**4. How women can drive meaningful CSR**

Going back to the women's liberation movement in the late 1960s and early 1970s, some CSR had already strayed into gender issues. Companies are increasingly addressing the issue of gender equality and incorporating it into their CSR programs. Gender inclusion in CSR can play a dynamic role in achieving gender equality in the workplace through activities, initiatives, strategies, and policies that provide female employees with equal access to job opportunities and equal treatment in the workplace. Likewise, gender equality is playing an increasingly important role at the firm level as a key factor in strengthening CSR strategies. Simultaneously, the involvement of women in leadership positions enhances CSR activity [4, 7]. From these evidences, it would seem that there is a link between the gender equality and CSR. As CSR performance are seen to be the consequence of board choices, a board's characteristics and attributes can be crucial in achieving better CSR results. Thus, it is vital to comprehend the theories and empirical evidences behind the impact of female directors and CSR.

Literatures have offered various reasons to support the notion that board gender diversity enhances the CSR. One of the most frequently used theories to explain the relationship between diversity boards and CSR is agency theory. It is predicated on the notion that there is a separation of ownership and control, which results in expenses associated with resolving conflicts of interest between a principal (i.e. shareholders) and an agent (i.e. a manager). To migrate agency costs, the firm incorporates a range of corporate governance mechanisms, including law, incentives, shareholder rights, and monitoring [32]. And the board of directors is recognized as one of the most important aspects for monitoring managerial behavior and mitigating the firm's agency problems. They are usually elected or appointed by shareholders and represent the company's shareholders. The board is responsible for making important strategic decisions, providing leadership, monitoring and supervising top management on behalf of shareholders.

In agency theory literatures, researchers often try to emphasize on the link between board composition and firm value, investment and firm's decisions. In recent years, researchers have increasingly focused their research on gender diversity. Based on social psychology theory, people often seek out people who have similar backgrounds, perspectives, and values, which are then reinforced in intragroup communication. The more we identify with our in-group, the more distant we feel from members of the out-group. Furthermore, when we strongly identify as an in-group member, we tend to elevate other members of the group. This phenomenon is commonly referred to as in-group favoritism or in-group bias, and it can be described as a privilege issue in which members are favored, while people in out-groups are discriminated against due to characteristics that they cannot change [33]. Sexual orientation, gender, ethnicity, race, religion, physical abilities, and age are examples of these characteristics. According to this viewpoint, researchers who believe in agency theory frequently assume male directors regard female directors as out-group members and may have a negative social bias regarding their appointment to board committee [34]. To put it another way, female directors are breaking away from the "old boys club"

and presenting a more independent point of view [2, 19]. This could also imply that firms with a high proportion of female directors have less conflict of interest between shareholders and managers, and the board is more likely to make decisions that benefit the firm in the long run.

Other researchers employ stereotypes concept to explain why different genders of directors behave and make decision differently. Stereotyping is a cognitive process in which a characteristic is associated with a group [35]. It is not an abnormality in human social behaviors and values because it is simply human nature to form opinions about other people and their actions based on our understanding and expectations of ourselves. For instance, the literature on gender stereotype often associated female with complex moral reasoning ability, risk-averse, empathy, caring, kindness, and interpersonal sensitivity. Men, on the other hand, are said to have attribute with agentic qualities such as assertive, ambitious, aggressive, independent, self-confidence, daring and competitive [27–29, 36]. In this regard, men and women differ both psychologically and cognitively. Thus, female directors may result in a greater level of compassion and sensitivity to CSR and stakeholder concerns.

In short, if CSR generates long-term value for stakeholders without endangering people, the environment or the economy, and female directors have the potential to reduce agency costs, firms with a high level of gender diversity may be more likely to improve CSD and CSR performance for the long-term benefit of all stakeholders [37].

Even though several researchers have used agency theory to explain the relationship between the gender diversity board and CSR, the theory appears to fall short of covering all aspects of this relationship. As previously stated, the agency cost theory employs social psychology concepts such as in-group bias and stereotype concept to support the logic underlying the possible positive relationship between female director and CSR. Likewise, Spitzeck [38] contends that the fact that companies are increasingly using CSR committees does not explain why they do so or how CSR governance structures might evolve, implying that the agency theory cannot fully explain the link between CSR and female directors. Similarly, Hussain [39] argued that no single theory fully accounts for all the hypothesized relationships.

Another important theory related to the arguments in favor of board gender diversity is the stakeholder theory. Though both agency and stakeholder theories advocate for the alignment of shareholder, stakeholder, and management interests, it is important to recognize that there are significant differences between the two theories. Unlike agency theory, stakeholder theory describes the composition of organizations as a group of different individual groups with diverse interests. These interests, when taken together, are a representation of the will of the firm. Business choices should take into account the interests of these collective groupings and also balance a multiplicity of stakeholders' interests that can affect or are affected by the achievement of an organization's objectives. Any disagreement between these groups reflects the loss of these interests. Therefore, the stakeholder engagement is crucially important for firms to justify the legitimacy of their operations [40]. Organizational legitimacy does not only ensure the inflow of capital, labor, and customers necessary for the company's viability but also could reduce the likelihood of other disruptive actions [41].

Companies usually attempt to gain legitimacy by disclosing social and environmental verifiable data and information [15] and CSD is a strategy that businesses can use to respond to societal expectations [16] and gain legitimacy of the company's activities in the eyes of stakeholders, who have a diverse expectation. It also signals to stakeholders that companies' actions are appropriate and desirable [17]. In other words, CSD is part of the dialog between the company and its stakeholders as it

provides information to stakeholders on the social and environmental impacts of corporate activities.

In recent year, CSR has become one of the major concerns for many companies and their managers. Gender equality and human rights are also two of the EU's founding values and the United Nations' guiding principles. Also, the corporate world's efforts to re-premiumize have compelled many European countries to implement gender quotas on corporate boards. Consequently, CSR has become the center of interest for both internal and external stakeholders. Unsurprisingly, many firms would choose to reveal social and environmental verified facts and information in order to balance the interests of a variety of stakeholders.

According to stakeholder theory, female directors are more likely to assist the business in developing an orientation toward the interests of diverse stakeholders [42]. Female directors are more likely to oriented toward philanthropic causes and typically have experience in nonprofit industries, firm with greater gender diversity have been shown to give more to community service organizations and programs [43]. In a similar vein, female directors are more likely to enhance stakeholder management by providing firms with relevant knowledge to enhance their ability manage these interests [44]. For instance, examining a sample of FTSE 350 firms, Jizi [8] shows that female board representation has a positive impact on CSR engagement and reporting as well as on the development of ethical policies. Rather than fulfilling the traditional monitoring function to resolving conflicts of interest between management and shareholders, female directors are more likely to exercise their power in ways that benefit a broader variety of stakeholder interests.

Another strand of research has used upper echelon theory-UET [45, 46] to explain the relation between female directors and CSR. According to the upper echelon theory-UET [45, 46], each decision maker brings a cognitive base and values to a decision, creating a screen between the situation and his/her eventual perception of it [46]. Thus, the more these values are shared by top management, the more likely that the directors to have effective and efficient information-sharing, joint decision making, and collaboration in order to formulate and implement good strategies and other decisions, including those for developing effective firms' environmental policy [47]. As mentioned earlier, female directors are said to possess a different set of psychological values and perceptions compared to their male counterparts. Female leaders are more likely to have a participative, democratic, and communal leadership style and more concerned about the environment. Accordingly, female directors may help boosting the firm's value in terms of environmental protection.

Likewise, a gender difference perspective can be deeply embedded in gender socialization theory, which states that socialization encourages and rewards different behaviors in men and women: individualistic and competitive behaviors in men versus cooperative and altruistic behaviors in women (for e.g. see [48, 49]). These diverse public roles and expectations give rise to career paths and leadership styles that differ depending on the gender of the leaders. As a result, women in leadership roles tend to take more participative and relationship development approaches, and they are more likely to pursue long-term strategies and stakeholder focused outcomes, which are critical to successful CSR practices [50, 51].

Given the theoretical justifications provided above, it is evident that female directors play an essential role in implementing CSR into a company's strategic objective. Nevertheless, no single theory fully explains for all the relationships. Indeed, the majority of ideas in favor of female directors are predicated on the premise of psychology idea that female directors have higher moral and communal attitudes than male

directors. As a result, female directors are more concerned with CSR and are more likely to launch CSR initiatives. However, facts might not always imply truth. The board decision is much more about individual traits. For instance, in recent qualitative study has revealed that, while female directors are highly impacted by their psychological perceptions that are aligned with CSR, a number of hurdles have been discovered for female directors to properly participate [52]. It is also argued that female executives must act like men in order to succeed. Female directors, in contrast to the female stereotype for the population, are more risk-taking than their male counterparts [4]. Furthermore, the preceding theories do not include the environmental context, which could be the important determinant of the relationship. For instance, in some countries, a female director's ability to influence on corporate strategy may be limited due to stereotyping challenges and culture. In short, the benefit of female director is prevalence in some extend, nonetheless the future investigations still needed.

In this stance, as part of corporate strategy, the board of directors play preliminary role in determining the socially responsible behavior of an organization [5, 53, 54], being relevant to this role of the fulfillment of social and environmental conscientiousness [55, 56].


4. Conclusion

Perhaps, larger amount of DHT is needed to stimulate AR-related DKK1 expression. High level of DKK1 was observed within 3–6 h after 50–100 nM DHT treatment previously [45]. In mice, DKK1 expression, similarly to BMP2/4 expression, is time-dependent in hair cycle and is changing according to hair follicle cycle phase [70]. We could not find severe relevant difference in the pattern of WNT signaling activity, which is not surprising, since fluctuations in these networks are closely associated with current hair cycle phase, which is, firstly, poorly studied in human, and secondly, impossible to verify in a culture of separated dermal papilla. SFRP1 is one of the significant overexpressed in dataset 1 genes that could stimulate WNT pathway. Other speculations we developed on the basis of pathway analysis of current micro-

Figure 6. Model of BMP, WNT, NOTCH, and AR-related pathway cross-talk in AGA dermal papilla cell.DKK4, EGR1, HEY1 and ID1-2 are underexpressed genes in nonbald DP cells 1–3 h after DHT treatment (datasets 5–6). The higher color saturation reflects the higher degree of differential expression ([0.5; 0.5] log-change interval, 0.05 p-value cutoff). SFRP1 and NR4A2 are underexpressed genes in bald DP cells 1 and 3 h after DHT treatment (datasets 3–4). BMP2, SMAD6-7, HEY1, and ID1-2 are overexpressed genes. DKK1 and DKK4 inhibit WNT due to feedback loop in WNT signaling. AR regulates WNT pathway through LATS2, EGR1, and CTNNB1. BMP signaling inhibits activity of WNT pathway. BMP signaling in general inhibits WNT signaling. NOTCH pathway has negative effect on AR expression via HEY1. EGR1 and HES1 have significant changes in expression at all time points assessed after DHT treatment, so their expression is shown

As shown in Figure 7, numerous feedback loops regulate BMP/WNT crass-talk themselves, as well as intersections with AR and other signaling pathways, such as SHH, NOTCH, or Hippo/

array data are shown (Figure 6).

YAP1 [73].

with heat maps.

160 Hair and Scalp Disorders

Hair follicle miniaturization in androgenic alopecia is complex process induced by AR influence on dermal papilla cell function. Signaling pathways are useful models of complicated protein interactions in dermal papilla cell during hair follicle cycle. Cross-talks between AR and TGFB, BMP, and WNT signaling are believed to promote shortage on amount and functional activity of dermal papilla cells in bald scalp in AGA. WNT signaling is considered to be the main trigger of telogen-anagen transition that drives repopulation of dermal papilla progenitor cells from bulge niche. Its impairment could potentially lead to AGA phenotype. Detailed understanding of molecular causes of hair loss in AGA is still has not been reached. New standardized studies of hair follicle cell cycle in human scalp are required to find humanspecific information about dermal papilla and stem cells maintenance during their lifetime renewals.

Pathway analysis of cDNA microarray data from cultured immortalized human DP cells from balding (frontal) AGA scalps reveals activation of chromatin remodeling and metaphaseanaphase transition pathways. Observed slight up-regulation of cell cycle inhibitor protein CDKN2A confirms other studies and indicates up-regulation of DNA repairing pathways. Expression of AR-related genes NOS3, EGR1, SMAD6, BTG2, and LATS2 was significantly down-regulated exclusively after DHT treatment in frontal DP cells compared to occipital bald cells from AGA scalp. Differential expression of TGFB1I1, TGFB2, THBS1, PTHLH, and ANGPT2 was not changed dramatically, but they appeared among top 25 regulators of underexpressed genes in bald DP cells samples. TGFB1I1 is a reported AR cofactor. Ligands TGFB2, THBS1, PTHLH, and ANGPT2 launch receptors signaling related to catagen progression (and ligand CTGF to telogen). MIR106A was found among the top of significant miRNA regulators of DE genes in bald DP cells.

After 1 h of DHT treatment, EGR1 and EGR2 genes reduced their expression by more than 70% in balding versus nonbalding scalp. SPRY1 and CEPBD and NR4A2 genes had also negatively differential expression trend in bald samples. CYB1P1 and CTGF had positively increased in bald samples versus nonbald after 1 h of DHT treatment. DKK1 as WNT inhibitor was underexpressed in nonbald DP samples 1 h after DHT treatment. DKK4 was, vice versa, overexpressed in nonbald DP cells in the first 1 h. NR4A2, EGR1, HES1, and NR4A2 had remarkable difference in expression between bald and nonbald DP cell samples. Unfortunately, role of these proteins was acquired in cancer-related studies that make them hardly applicable to the hair follicle research. Further investigation is required.

The obtained results suggest that AGA dermal papilla in frontal scalp area differs from occipital ones unlikely due to downregulation of proliferation and increased expression of catagen triggers, but rather due to reduced expression of anagen triggers.


**2. Contributions**

The Special Issue "Failure Analysis of Biometals" comprises ten original research articles [1–10] covering a great common range of metallic biomaterials (Ti alloys, CoCrMo alloys, Mg alloys, NiTi alloys) and their failure mechanisms (corrosion, fatigue, fracture, and fretting wear) that commonly occur in medical implants and surgical instruments. This collection of studies also includes two review papers [9,10]: the corrosion behaviour of new generation low modulus titanium alloys for implant applications, and the three-dimensional (3D) printed acetabular cups for hip replacement implants reviewing the clinical use of 3D printing in orthopaedics.

Starting with research articles, Alqedairi et al. [1] studied the cyclic fatigue behaviour of endodontic rotary files made of nickel titanium (NiTi) alloys. Owing to the cyclic rotation of this dental instrument within the curved canal, fatigue fracture can occur over time. The failure can be either flexural or torsional in its loading nature. Three different rotary instruments including ProTaper Universal (PTU), ProTaper Gold (PTG), and ProTaper Next (PTN) were assessed in this work. Artificial canals were machined in stainless steel blocks. Fifteen rotary instruments of each type (five types of PTU and PTG and three types of PTN) were rotated with 300 rpm until fracture. Weibull reliability analysis and the probabilities of survival calculated for the PTU, PTG, and PTN instruments showed the PTG series to offer a higher reliability than the PTU and PTN series. The PTG instruments were found to have a superior cyclic fatigue behaviour when compared with the PTU and PTN series. The higher fatigue resistance of the PTG and PTN instruments was attributed to the thermomechanical treatment performed on the NiTi alloy of these instruments. Fatigue is understood to be the most common failure mechanism in rotary instruments; therefore, international standards for testing against fatigue failure of these medical devices should be established in order to reduce considerable differences in their behaviour.

In a two-part study conducted by Goswami and co-authors [2,3], the failure of a stainless steel 316 L proximal humerus internal locking system (PHILOS) plate and screws, which had been used for a pantalar arthrodesis, was investigated. The research employed both experimental (SEM/EDS, microstructural analysis using electron backscatter di ffraction (EBSD), hardness tests, and fractography) and computational (finite element modelling) approaches. The results of fractography, particularly SEM investigations, indicated the occurrence of corrosion fatigue failure initiated by crack initiations in the distal region of the steel plate, leading to crack propagations towards the proximal region and a final brittle fracture. Crack initiations in the plate were reported to be the result of the inclusions and corrosion pits. The failure of the screws was because of overloading, which occurred ahead of the plate from the proximal end. Finite element analysis (FEA) on the implant system, implemented using ANSYS, showed increased von Mises stresses in the cortical screws as the angle between the screws and the plates increased. Moreover, the stress magnitudes were found to be lower (by 25.5%) in the locking screws when compared with the cortical screws, which was understood to be because of the fixed angles of the locking screws onto the plate (less range of motion).

Returning the focus again to nickel titanium alloys in this Special Issue, the in vitro corrosion assessment of porous NiTi structures was studied by Ibrahim et al. [4] for bone fixation applications. The structures were fabricated by additive manufacturing, which is expanding very rapidly nowadays. It is, however, noted that additively manufactured NiTi structures still have some issues such as poor surface quality and presence of impurities and defects. Employing the selective laser melting (SLM) technique, NiTi samples in both the bulk and porous forms (porosity levels of 15–50%) were fabricated. The electrochemical corrosion characteristics of these SLM NiTi samples were found to be similar to those of conventionally fabricated NiTi samples. The 50% porous structures showed the highest Ni ion release level owing to their biggest surface area exposed to the corrosive environment. The main finding in this work was that the SLM manufacturing process employed to fabricate NiTi structures for bone fixation applications did not cause a significant deterioration in their corrosion resistance.

In medical implants, where there is contact between the implant material and living bone, a stable implant–bone interface is essential for clinical success of fixation by osseointegration and bone ingrowth. A second important factor vital for achieving success is that there be a minimal mismatch between the mechanical properties of the implanted prostheses and the host skeleton. If there is a substantial discrepancy between the said properties, significant stress shielding can occur, causing adverse e ffects on the implant and/or the host skeleton. Additive manufacturing can be wisely employed to fabricate sti ffness-modulated implants to minimise stress shielding failure, which is one of the interesting areas of research at present. In a study by Jahadakbar et al. [5], bone fixation plates were designed and manufactured out of NiTi alloys using additive manufacturing (SLM method), such that the sti ffness of the Nitinol plates was modulated. Five di fferent porosity levels (17%, 20%, 24%, 27%, and 30%) as well as a bulk plate (0% porosity) were designed and analysed using finite element modelling (Abaqus software), showing a good agreement with the experimental results of mechanical testing. Following the model verification, Ni-rich fixation plates were manufactured, offering a superelastic behaviour. Di fferential scanning calorimetry (DSC) was employed in order to identify the transformation temperatures (TTs) from −90 to 100 ◦C in a nitrogen atmosphere. The DSC results showed a small variation in the transformation temperature of di fferent points in the fixation plates owing to the various thermal histories that the complex plates experienced during the additive manufacturing process. A post-processing heat treatment may thus be required in order to achieve homogeneity in the as-fabricated parts.

Magnesium (Mg)-based alloys exhibit biodegradable and biocompatible characteristics, enabling them to be used in degradable implants that can dissolve in the body after the treatment of a medical condition. However, there are still a number of aspects that need to be further researched and addressed (e.g., mechanical and corrosion properties and degradation behaviour). Taking into account that calcium (Ca) is an important element in the bone structure, Chen et al. [6] developed a magnesium-based alloy (Mg-1Ca-0.5Zr) using casting. Zirconium (Zr) was also added to enhance the grain refinement in the alloy. The main aim was to assess the mechanical properties and surface corrosion mechanism in the developed alloy. A heat treatment (400 ◦C for 8 h, followed by quenching in water) was applied to the alloy to improve its mechanical properties. In addition to tensile tests, erosion wear tests were performed on the alloy samples with and without the heat treatment. Moreover, a potentiodynamic polarization test was performed on these samples and a pure Mg sample. The heat treatment was found to enhance the ductility and reduce the corrosion rate of the alloy. Moreover, it was reported that the Mg alloy subjected to the heat treatment formed a protective calcium phosphate film when immersed and tested in a simulated body fluid. This protective layer decreased the corrosion rate considerably.

In an investigation on retrieved total hip replacement implants, particularly the metallic taper junction known as the head–neck junction, corrosion damage to the neck part of 137 femoral stem implants was analysed using the Goldberg's scoring method [7]. The studied stems were made of three biometals including CoCrMo, stainless steel (SS), and titanium alloy. The neck surface was divided into eight distinct zones to statistically study the distribution and severity of corrosion damage. The distal region was found to have more corrosion damage compared with the proximal region of the neck. The most severe corrosion also occurred in the medial distal zone. This study suggests that retrieval studies of head–neck taper junctions should assess the corrosion damage in various zones of metallic implants separately.

In the same area of the head–neck taper junction in hip implants, Fallahnezhad et al. [8] studied material loss as a result of fretting wear with a focus on the role of assembly force (impaction force applied by surgeons to assemble the junction intraoperatively). Both the head and neck components were made out of CoCrMo alloy, with an angular mismatch of 0.01◦. Developing an adaptive finite element model for fretting wear, four assembly forces (2, 3, 4, and 5 kN) were applied to the junction followed by a walking gait loading (1,025,000 cycles). The results showed the direct e ffect of assembly force; the higher the force, the greater the material removal owing to fretting wear. It was discussed that a high assembly force can generally improve the stability of the junction; however, it may also enhance the wear damage to the material. This study did not include corrosion in the simulations; thus, further research is suggested to create novel models to capture both fretting wear and corrosion simultaneously.

In a review on the clinical use of 3D printing (additive manufacturing) in orthopaedics, Dall'Ava et el. [9] focused on titanium acetabular cup implants used in total hip replacement, where they compared 3D printing with conventional manufacturing. This review defined the rationale of additively manufactured acetabular cups from both the clinical and engineering perspectives. A number of interesting aspects associated with the topic were discussed and summarised. These include the key manufacturing-related variables that can have an influence on the characteristics and properties of the fabricated implants, and the limitations associated with this manufacturing technology. Additively manufactured titanium cups have presented promising early clinical outcomes. It is, however, important to note that more detailed studies are still needed to look at their failure and long-term performance in the body.

Finally, this Special Issue presents a review article [10] on the corrosion behaviour of new generation titanium alloys (β-type o ffering low Young's modulus) that can be desirably used for medical implants.

There are some existing concerns about the toxicity of the two alloying elements (aluminium and vanadium) in the most commonly used titanium alloy in medical applications (Ti-6Al-4V). Furthermore, the sti ffness (Young's modulus) of this alloy, which is approximately 110 GPa, is much higher than the typical sti ffness of the bone (10–30 GPa). This can result in stress shielding under the loads of physical activities and, consequently, prosthesis loosening, bone loss, and fracture failure. To address these issues, extensive research has been done to develop new generation β-type titanium alloys with lower levels of sti ffness. These alloys contain beta-stabilising elements, for example, Nb, Ta, and Zr, which are also non-allergic and non-toxic. Although there has been a lot of work around the

improvement of mechanical properties in these alloys, their corrosion behaviour still needs further research (given that the implant working environment is corrosive). In this article, Afzali et al. [10] reviewed and discussed a number of key factors (fabrication process, chemical composition, passive layer, mechanical treatments, body electrolyte properties, and constituent phases) influencing the corrosion behaviour/resistance of new generation titanium alloys. The e ffects of α and β phases and their dissolution rates on the oxide layer and corrosion behaviour were also reviewed. It was recommended that the microstructure of these new generation alloys should contain suitable amounts of α and β phases to achieve a high corrosion resistance, as well as a stable oxide layer.


**3. Material and methods**

484 Advances in Cancer Therapy

available body of evidence suggests that these AMLs are unlikely to profit from an alloSCT

Thus, it could be concluded that patients with AML with t(8;21), AML with inv(16)/t(16;16), AML with *NPM1*mut/ *FLT3*-ITDneg and AML with *CEBPA* mutations should not considered for alloSCT. Nevertheless, it is important to stress that sufficient evidence so far exist only for the AML with recurrent cytogenetic abnormalities and the benefits for the two additional genetic low-risk AML entities should be validated in the larger studies in the

 In the near future these results will also need to be considered more specifically in the light of the extended scale of allogeneic stem cell transplantation strategies with respect to reduced-intensity conditioning regimens and in relationship to different transplant sources and donor types (matched unrelated and haploidentical donors, umbilical stem cell

Correct diagnosis of the diverse subtypes of acute myeloid leukemia (AML) and acute lymphoblastic leukemia (ALL) play a central role for individual clinical risk stratification and therapeutic decisions. Modern therapeutic concepts of AML are based on individual risk stratification at diagnosis and during follow-up. The ultimate test of any disease diagnostic algorithm approach is its usefulness in guiding the selection of effective treatment strategies. (Haferlach et al., 2007, Löwenberg et al, 2008b) As discussed above, cytogenetic as well as various genomic markers (gene mutations, gene overexpression) may provide input for algorithms for remission induction and post-remission treatment decisions. At the same time, it remains appropriate to realize that prognostic factors in fact remain a moving target and they are only relevant to therapies available at a given time. Algorithms that provide a basis for risk-adapted therapeutic choices may include immunological markers, cytogenetic factors, molecular markers as well as clinical parameters (e.g., age, attainment of an early or late complete remission) and hematological

On the other hand, the more carefully AML is studied, the clearer it becomes that there is considerable heterogeneity between cases with respect to morphology, immunological phenotype, associated cytogenetic and molecular abnormalities and, more recently, patterns of gene expression. This is reflected in the substantially different responses to treatment. Some entities are becoming so distinct that they are regarded as different diseases with

In order to improve and simplify the diagnosis and management of AML patients that are diagnosed and treated at the at the University Clinic of Hematology-Skopje we conducted a prospective study to establish and standardize a diagnostic algorithm based on minimal screening tests which will facilitate risk adapted therapy for each single AML patient. The aims of our study were: first, to establish the correct lineage assignment of the blast cells, second, to evaluate the incidence of the favorable genetic markers PML/RAR, AML1/ETO and CBF/MYH11 among the AML cases, then to correlate the obtained results with the patient age, comorbidities, and performance status and consecutively to select the effective

determinants (e.g., secondary AML, white blood cell count at diagnosis).

treatment strategy for each single acute leukemia patient.

(Schlenk et al., 2008).

future (Schlenk et al., 2008).

grafts).(Löwenberg et al, 2008a)

specific approaches to treatment.

**2. Motivation and aim of the study**

#### **3.1 Patients and samples**

A total of 76 adult (>15 years) patients (from initially 77 tested) with acute leukemia who were consecutively admitted at the Clinic of Hematology-Skopje from January through December 2008 were enrolled in this study. The median age of the patients (41 men, 35 women) was 52 + 18.66 years (range 16-80), and most of the patients 37 (48.7%) were between 55 and 75 years old. The diagnosis was made by standard morphological examination and cytochemical analyses of bone marrow smears according to the criteria established by the FAB Cooperative Study Group (Bennet et al, 1997) and confirmed by immunophenotyping of bone marrow aspirates and/or peripheral blood samples (Bain et al., 2002; Bene et al., 1995 ) following the criteria of the European Group for the Immunological Classification of leukemias (EGIL) and the British Committee for Standards in Hematology (BCSH) (Bain et al., 2002; Bene et al., 1995). Consecutively, patients were further stratified in the adequate genetic AML entities according to the results of the molecular analyses. The samples contained more than 20% of blast cells (most of which had more than 50%). All patients were tested for the presence of the fusion transcript of the mayor recurrent cytogenteic abnormalities in AML (PML/RAR, AML1/ETO, CBF/MYH11) by RT-PCR, according to standard procedures. (Gabert et al.2003; Beillardet al, 2003).

Fig. 1. Average age of the patients in the study group

Immunophenotyping of the Blast Cells in Correlations with

Jose.CA.USA) ((Bain et al., 2002).

al., 2008)

al., 2002).

**3.5 Molecular analysis**

and probe are shown in Table 2.

**3.6 Statistical analysis**

the Molecular Genetics Analyses for Diagnostic and Clinical Stratification of Patients… 487

For the detection of cytoplasmic and nuclear antigens we used commercially available permeabilization/fixation solutions (FACS Permeabilization solution BDBiosciencies;San

We incubated 100ml of specimens (peripheral blood or bone marrow) and appropriate McAb for 15min at room temperature, than added the permeabilization solutions and/or lysing solution and repeated the incubation procedure. After the incubation we washed the samples three times with PBS-A, than re-suspended the cell with isotones solution and acquired the data. As control we used a lysed, but unstained sample (Panovska-Stavridis et

Acquired data were analyzed with software by using CD45 gating strategy (Borowitz et al, 1993). This technique involves incubation of all samples with fluorochrome - labeled CD45 McAb and with the McAb for which reactivity needs to be established with an alternative fluorochrome). In the initial step of the analyses, gating was set up on a CD45-positive versus light side-scatter dot plot. The procedure allowed the discrimination between the blast cell population and normal cells and the exclusion of platelets and debris. Thereafter, if necessary, it was possible to perform another gating, to separate the cells positive with the McAb under study (Borowitz et al, 1993). Leukemias were first screened by the primary panel and, if necessary, further characterized by the McAb of the secondary panel (Bain et

Antigen expression was considered positive if 20% or more blast cells reacted with a particular antibody, except reactivity of blasts cells with MPO. It was considered positive if 10% or more of mononuclear cells were MPO positive (Bain et al., 2002; Döhner et al., 2010).

Mononuclear cell preparation, RNA isolation, and cDNA synthesis were performed at the Department of Molecular Biology, Immunology and Pharmacogenetics, Faculty of Pharmacy-Skopje, according to standard procedures. Aliquots of 5 μL of cDNA (100 ng RNA equivalent) were used for Real-time Quantitative polymerase chain reaction (RQ-PCR) with primers and dual-labeled probes as described by Gabert et al. Positions and nucleotide

The RQ-PCR reaction was performed in a 25-μl reaction volume using 12.5 μl (1x) Master Mix (Applied Biosystems), 300nM primers and 200nM probes on a Mx3005P(TM) QPCR System (Stratagene) under the following conditions: 95C for 10min, followed by 50 cycles of 95C for 15s, 50C for 1min. In order to correct variations in RNA quality and quantity and to calculate the sensitivity of each measurement, a control gene (CG) transcript was amplified in parallel to the fusion gene (FG) transcript. Since ABL (Abelson) gene transcript expression did not differ significantly between normal and leukemic samples, ABL was used as a control gene in this study (24). Positions and nucleotide sequences of the primers

Statistical analyses were performed by using the statistical analyses software SPSS 18.0 and by applying the Descriptive statistics (cross tabulation, frequencies, descriptive ratio statistics), bivariate statistics (means, t-test, correlation (bivariate, partial, distances), nonparametric tests and linear regression statistical methods. The Level of probability for obtaining the null hypothesis, in accordance with the international conventions for bio-

sequences of the primers and probes are shown in Table 2.

medical sciences was 0.05 or 0.01 (Armitage et al., 2002).


**4. Improvement of the antiproliferative effect of vitamin K2**

#### **4.1. A novel chemosynthetic vitamin K derivative**

Carr et al. demonstrated that a new, chemically synthesized vitamin K analog, compound 5 (Cpd5), inhibited Cdc25A phosphatase activity and particularly reduced HCC cell growth through arrest of the G1/S phase of the cell cycle. Inhibition of Cdc25A by Cpd5 results in prolonged tyrosine phosphorylation and activation of ERK1/2, which could be triggered by upstream epidermal growth factor receptor signaling pathway molecules [23]. Suhara et al. synthesized vitamin K2 analogues with hydroxyl or phenyl groups at the ω-terminal of the side chain, and with dual side chains at the C-2 and C-3 positions. They found that modifying the side chain of vitamin K affects the SXR-mediated transcriptional activity [24, 25]. The novel biological activities of MK-4 include tumor suppressive effects related to gene transcription through the SXR. The new derivatives of MK-4 have shown some efficacy against HCC, but a lengthy development process is still necessary to yield safe and effective clinical products.

in HCC. MKH is mainly generated by a vitamin K 2, 3-epoxide reductase complex subunit 1 like-1 (VKORC1L1) in mammalian cells [38]. Key reasons for further development are that MK-4 delivery to HCC cells is poor and reductive activation of MK-4 is low in HCC cells. We hypothesized that effective delivery of MKH into HCC cells is essential in regulating HCC growth and metastasis. However, MKH cannot be used as a therapeutic agent, because it is easy to be oxidized and converted to MK-4. In addition, the production of MKH depends on VKORC1L1 activity in HCC cells. As such, MK-4 does not display sufficient antitumor activity

Enhanced Intracellular Delivery and Improved Antitumor Efficacy of Menaquinone-4

http://dx.doi.org/10.5772/63343

315

In light of these findings, we synthesized three types of *N,N*-dimethylglycine esters of MKH (MKH-1-DMG, MKH-4-DMG, and MKH-bis-DMG) and assessed their potential as watersoluble prodrugs for a bioreductive activation-independent delivery system of MKH [33–35]

**Figure 1.** Structure of the MKH derivatives and schematic illustration of the concept of the MKH delivery system. (A) Chemical structures of menaquinone-4 (MK-4), menahydroquinone-4 (MKH) and MKH *N,N*-dimethylglycinate (MKH–DMG). (B) Schematic illustration of the vitamin K cycle and the concept of the MKH delivery system. Adapted

even at high doses.

(**Figure 1**).

from Ref. [32].

#### **4.2. Synergistic drug combinations**

Yoshiji et al. [26, 27] reported that combined treatment with MK-4 and angiotensin-converting enzyme inhibitor significantly suppressed experimental hepatocarcinogenesis. Further to this, they reported that the combined treatment with MK-4 and angiotensin-converting enzyme inhibitor may suppress the cumulative recurrence of HCC after the curative therapy, at least partly through suppression of the vascular endothelial growth-mediated neovascularization. Kanamori et al. [28] demonstrated that a combination of MK-4 and acyclic retinoid synergistically inhibited the growth of Huh7 HCC cells by increasing apoptosis. When combined with acyclic retinoid, MK-4 synergistically inhibits Ras activation and inhibits phosphorylation of retinoid X receptor α. Zhang H et al. [29] showed that MK-4 enhanced the inhibition of 5 fluorouracil-induced cell growth in HepG2, Huh7, HLE, and Hep3B HCC cells, and via G1 cell-cycle arrest through induced expression of p21 and p27 and inhibited expression of cyclin D1. Zhang et al. reported that a combination of MK-4 and sorafenib work synergistically to inhibit growth of HepG2, Hep3B, and HuH7 HCC cells. They also demonstrated that the levels of cyclin D1 expression are clearly reduced in HepG2 cells treated with a combination treatment of MK-4 and sorafenib [30]. A clinical trial to test the efficacy of the combination of MK-4 and sorafenib in HCC was attempted [31].

#### **4.3. Delivery of menahydroquinone-4, the active form of MK-4**

We have previously synthesized the ester derivatives of menahydroquinone-4 (MKH), the fully reduced form of MK-4, and revealed their effective antiproliferative activity against HCC cell lines [32–35]. The menaquinone (MK-4 to MK-10) concentrations were significantly lower in HCC tissues, from patients with or without increased plasma concentrations of DCP, than in the surrounding normal liver tissue. There was no significant difference between PK and PK epoxide concentration in HCC tissues without increased plasma concentrations of DCP and normal liver tissue [36]. Furthermore, the rate of uptake into MH7777 cells (vitamin K2 sensitive HCC) was lower than for normal hepatocytes. In addition, the rate of uptake into H4IIE cells (vitamin K2-resistant HCC) was negligible compared with that for MH7777 cells and normal hepatocytes. Further, hepatocytes from diethylnitrosamine-induced liver nodules exhibited a significantly lower rate of vitamin K2 uptake than that for normal hepatocytes [37].

MKH acts as a cofactor for *γ*-glutamyl carboxylase (GGCX), which catalyzes the carboxylation of specific Glu residues (*γ*-carboxylation) of substrate proteins such as prothrombin. Thus, decreased MKH availability in HCC cells is a possible causative mechanism of DCP production in HCC. MKH is mainly generated by a vitamin K 2, 3-epoxide reductase complex subunit 1 like-1 (VKORC1L1) in mammalian cells [38]. Key reasons for further development are that MK-4 delivery to HCC cells is poor and reductive activation of MK-4 is low in HCC cells. We hypothesized that effective delivery of MKH into HCC cells is essential in regulating HCC growth and metastasis. However, MKH cannot be used as a therapeutic agent, because it is easy to be oxidized and converted to MK-4. In addition, the production of MKH depends on VKORC1L1 activity in HCC cells. As such, MK-4 does not display sufficient antitumor activity even at high doses.

prolonged tyrosine phosphorylation and activation of ERK1/2, which could be triggered by upstream epidermal growth factor receptor signaling pathway molecules [23]. Suhara et al. synthesized vitamin K2 analogues with hydroxyl or phenyl groups at the ω-terminal of the side chain, and with dual side chains at the C-2 and C-3 positions. They found that modifying the side chain of vitamin K affects the SXR-mediated transcriptional activity [24, 25]. The novel biological activities of MK-4 include tumor suppressive effects related to gene transcription through the SXR. The new derivatives of MK-4 have shown some efficacy against HCC, but a lengthy development process is still necessary to yield safe and effective clinical products.

Yoshiji et al. [26, 27] reported that combined treatment with MK-4 and angiotensin-converting enzyme inhibitor significantly suppressed experimental hepatocarcinogenesis. Further to this, they reported that the combined treatment with MK-4 and angiotensin-converting enzyme inhibitor may suppress the cumulative recurrence of HCC after the curative therapy, at least partly through suppression of the vascular endothelial growth-mediated neovascularization. Kanamori et al. [28] demonstrated that a combination of MK-4 and acyclic retinoid synergistically inhibited the growth of Huh7 HCC cells by increasing apoptosis. When combined with acyclic retinoid, MK-4 synergistically inhibits Ras activation and inhibits phosphorylation of retinoid X receptor α. Zhang H et al. [29] showed that MK-4 enhanced the inhibition of 5 fluorouracil-induced cell growth in HepG2, Huh7, HLE, and Hep3B HCC cells, and via G1 cell-cycle arrest through induced expression of p21 and p27 and inhibited expression of cyclin D1. Zhang et al. reported that a combination of MK-4 and sorafenib work synergistically to inhibit growth of HepG2, Hep3B, and HuH7 HCC cells. They also demonstrated that the levels of cyclin D1 expression are clearly reduced in HepG2 cells treated with a combination treatment of MK-4 and sorafenib [30]. A clinical trial to test the efficacy of the combination of

We have previously synthesized the ester derivatives of menahydroquinone-4 (MKH), the fully reduced form of MK-4, and revealed their effective antiproliferative activity against HCC cell lines [32–35]. The menaquinone (MK-4 to MK-10) concentrations were significantly lower in HCC tissues, from patients with or without increased plasma concentrations of DCP, than in the surrounding normal liver tissue. There was no significant difference between PK and PK epoxide concentration in HCC tissues without increased plasma concentrations of DCP and normal liver tissue [36]. Furthermore, the rate of uptake into MH7777 cells (vitamin K2 sensitive HCC) was lower than for normal hepatocytes. In addition, the rate of uptake into H4IIE cells (vitamin K2-resistant HCC) was negligible compared with that for MH7777 cells and normal hepatocytes. Further, hepatocytes from diethylnitrosamine-induced liver nodules exhibited a significantly lower rate of vitamin K2 uptake than that for normal hepatocytes [37].

MKH acts as a cofactor for *γ*-glutamyl carboxylase (GGCX), which catalyzes the carboxylation of specific Glu residues (*γ*-carboxylation) of substrate proteins such as prothrombin. Thus, decreased MKH availability in HCC cells is a possible causative mechanism of DCP production

**4.2. Synergistic drug combinations**

314 Vitamin K2 - Vital for Health and Wellbeing

MK-4 and sorafenib in HCC was attempted [31].

**4.3. Delivery of menahydroquinone-4, the active form of MK-4**

In light of these findings, we synthesized three types of *N,N*-dimethylglycine esters of MKH (MKH-1-DMG, MKH-4-DMG, and MKH-bis-DMG) and assessed their potential as watersoluble prodrugs for a bioreductive activation-independent delivery system of MKH [33–35] (**Figure 1**).

**Figure 1.** Structure of the MKH derivatives and schematic illustration of the concept of the MKH delivery system. (A) Chemical structures of menaquinone-4 (MK-4), menahydroquinone-4 (MKH) and MKH *N,N*-dimethylglycinate (MKH–DMG). (B) Schematic illustration of the vitamin K cycle and the concept of the MKH delivery system. Adapted from Ref. [32].


**Abstract**

The failure of systems to meet the specified requirements may have adverse effects on their integrity and reliability. The systems could be mechanical, electrical, structural, telecommunications, or electronic that are designed and built to satisfy certain technical specifications and operational requirements. Failure does not necessarily mean the occurrence of a disaster or damage to the system, but also the degraded performance of such systems is considered a failure. One of the essential indicators of the performance and reliability of a system is the probability of failure which is computed by probabilistic methods. One of these methods is the first-order reliability method (FORM). Using FORM to estimate the probability of failure of systems having a nonlinear or a higher-order performance function may provide inaccurate results that may lead to misleading conclusions. To resolve this issue, the second-order reliability method (SORM) is recommended to estimate the probability of failure. This chapter presents commonly used probabilistic approximation methods to estimate the probability of failure for nonlinear performance functions. Illustrative examples to demonstrate the application of these methods are provided at the end of the chapter.

**Keywords:** second-order approximation, failure analysis, probabilistic methods in engineering, second-order reliability, reliability engineering

## **1. Introduction**

Probability of failure can be calculated using analytical as well as simulation methods. Simulation methods such as the Monte Carlo technique can be used to calculate the probability of failure. It is an essential step in the calculation process to validate the analytical results. One of the analytical methods to compute the probability of failure is the first-order reliability method (FORM) which is based on the first-order expansion of the Taylor series. This method may provide inaccurate results when approximating nonlinear or higher-order performance functions. It should be noted that the performance function can be referred to as the limit state function (LSF).

The second-order reliability method, abbreviated as SORM, is used to resolve the nonlinearity issues of the performance function or the LSF. It uses the second-order expansion of the Taylor series to include the curvature of the LSF in the calculation to achieve better accuracy of the results.

**Figure 1.** *Representation of the linear and nonlinear LSF.*

**Figure 1** shows the failure regions for linear and nonlinear LSFs. It also shows the most probable point of failure (MPPF) as the tangent point on the limit-state surface, *Z U* ∗ ð Þ¼ 0, and the reliability index, *β*, is the shortest distance from the origin to the limit-state surface.

Breitung [1], Hohenbichler and Rackwitz [2], Tvedt [3, 4], and many other researchers came up with methods to calculate the probability of failure for nonlinear second-order LSFs. These methods use *β* obtained by FORM and the principal curvature, *ki*, of the limit state surface in the calculation. Essentially, *ki* estimates how much the curve diverges from the straight line, and it is an essential part of the calculation process. It is added to the calculation to provide a more accurate estimate of the probability of failure obtained by FORM. This chapter presents commonly used techniques that provide solutions to nonlinear second-order LSFs. The focus of this chapter is on SORM with the assumption that the random variables of the LSF are uncorrelated. A brief discussion about SORM and the commonly used methods for calculating the probability of failure are provided in the next section. Numerical examples are provided in Section 3 to demonstrate the calculation steps and provide in-depth understanding of these methods. Appendix A provides an overview of FORM and the steps involved in its calculation.

## **2. Second Order Reliability Method (SORM)**

The LSF may exhibit nonlinearity when random variables have a non-normal distribution or a nonlinear relationship. Moreover, transforming uncorrelated variables to correlated variables may present nonlinearity in the LSF. SORM is used to deal with the nonlinearity issues to provide more accurate results. It uses the second-order Taylor series expansion of the LSF at the most probable point of failure (MPPF). The vectorized form of the second-order Taylor Series expansion is used to express the LSF around the MPPF as:

$$\mathbf{Z}(\mathbf{U}) \approx \mathbf{Z}(\mathbf{U}^\*) + \nabla \mathbf{Z}(\mathbf{U}^\*)^t (\mathbf{U} - \mathbf{U}^\*) + \mathbf{0}.\\ \mathbf{5}(\mathbf{U} - \mathbf{U}^\*)^t \nabla^2 \mathbf{Z}(\mathbf{U}^\*)(\mathbf{U} - \mathbf{U}^\*) \tag{1}$$

*Perspective Chapter: Probabilistic Modeling of Failure – Nonlinear Approximation DOI: http://dx.doi.org/10.5772/intechopen.109266*

where:

Since *U* ∗ is located on the limit-state surface, *Z U* ∗ ð Þ equals zero. *Z U*ð Þ is normalized by dividing Eq. (1) by j∇*Z U* ∗ ð Þj which yields:

$$Z\_{norm}(U) \approx \frac{\nabla Z(U^\*)^t (U - U^\*)}{|\nabla Z(U^\*)|} + \frac{\mathbf{0.5}(U - U^\*)^t \nabla^2 Z(U^\*)(U - U^\*)}{|\nabla Z(U^\*)|}\tag{2}$$

where j∇*Z U* ∗ ð Þj is the length of the gradient vector of the LSF at the MPPF and is equal to:

$$|\nabla Z(U^\*)| = \sqrt{\sum\_{i=1}^n \left(\frac{\partial Z(U^\*)}{\partial u\_i}\right)^2} \tag{3}$$

The approximation of LSF at the MPPF is expressed as:

$$Z\_{norm}(U) \approx \frac{\nabla Z(U^\*)^t (U - U^\*)}{|\nabla Z(U^\*)|} + \frac{\mathbf{0.5} (U - U^\*)^t \nabla^2 Z(U^\*) (U - U^\*)}{|\nabla Z(U^\*)|} = \mathbf{0} \tag{4}$$

From the first-order reliability approximation (FORM approximation) we have:

$$\frac{\nabla Z(U^\*)^t U^\*}{|\nabla Z(U^\*)|} = -\beta \tag{5}$$

$$\frac{\nabla Z(U^\*)}{|\nabla Z(U^\*)|} = a \tag{6}$$

*α: the directional cosines, α*1, … , *αn the components of the unit gradient vector α: These are the cosines of the angles between the vector β and the axes.*

Denoting ∇2*Z U* ∗ ð Þ j∇*Z U* ∗ ð Þj as *H:*

$$\mathbf{H} = \frac{\nabla^2 \mathbf{Z}(\mathbf{U}^\*)}{|\nabla \mathbf{Z}(\mathbf{U}^\*)|} \tag{7}$$

**Figure 2.** *Second-order failure surface shown in the rotated coordinates.*

Using Eqs. (5) and (7), Eq. (4) yields:

$$Z\_{norm}(U) \approx \frac{\nabla Z(U^\*)^t U}{|\nabla Z(U^\*)|} + \beta + 0.5(U - U^\*)^t \mathbf{H}(U - U^\*) = \mathbf{0} \tag{8}$$

To go further with the solution of SORM, the *U* variables are transformed into a new set of standard normal random variables denoted as *U*0 *i* where the axis of the last variable coincides with the *β* vector as indicated in **Figure 2** [5–7].

This is an orthogonal transformation where:

$$\mathbf{U}' = \mathbf{R}\mathbf{U} \tag{9}$$

*R*: orthogonal *n* by *n* rotation matrix, where *n* is the number of variables. Since *R* is an orthogonal rotation matrix, *R*�**1** ¼ *Rt*

$$\mathbf{U} = \mathbf{R}^t \mathbf{U}'\tag{10}$$

The matrix *R* is constructed in two steps. First, define an initial matrix, *R***01** that consists of rows representing the unit vectors of the axes of the input variables. Since, we want to make *β* vector coincide with the axis of the last variable, we can simply substitute the last row of matrix *R***01** with the directional cosines of *β*. The following row vectors are the components of the *R***01** matrix:

*r*01 ¼ ½ � 100 *:::::::::* 0 *r*02 ¼ ½ � 010 *:::::::::* 0 *: : :* (11)

$$r\_{0\mathfrak{n}} = \left[ -\frac{\partial Z(U^\*)\_{\rangle\_{\partial U\_1}}}{|\nabla Z(U^\*)|} \quad . \quad . \quad . \quad . -\frac{\partial Z(U^\*)\_{\rangle\_{\partial U\_\bullet}}}{|\nabla Z(U^\*)|} \right]$$

*Perspective Chapter: Probabilistic Modeling of Failure – Nonlinear Approximation DOI: http://dx.doi.org/10.5772/intechopen.109266*

The Gram-Schmidt process is applied to orthogonalize the matrix [8]. The next step is to orthonormalize the matrix by dividing each element of the matrix by the length of its corresponding row vector.

Perform the Gram-Schmidt process for the matrix **R01** using Eqs. (12) and (13), in reverse order:

$$r\_n = r\_{0n} \tag{12}$$

$$r\_i = r\_{0i} - \sum\_{j=i+1}^{n} \frac{r\_j r\_{0i}^t}{r\_j r\_j^t} r\_j \quad i = 1, 2, \dots, n-1 \tag{13}$$

Then orthonormalization is performed for each vector to produce the matrix *R*. Using Eq. (10), Eq. (8) yields:

$$Z\_{norm}(U') \approx -U\_n' + \beta + 0.5(U' - U'^\*)^t \mathbf{RHR}^t (U' - U'^\*) = \mathbf{0} \tag{14}$$

The *H* matrix is the second-order derivative of the LSF at the design point *U* ∗ divided by the length of the gradient vector, j∇*Z U* ∗ ð Þj, and is formulated as:

$$H = \frac{1}{|\nabla Z(\boldsymbol{U}^{\prime\prime})|} \begin{bmatrix} \frac{\partial^2 Z(\boldsymbol{U}^{\prime\prime})}{\partial \boldsymbol{u}\_1^2} & \frac{\partial^2 Z(\boldsymbol{U}^{\prime\prime})}{\partial \boldsymbol{u}\_1 \partial \boldsymbol{u}\_2} & \cdots & \frac{\partial^2 Z(\boldsymbol{U}^{\prime\prime})}{\partial \boldsymbol{u}\_1 \partial \boldsymbol{u}\_n} \\\\ \frac{\partial^2 Z(\boldsymbol{U}^{\prime\prime})}{\partial \boldsymbol{u}\_2 \partial \boldsymbol{u}\_1} & \ddots & \ddots & \ddots & \vdots \\\\ \vdots & \ddots & \ddots & \ddots & \ddots \\\\ \frac{\partial^2 Z(\boldsymbol{U}^{\prime\prime})}{\partial \boldsymbol{u}\_n \partial \boldsymbol{u}\_1} & \cdots & \cdots & \frac{\partial^2 Z(\boldsymbol{U}^{\prime\prime})}{\partial \boldsymbol{u}\_n^2} \end{bmatrix} \tag{15}$$

Denoting *RHRt* as *B*, Eq. (14) becomes:

$$Z\_{norm}(U') \approx -U\_n' + \beta + \mathbf{0.5(U'^t \mathbf{B} U')} = \mathbf{0} \tag{16}$$

Then the solution of the second-order approximation is given as:

$$U\_n' = \beta + 0.5(\mathbf{U}^t \mathbf{B} \mathbf{U}') \tag{17}$$

*B* is an *(n* � *1)* by *(n* � *1)* matrix and *U*0 is a *1* by *(n* � *1)* vector. The eigenvalues of matrix *B* are computed to obtain the main curvatures, *ki* 0 *s* , of the LSF at the MPPF. These *ki* 0 *s* are used in the calculation of the probability of failure for nonlinear LSF as indicated in the next subsection [5–7]. The 0*:*5 *U*0*t BU*0 � � term is simplified to *n*

0*:*5 P�1 *i*¼1 *kiU*02 *i* . Thus, Eq. (17) is expressed in terms of the curvatures, *ki* 0 *s*, as:

$$U\_n' = \beta + 0.5 \sum\_{i=1}^{n-1} k\_i U\_i^2 \tag{18}$$

If *ki* >0, the LSF will have a convex failure region, and if *ki* < 0, it will have a concave failure region as shown in **Figure 3**. The figure shows the failure surfaces for an arbitrary LSF with three random variables.

#### **Figure 3**.

*Convex and concave failure regions for an arbitrary LSF with three random variables.*

The calculation of ∇*Z U* ∗ ð Þ and ∇2 *Z U* ∗ ð Þ can be performed in the X domain using the following equations:

$$\nabla Z\left(U^{\prime \*}\right) = \frac{\partial Z\left(U^{\prime \*}\right)}{\partial u\_i} = \frac{\partial Z(X^\*)}{\partial \varkappa\_i} \sigma\_{xi} \tag{19}$$

$$\nabla^2 Z(\mathbf{U}^{\prime \*}) = \frac{\partial^2 Z(\mathbf{U}^{\prime \*})}{\partial u\_i^2} = \frac{\partial^2 Z(\mathbf{X}^\*)}{\partial \mathbf{x}\_i^2} \sigma\_{\mathbf{x}\_i}^2 \tag{20}$$

where *σxi* is the standard deviation of the random variable *xi*.

To summarize, the main objective of using SORM is to improve the estimate of the probability of failure that was obtained by FORM. The calculation includes three key steps. These are, the calculation of the reliability index and its associated design points either in the U domain or X domain using FORM, determining the principal curvatures, and determining the probability of failure using one of the commonly used methods for nonlinear LSFs.

#### **2.1 Probability of failure**

Breitung, Hohenbichler, and Tvedt methods use a correction factor which is expressed in terms of *ki* and *β* to adjust the probability of failure obtained by FORM. The probability of failure for nonlinear LSF is then formulated as:

$$P\_f = \Phi(-\beta)(\mathcal{CF}) \tag{21}$$

The first term is the cumulative distribution function (cdf) of beta, *CF* is the SORM correction factor and *β* is the reliability index obtained by FROM.

#### *2.1.1 Breitung formulation*

Breitung formulation is expressed as:

*Perspective Chapter: Probabilistic Modeling of Failure – Nonlinear Approximation DOI: http://dx.doi.org/10.5772/intechopen.109266*

$$P\_{f\_{Brewing}} = \Phi(-\beta) \prod\_{i=1}^{n-1} \frac{1}{\sqrt{(1+\beta k\_i)}} \tag{22}$$

#### *2.1.2 Hohenbichler formulation*

Hohenbichler formulation is expressed in terms of the cdf of beta (the first term), probability density function of beta (pdf) which is the upper term in the denominator and the cdf of beta which is the lower term in the denominator as:

$$P\_{f\_{Holenbichler}} = \Phi(-\beta) \prod\_{i=1}^{n-1} \frac{1}{\sqrt{\left(\mathbf{1} + \frac{\phi(-\beta)}{\Phi(-\beta)} k\_i\right)}} \tag{23}$$

#### *2.1.3 Tvedt formulation*

Tvedt formulation has three parts as indicated below:

$$P\_{f\_{Tend}} = \Phi(-\beta)f1 + f2 + f3 \tag{24}$$

$$f1 = \prod\_{i=1}^{n-1} \frac{1}{\sqrt{(1+\beta k\_i)}}$$

$$f2 = (\beta \Phi(-\beta) - \phi(-\beta)) \left( f1 - \left(\prod\_{i=1}^{n-1} \frac{1}{\sqrt{(1+(1+\beta)k\_i)}}\right)\right)$$

$$f3 = (1+\beta)(\beta \Phi(-\beta) - \phi(-\beta)) \left( f1 - \text{Real} \left(\prod\_{i=1}^{n-1} \frac{1}{\sqrt{(1+(j+\beta)k\_i)}}\right)\right)$$

#### **2.2 Calculation steps**

$$\sqrt{\sum\_{i=1}^{n} \left(\frac{\partial Z(U^\*)}{\partial u\_i}\right)^2}$$

## **3. Examples**

#### **3.1 Example 1**

The limit state function for a system has been formulated as:

*Z X*ð Þ¼ 2*:*5 *x*2 1 � *x*2 2 � 0*:*167 where *x*1 � *N*ð Þ 2, 0*:*2 , *x*2 � *N*ð Þ 2, 0*:*32 *:*

**Table 1.** *MCS results – Example 1.0, Part C.*

#### **Solution**

#### *Part A*

Determine *β* and the MPPF in the *X* space and the *U* space. FORM is used to solve part A. The computation continues until the solution converges with a tolerance error of 0.0001 j*βi*�*βi*�1j *βi*�1 <0*:*0001 *:*

*Assume the initial MPPF as the given mean of each variable. Transform the LSF Z*ð Þ *X to Z*ð Þ *U :*

$$Z(\mathbf{U}) = 2.5(0.2\mu\_1 + 2)^2 - (0.32\mu\_2 + 2)^2 - 0.167$$

*Compute the initial values of the design point in U space:*

$$\begin{aligned} \mathbf{x}\_1 &= \mu\_{\mathbf{x}\_1} = 2\\ \mathbf{x}\_2 &= \mu\_{\mathbf{x}\_2} = 2 \end{aligned}$$

$$\begin{aligned} \mu\_1 &= \frac{\mathbf{x}\_1 - \mu\_{\mathbf{x}\_1}}{\sigma\_{\mathbf{x}1}} = \frac{2 - 2}{0.2} = \mathbf{0} \\ \mu\_2 &= \frac{\mathbf{x}\_2 - \mu\_{\mathbf{x}\_2}}{\sigma\_{\mathbf{x}2}} = \frac{2 - 2}{0.32} = \mathbf{0} \end{aligned}$$

*Perspective Chapter: Probabilistic Modeling of Failure – Nonlinear Approximation DOI: http://dx.doi.org/10.5772/intechopen.109266*

*Compute the initial estimate of Z U*ð Þ:

$$Z(0,0) = 5.83300$$

*Compute the partial derivatives of the LSF:*

$$\frac{\partial Z(U^\*)}{\partial u\_1} = 0.2u\_1 + 2 = 2$$

$$\frac{\partial Z(U^\*)}{\partial u\_2} = -0.2048u\_2 - 1.28 = -1.28$$

*Compute the standard deviation of the LSF*:

$$\sigma\_{\overline{z}} = \sqrt{\sum\_{i=1}^{n} \left(\frac{\partial Z(U^\*)}{\partial u\_i}\right)^2} = \sqrt{(2)^2 + (-1.28)^2} = 2.3745$$

*Compute the initial reliability index β:*

$$
\beta = \frac{\mu\_x}{\sigma\_x} = \frac{5.8330}{2.3745} = 2.4565
$$

*Compute the directional cosines αi*:

$$a\_1 = -\frac{\left(\frac{\partial Z(U^\*)}{\partial u\_1}\right)}{\sqrt{\sum\_{i=1}^n \left(\frac{\partial Z(U^\*)}{\partial u\_i}\right)^2}} = -\frac{2}{2.3745} = -0.8423$$

$$a\_2 = -\frac{\left(\frac{\partial Z(U^\*)}{\partial u\_2}\right)}{\sqrt{\sum\_{i=1}^n \left(\frac{\partial Z(U^\*)}{\partial u\_i}\right)^2}} = -\frac{-1.28}{2.3745} = 0.5391$$

#### *Iteration 1*

*Determine the new MPPF/design point in U space and X space:*

$$u\_1 = \beta a\_1 = 2.4565 \ast (-0.8423) = -2.0690$$

$$u\_2 = \beta a\_2 = 2.4565 \ast (0.5391) = 1.3242$$

$$\alpha\_1 = \beta a\_1 \sigma\_{\ge 1} + \mu\_{\ge 1} = u\_1 \sigma\_{\ge 1} + \mu\_{\ge 1} = -2.0690 \ast 0.2 + 2 = 1.5862$$

$$\alpha\_2 = \beta a\_2 \sigma\_{\ge 1} + \mu\_{\ge 1} = u\_2 \sigma\_{\ge 1} + \mu\_{\ge 1} = 1.3242 \ast 0.32 + 2 = 2.4237$$

*Compute the LSF in terms of the new design points:*

$$Z(U^\*) = Z(-2.0690, 1.3242\ ) = 0.2485$$

*Compute the partial derivatives and the standard deviation of the LSF at the new design points:*

*The partial derivatives:*

$$\frac{\partial Z(U^\*)}{\partial u\_1} = 1.5862$$

*Failure Analysis – Structural Health Monitoring of Structure and Infrastructure Components*

$$\frac{\partial Z(U^\*)}{\partial u\_2} = -1.5512$$

*The standard deviation of Z:*

$$
\sigma\_x = \sqrt{(1.5862)^2 + (-1.5512)^2} = 2.2186
$$

*Compute the new β in terms of the new design points:*

$$\beta = \frac{Z(\mathbf{U}^\*) - \sum\_{i=1}^n \frac{dZ(U)}{du\_i} \left(\mu\_i^\*\right)}{\sqrt{\sum\_{i=1}^n \left(\frac{dZ(U^\*)}{du\_i}\right)^2}}$$

$$= \frac{0.2485 - (1.5862 \ast (-2.0690) + (-1.5512 \ast 1.3242))}{2.2186}$$

$$\beta = 2.5171$$

We examine the tolerance:

$$\frac{|\beta\_i - \beta\_{i-1}|}{\beta\_{i-1}} = \frac{|2.5171 - 2.45650|}{2.45650} = 0.02466$$

$$0.02466 > 0.0001$$

Since the error is greater than the established tolerance *(0.0001)*, the computation should continue until convergence is obtained.

*The next step is to compute the directional cosines:*

$$a\_1 = -\frac{1.5862}{2.2186} = -0.7150$$

$$a\_2 = -\frac{-1.5512}{2.2186} = 0.6992$$

*The computation continues until β converges following the steps mentioned in Appendix A. Once convergence is obtained the probability of failure is calculated.*

**Table 2** shows the results for the next iterations. As the table indicates, convergence occurred at the third iteration with an error less than the tolerance.

$$\frac{|\beta\_i - \beta\_{i-1}|}{\beta\_{i-1}} = \frac{|2.51171 - 2.51170|}{2.51170} = 0.000003981$$

$$0.000003981 \ll 0.0001$$

*β* is calculated to be *2.51171* and the MPPF is located at *U(-1.7758, 1.7762)* and *X(1.6448, 2.5684)*.

The probability of failure is computed in terms of the final value of *β* :

$$P\_f = \Phi(-\beta) = \mathbf{1} - \Phi(\beta) = \mathbf{1} - \Phi(2.51171) = 0.006007$$

*Perspective Chapter: Probabilistic Modeling of Failure – Nonlinear Approximation DOI: http://dx.doi.org/10.5772/intechopen.109266*

**Table 2.**

*Summary of results – Example 1.0, Part A.*

### *Part B*

Determine the probability of failure using SORM by applying *Breitung*, *Hohenbichler,* and *Tvedt* methods.

*Determine the length of the gradient vector at the MPPF: The partial derivatives at the third iteration are:*

$$\frac{\partial Z(U^\*)}{\partial u\_1} = 1.6448$$

$$\frac{\partial Z(U^\*)}{\partial u\_2} = -1.6438$$

$$|\nabla Z(U^\*)| = \sqrt{\sum\_{i=1}^n \left(\frac{\partial Z(U^\*)}{\partial u\_i}\right)^2} = \sqrt{\left(1.6448\right)^2 + \left(-1.6438\right)^2} = 2.3254$$

*Construct the matrix* **R01** *using Eq. (11):*

$$\mathbf{R\_{01}} = \begin{bmatrix} 1 & 0\\ -\frac{dZ(U^\*)\_{\acute{\beta}\mathbf{u}\_1}}{|\nabla Z(U^\*)|} & -\frac{dZ(U^\*)\_{\acute{\beta}\mathbf{u}\_2}}{|\nabla Z(U^\*)|} \end{bmatrix} = \begin{bmatrix} \mathbf{1} & \mathbf{0} \\ -\mathbf{0.7073} & \mathbf{0.7069} \end{bmatrix}.$$

The second row is the directional cosines of the reliability index, *β*, at the MPPF. *Perform Gram–Schmidt orthogonalization for the matrix R***01** *using Eqs.* (12) and (13)*, and perform orthonormalization of each row vector to come up with the matrix R:*

$$r\_n = r\_{02}$$

The last element of the matrix is calculated as:

$$r\_i = r\_{0i} - \sum\_{j=i+1}^{n} \frac{r\_j r\_{0i}^t}{r\_j r\_j^t} r\_j$$

*i=2*

$$r\_2 = r\_{02} = [-0.7073 \ 0.7069]$$

*i=1*

$$r\_1 = r\_{01} - \sum\_{j=i+1}^{n} \frac{r\_j r\_{02}^t}{r\_j r\_j^t} r\_j = \begin{bmatrix} 1 & 0 \end{bmatrix} - \frac{r\_1 r\_{02}^t}{r\_1 r\_1^t} r\_1$$

$$= \begin{bmatrix} 1 & 0 \end{bmatrix} - \frac{[-0.7073 & 0.7069] \begin{bmatrix} 1 \\ 0 \end{bmatrix}}{[-0.7073 & 0.7069] \begin{bmatrix} -0.7073 \\ 0.7069 \end{bmatrix}} [-0.7073 & 0.7069]$$

$$r\_1 = [0.4997 & 0.5000]$$

Normalizing the elements of the row vector, *r*2 becomes:

$$r\_2 = \frac{[0.4997 \, 0.50000]}{\sqrt{0.4997^2 + 0.5000^2}} = [0.7073 \, 0.7069]$$

The matrix *R* becomes:

$$\mathbf{R} = \begin{bmatrix} \mathbf{0.7069} & \mathbf{0.7073} \\ -\mathbf{0.7073} & \mathbf{0.7069} \end{bmatrix}$$

*Compute the second-order derivative of the LSF at the design point, U* ∗ *, using Equation (20) to obtain the H matrix:*

$$H = \frac{1}{|\nabla Z(U^\*)|} \begin{bmatrix} \frac{\partial^2 Z(U^\*)}{\partial u\_1^2} & \frac{\partial^2 Z(U^\*)}{\partial u\_1 \partial u\_2} \\ \frac{\partial^2 Z(U^\*)}{\partial u\_2 \partial u\_1} & \frac{\partial^2 Z(U^\*)}{\partial u\_2^2} \end{bmatrix} = \frac{1}{2.3254} \begin{bmatrix} 0.2 & 0 \\ 0 & -0.2048 \end{bmatrix} = \begin{bmatrix} 0.0860 & 0 \\ 0 & -0.0881 \end{bmatrix}$$

Compute *RHRt* matrix: *RHRt* **=** �0*:*0011 �0*:*0870 �0*:*0870 �0*:*0010 � �

*Compute the eigenvalues of the matrix B to obtain principal curvatures (ki's):*

The principal curvatures, *ki* 0 *s*, are computed by solving the eigenvalues of *RHRt* . The last column and last row are dropped from the above matrix, then the eigenvalues are obtained to determine the *ki* values. From the above matrix, there is only one *k* at the MPPF, with a value of *(*�*0.0011)*.

*Calculate the probability of failure Pf using Eqs. (22)–(24):*

*Breitung method*

$$P\_{f\_{Bereiting}} = \Phi(-\beta) \prod\_{i=1}^{n-1} \frac{1}{\sqrt{(1+\beta k\_i)}}$$

*Perspective Chapter: Probabilistic Modeling of Failure – Nonlinear Approximation DOI: http://dx.doi.org/10.5772/intechopen.109266*

$$=\Phi(-2.5117)\frac{1}{\sqrt{(1+2.5117)(-0.0011)}}$$

$$P\_{f\_{Breatung}} = 0.00601565$$

*Hohenbichler method*

$$P\_{f\_{Hoble,ieller}} = \Phi(-\beta) \prod\_{i=1}^{n-1} \frac{1}{\sqrt{\left(1 + \frac{\phi(-\beta)}{\Phi(-\beta)} k\_i\right)}}$$

$$= \Phi(-2.5117) \frac{1}{\sqrt{\left(1 + \frac{\phi(-2.5117)}{\Phi(-2.5117)} (-0.0011)\right)}}$$

$$P\_{f\_{Huberible}} = 0.00601671$$

*Tvedt method*

$$f1 = \prod\_{i=1}^{n-1} \frac{1}{\sqrt{(1+\beta k\_i)}} = 1.00137$$

$$f2 = (\beta \Phi(-\beta) - \phi(-\beta)) \left(f1 - \left(\prod\_{i=1}^{n-1} \frac{1}{\sqrt{(1+(1+\beta)k\_i)}}\right)\right)$$

$$= (-0.00193)(1.00137 - (1.00192)) = 1.05670E - 06$$

$$f2 = 1.05670E - 06$$

$$f3 = (1+\beta)(\beta \Phi(-\beta) - \phi(-\beta)) \left(f1 - \text{Real}\left(\prod\_{i=1}^{n-1} \frac{1}{\sqrt{(1+(j+\beta)k\_i)}}\right)\right)$$

$$= (1+2.5117)(2.5117(\Phi(-2.5117)) - \phi(-2.5117))$$

$$\left(1.00137 - \text{Real}\left(\frac{1}{\sqrt{(1+(j+2.5117))(-0.0011)}}\right)\right)$$

$$f3 = -3.03430E - 9$$

$$P\_{f\_{T\text{mid}}} = \Phi(-\beta)f1 + f2 + f3$$

$$= (6.00743E - 3)(1.00137) + (1.05670E - 06) + (-3.03430E - 9)$$

$$P\_{f\_{T\text{mid}}} = 6.01671E - 03$$

See **Table 3**. *Part C*

Compare the results of parts *A* and *B* with results obtained by Monte Carlo simulation:

The simulation was conducted using *2e5* and *1e6* simulation cycles as shown in **Table 1**. The results obtained by *Breitung*,*Tvedt,* and *Hohenbichler* methods are close to the Monte Carlo simulation results when using 1e6 simulation cycles.

#### **Table 3.**

*Summary of SORM results – Example 1.0 Part B.*