A comparison of Markov and generalized linear models of hospital mortality

This paper reports a comparison of three modeling strategies for the analysis of hospital mortality in a sample of general medicine inpatients in a Department of Veterans Affairs medical center. Logistic regression, a Markov chain model, and longitudinal logistic regression were evaluated on predictive performance as measured by the c-index and on accuracy of expected numbers of deaths compared to observed. The logistic regression used patient information collected at admission; the Markov model was comprised of two absorbing states for discharge and death and three transient states reflecting increasing severity of illness as measured by laboratory data collected during the hospital stay; longitudinal regression employed Generalized Estimating Equations (GEE) to model covariance structure for the repeated binary outcome. Results showed that the logistic regression predicted hospital mortality as well as the alternative methods but was limited in scope of application. The Markov chain provides insights into how day to day changes of illness severity lead to discharge or death. The longitudinal logistic regression showed that increasing illness trajectory is associated with hospital mortality. The conclusion is reached that for standard applications in modeling hospital mortality, logistic regression is adequate, but for new challenges facing health services research today, alternative methods are equally predictive, practical, and can provide new insights. ^

Longitudinal categorical data analysis: A continuous -time Markov chain approach

In this dissertation, we propose a continuous-time Markov chain model to examine the longitudinal data that have three categories in the outcome variable. The advantage of this model is that it permits a different number of measurements for each subject and the duration between two consecutive time points of measurements can be irregular. Using the maximum likelihood principle, we can estimate the transition probability between two time points. By using the information provided by the independent variables, this model can also estimate the transition probability for each subject. The Monte Carlo simulation method will be used to investigate the goodness of model fitting compared with that obtained from other models. A public health example will be used to demonstrate the application of this method. ^

Risk of Second Malignant Neoplasms Following Proton Arc Therapy and Volumetric Modulated Arc Therapy for Prostate Cancer

The risk of second malignant neoplasms (SMNs) following prostate radiotherapy is a concern due to the large population of survivors and decreasing age at diagnosis. It is known that parallel-opposed beam proton therapy carries a lower risk than photon IMRT. However, a comparison of SMN risk following proton and photon arc therapies has not previously been reported. The purpose of this study was to predict the ratio of excess relative risk (RRR) of SMN incidence following proton arc therapy to that after volumetric modulated arc therapy (VMAT). Additionally, we investigated the impact of margin size and the effect of risk-minimized proton beam weighting on predicted RRR. Physician-approved treatment plans were created for both modalities for three patients. Therapeutic dose was obtained with differential dose-volume histograms from the treatment planning system, and stray dose was estimated from the literature or calculated with Monte Carlo simulations. Then, various risk models were applied to the total dose. Additional treatment plans were also investigated with varying margin size and risk-minimized proton beam weighting. The mean RRR ranged from 0.74 to 0.99, depending on risk model. The additional treatment plans revealed that the RRR remained approximately constant with varying margin size, and that the predicted RRR was reduced by 12% using a risk-minimized proton arc therapy planning technique. In conclusion, proton arc therapy was found to provide an advantage over VMAT in regard to predicted risk of SMN following prostate radiotherapy. This advantage was independent of margin size and was amplified with risk-optimized proton beam weighting.

Assessment of Collimator Jaw Optimization in Reducing Normal Tissue Irradiation with Intensity Modulated Radiation Therapy

Purpose: To evaluate normal tissue dose reduction in step-and-shoot intensity-modulated radiation therapy (IMRT) on the Varian 2100 platform by tracking the multileaf collimator (MLC) apertures with the accelerator jaws. Methods: Clinical radiation treatment plans for 10 thoracic, 3 pediatric and 3 head and neck patients were converted to plans with the jaws tracking each segment’s MLC apertures. Each segment was then renormalized to account for the change in collimator scatter to obtain target coverage within 1% of that in the original plan. The new plans were compared to the original plans in a commercial radiation treatment planning system (TPS). Reduction in normal tissue dose was evaluated in the new plan by using the parameters V5, V10, and V20 in the cumulative dose-volume histogram for the following structures: total lung minus GTV (gross target volume), heart, esophagus, spinal cord, liver, parotids, and brainstem. In order to validate the accuracy of our beam model, MLC transmission measurements were made and compared to those predicted by the TPS. Results: The greatest change between the original plan and new plan occurred at lower dose levels. The reduction in V20 was never more than 6.3% and was typically less than 1% for all patients. The reduction in V5 was 16.7% maximum and was typically less than 3% for all patients. The variation in normal tissue dose reduction was not predictable, and we found no clear parameters that indicated which patients would benefit most from jaw tracking. Our TPS model of MLC transmission agreed with measurements with absolute transmission differences of less than 0.1 % and thus uncertainties in the model did not contribute significantly to the uncertainty in the dose determination. Conclusion: The amount of dose reduction achieved by collimating the jaws around each MLC aperture in step-and-shoot IMRT does not appear to be clinically significant.

EVALUATION OF INTENSITY MODULATED RADIATION THERAPY (IMRT) DELIVERY ERROR DUE TO IMRT TREATMENT PLAN COMPLEXITY AND IMPROPERLY MATCHED DOSIMETRY DATA

Intensity modulated radiation therapy (IMRT) is a technique that delivers a highly conformal dose distribution to a target volume while attempting to maximally spare the surrounding normal tissues. IMRT is a common treatment modality used for treating head and neck (H&N) cancers, and the presence of many critical structures in this region requires accurate treatment delivery. The Radiological Physics Center (RPC) acts as both a remote and on-site quality assurance agency that credentials institutions participating in clinical trials. To date, about 30% of all IMRT participants have failed the RPC’s remote audit using the IMRT H&N phantom. The purpose of this project is to evaluate possible causes of H&N IMRT delivery errors observed by the RPC, specifically IMRT treatment plan complexity and the use of improper dosimetry data from machines that were thought to be matched but in reality were not. Eight H&N IMRT plans with a range of complexity defined by total MU (1460-3466), number of segments (54-225), and modulation complexity scores (MCS) (0.181-0.609) were created in Pinnacle v.8m. These plans were delivered to the RPC’s H&N phantom on a single Varian Clinac. One of the IMRT plans (1851 MU, 88 segments, and MCS=0.469) was equivalent to the median H&N plan from 130 previous RPC H&N phantom irradiations. This average IMRT plan was also delivered on four matched Varian Clinac machines and the dose distribution calculated using a different 6MV beam model. Radiochromic film and TLD within the phantom were used to analyze the dose profiles and absolute doses, respectively. The measured and calculated were compared to evaluate the dosimetric accuracy. All deliveries met the RPC acceptance criteria of ±7% absolute dose difference and 4 mm distance-to-agreement (DTA). Additionally, gamma index analysis was performed for all deliveries using a ±7%/4mm and ±5%/3mm criteria. Increasing the treatment plan complexity by varying the MU, number of segments, or varying the MCS resulted in no clear trend toward an increase in dosimetric error determined by the absolute dose difference, DTA, or gamma index. Varying the delivery machines as well as the beam model (use of a Clinac 6EX 6MV beam model vs. Clinac 21EX 6MV model), also did not show any clear trend towards an increased dosimetric error using the same criteria indicated above.

Verification of intensity modulated stereotactic radiotherapy using Monte Carlo calculations and EPID dosimetry

The purpose of this work was to develop a comprehensive IMSRT QA procedure that examined, using EPID dosimetry and Monte Carlo (MC) calculations, each step in the treatment planning and delivery process. These steps included verification of the field shaping, treatment planning system (RTPS) dose calculations, and patient dose delivery. Verification of each step in the treatment process is assumed to result in correct dose delivery to the patient. ^ The accelerator MC model was verified against commissioning data for field sizes from 0.8 × 0.8 cm 2 to 10 × 10 cm 2. Depth doses were within 2% local percent difference (LPD) in low gradient regions and 1 mm distance to agreement (DTA) in high gradient regions. Lateral profiles were within 2% LPD in low gradient regions and 1 mm DTA in high gradient regions. Calculated output factors were within 1% of measurement for field sizes ≥1 × 1 cm2. ^ The measured and calculated pretreatment EPID dose patterns were compared using criteria of 5% LPD, 1 mm DTA, or 2% of central axis pixel value with ≥95% of compared points required to pass for successful verification. Pretreatment field verification resulted in 97% percent of the points passing. ^ The RTPS and Monte Carlo phantom dose calculations were compared using 5% LPD, 2 mm DTA, or 2% of the maximum dose with ≥95% of compared points required passing for successful verification. RTPS calculation verification resulted in 97% percent of the points passing. ^ The measured and calculated EPID exit dose patterns were compared using criteria of 5% LPD, 1 mm DTA, or 2% of central axis pixel value with ≥95% of compared points required to pass for successful verification. Exit dose verification resulted in 97% percent of the points passing. ^ Each of the processes above verified an individual step in the treatment planning and delivery process. The combination of these verification steps ensures accurate treatment delivery to the patient. This work shows that Monte Carlo calculations and EPID dosimetry can be used to quantitatively verify IMSRT treatments resulting in improved patient care and, potentially, improved clinical outcome. ^

Simulation study of joint transition and Weibull survival model with shared parameters

This study investigates a theoretical model where a longitudinal process, that is a stationary Markov-Chain, and a Weibull survival process share a bivariate random effect. Furthermore, a Quality-of-Life adjusted survival is calculated as the weighted sum of survival time. Theoretical values of population mean adjusted survival of the described model are computed numerically. The parameters of the bivariate random effect do significantly affect theoretical values of population mean. Maximum-Likelihood and Bayesian methods are applied on simulated data to estimate the model parameters. Based on the parameter estimates, predicated population mean adjusted survival can then be calculated numerically and compared with the theoretical values. Bayesian method and Maximum-Likelihood method provide parameter estimations and population mean prediction with comparable accuracy; however Bayesian method suffers from poor convergence due to autocorrelation and inter-variable correlation. ^

Estimating parameters in Markov models for longitudinal studies with missing data or surrogate outcomes

The discrete-time Markov chain is commonly used in describing changes of health states for chronic diseases in a longitudinal study. Statistical inferences on comparing treatment effects or on finding determinants of disease progression usually require estimation of transition probabilities. In many situations when the outcome data have some missing observations or the variable of interest (called a latent variable) can not be measured directly, the estimation of transition probabilities becomes more complicated. In the latter case, a surrogate variable that is easier to access and can gauge the characteristics of the latent one is usually used for data analysis. ^ This dissertation research proposes methods to analyze longitudinal data (1) that have categorical outcome with missing observations or (2) that use complete or incomplete surrogate observations to analyze the categorical latent outcome. For (1), different missing mechanisms were considered for empirical studies using methods that include EM algorithm, Monte Carlo EM and a procedure that is not a data augmentation method. For (2), the hidden Markov model with the forward-backward procedure was applied for parameter estimation. This method was also extended to cover the computation of standard errors. The proposed methods were demonstrated by the Schizophrenia example. The relevance of public health, the strength and limitations, and possible future research were also discussed. ^

Proton therapy versus intensity modulated x-ray therapy in the treatment of prostate cancer: Estimating secondary cancer risks

External beam radiation therapy is used to treat nearly half of the more than 200,000 new cases of prostate cancer diagnosed in the United States each year. During a radiation therapy treatment, healthy tissues in the path of the therapeutic beam are exposed to high doses. In addition, the whole body is exposed to a low-dose bath of unwanted scatter radiation from the pelvis and leakage radiation from the treatment unit. As a result, survivors of radiation therapy for prostate cancer face an elevated risk of developing a radiogenic second cancer. Recently, proton therapy has been shown to reduce the dose delivered by the therapeutic beam to normal tissues during treatment compared to intensity modulated x-ray therapy (IMXT, the current standard of care). However, the magnitude of stray radiation doses from proton therapy, and their impact on this incidence of radiogenic second cancers, was not known. ^ The risk of a radiogenic second cancer following proton therapy for prostate cancer relative to IMXT was determined for 3 patients of large, median, and small anatomical stature. Doses delivered to healthy tissues from the therapeutic beam were obtained from treatment planning system calculations. Stray doses from IMXT were taken from the literature, while stray doses from proton therapy were simulated using a Monte Carlo model of a passive scattering treatment unit and an anthropomorphic phantom. Baseline risk models were taken from the Biological Effects of Ionizing Radiation VII report. A sensitivity analysis was conducted to characterize the uncertainty of risk calculations to uncertainties in the risk model, the relative biological effectiveness (RBE) of neutrons for carcinogenesis, and inter-patient anatomical variations. ^ The risk projections revealed that proton therapy carries a lower risk for radiogenic second cancer incidence following prostate irradiation compared to IMXT. The sensitivity analysis revealed that the results of the risk analysis depended only weakly on uncertainties in the risk model and inter-patient variations. Second cancer risks were sensitive to changes in the RBE of neutrons. However, the findings of the study were qualitatively consistent for all patient sizes and risk models considered, and for all neutron RBE values less than 100. ^

A multivariate frailty model for disease recurrences and survival

A multivariate frailty hazard model is developed for joint-modeling of three correlated time-to-event outcomes: (1) local recurrence, (2) distant recurrence, and (3) overall survival. The term frailty is introduced to model population heterogeneity. The dependence is modeled by conditioning on a shared frailty that is included in the three hazard functions. Independent variables can be included in the model as covariates. The Markov chain Monte Carlo methods are used to estimate the posterior distributions of model parameters. The algorithm used in present application is the hybrid Metropolis-Hastings algorithm, which simultaneously updates all parameters with evaluations of gradient of log posterior density. The performance of this approach is examined based on simulation studies using Exponential and Weibull distributions. We apply the proposed methods to a study of patients with soft tissue sarcoma, which motivated this research. Our results indicate that patients with chemotherapy had better overall survival with hazard ratio of 0.242 (95% CI: 0.094 - 0.564) and lower risk of distant recurrence with hazard ratio of 0.636 (95% CI: 0.487 - 0.860), but not significantly better in local recurrence with hazard ratio of 0.799 (95% CI: 0.575 - 1.054). The advantages and limitations of the proposed models, and future research directions are discussed. ^

A Bayesian approach to estimating the regression coefficients of a multinomial logit model

A Bayesian approach to estimation of the regression coefficients of a multinominal logit model with ordinal scale response categories is presented. A Monte Carlo method is used to construct the posterior distribution of the link function. The link function is treated as an arbitrary scalar function. Then the Gauss-Markov theorem is used to determine a function of the link which produces a random vector of coefficients. The posterior distribution of the random vector of coefficients is used to estimate the regression coefficients. The method described is referred to as a Bayesian generalized least square (BGLS) analysis. Two cases involving multinominal logit models are described. Case I involves a cumulative logit model and Case II involves a proportional-odds model. All inferences about the coefficients for both cases are described in terms of the posterior distribution of the regression coefficients. The results from the BGLS method are compared to maximum likelihood estimates of the regression coefficients. The BGLS method avoids the nonlinear problems encountered when estimating the regression coefficients of a generalized linear model. The method is not complex or computationally intensive. The BGLS method offers several advantages over Bayesian approaches. ^

Bayesian and survival model for tumor relapse status and disease-specific survival, with applications to breast cancer

Breast cancer is the most common non-skin cancer and the second leading cause of cancer-related death in women in the United States. Studies on ipsilateral breast tumor relapse (IBTR) status and disease-specific survival will help guide clinic treatment and predict patient prognosis.^ After breast conservation therapy, patients with breast cancer may experience breast tumor relapse. This relapse is classified into two distinct types: true local recurrence (TR) and new ipsilateral primary tumor (NP). However, the methods used to classify the relapse types are imperfect and are prone to misclassification. In addition, some observed survival data (e.g., time to relapse and time from relapse to death)are strongly correlated with relapse types. The first part of this dissertation presents a Bayesian approach to (1) modeling the potentially misclassified relapse status and the correlated survival information, (2) estimating the sensitivity and specificity of the diagnostic methods, and (3) quantify the covariate effects on event probabilities. A shared frailty was used to account for the within-subject correlation between survival times. The inference was conducted using a Bayesian framework via Markov Chain Monte Carlo simulation implemented in softwareWinBUGS. Simulation was used to validate the Bayesian method and assess its frequentist properties. The new model has two important innovations: (1) it utilizes the additional survival times correlated with the relapse status to improve the parameter estimation, and (2) it provides tools to address the correlation between the two diagnostic methods conditional to the true relapse types.^ Prediction of patients at highest risk for IBTR after local excision of ductal carcinoma in situ (DCIS) remains a clinical concern. The goals of the second part of this dissertation were to evaluate a published nomogram from Memorial Sloan-Kettering Cancer Center, to determine the risk of IBTR in patients with DCIS treated with local excision, and to determine whether there is a subset of patients at low risk of IBTR. Patients who had undergone local excision from 1990 through 2007 at MD Anderson Cancer Center with a final diagnosis of DCIS (n=794) were included in this part. Clinicopathologic factors and the performance of the Memorial Sloan-Kettering Cancer Center nomogram for prediction of IBTR were assessed for 734 patients with complete data. Nomogram for prediction of 5- and 10-year IBTR probabilities were found to demonstrate imperfect calibration and discrimination, with an area under the receiver operating characteristic curve of .63 and a concordance index of .63. In conclusion, predictive models for IBTR in DCIS patients treated with local excision are imperfect. Our current ability to accurately predict recurrence based on clinical parameters is limited.^ The American Joint Committee on Cancer (AJCC) staging of breast cancer is widely used to determine prognosis, yet survival within each AJCC stage shows wide variation and remains unpredictable. For the third part of this dissertation, biologic markers were hypothesized to be responsible for some of this variation, and the addition of biologic markers to current AJCC staging were examined for possibly provide improved prognostication. The initial cohort included patients treated with surgery as first intervention at MDACC from 1997 to 2006. Cox proportional hazards models were used to create prognostic scoring systems. AJCC pathologic staging parameters and biologic tumor markers were investigated to devise the scoring systems. Surveillance Epidemiology and End Results (SEER) data was used as the external cohort to validate the scoring systems. Binary indicators for pathologic stage (PS), estrogen receptor status (E), and tumor grade (G) were summed to create PS+EG scoring systems devised to predict 5-year patient outcomes. These scoring systems facilitated separation of the study population into more refined subgroups than the current AJCC staging system. The ability of the PS+EG score to stratify outcomes was confirmed in both internal and external validation cohorts. The current study proposes and validates a new staging system by incorporating tumor grade and ER status into current AJCC staging. We recommend that biologic markers be incorporating into revised versions of the AJCC staging system for patients receiving surgery as the first intervention.^ Chapter 1 focuses on developing a Bayesian method to solve misclassified relapse status and application to breast cancer data. Chapter 2 focuses on evaluation of a breast cancer nomogram for predicting risk of IBTR in patients with DCIS after local excision gives the statement of the problem in the clinical research. Chapter 3 focuses on validation of a novel staging system for disease-specific survival in patients with breast cancer treated with surgery as the first intervention. ^

Logistic regression with Markov chains as covariates

The tobacco-specific nitrosamine 4-(methylnitrosamino)-1-(3-pyridyl)-1-butanone (NNK) is an obvious carcinogen for lung cancer. Since CBMN (Cytokinesis-blocked micronucleus) has been found to be extremely sensitive to NNK-induced genetic damage, it is a potential important factor to predict the lung cancer risk. However, the association between lung cancer and NNK-induced genetic damage measured by CBMN assay has not been rigorously examined. ^ This research develops a methodology to model the chromosomal changes under NNK-induced genetic damage in a logistic regression framework in order to predict the occurrence of lung cancer. Since these chromosomal changes were usually not observed very long due to laboratory cost and time, a resampling technique was applied to generate the Markov chain of the normal and the damaged cell for each individual. A joint likelihood between the resampled Markov chains and the logistic regression model including transition probabilities of this chain as covariates was established. The Maximum likelihood estimation was applied to carry on the statistical test for comparison. The ability of this approach to increase discriminating power to predict lung cancer was compared to a baseline "non-genetic" model. ^ Our method offered an option to understand the association between the dynamic cell information and lung cancer. Our study indicated the extent of DNA damage/non-damage using the CBMN assay provides critical information that impacts public health studies of lung cancer risk. This novel statistical method could simultaneously estimate the process of DNA damage/non-damage and its relationship with lung cancer for each individual.^