bias the subsequent ligand engagement/s with the dimerized/ oligomerized receptors. This "subliminal immune suppression" in G allele–bearing individuals can have far reaching impact on onset of diseases such as cancer and obesity (both have an element inflammation) and vaccination for infectious diseases and even dreaded diseases as cancer. This immune suppression will certainly lower the individual's risk for autoimmune disorders such as rheumatoid arthritis, lupus, etc. The relationship between the μOR-mediated cell signaling and impact of stimulation of MOR as partner receptors, which influences binding of the second ligand in immunocytes and thereby the outcome on immune cells function in mounting and regulating the immune response/s, needs more detailed

the pathogen or an autoimmune response to an antigen.

**9. Conclusion and future perspectives**

of

*DOI: http://dx.doi.org/10.5772/intechopen.81568*

#### *Immune Cell Activation: Stimulation, Costimulation, and Regulation of Cellular Activation DOI: http://dx.doi.org/10.5772/intechopen.81568*

circulating IgG, and the GG individuals showed the lowest with AG having intermediate values [94]. The immunosuppressive effects documented in opiate naïve individuals can be attributed to altered regulation of PKA and pERK1/2 due to the levels of endogenous opioid. In addition to the absence of G genotype in the immortalized cell lines and based on the cell culture data showing cell cycle arrest observed in the present study (in A549 and HepG2 cell lines), we hypothesize that coactivation of μOR in presence of 118G allele leads to a suppression/arrest of cell division.

To test this hypothesis, another pilot study was carried out in which healthy opiate naïve volunteers were enrolled and the cell count for circulating lymphocyte subsets was studied as a measure of immune competence. Genotypic association studies showed a correlation between the immune cell numbers. Total lymphocyte count showed a significant lowering in cell numbers in 118G-allele-bearing individuals when compared to 118A-bearing individuals. However, cell numbers in all individuals remained within the documented normal range of 500–4000 cell/ml. The GG allele individuals showed significantly lower cell count, averaging 490, which differed markedly from cell numbers observed in AA-bearing individuals with mean numbers of 1976 cells/ml, (p = 0.008) and a correlation factor, r2 of 0.79 between the genotype and average cell numbers. Our data show a significant lowering in all immunocytes, namely leucocyte populations (CD45+ve cells), B lymphocytes (CD10+ve cells), T lymphocytes (CD3, CD4, and CD8+ve cells), NK cells (CD56+ve cells), activated monocytes (CD 11b+ve cells), and mesenchymal progenitors in GG-bearing individuals when compared to AA-allele-bearing individuals but not always in comparison with those bearing the AG allele. This baseline lowering of cell numbers in GG-bearing individuals supports the hypothesis that GG genotype suppresses cell division, and since mounting of a successful immune response and/or overstimulation of immune system, as in case of patients with autoimmune disorders, depends on activation of both innate and adaptive immune responses, the 118G-bearing individuals would be prone to immune suppression due to lack of amplification by selective cell division, a critical step in elimination of the pathogen or an autoimmune response to an antigen.


**1. Introduction**

An instantaneous plane strain rigid plastic solution is obtained for compression of an infinite wedge of orthotropic material confined between two rough plates, inclined at angle 2α, and which intersect in a line. This boundary value problem is ideal for studying qualitative mathematical properties of boundary value problems, including constitutive equations and boundary conditions. For, exact analytical or semi-analytical solutions can be found for many constitutive equations. In particular, such solutions have been presented in [1,2] for isotropic viscoplastic materials and in [3] for the double slip and rotation model. A description of this model can be found in [4].

The present paper provides an analytic solution for rigid plastic orthotropic material. It is assumed that the principal axes of anisotropy are straight lines through the apex of the wedge and orthogonal curves, which are of course circular arcs. This type of orthotropy is of practical interest [5–8] among many others. The paper focuses on qualitative features of the solution such as non-existence of the solution, singularity in the stress and velocity fields, appearance of a rigid region near the plates and transition between the regimes of sticking and sliding. The effect of plastic anisotropy on these features is discussed.

The stress and velocity fields are singular if the regime of sliding occurs in the case of the maximum friction law. A detailed asymptotic analysis of the solution is performed for this case. In particular, it is shown that the asymptotic behavior of the solution is in agreement with the general theory developed in [9].

An applied aspect of the solution found, is that it can be used in conjunction with the method for analysis and the design of flat-rolling proposed in [10]. It is known that solutions, found by means of this method, show a good comparison with experiment [11–13], and are used for verifying solutions found by means of other approximate methods [14,15]. The importance of developing fast approximate methods for the analysis and design of the process of rolling has been emphasized in [16].

### **2. Statement of the Problem**

Two semi-infinite rough plates rotate towards each other with angular velocity of magnitude *ω* about an axis *O* and compress a wedge of polar orthotropic material. The plates are inclined to each other at an angle 2 α (Figure 1). The boundary value problem consists of the instantaneous plane strain deformation of the wedge. The problem is solved in a system of plane polar coordinates (*r*, *θ*) with its origin at *O* and with *θ* = 0, taken as the perpendicular bisector of the angle 2 α. It is assumed that the principal axes of anisotropy coincide with coordinate curves of the coordinate system chosen. Then, *θ* = 0 is an axis of symmetry for the flow and it is sufficient to find the solution in the region *θ* ≥ 0. The components of the stress tensor referred to the polar coordinate system are denoted as *σrr*, *σθθ* and *σrθ*; and the components of the velocity vector as *ur* and *uθ*. There is no material flux through *O*.

**Figure 1.** Geometry of the boundary value problem.

Therefore, the radial velocity should satisfy the following condition:

$$
u\_{\mathcal{I}} = 0\tag{1}$$

at *r* = 0. By symmetry,

$$u\_{\theta} = 0\tag{2}$$

and

$$
\sigma\_{r\emptyset} = 0 \tag{3}
$$

at *θ* = 0. The circumferential velocity should also satisfy the condition:

$$
\mu\_{\theta} = -\omega r \tag{4}
$$

at *θ* = *α*. Finally, the friction law is taken in the form:

$$\begin{cases} \ u\_r = 0 \text{if} |\sigma\_{r\theta}| \le \tau\_f \\ \ \sigma\_{r\theta} = -\tau\_f \text{otherwise} \end{cases} \tag{5}$$

at *θ* = *α*. Here *τf* > 0 denotes the frictional stress at sliding. The magnitude of *τf* will be specified later. The sense of *σrθ* in (5) is dictated by the condition that *ur* ≥ 0 at the plate.

It is assumed that the material obeys Hill's quadratic yield criterion [17] and its associated flow rule. The elastic portion of strain is neglected. In the case of plane strain deformation of a polar orthotropic material, whose principal axes of anisotropy coincide with the coordinate curves of the polar coordinate system, the constitutive equations of the model are:

$$\frac{\left(\sigma\_{rr} - \sigma\_{\theta\theta}\right)^2}{4(1-c)} + \sigma\_{r\theta}^2 = T^2 \tag{6}$$

and

$$
\zeta\_{\tau\tau}^{\tau} = \lambda \frac{\left(\sigma\_{\tau\tau} - \sigma\_{\theta\theta}\right)}{2(1-c)}, \quad \zeta\_{\theta\theta}^{\tau} = -\lambda \frac{\left(\sigma\_{\tau\tau} - \sigma\_{\theta\theta}\right)}{2(1-c)}, \zeta\_{r\theta}^{\tau} = \lambda \sigma\_{r\theta} \tag{7}
$$

where (6) in the yield criterion and (7) is the associated flow rule. The quantity *T* is the shear yield stress in the coordinate system chosen, *c* is a constitutive parameter, *λ* is a non-negative multiplier, *ξrr*, *ξθθ*, *ξrθ* denote the components of the strain rate tensor. The parameter *c* can be expressed in terms of the yield stresses in the directions of the principal axes of anisotropy and can vary (theoretically) in the range − ∞ < *c* < 1 [17]. Eliminating *λ* between the equations in (7) yields:

$$
\xi\_{rr}^{\mathfrak{x}} + \xi\_{\theta\theta}^{\mathfrak{x}} = 0, \quad \frac{\mathfrak{z}\_{r\theta}}{\mathfrak{z}\_{rr}^{\mathfrak{x}} - \xi\_{\theta\theta}^{\mathfrak{x}}} = \frac{(1 - c)\sigma\_{r\theta}}{\sigma\_{rr} - \sigma\_{\theta\theta}}. \tag{8}
$$

It is evident that the first equation here is the equation of incompressibility. The strain rate components are expressed in terms of the velocity components as

$$\xi\_{r\tau}^{\tau} = \frac{\partial u\_r}{\partial r}, \quad \xi\_{\theta\theta} = \frac{1}{r}\frac{\partial u\_\theta}{\partial \theta} + \frac{u\_r}{r}, \quad \xi\_{r\theta} = \frac{1}{2}\left(\frac{1}{r}\frac{\partial u\_r}{\partial \theta} + \frac{\partial u\_\theta}{\partial r} - \frac{u\_\theta}{r}\right). \tag{9}$$

The system of Equations (6), (8) and (9) are supplemented by the stress equilibrium equations:

$$\frac{\partial \sigma\_{rr}}{\partial r} + \frac{1}{r} \frac{\partial \sigma\_{r\theta}}{\partial \theta} + \frac{\sigma\_{rr} - \sigma\_{\theta\theta}}{r} = 0, \quad \frac{\partial \sigma\_{r\theta}}{\partial r} + \frac{1}{r} \frac{\partial \sigma\_{\theta\theta}}{\partial \theta} + \frac{2\sigma\_{r\theta}}{r} = 0. \tag{10}$$

In total, there are five unknowns (three components of the stress tensor and two components of the velocity vector). The equations to solve are (6), (8) and (10). It is understood here that the components of the strain rate tensor in (8) should be eliminated by means of (9). The solution should satisfy the conditions (1) to (5).

### **3. General Stress Solution**

The yield criterion (6) is satisfied by the following substitution:

$$
\sigma\_{rr} = \sigma + T\sqrt{1-c}\cos 2\varphi, \quad \sigma\_{\theta\theta} = \sigma - T\sqrt{1-c}\cos 2\varphi, \quad \sigma\_{r\theta} = -T\sin 2\varphi \tag{11}
$$

where *σ* and *ϕ* are new unknown functions of *r* and *θ*. The direction of flow dictates that *σrθ* ≤ 0 and *σrr* − *σθθ* ≥ 0. Then, it is immediate from (11) that:

$$0 \le \varphi \le \frac{\pi}{4}.\tag{12}$$

Using (11) and (12) the boundary condition (3) transforms to:

$$
\varphi = 0 \tag{13}
$$

at *θ* = 0. Substituting (11) into (10) gives:

$$\begin{cases} \frac{\partial \sigma}{\partial r} - 2T\sqrt{1-\varepsilon}\sin 2\varrho \frac{\partial \varrho}{\partial r} - \frac{2T\cos 2\varrho}{r} \frac{\partial \varrho}{\partial \theta} + \frac{2T\sqrt{1-\varepsilon}\cos 2\varrho}{r} = 0, \\\ -2T\cos 2\varrho \frac{\partial \varrho}{\partial r} + \frac{\partial \sigma}{r\partial \theta} - \frac{2T\sqrt{1-\varepsilon}\sin 2\varrho}{r} \frac{\partial \varrho}{\partial \theta} - \frac{2T\sin 2\varrho}{r} = 0. \end{cases} \tag{14}$$

A standard assumption made in similar problems of the classical theory of plasticity is that *ϕ* is independent of *r* [17]. In this case, the equations in (14) become:

$$\frac{\partial \sigma}{\partial r} - \frac{2T \cos 2\varrho}{r} \frac{d\varrho}{d\theta} + \frac{2T \sqrt{1 - c} \cos 2\varrho}{r} = 0, \quad \frac{\partial \sigma}{2T \partial \theta} - \sqrt{1 - c} \sin 2\varrho \frac{d\varrho}{d\theta} - \sin 2\varrho = 0. \tag{15}$$

The first equation can be immediately integrated to give:

$$\frac{\sigma}{2T} = \left(\frac{d\varphi}{d\theta} - \sqrt{1 - c}\right) \cos 2\varphi \ln\left(\frac{r}{r\_0}\right) + \frac{\sigma\_0(\theta)}{2T}.\tag{16}$$

Here *r*0 is a constant introduced for convenience and *σ*0(*θ*) is an arbitrary function of *θ*. Substituting (16) into the second equation in (15) yields:

$$\frac{d}{d\theta}\left[\left(\frac{d\rho}{d\theta} - \sqrt{1 - c}\right)\cos 2\rho\right] \ln\left(\frac{r}{r\_0}\right) = -\sqrt{1 - c}\sin 2\rho \frac{d\rho}{d\theta} + \sin 2\rho - \frac{d\sigma\_0}{2Td\theta}.\tag{17}$$

Since the right-hand side of this equation is independent of *r*, the coefficient of ln(*r*/*r*0) on the left-hand side must vanish. Then, Equation (17) results in the following two equations:

$$
\left(\frac{d\varphi}{d\theta} - \sqrt{1 - c}\right)\cos 2\varphi = K\rho\sqrt{1 - c}, \quad \frac{d\sigma\_0}{2Td\theta} = \left(-\sqrt{1 - c}\frac{d\varphi}{d\theta} + 1\right)\sin 2\varphi. \tag{18}
$$

Here *K*0 is a constant of integration. Equation (16) becomes:

$$\frac{\sigma}{2T} = K\_0 \sqrt{1 - c} \ln\left(\frac{r}{r\_0}\right) + \frac{\sigma\_0(\theta)}{2T}.\tag{19}$$

The second equation in (18) can be rewritten as:

$$\frac{d\sigma\_0}{2Td\rho} = \left(-\sqrt{1-c} + \frac{d\theta}{d\rho}\right)\sin 2\rho.$$

Eliminating in this equation the derivative *dθ*/*dϕ* by means of the first equation in (18) leads to:

$$
\frac{d\sigma\_0}{2Td\rho} = \frac{1}{\sqrt{1-c}} \left[ 1 - c + \frac{\cos 2\varrho}{(K\_0 + \cos 2\varrho)} \right] \sin 2\varrho.
$$

$$
\frac{\sigma\_0}{2T} = \frac{1}{2\sqrt{1-c}} [-\cos 2\varrho + K\_0 \ln(K\_0 + \cos 2\varrho) + K\_1] \tag{20}
$$

gives:

where *K*1 is constant of integration.

It is seen from (12) and (13) that *dϕ*/*dθ* > 0 at *ϕ* = 0. Therefore, it follows from the first equation in (18) that

$$K\_0 > -1.\tag{21}$$

The first equation in (18) can be integrated to give:

$$\theta \sqrt{1-c} = \varrho - K\_0 \text{arctanh}\left[\sqrt{\frac{1-K\_0}{1+K\_0}} \tan \varrho t\right] \left(1 - K\_0^2\right)^{-1/2} \tag{22}$$

if |*K*0| < 1,

Integrating

$$\theta \sqrt{1-c} = \phi - K\_0 \arctan\left[\sqrt{\frac{K\_0 - 1}{K\_0 + 1}} \tan \phi\right] \left(K\_0^2 - 1\right)^{-1/2} \tag{23}$$

if *K*0 > 1, and

$$
\theta \sqrt{1-c} = \varphi - \frac{\tan \varphi}{2} \tag{24}
$$

if *K*0 = 1. The solution for an important special case of (22), *K*0 = 0, is represented as

$$
\theta \sqrt{1-c} = \varphi.\tag{25}
$$

The constant *K*0 cannot be determined without the solution for velocity.

### **4. General Velocity Solution**

The velocity components may be represented as:

$$u\_r = \frac{\omega r}{2} \frac{d\lg(\varphi)}{d\theta} \quad \text{and} \qquad u\_\theta = -\omega r \lg(\varphi) \,. \tag{26}$$

The condition (1) and the first equation in (8) are then automatically satisfied for any choice of the function *g*(*ϕ*). Equations (9) and (26) combine to give:

$$\xi\_{rr} = \frac{\omega}{2} \frac{d\mathfrak{g}(\mathfrak{q})}{d\theta}, \quad \tilde{\xi}\_{\theta\theta} = -\frac{\omega}{2} \frac{d\mathfrak{g}(\mathfrak{q})}{d\theta}, \quad \tilde{\xi}\_{r\theta} = \frac{\omega}{4} \frac{d^2\mathfrak{g}(\mathfrak{q})}{d\theta^2}. \tag{27}$$

Substituting (11) and (27) in the second equation in (8) yields:

$$\frac{d^2\mathcal{S}}{d\theta^2} + 2\sqrt{1-c}\tan 2\varphi \frac{d\mathcal{g}}{d\theta} = 0\tag{28}$$

or

$$\frac{dG}{d\theta} + 2\sqrt{1 - c}\tan 2\varphi G = 0\tag{29}$$

where *G* = *dg*/*dθ*. Replacing in (29) differentiation with respect to *θ* with differentiation with respect to *ϕ* by means of the first equation in (18) results in:

$$\frac{dG}{d\varphi} = -\frac{2\sin 2\varphi}{(K\_0 + \cos 2\varphi)} G. \tag{30}$$

Integrating gives

$$G = G\_0 (K\_0 + \cos 2\varphi). \tag{31}$$

Here *G*0 is constant of integration. The definition for G and (31) combine to give:

$$\frac{d\mathbf{g}}{d\theta} = \mathbf{G}\_0 (\mathbf{K}\_0 + \cos 2\varphi). \tag{32}$$

Replacing here differentiation with respect to *θ* with differentiation with respect to *ϕ* by means of the first equation in (18) results in:

$$\frac{d\mathcal{S}}{d\boldsymbol{\varrho}} = \frac{G\_0 \cos 2\boldsymbol{\varrho}}{\sqrt{1-\boldsymbol{c}}}.\tag{33}$$

It is seen from (2), (13) and (26) that *g* = 0 at *ϕ* = 0. The solution of Equation (32) satisfying this condition is:

$$g = \frac{G\_0 \sin 2\varphi}{2\sqrt{1-\varepsilon}}.\tag{34}$$

Substituting (33) into (26) and then the resulting expression for the circumferential velocity into (4) yields:

$$G\_0 \sin 2\varphi\_w = 2\sqrt{1-c} \tag{35}$$

where *ϕw* is the value of *ϕ* at *θ* = *α*. The dependence of *ϕw* on *α* follows from the solution of the first equation in (18).

To complete the solution of the boundary value problem, it is necessary to satisfy the boundary condition (5).

### **5. Solution of the Boundary Value Problem**

The boundary condition (5) comprises two friction regimes, sticking and sliding. These regimes should be treated separately.

### *5.1. Regime of Sticking*

In this regime, the boundary condition (5) becomes *ur* = 0 at *θ* = *α*. It is seen from the definition for G and (26) that this condition is equivalent to the condition *G* = 0 at *θ* = *α*. Then, it follows from (31) that:

$$\mathbb{K}\_0 = -\cos 2\varphi\_w. \tag{36}$$

In this case the dependence of *θ* on *ϕ* is given by (22). Eliminating *K*0 in (22) by means of (36), it is possible to find that the argument of the inverse hyperbolic tangent function is equal to 1 at *θ* = *α*. Therefore, the left-hand side of (22) approaches infinity (or negative infinity) unless *K*0 = 0. In the latter case, it is more convenient to use the solution (25). It follows from this solution, (35) and (36) that

$$\varphi\_{\rm tr} = \frac{\pi}{4}, \quad \mathcal{G}\_0 = 2\sqrt{1-c} \text{ and } a = \frac{\pi}{4\sqrt{1-c}} = a\_{\rm cr}. \tag{37}$$

The solution at sticking is possible only if *α* and c satisfy the third equation. Another restriction on the existence of the solution at sticking is that the shear stress at *θ* = *α* is less or equal to *τf* involved in (5). It is seen from the first equation in (37) and (11) that *σrθ* = − *T* at *θ* = *α* if the regime of sticking occurs. Since *T* is the maximum possible value of the shear stress in the polar coordinate system, a necessary condition for the existence of the regime of sticking is that *τf* = *T*. If *τf* < *T* then no solution at sticking exists.

Assume that *τf* = *T*. The relation between *α* and *c* in (37) has been derived assuming that plastic yielding occurs in the region 0 ≤ *θ* ≤ *α*. In the case of rigid/plastic solids, rigid regions may appear. In the case under consideration, the solution at sticking is possible if *α* > *αcr* and the material in the region *α* ≥ *θ* ≥ *αcr* is rigid. It worthy of note that the stress solution at *K*0 = 0 given in Section 3 is valid in the rigid region. Therefore, the yield criterion is not violated in the range *α* ≥ *θ* ≥ *αcr* and the solution is complete.

### *5.2. Regime of Sliding*

It is convenient to consider two cases, *τf* = *T* and *τf* < *T*, separately. Assume that *τf* = *T* and *α* < *αcr*. Then, no solution at sticking exists and it is necessary to find the solution at sliding. It follows from (5), (11) and (35) that the first and second equations in (37) are valid. The equation for determining *K*0 follows from (22) or (23). It is however convenient to start with the special case *K*0 = 1. In this case Equation (24) is valid. Therefore, this special case occurs only if *α* and c satisfy the following equation:

$$\alpha = \frac{(\pi - 2)}{4\sqrt{1 - c}} = \alpha\_5. \tag{38}$$

It is evident from (37) and (38) that *αs* < *αcr*. Equation (22) is valid in the range *αs* < *α* < *αcr*. In this case, the equation for *K*0 is

$$\ln\sqrt{1-c} = \frac{\pi}{4} - K\_0 \operatorname{arctanh}\left[\sqrt{\frac{1-K\_0}{1+K\_0}}\right] \left(1-K\_0^2\right)^{-1/2}.\tag{39}$$

Equation (23) is valid in the range 0 < *α* < *αs*. In this case, the equation for *K*0 is

$$
\alpha \sqrt{1-c} = \frac{\pi}{4} - K\_0 \arctan\left[\sqrt{\frac{K\_0 - 1}{K\_0 + 1}}\right] \left(K\_0^2 - 1\right)^{-1/2}.\tag{40}
$$

Equations (39) and (40) should be solved numerically.

Prandtl's friction law reads *τf* = *mT* where 0 ≤ *m* ≤ 1. The case *m* = 1 has been treated above. Therefore, assume that *m* < 1. In this case, no solution at sticking exists. The friction law (5) becomes *σrθ* = −*mT* at *θ* = *α* (or *ϕ* = *ϕw*). Then, it follows from (11) that:

$$
\varphi\_{\mathcal{O}^{\mathcal{O}}} = \frac{1}{2} \arcsin m. \tag{41}
$$

The value of *G*0 is found from (35) and (41) as:

$$G\_0 = \frac{2\sqrt{1-c}}{m}.\tag{42}$$

The equation for determining *K*0 follows from (22) or (23). As before, it is more convenient to consider special cases first. The values of *αcr* and *αs* are now determined from (24) and (25) as:

$$
\mu\_{\mathcal{U}} = \frac{\varrho\_w}{\sqrt{1-\mathcal{c}}} \quad \text{and} \quad \mu\_s = \left(\varrho\_w - \frac{\tan \varrho\_w}{2}\right) \frac{1}{\sqrt{1-\mathcal{c}}}.\tag{43}
$$

In these equations, *ϕw* should be eliminated by means of (41). Equation (22) is valid in the ranges *αs* < *α* < *αcr* and *α* > *αcr*. In this case, the equation for *K*0 is:

$$\alpha \sqrt{1-c} = \rho\_w - K\_0 \operatorname{arctanh} \left[ \sqrt{\frac{1-K\_0}{1+K\_0}} \tan \phi\_w \right] \left( 1 - k\_0^2 \right)^{-1/2}. \tag{44}$$

The value of *K*0 is positive in the range *αs* < *α* < *αcr* and negative in the range *α* > *αcr*. Equation (23) is valid in the ranges 0 < *α* < *αs*. In this case, the equation for *K*0 is:

$$a\sqrt{1-c} = q\_{\overline{w}} - K\_0 \arctan\left[\sqrt{\frac{K\_0 - 1}{K\_0 + 1}} \tan \varphi\_w\right] \left(K\_0^2 - 1\right)^{-1/2}.\tag{45}$$


*5.1. Use Case Multirotor Drone*

The validation of the proposed sizing approach is conducted using data from a reduced-scale multirotor drone with a GTOW of 15 kg. An overview of the drone, composed of eight propulsion chains with U7-V2 420 KV motors and P18×6.1 propellers, is presented in Figure 9. Each set of four propulsion chains is powered by an 6S1P LiPo battery. Detailed specifications of the drone can be found in Table A3 in the Appendix A. It is worth noting that the sizing methodology does not consider the coupling effect of the coaxial configuration. This effect is disregarded for the drone used in the validation step of the sizing methodology. According to [38], at the scale level of the validation drone, the coupling effect of the coaxial configuration on propulsion efficiency does not exceed 6%. This demonstrates the effectiveness of the proposed sizing methodology. To demonstrate the efficacy of the sizing methodology, a simulation of the propulsion chain sizing approach is carried out using MATLAB code. A database comprising parameters of 45 randomly selected electric motor examples, including the motor used in the drone shown in Figure 9, is created based on data provided by T-motor. Flight mission data specific to this drone is also incorporated.

**Figure 9.** multirotor eVTOL drone.