On the Axiomatic Systems of Steenrod Homology Theory of Compact Spaces

On the category of compact metric spaces an exact homology theory was defined and its relation to the Vietoris homology theory was studied by N. Steenrod [S]. In particular, the homomorphism from the Steenrod homology groups to the Vietoris homology groups was defined and it was shown that the kernel of the given homomorphism are homological groups, which was called weak homology groups [S], [E]. The Steenrod homology theory on the category of compact metric pairs was axiomatically described by J.Milnor. In [Mil] the uniqueness theorem is proved using the Eilenberg-Steenrod axioms and as well as relative homeomorphism and clusres axioms. J. Milnor constructed the homology theory on the category $Top^2_C$ of compact Hausdorff pairs and proved that on the given category it satisfies nine axioms - the Eilenberg-Steenrod, relative homeomorphis and cluster axioms (see theorem 5 in [Mil]). Besides, using the construction of weak homology theory, J.Milnor proved that constructed homology theory satisfies partial continuity property on the subcategory $Top^2_{CM}$ (see theorem 4 in [Mil]) and the universal coefficient formula on the category $Top^2_C$ (see Lemma 5 in [Mil]). On the category of compact Hausdorff pairs, different axiomatic systems were proposed by N. Berikashvili [B1], [B2], H.Inasaridze and L. Mdzinarishvili [IM], L. Mdzinarishvili [M] and H.Inasaridze [I], but there was not studied any connection between them. The paper studies this very problem. In particular, in the paper it is proved that any homology theory in Inasaridze sense is the homology theory in the Berikashvili sense, which itself is the homology theory in the Mdzinarishvili sense. On the other hand, it is shown that if a homology theory in the Mdzinarishvili sense is exact functor of the second argument, then it is the homology in the Inasaridze sense.Comment: 13 page

Weibull-like Model of Cancer Development in Aging

Mathematical modeling of cancer development is aimed at assessing the risk factors leading to cancer. Aging is a common risk factor for all adult cancers. The risk of getting cancer in aging is presented by a hazard function that can be estimated from the observed incidence rates collected in cancer registries. Recent analyses of the SEER database show that the cancer hazard function initially increases with the age, and then it turns over and falls at the end of the lifetime. Such behavior of the hazard function is poorly modeled by the exponential or compound exponential-linear functions mainly utilized for the modeling. In this work, for mathematical modeling of cancer hazards, we proposed to use the Weibull-like function, derived from the Armitage-Doll multistage concept of carcinogenesis and an assumption that number of clones at age t developed from mutated cells follows the Poisson distribution. This function is characterized by three parameters, two of which (r and λ) are the conventional parameters of the Weibull probability distribution function, and an additional parameter (C0) that adjusts the model to the observational data. Biological meanings of these parameters are: r—the number of stages in carcinogenesis, λ—an average number of clones developed from the mutated cells during the first year of carcinogenesis, and C0—a data adjustment parameter that characterizes a fraction of the age-specific population that will get this cancer in their lifetime. To test the validity of the proposed model, the nonlinear regression analysis was performed for the lung cancer (LC) data, collected in the SEER 9 database for white men and women during 1975–2004. Obtained results suggest that: (i) modeling can be improved by the use of another parameter A- the age at the beginning of carcinogenesis; and (ii) in white men and women, the processes of LC carcinogenesis vary by A and C0, while the corresponding values of r and λ are nearly the same. Overall, the proposed Weibull-like model provides an excellent fit of the estimates of the LC hazard function in aging. It is expected that the Weibull-like model can be applicable to fit estimates of hazard functions of other adult cancers as well

Estimation of Hazard Functions in the Log-Linear Age-Period-Cohort Model: Application to Lung Cancer Risk Associated with Geographical Area

An efficient computing procedure for estimating the age-specific hazard functions by the log-linear age-period-cohort (LLAPC) model is proposed. This procedure accounts for the influence of time period and birth cohort effects on the distribution of age-specific cancer incidence rates and estimates the hazard function for populations with different exposures to a given categorical risk factor. For these populations, the ratio of the corresponding age-specific hazard functions is proposed for use as a measure of relative hazard. This procedure was used for estimating the risks of lung cancer (LC) for populations living in different geographical areas. For this purpose, the LC incidence rates in white men and women, in three geographical areas (namely: San Francisco-Oakland, Connecticut and Detroit), collected from the SEER 9 database during 1975–2004, were utilized. It was found that in white men the averaged relative hazard (an average of the relative hazards over all ages) of LC in Connecticut vs. San Francisco-Oakland is 1.31 ± 0.02, while in Detroit vs. San Francisco-Oakland this averaged relative hazard is 1.53 ± 0.02. In white women, analogous hazards in Connecticut vs. San Francisco-Oakland and Detroit vs. San Francisco-Oakland are 1.22 ± 0.02 and 1.32 ± 0.02, correspondingly. The proposed computing procedure can be used for assessing hazard functions for other categorical risk factors, such as gender, race, lifestyle, diet, obesity, etc

Breast cancer incidence in black and white women stratified by estrogen and progesterone receptor statuses.

BACKGROUND: There is increasing evidence that breast cancer is a heterogeneous disease presented by different phenotypes and that white women have a higher breast cancer incidence rate, whereas black women have a higher mortality rate. It is also well known that white women have lower incidence rates than black women until approximately age 40, when rate curves cross over and white women have higher rates. The goal of this study was to validate the risk of white and black women to breast cancer phenotypes, stratified by statuses of the estrogen (ER) and progesterone (PR) receptors. METHODOLOGY/PRINCIPAL FINDINGS: SEER17 data were fractioned by receptor status into [ER+, PR+], [ER-, PR-], [ER+, PR-], and [ER-, PR+] phenotypes. It was shown that in black women compared to white women, cumulative age-specific incidence rates are: (i) smaller for the [ER+, PR+] phenotype; (ii) larger for the [ER-, PR-] and [ER-, PR+] phenotypes; and (iii) almost equal for the [ER+, PR-] phenotype. Clemmesen\u27s Hook, an undulation unique to women\u27s breast cancer age-specific incidence rate curves, is shown here to exist in both races only for the [ER+, PR+] phenotype. It was also shown that for all phenotypes, rate curves have additional undulations and that age-specific incidence rates are nearly proportional in all age intervals. CONCLUSIONS/SIGNIFICANCE: For black and white women, risk for the [ER+, PR+], [ER-, PR-] and [ER-, PR+] phenotypes are race dependent, while risk for the [ER+, PR-] phenotype is almost independent of race. The processes of carcinogenesis in aging, leading to the development of each of the considered breast cancer phenotypes, are similar in these racial groups. Undulations exhibited on the curves of age-specific incidence rates of the considered breast cancer phenotypes point to the presence of several subtypes (to be determined) of each of these phenotypes

A Novel Approach for Analysis of the Log-Linear Age-Period-Cohort Model: Application to Lung Cancer Incidence

A simple, computationally efficient procedure for analyses of the time period and birth cohort effects on the distribution of the age-specific incidence rates of cancers is proposed. Assuming that cohort effects for neighboring cohorts are almost equal and using the Log-Linear Age-Period-Cohort Model, this procedure allows one to evaluate temporal trends and birth cohort variations of any type of cancer without prior knowledge of the hazard function. This procedure was used to estimate the influence of time period and birth cohort effects on the distribution of the age-specific incidence rates of first primary, microscopically confirmed lung cancer (LC) cases from the SEER9 database. It was shown that since 1975, the time period effect coefficients for men increase up to 1980 and then decrease until 2004. For women, these coefficients increase from 1975 up to 1990 and then remain nearly constant. The LC birth cohort effect coefficients for men and women increase from the cohort of 1890–94 until the cohort of 1925–29, then decrease until the cohort of 1950–54 and then remain almost unchanged. Overall, LC incidence rates, adjusted by period and cohort effects, increase up to the age of about 72–75, turn over, and then fall after the age of 75–78. The peak of the adjusted rates in men is around the age of 77–78, while in women, it is around the age of 72–73. Therefore, these results suggest that the age distribution of the incidence rates in men and women fall at old ages

A Generalized Beta model for the age distribution of cancers: application to pancreatic and kidney cancer.

The relationships between cancer incidence rates and the age of patients at cancer diagnosis are a quantitative basis for modeling age distributions of cancer. The obtained model parameters are needed to build rigorous statistical and biological models of cancer development. In this work, a new mathematical model, called the Generalized Beta (GB) model is proposed. Confidence intervals for parameters of this model are derived from a regression analysis. The GB model was used to approximate the incidence rates of the first primary, microscopically confirmed cases of pancreatic cancer (PC) and kidney cancer (KC) that served as a test bed for the proposed approach. The use of the GB model allowed us to determine analytical functions that provide an excellent fit for the observed incidence rates for PC and KC in white males and females. We make the case that the cancer incidence rates can be characterized by a unique set of model parameters (such as an overall cancer rate, and the degree of increase and decrease of cancer incidence rates). Our results suggest that the proposed approach significantly expands possibilities and improves the performance of existing mathematical models and will be very useful for modeling carcinogenic processes characteristic of cancers. To better understand the biological plausibility behind the aforementioned model parameters, detailed molecular, cellular, and tissue-specific mechanisms underlying the development of each type of cancer require further investigation. The model parameters that can be assessed by the proposed approach will complement and challenge future biomedical and epidemiological studies

Extension of Cox Proportional Hazard Model for Estimation of Interrelated Age-Period-Cohort Effects on Cancer Survival

In the frame of the Cox proportional hazard (PH) model, a novel two-step procedure for estimating age-period-cohort (APC) effects on the hazard function of death from cancer was developed. In the first step, the procedure estimates the influence of joint APC effects on the hazard function, using Cox PH regression procedures from a standard software package. In the second step, the coefficients for age at diagnosis, time period and birth cohort effects are estimated. To solve the identifiability problem that arises in estimating these coefficients, an assumption that neighboring birth cohorts almost equally affect the hazard function was utilized. Using an anchoring technique, simple procedures for obtaining estimates of interrelated age at diagnosis, time period and birth cohort effect coefficients were developed

A Heuristic Solution of the Identifiability Problem of the Age-Period-Cohort Analysis of Cancer Occurrence: Lung Cancer Example

Background: The Age–Period–Cohort (APC) analysis is aimed at estimating the following effects on disease incidence: (i) the age of the subject at the time of disease diagnosis; (ii) the time period, when the disease occurred; and (iii) the date of birth of the subject. These effects can help in evaluating the biological events leading to the disease, in estimating the influence of distinct risk factors on disease occurrence, and in the development of new strategies for disease prevention and treatment. Methodology/Principal Findings: We developed a novel approach for estimating the APC effects on disease incidence rates in the frame of the Log-Linear Age-Period-Cohort (LLAPC) model. Since the APC effects are linearly interdependent and cannot be uniquely estimated, solving this identifiability problem requires setting four redundant parameters within a set of unknown parameters. By setting three parameters (one of the time-period and the birth-cohort effects and the corresponding age effect) to zero, we reduced this problem to the problem of determining one redundant parameter and, used as such, the effect of the time-period adjacent to the anchored time period. By varying this identification parameter, a family of estimates of the APC effects can be obtained. Using a heuristic assumption that the differences between the adjacent birth-cohort effects are small, we developed a numerical method for determining the optimal value of the identification parameter, by which a unique set of all APC effects is determined and the identifiability problem is solved

Ginkgo Extract EGb761 Confers Neuroprotection by Reduction of Glutamate Release in Ischemic Brain

Purpose - Ginkgo extract EGb761 has shown anti-edema and anti-ischemic effects in various experimental models. In the present study, we demonstrate neuroprotective effects of EGb761 in experimental stroke while monitoring brain metabolism by microdialysis. Methods - We have used oxygen-glucose deprivation in brain slices in vitro and middle cerebral artery occlusion (MCAO) in vivo to induce ischemia in mouse brain. We used microdialysis in mouse striatum to monitor extracellular concentrations of glucose and glutamate. Results - In vitro, EGb761 reduced ischemia-induced cell swelling in hippocampal slices by 60%. In vivo, administration of EGb761 (300 mg/kg) reduced cell degeneration and edema formation after MCAO by 35-50%. Immediately following MCAO, striatal glucose levels dropped to 25% of controls, and this reduction was not significantly affected by EGb761. Striatal glutamate levels, in contrast, increased 15-fold after MCAO; after pretreatment with EGb761, glutamate levels only increased by 4-5fold. Conclusions - We show that pretreatment with EGb761 strongly reduces cellular edema formation and neurodegeneration under conditions of ischemia. The mechanism of action seems to be related to a reduction of excitotoxicity, because ischemia-induced release of glutamate was strongly suppressed. Ginkgo extracts such as EGb761 may be valuable to prevent ischemia-induced damage in stroke-prone patients. This article is open to POST-PUBLICATION REVIEW. Registered readers (see “For Readers”) may comment by clicking on ABSTRACT on the issue’s contents page