Novel analytical and numerical methods for solving fractional dynamical systems

During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations.

Melanopsin expressing intrinsically photosensitive Retinal Ganglion cells in retinal disease

Melanopsin containing intrinsically photosensitive Retinal Ganglion Cells (ipRGCs) are a class of photoreceptors with established roles in non-image forming processes. Their contributions to image forming vision may include the estimation of brightness. Animal models have been central for understanding the physiological mechanisms of ipRGC function and there is evidence of conservation of function across species. ipRGCs can be divided into 5 ganglion cell subtypes that show morphological and functional diversity. Research in humans has established that ipRGCs signal environmental irradiance to entrain the central body clock to the solar day for regulating circadian processes and sleep. In addition, ipRGCs mediate the pupil light reflex (PLR), making the PLR a readily accessible behavioural marker of ipRGC activity. Less is known about ipRGC function in retinal and optic nerve disease, with emerging research providing insight into their function in diabetes, retinitis pigmentosa, glaucoma and hereditary optic neuropathy. We briefly review the anatomical distributions, projections and basic physiological mechanisms of ipRGCs, their proposed and known functions in animals and humans with and without eye disease. We introduce a paradigm for differentiating inner and outer retinal inputs to the pupillary control pathway in retinal disease and apply this paradigm to patients with age-related macular degeneration (AMD). In these cases of patients with AMD, we provide the initial evidence that ipRGC function is altered, and that the dysfunction is more pronounced in advanced disease. Our perspective is that with refined pupillometry paradigms, the pupil light reflex can be extended to AMD assessment as a tool for the measurement of inner and outer retinal dysfunction.

Quality of life and genetics in men with prostate cancer

As family history has been established as a risk factor for prostate cancer, attempts have been made to isolate predisposing genetic variants that are related to hereditary prostate cancer. With many genetic variants still to be identified and investigated, it is not yet possible to fully understand the impact of genetic variants on prostate cancer development. The high survival rates among men with prostate cancer have meant that other issues, such as quality of life (QoL), have also become important. Through their effect on a person’s health, a range of inherited genetic variants may potentially influence QoL in men with prostate cancer, even prior to treatment. Until now, limited research has been conducted on the relationship between genetics and QoL. Thus, this study contributes to an emerging field by aiming to identify certain genetic variants related to the QoL found in men with prostate cancer. It is hoped that this study may lead to future research that will identify men who have an increased risk of a poor QoL following prostate cancer treatment, which will aid in developing treatments that are individually tailored to support them. Previous studies have established that genetic variants of Vascular Endothelial Growth Factor (VEGF) and Insulin-like Growth Factor 1 (IGF-1) may play a role in prostate cancer development. VEGF and IGF-1 have also been reported to be associated with QoL in people with ovarian cancer and colorectal cancer, respectively. This study completed a series of secondary analyses using two major data-sets (from 850 men newly diagnosed with prostate cancer, and approximately 550 men from the general Queensland population), in which genetic variants of VEGF and IGF-1 were investigated for associations with prostate cancer susceptibility and QoL. The first aim of this research was to investigate genetic variants in the VEGF and IGF-I gene for an association with the risk of prostate cancer. It was found that one IGF-1 genetic variant (rs35765) had a statistically significant association with prostate cancer (p = 0.04), and one VEGF genetic variant (rs2146323) had a statistically significant association with advanced prostate cancer (p = 0.02). The estimates suggest that carriers of the CA and AA genotype for rs35765 may have a reduced risk of developing prostate cancer (Odds Ratio (OR) = 0.72, 95% Confidence Interval (CI) = 0.55, 0.95, OR = 0.60, 95% CI = 0.26, 1.39, respectively). Meanwhile, carriers of the CA and AA genotype for rs2146323 may be at increased risk of advanced prostate cancer, which was determined by a Gleason score of above 7 (OR = 1.72, 95% CI = 1.12, 2.63, OR = 1.90, 95% CI = 1.08, 3.34, respectively). Utilising the widely used short-form health survey, the SF-36v2, the second aim of this study was to investigate the relationship between prostate cancer and QoL prior to treatment. Assessing QoL at this time-point was important as little research has been conducted to evaluate if prostate cancer affects QoL regardless of treatment. The analyses found that mean SF-36v2 scale scores related to physical health were higher by at least 0.3 Standard Deviations (SD) among men with prostate cancer than the general population comparison group. This difference was considered clinically significant (defined by group differences in mean SF-36v2 scores by at least 0.3 SD). These differences were also statistically significant (p<0.05). Mean QoL scale scores related to mental health were similar between men with prostate cancer and those from the general population comparison group. The third aim of this study was to investigate genetic variants in the VEGF and IGF-1 gene for an association with QoL in prostate cancer patients prior to their treatment. It was essential to evaluate these relationships prior to treatment, before the involvement of these genes was potentially interrupted by treatment. The analyses found that some genetic variants had a small clinically significant association (0.3 SD) to some QoL domains experienced by these men. However, most relationships were not statistically significant (p>0.05). Most of the associations found identified that a small sub-group of men with prostate cancer (approximately 2%) reported, on average, a slightly better QoL than the majority of the prostate cancer patients. The fourth aim of this research was to investigate whether associations between genetic variants in VEGF and IGF-1 and QoL were specific to men with prostate cancer, or were also applicable to the general male population. It was found that twenty out of one-hundred relationships between the genetic variants of VEGF and IGF-1 and QoL health-measures and scales examined differed between these groups. In the majority of the relationships involving VEGF SNPs that differed, a clinically significant difference (0.3 or more SD) between mean scores among the genotype groups in prostate cancer patients was found, while mean scores among men from the general-population comparison group were similar. For example, prostate cancer participants who carried at least one T allele (CT or TT genotype) for rs3024994 had a clinically significant higher (0.3 SD) mean QoL score in terms of the role-physical scale, than participants who carried the CC genotype. This was not seen among men from the general population sample, as the mean score was similar between genotype groups. The opposite was seen in regards to the IGF-1 SNPs examined. Overall, these relationships were not considered to directly impact on the clinical options for men with prostate cancer. As this study utilised secondary data from two separate studies, there are a number of important limitations that should be acknowledged including issues of multiple comparisons, power, and missing or unavailable data. It is recommended that this study be replicated as a better-designed study that takes greater consideration of the many factors involved in prostate cancer and QoL. Investigation into other genetic variants of VEGF or IGF-1 is also warranted, as is consideration of other genes and their relationship with QoL. Through identifying certain genetic variants that have a modest association to prostate cancer, this project adds to the knowledge surrounding VEGF and IGF-1 and their role in prostate cancer susceptibility. Importantly, this project has also introduced the potential role genetics plays in QoL, through investigating the relationships between genetic variants of VEGF and IGF-1 and QoL.

Two novel mutations and a previously unreported intronic polymorphism in the NOTCH3 gene

Cerebral autosomal dominant arteriopathy with subcortical infarcts and leukoencephalopathy (CADASIL) is a hereditary disease of small vessel caused by mutations in the NOTCH3 gene (NCBI Gene ID: 4854) located on chromosome 19p13.1. NOTCH3 consists of 33 exons which encode a protein of 2321 amino acids. Exons 3 and 4 were found to be mutation hotspots, containing more than 65% of all CADASIL mutations. We performed direct sequencing on an ABI 3130 Genetic Analyser to screen for mutations and polymorphisms on 300 patients who were clinically suspected to have CADASIL. First, exons 3 and 4 were screened in NOTCH3 and if there were no variations found, then extended CADASIL testing (exons 2, 11, 18 and 19) was offered to patients. Here we report two novel non-synonymous mutations identified in the NOTCH3 gene. The first mutation, located in exon 4 was found in a 49-year-old female and causes an alanine to valine amino acid change at position 202 (605C > T). The second mutation, located in exon 11, was found in a 66-year-old female and causes a cysteine to arginine amino acid change at position 579 (1735T > C). We also report a 46-year-old male with a known polymorphism Thr101Thr (rs3815188) and an unreported polymorphism NM_000435.2:c.679+60G>A observed in intron 4 of the NOTCH3 gene. Although Ala202Ala (rs1043994) is a common polymorphism in the NOTCH3 gene, our reported novel mutation (Ala202Val) causes an amino acid change at the same locus. Our other reported mutation (Cys579Arg) correlates well with other known mutations in NOTCH3, as the majority of the CADASIL-associated mutations in NOTCH3 generally occur in the EGF-like (epidermal growth factor-like) repeat domain, causing a change in the number of cysteine residues. The intronic polymorphism NM_000435.2:c.679+60G>A lies close to the intron–exon boundary and may affect the splicing mechanism in the NOTCH3 gene.

Regional chromosomal assignment of human renin gene to 1q12→qter and use in linkage studies in Charcot-Marie-Tooth disease

The gene for renin, previously mapped to human chromosome 1, was further localized to 1q12 → qter using human-mouse somatic cell hybrid DNAs. The renin DNA probe used (λ HR5) could detect a HindIII restriction fragment length polymorphism. When used in studies of 12 informative families, no linkage could be found between the renin and Charcot-Marie-Tooth disease. Furthermore, an association of any renin allele with hypertension was not apparent.

Heterogeneity evidence and linkage studies on Charcot-Marie-Tooth disease

Charcot-Marie-Tooth neuropathy type 1 (CMT1) is an autosomal dominant disorder originally localized to chromosome 1 by linkage to the Duffy blood group. Studies have since shown that the disorder may be heterogeneous, as not all families show this linkage. We tested genetic heterogeneity by the HOMOG computer program in 15 CMT1 pedigrees informative for Duffy. We detected no evidence for heterogeneity in this sample, but when we combined results with previously published lod scores, heterogeneity was statistically significant. Twelve of the 15 families studied did not show linkage to Duffy. We found six of these families to be informative for a chromosome 19 marker, apolipoprotein CII(ApoC2). Despite a previous report showing probable linkage of a non-Duffy-linked CMT1 pedigree to two chromosome 19 markers, we did not detect significant linkage of ApoC2 to CMT1 in these families.

Isolation and use of chromosome 1 probes for linkage studies on Charcot-Marie-Tooth disease

Nine probes were isolated from a human chromosome 1 enriched library and mapped to regions of chromosome 1 using somatic cell hybrid lines. One clone, LR67, which mapped 1q12→q23 detected a BglI RFLP. This probe, as well as 4 other known chromosome 1 markers, α-spectrin, Factor XIIIB, DR10 and DR78, were used for linkage studies in 15 Charcot-Marie-Tooth disease (CMT1) families. Close linking of CMT1 to any of the 5 markers was not indicated. Total lod scores excluded linkage of CMT1 to LR67 and to DR10 at 5 cM or less, to DR78 and 10 cM or less, α-spectrin at 15 cM or less and Factor XIIIB at 20 cM or less. Possible linkage, however, was shown between LR67 and CMT1 at a distance of 30 cM. Also linkage at a distance of 5 cM was detected between this probe and α-spectrin.

Association and linkage analyses of restriction fragment length polymorphisms for the human renin and antithrombin III genes in essential hypertension

Essential hypertension is a highly hereditable disorder in which genetic influences predominate over environmental factors. The molecular genetic profiles which predispose to essential hypertension are not known. In rats with genetic hypertension, there is some recent evidence pointing to linkage of renin gene alleles with blood pressure. The genes for renin and antithrombin III belong to a conserved synteny group which, in humans, spans the q21.3-32.3 region of chromosome I and, in rats, is linkage group X on chromosome 13. The present study examined the association of particular human renin gene (REN) and antithrombin III gene (AT3) polymorphisms with essential hypertension by comparing the frequency of specific alleles for each of these genes in 50 hypertensive offspring of hypertensive parents and 91 normotensive offspring of normotensive parents. In addition, linkage relationships were examined in hypertensive pedigrees with multiple affected individuals. Alleles of a REN HindIII restriction fragment length polymorphism (RFLP) were detected using a genomic clone, λHR5, to probe Southern blots of HindIII-cut leucocyte DNA, and those for an AT3 Pstl RFLP were detected by phATIII 113 complementary DNA probe. The frequencies of each REN allele in the hypertensive group were 0.76 and 0.24 compared with 0.74 and 0.26 in the normotensive group. For AT3, hypertensive allele frequencies were 0.49 and 0.51 compared with normotensive values of 0.54 and 0.46. These differences were not significant by χ2 analysis (P > 0.2). Linkage analysis of a family (data from 16 family members, 10 of whom were hypertensive), informative for both markers, without an age-of-onset correction, and assuming dominant inheritance of hypertension, complete penetrance and a disease frequency of 20%, did not indicate linkage of REN with hypertension, but gave a positive, although not significant, logarithm of the odds for linkage score of 0.784 at a recombination fraction of 0 for AT3 linkage to hypertension. In conclusion, the present study could find no evidence for an association of a REN HindIII RFLP with essential hypertension or for a linkage of the locus defined by this RFLP in a family segregating for hypertension. In the case of an AT3 Pstl RFLP, although association analysis was negative, linkage analysis suggested possible involvement (odds of 6:1 in favour) of a gene located near the 1q23 locus with hypertension in one informative family.

Assessment of damages for wrongful birth and consolidation in advance care directives

The recent decision of Waller v James involved a claim by the plaintiff parents for damages for wrongful birth against the defendant doctor, Dr James, a gynaecologist with a practice in infertility and IVF procedures, who had been consulted by the plaintiffs. The second plaintiff, Mr Waller suffered an inherited anti-thrombin deficiency (ATD), a condition which results in a propensity for the blood to clot, at least in adults. Dr James subsequently recommended IVF treatment. The first plaintiff, Mrs Waller became pregnant after the first cycle of IVF treatment. Her son Keeden was born on 10 August 2000 with a genetic anti-thrombin deficiency. Keeden was released from hospital on 14 August 2000. However, he was brought back to the hospital the next day with cerebral thrombosis (CSVT). As a result of the thrombosis, he suffered permanent brain damage, cerebral palsy and related disabilities. The plaintiffs alleged that the defendant was in breach of contract and his common law duty of care to the plaintiffs in failing to inform them, or cause them to be informed, of the hereditary aspects of ATD. They further alleged that, had they been properly informed, they would not have proceeded to conceive a child using the male plaintiff’s sperm and therefore avoided the harm that had befallen them. The plaintiffs claimed damages to compensate them for their losses, including psychiatric and physical injuries and the costs of having, raising and caring for Keeden. The defendant was held to be not liable in negligence by Justice Hislop of the Supreme Court of New South Wales because a finding was made on medical causation which was adverse to the plaintiffs claim.

The hereditary transmission of the glutathione transferase hGSTT1-1 conjugator phenotype in a large family

The polymorphism of human glutathione transferase hGSTT1-1 is expressed in three phenotypes. Experimentally, individuals can be classified as non-conjugators, low conjugators and 'high' conjugators depending on the enzyme activity in blood towards methylene chloride using a gas chromatographic assay. Non-conjugators do not have a functional hGSTT1 gene; however, little is known about the molecular basis of the three conjugator phenotypes. The higher hGSTT1-1 activity in high conjugators may be the result of enzyme induction or be genetically determined. Twenty-nine members of a large family, including three generations were phenotyped and genotyped with respect to hGSTT1-1. The hGSTT1-1 enzyme activity of high conjugators was twice as high as that of low conjugators. The distribution of hGSTT1-1 phenotypes strongly indicates a Mendelian intermediary inheritance, in which a gene-dosage effect results in a doubled enzyme expression in the presence of two functional alleles. The Mendelian intermediary inheritance is further supported by the findings of a semiquantitative polymerase chain reaction method designed to distinguish the three genotypes of hGSTT1 for rapid screening of large study groups.

Super DNAging - New insights into DNA integrity, genome stability and telomeres in the oldest old

Reductions in DNA integrity, genome stability, and telomere length are strongly associated with the aging process, age-related diseases as well as the age-related loss of muscle mass. However, in people reaching an age far beyond their statistical life expectancy the prevalence of diseases, such as cancer, cardiovascular disease, diabetes or dementia, is much lower compared to “averagely” aged humans. These inverse observations in nonagenarians (90–99 years), centenarians (100–109 years) and super-centenarians (110 years and older) require a closer look into dynamics underlying DNA damage within the oldest old of our society. Available data indicate improved DNA repair and antioxidant defense mechanisms in “super old” humans, which are comparable with much younger cohorts. Partly as a result of these enhanced endogenous repair and protective mechanisms, the oldest old humans appear to cope better with risk factors for DNA damage over their lifetime compared to subjects whose lifespan coincides with the statistical life expectancy. This model is supported by study results demonstrating superior chromosomal stability, telomere dynamics and DNA integrity in “successful agers”. There is also compelling evidence suggesting that life-style related factors including regular physical activity, a well-balanced diet and minimized psycho-social stress can reduce DNA damage and improve chromosomal stability. The most conclusive picture that emerges from reviewing the literature is that reaching “super old” age appears to be primarily determined by hereditary/genetic factors, while a healthy lifestyle additionally contributes to achieving the individual maximum lifespan in humans. More research is required in this rapidly growing population of super old people. In particular, there is need for more comprehensive investigations including short- and long-term lifestyle interventions as well as investigations focusing on the mechanisms causing DNA damage, mutations, and telomere shortening.