Statistical and econometric analysis of the treatment of HIV/AIDS

The epidemic of HIV/AIDS in the United States is constantly changing and evolving, starting from patient zero to now an estimated 650,000 to 900,000 Americans infected. The nature and course of HIV changed dramatically with the introduction of antiretrovirals. This discourse examines many different facets of HIV from the beginning where there wasn't any treatment for HIV until the present era of highly active antiretroviral therapy (HAART). By utilizing statistical analysis of clinical data, this paper examines where we were, where we are and projections as to where treatment of HIV/AIDS is headed.

Chapter Two describes the datasets that were used for the analyses. The primary database utilized was collected by myself from an outpatient HIV clinic. The data included dates from 1984 until the present. The second database was from the Multicenter AIDS Cohort Study (MACS) public dataset. The data from the MACS cover the time between 1984 and October 1992. Comparisons are made between both datasets.

Chapter Three discusses where we were. Before the first anti-HIV drugs (called antiretrovirals) were approved, there was no treatment to slow the progression of HIV. The first generation of antiretrovirals, reverse transcriptase inhibitors such as AZT (zidovudine), DDI (didanosine), DDC (zalcitabine), and D4T (stavudine) provided the first treatment for HIV. The first clinical trials showed that these antiretrovirals had a significant impact on increasing patient survival. The trials also showed that patients on these drugs had increased CD4+ T cell counts. Chapter Three examines the distributions of CD4 T cell counts. The results show that the estimated distributions of CD4 T cell counts are distinctly non-Gaussian. Thus distributional assumptions regarding CD4 T cell counts must be taken, into account when performing analyses with this marker. The results also show the estimated CD4 T cell distributions for each disease stage: asymptomatic, symptomatic and AIDS are non-Gaussian. Interestingly, the distribution of CD4 T cell counts for the asymptomatic period is significantly below that of the CD4 T cell distribution for the uninfected population suggesting that even in patients with no outward symptoms of HIV infection, there exists high levels of immunosuppression.

Chapter Four discusses where we are at present. HIV quickly grew resistant to reverse transcriptase inhibitors which were given sequentially as mono or dual therapy. As resistance grew, the positive effects of the reverse transcriptase inhibitors on CD4 T cell counts and survival dissipated. As the old era faded a new era characterized by a new class of drugs and new technology changed the way that we treat HIV-infected patients. Viral load assays were able to quantify the levels of HIV RNA in the blood. By quantifying the viral load, one now had a faster, more direct way to test antiretroviral regimen efficacy. Protease inhibitors, which attacked a different region of HIV than reverse transcriptase inhibitors, when used in combination with other antiretroviral agents were found to dramatically and significantly reduce the HIV RNA levels in the blood. Patients also experienced significant increases in CD4 T cell counts. For the first time in the epidemic, there was hope. It was hypothesized that with HAART, viral levels could be kept so low that the immune system as measured by CD4 T cell counts would be able to recover. If these viral levels could be kept low enough, it would be possible for the immune system to eradicate the virus. The hypothesis of immune reconstitution, that is bringing CD4 T cell counts up to levels seen in uninfected patients, is tested in Chapter Four. It was found that for these patients, there was not enough of a CD4 T cell increase to be consistent with the hypothesis of immune reconstitution.

In Chapter Five, the effectiveness of long-term HAART is analyzed. Survival analysis was conducted on 213 patients on long-term HAART. The primary endpoint was presence of an AIDS defining illness. A high level of clinical failure, or progression to an endpoint, was found.

Chapter Six yields insights into where we are going. New technology such as viral genotypic testing, that looks at the genetic structure of HIV and determines where mutations have occurred, has shown that HIV is capable of producing resistance mutations that confer multiple drug resistance. This section looks at resistance issues and speculates, ceterus parabis, where the state of HIV is going. This section first addresses viral genotype and the correlates of viral load and disease progression. A second analysis looks at patients who have failed their primary attempts at HAART and subsequent salvage therapy. It was found that salvage regimens, efforts to control viral replication through the administration of different combinations of antiretrovirals, were not effective in 90 percent of the population in controlling viral replication. Thus, primary attempts at therapy offer the best change of viral suppression and delay of disease progression. Documentation of transmission of drug-resistant virus suggests that the public health crisis of HIV is far from over. Drug resistant HIV can sustain the epidemic and hamper our efforts to treat HIV infection. The data presented suggest that the decrease in the morbidity and mortality due to HIV/AIDS is transient. Deaths due to HIV will increase and public health officials must prepare for this eventuality unless new treatments become available. These results also underscore the importance of the vaccine effort.

The final chapter looks at the economic issues related to HIV. The direct and indirect costs of treating HIV/AIDS are very high. For the first time in the epidemic, there exists treatment that can actually slow disease progression. The direct costs for HAART are estimated. It is estimated that the direct lifetime costs for treating each HIV infected patient with HAART is between $353,000 to $598,000 depending on how long HAART prolongs life. If one looks at the incremental cost per year of life saved it is only $101,000. This is comparable with the incremental costs per year of life saved from coronary artery bypass surgery.

Policy makers need to be aware that although HAART can delay disease progression, it is not a cure and HIV is not over. The results presented here suggest that the decreases in the morbidity and mortality due to HIV are transient. Policymakers need to be prepared for the eventual increase in AIDS incidence and mortality. Costs associated with HIV/AIDS are also projected to increase. The cost savings seen recently have been from the dramatic decreases in the incidence of AIDS defining opportunistic infections. As patients who have been on HAART the longest start to progress to AIDS, policymakers and insurance companies will find that the cost of treating HIV/AIDS will increase.

Studies of the equation of state and elasticity of mantle minerals

Because the Earth’s upper mantle is inaccessible to us, in order to understand the chemical and physical processes that occur in the Earth’s interior we must rely on both experimental work and computational modeling. This thesis addresses both of these geochemical methods. In the first chapter, I develop an internally consistent comprehensive molar volume model for spinels in the oxide system FeO-MgO-Fe₂O₃-Cr₂O₃-Al₂O₃-TiO₂. The model is compared to the current MELTS spinel model with a demonstration of the impact of the model difference on the estimated spinel-garnet lherzolite transition pressure. In the second chapter, I calibrate a molar volume model for cubic garnets in the system SiO₂-Al₂O₃-TiO₂-Fe₂O₃-Cr₂O₃-FeO-MnO-MgO-CaO-Na₂O. I use the method of singular value analysis to calibrate excess volume of mixing parameters for the garnet model. The implications the model has for the density of the lithospheric mantle are explored. In the third chapter, I discuss the nuclear inelastic X-ray scattering (NRIXS) method, and present analysis of three orthopyroxene samples with different Fe contents. Longitudinal and shear wave velocities, elastic parameters, and other thermodynamic information are extracted from the raw NRIXS data.

New insights into the formation and modification of carbonate-bearing minerals and methane gas in geological systems using multiply substituted isotopologues

This thesis describes the use of multiply-substituted stable isotopologues of carbonate minerals and methane gas to better understand how these environmentally significant minerals and gases form and are modified throughout their geological histories. Stable isotopes have a long tradition in earth science as a tool for providing quantitative constraints on how molecules, in or on the earth, formed in both the present and past. Nearly all studies, until recently, have only measured the bulk concentrations of stable isotopes in a phase or species. However, the abundance of various isotopologues within a phase, for example the concentration of isotopologues with multiple rare isotopes (multiply substituted or 'clumped' isotopologues) also carries potentially useful information. Specifically, the abundances of clumped isotopologues in an equilibrated system are a function of temperature and thus knowledge of their abundances can be used to calculate a sample’s formation temperature. In this thesis, measurements of clumped isotopologues are made on both carbonate-bearing minerals and methane gas in order to better constrain the environmental and geological histories of various samples.

Clumped-isotope-based measurements of ancient carbonate-bearing minerals, including apatites, have opened up paleotemperature reconstructions to a variety of systems and time periods. However, a critical issue when using clumped-isotope based measurements to reconstruct ancient mineral formation temperatures is whether the samples being measured have faithfully recorded their original internal isotopic distributions. These original distributions can be altered, for example, by diffusion of atoms in the mineral lattice or through diagenetic reactions. Understanding these processes quantitatively is critical for the use of clumped isotopes to reconstruct past temperatures, quantify diagenesis, and calculate time-temperature burial histories of carbonate minerals. In order to help orient this part of the thesis, Chapter 2 provides a broad overview and history of clumped-isotope based measurements in carbonate minerals.

In Chapter 3, the effects of elevated temperatures on a sample’s clumped-isotope composition are probed in both natural and experimental apatites (which contain structural carbonate groups) and calcites. A quantitative model is created that is calibrated by the experiments and consistent with the natural samples. The model allows for calculations of the change in a sample’s clumped isotope abundances as a function of any time-temperature history.

In Chapter 4, the effects of diagenesis on the stable isotopic compositions of apatites are explored on samples from a variety of sedimentary phosphorite deposits. Clumped isotope temperatures and bulk isotopic measurements from carbonate and phosphate groups are compared for all samples. These results demonstrate that samples have experienced isotopic exchange of oxygen atoms in both the carbonate and phosphate groups. A kinetic model is developed that allows for the calculation of the amount of diagenesis each sample has experienced and yields insight into the physical and chemical processes of diagenesis.

The thesis then switches gear and turns its attention to clumped isotope measurements of methane. Methane is critical greenhouse gas, energy resource, and microbial metabolic product and substrate. Despite its importance both environmentally and economically, much about methane’s formational mechanisms and the relative sources of methane to various environments remains poorly constrained. In order to add new constraints to our understanding of the formation of methane in nature, I describe the development and application of methane clumped isotope measurements to environmental deposits of methane. To help orient the reader, a brief overview of the formation of methane in both high and low temperature settings is given in Chapter 5.

In Chapter 6, a method for the measurement of methane clumped isotopologues via mass spectrometry is described. This chapter demonstrates that the measurement is precise and accurate. Additionally, the measurement is calibrated experimentally such that measurements of methane clumped isotope abundances can be converted into equivalent formational temperatures. This study represents the first time that methane clumped isotope abundances have been measured at useful precisions.

In Chapter 7, the methane clumped isotope method is applied to natural samples from a variety of settings. These settings include thermogenic gases formed and reservoired in shales, migrated thermogenic gases, biogenic gases, mixed biogenic and thermogenic gas deposits, and experimentally generated gases. In all cases, calculated clumped isotope temperatures make geological sense as formation temperatures or mixtures of high and low temperature gases. Based on these observations, we propose that the clumped isotope temperature of an unmixed gas represents its formation temperature — this was neither an obvious nor expected result and has important implications for how methane forms in nature. Additionally, these results demonstrate that methane-clumped isotope compositions provided valuable additional constraints to studying natural methane deposits.

A generalized treatment of the order-disorder transformation in alloys and its effects on their magnetic properties

A theory of the order-disorder transformation is developed in complete generality. The general theory is used to calculate long range order parameters, short range order parameters, energy, and phase diagrams for a face centered cubic binary alloy. The theoretical results are compared to the experimental determination of the copper-gold system, Values for the two adjustable parameters are obtained.

An explanation for the behavior of magnetic alloys is developed, Curie temperatures and magnetic moments of the first transition series elements and their alloys in both the ordered and disordered states are predicted. Experimental agreement is excellent in most cases. It is predicted that the state of order can effect the magnetic properties of an alloy to a considerable extent in alloys such as Ni₃Mn. The values of the adjustable parameter used to fix the level of the Curie temperature, and the adjustable parameter that expresses the effect of ordering on the Curie temperature are obtained.

The interplay of localization and interactions in quantum many-body systems

Disorder and interactions both play crucial roles in quantum transport. Decades ago, Mott showed that electron-electron interactions can lead to insulating behavior in materials that conventional band theory predicts to be conducting. Soon thereafter, Anderson demonstrated that disorder can localize a quantum particle through the wave interference phenomenon of Anderson localization. Although interactions and disorder both separately induce insulating behavior, the interplay of these two ingredients is subtle and often leads to surprising behavior at the periphery of our current understanding. Modern experiments probe these phenomena in a variety of contexts (e.g. disordered superconductors, cold atoms, photonic waveguides, etc.); thus, theoretical and numerical advancements are urgently needed. In this thesis, we report progress on understanding two contexts in which the interplay of disorder and interactions is especially important.

The first is the so-called “dirty” or random boson problem. In the past decade, a strong-disorder renormalization group (SDRG) treatment by Altman, Kafri, Polkovnikov, and Refael has raised the possibility of a new unstable fixed point governing the superfluid-insulator transition in the one-dimensional dirty boson problem. This new critical behavior may take over from the weak-disorder criticality of Giamarchi and Schulz when disorder is sufficiently strong. We analytically determine the scaling of the superfluid susceptibility at the strong-disorder fixed point and connect our analysis to recent Monte Carlo simulations by Hrahsheh and Vojta. We then shift our attention to two dimensions and use a numerical implementation of the SDRG to locate the fixed point governing the superfluid-insulator transition there. We identify several universal properties of this transition, which are fully independent of the microscopic features of the disorder.

The second focus of this thesis is the interplay of localization and interactions in systems with high energy density (i.e., far from the usual low energy limit of condensed matter physics). Recent theoretical and numerical work indicates that localization can survive in this regime, provided that interactions are sufficiently weak. Stronger interactions can destroy localization, leading to a so-called many-body localization transition. This dynamical phase transition is relevant to questions of thermalization in isolated quantum systems: it separates a many-body localized phase, in which localization prevents transport and thermalization, from a conducting (“ergodic”) phase in which the usual assumptions of quantum statistical mechanics hold. Here, we present evidence that many-body localization also occurs in quasiperiodic systems that lack true disorder.

I. Quantum mechanics of one-dimensional two-particle models. II. A quantum mechanical treatment of inelastic collisions

Part I

Solutions of Schrödinger’s equation for system of two particles bound in various stationary one-dimensional potential wells and repelling each other with a Coulomb force are obtained by the method of finite differences. The general properties of such systems are worked out in detail for the case of two electrons in an infinite square well. For small well widths (1-10 a.u.) the energy levels lie above those of the noninteresting particle model by as much as a factor of 4, although excitation energies are only half again as great. The analytical form of the solutions is obtained and it is shown that every eigenstate is doubly degenerate due to the “pathological” nature of the one-dimensional Coulomb potential. This degeneracy is verified numerically by the finite-difference method. The properties of the square-well system are compared with those of the free-electron and hard-sphere models; perturbation and variational treatments are also carried out using the hard-sphere Hamiltonian as a zeroth-order approximation. The lowest several finite-difference eigenvalues converge from below with decreasing mesh size to energies below those of the “best” linear variational function consisting of hard-sphere eigenfunctions. The finite-difference solutions in general yield expectation values and matrix elements as accurate as those obtained using the “best” variational function.

The system of two electrons in a parabolic well is also treated by finite differences. In this system it is possible to separate the center-of-mass motion and hence to effect a considerable numerical simplification. It is shown that the pathological one-dimensional Coulomb potential gives rise to doubly degenerate eigenstates for the parabolic well in exactly the same manner as for the infinite square well.

Part II

A general method of treating inelastic collisions quantum mechanically is developed and applied to several one-dimensional models. The formalism is first developed for nonreactive “vibrational” excitations of a bound system by an incident free particle. It is then extended to treat simple exchange reactions of the form A + BC →AB + C. The method consists essentially of finding a set of linearly independent solutions of the Schrödinger equation such that each solution of the set satisfies a distinct, yet arbitrary boundary condition specified in the asymptotic region. These linearly independent solutions are then combined to form a total scattering wavefunction having the correct asymptotic form. The method of finite differences is used to determine the linearly independent functions.

The theory is applied to the impulsive collision of a free particle with a particle bound in (1) an infinite square well and (2) a parabolic well. Calculated transition probabilities agree well with previously obtained values.

Several models for the exchange reaction involving three identical particles are also treated: (1) infinite-square-well potential surface, in which all three particles interact as hard spheres and each two-particle subsystem (i.e. BC and AB) is bound by an attractive infinite-square-well potential; (2) truncated parabolic potential surface, in which the two-particle subsystems are bound by a harmonic oscillator potential which becomes infinite for interparticle separations greater than a certain value; (3) parabolic (untruncated) surface. Although there are no published values with which to compare our reaction probabilities, several independent checks on internal consistency indicate that the results are reliable.

A mathematical and experimental investigation of microcycle spectral estimates of semiconductor flicker noise

The experimental portion of this thesis tries to estimate the density of the power spectrum of very low frequency semiconductor noise, from 10^-6.3 cps to 1. cps with a greater accuracy than that achieved in previous similar attempts: it is concluded that the spectrum is 1/f^α with α approximately 1.3 over most of the frequency range, but appearing to have a value of about 1 in the lowest decade. The noise sources are, among others, the first stage circuits of a grounded input silicon epitaxial operational amplifier. This thesis also investigates a peculiar form of stationarity which seems to distinguish flicker noise from other semiconductor noise.

In order to decrease by an order of magnitude the pernicious effects of temperature drifts, semiconductor "aging", and possible mechanical failures associated with prolonged periods of data taking, 10 independent noise sources were time-multiplexed and their spectral estimates were subsequently averaged. If the sources have similar spectra, it is demonstrated that this reduces the necessary data-taking time by a factor of 10 for a given accuracy.

In view of the measured high temperature sensitivity of the noise sources, it was necessary to combine the passive attenuation of a special-material container with active control. The noise sources were placed in a copper-epoxy container of high heat capacity and medium heat conductivity, and that container was immersed in a temperature controlled circulating ethylene-glycol bath.

Other spectra of interest, estimated from data taken concurrently with the semiconductor noise data were the spectra of the bath's controlled temperature, the semiconductor surface temperature, and the power supply voltage amplitude fluctuations. A brief description of the equipment constructed to obtain the aforementioned data is included.

The analytical portion of this work is concerned with the following questions: what is the best final spectral density estimate given 10 statistically independent ones of varying quality and magnitude? How can the Blackman and Tukey algorithm which is used for spectral estimation in this work be improved upon? How can non-equidistant sampling reduce data processing cost? Should one try to remove common trands shared by supposedly statistically independent noise sources and, if so, what are the mathematical difficulties involved? What is a physically plausible mathematical model that can account for flicker noise and what are the mathematical implications on its statistical properties? Finally, the variance of the spectral estimate obtained through the Blackman/Tukey algorithm is analyzed in greater detail; the variance is shown to diverge for α ≥ 1 in an assumed power spectrum of k/|f|^α, unless the assumed spectrum is "truncated".