Handbook of Cancer Models with Applications

The following sections are included:

• CONTENTS

• CONTRIBUTORS

• PREFACE

Human solid tumors are believed to be caused by a sequence of genetic abnormalities arising in normal and premalignant cells. The understanding of these sequences is important for improving cancer treatment. Models for the occurrence of the abnormalities include linear structure and a recently proposed tree-based structure. We will describe the oncogenetic tree model and an efficient algorithm for its estimation. We also discuss methods for estimating the reliability and goodness-of-fit of this reconstruction. An R package “Oncotree” implementing the described methodology is available from the authors.

The hazard function is an important tool in epidemiologic research. Mathematical cancer models are often applied in epidemiologic research via the hazard functions induced by such models. This chapter examines some of the existing methods for computing hazard functions induced by a wide class of stochastic cancer models.

In this chapter we survey recent studies by molecular biologists and cancer geneticists and propose some general stochastic models of carcinogenesis. To develop analytic results for these stochastic models, because the traditional Markov theory approach becomes too complicated to be of much use, in this chapter we propose an alternative approach through stochastic equations. Given observed cancer incidence data, we further combine these stochastic models with statistical models to develop state space models for carcinogenesis. By using these state space models, we then develop a generalized Bayesian method and a predicted inference procedure to estimate the unknown parameters and to predict state variables via multi-level Gibbs sampling procedures. In this chapter we use the extended multi-event model and mixture model as examples to illustrate our modeling approach and some basic theories.

Epidemiological analyses using the two-stage clonal expansion (TSCE) model indicate that low-dose radiation and exposure to some chemicals, such as arsenic or tobacco smoke, operate through several mechanisms during carcinogenesis. Typically, the most significant effect of these exposures is to increase the promotion of initiated cells. The estimated increases in promotion are small in absolute terms, yet highly significant, suggesting that initiated cells are almost, but not completely, under homeostatic regulation that serves to maintain the size of adult tissues and organs. To understand these effects better, we introduce a combined cell cycle and multistage carcinogenesis model with both deterministic and stochastic components. The model incorporates homeostatic regulation of cell cycle progression based on the difference between the total mean (deterministic) number and target number of stem cells. There are several reasons for extending carcinogenesis models to represent cell cycle states. First, the early mutations in many cancers appear to involve proteins associated with checkpoint control or other aspects of cell cycle regulation. Second, the sensitivity of cells exposed to radiation or chemicals changes markedly throughout the cell cycle. Modeling of damage and repair processes along with tissue homeostasis may be key to understanding the mechanisms that underlie promotion and other exposure-response phenomena. For example, loss of a checkpoint protein in initiated cells may shorten the cell cycle period and provide a slight growth advantage compared to normal cells in replacing cells damaged by endogenous processes, radiation, or chemical exposure.

A variety of quasi-mechanistic models of carcinogenesis are reviewed, and in particular, the multi-stage model of Armitage and Doll and the two-mutation model of Moolgavkar, Venzon, and Knudson, in the light of the known effects of radiation on cancer risk. Both the latter models, and various generalizations of them also, are capable of describing at least qualitatively many of the observed patterns of excess cancer risk following ionizing radiation exposure. However, there are certain inconsistencies with the biological and epidemiological data both for the multi-stage model and the two-mutation model. In particular, there are indications that the two-mutation model is not totally suitable for describing the pattern of excess risk for solid cancers that is often seen after exposure to radiation, although leukemia may be better fitted by this type of model. Generalizations of the model of Moolgavkar, Venzon, and Knudson which require three or more mutations, and models allowing for genomic instability, are easier to reconcile with the epidemiological and biological data relating to solid cancers.

The chapter deals with mechanistic biologically motivated modeling of metastasis. A general methodology for such a modeling is outlined, and a comprehensive mathematical model of individual natural history of metastatic cancer allowing for interaction between the primary tumor and metastases is formulated. This model is applied to computing the distribution of the sizes of detectable (or all) metastases in a given host site at any time post-diagnosis. A parametric version of the model for exponentially growing primary and secondary tumors and exponentially distributed metastasis promotion times is fully developed, and identifiability properties of this model are established.

Recent developments in the prevention, detection, treatment, and management of cancer are highlighted. The significance and importance of mathematical modeling of cancer is discussed. Some mathematical models of cancer are discussed, reviewed, appraised, and related to clinical studies, with a special focus on the cancers of the disseminated type. Special emphasis is placed on mathematical models of acute leukemia, being a dangerous type of cancer. Insights and predictions brought on by the models are discussed and placed within clinical context. These insights and predictions are also placed within the framework of the growing relevance of mathematical models to clinical studies and investigations. The models are used to demonstrate and spotlight the importance of looking at cancer treatment as a formal optimization problem.

Although cancer has been considered a disease of deoxyribonucleic acid (DNA), the fundamental dogma that cancer is the result of cumulative mutations that alter specific locations in a cell's DNA is facing challenges. Several major hypotheses of carcinogenesis associated with epi-genetic factors are reviewed. Warburg proposed that cancer is caused by irreversible damage to the aerobic respiration capacity of normal cells; Pauling suggested that a deficiency of vitamin C was a cause of carcinogenesis; and Szent-Györgyi proposed a bioelectrical conductivity hypothesis. Deficiency of micronutrients such as folic acid, vitamin B₆, B₁₂, choline, methionine, iron, and zinc are factors that cause cancer. A decrease of cellular nicotinamide adenine dinucleotides concentration and niacin deficiency are also suggested to be involved in the evolution of cancer formation. Dysfunctional gap junctional intercellular communication (GJIC) caused by environmental carcinogens was proposed to be a cause of cancer. The alterations in DNA methylation is also considered to be a cause of carcinogenesis. Chromosomal abnormalities and viral infections also cause cancers. Each hypothesis, which has survived long scrutiny and many trials, led to a valuable study of a particular facet in the complicated processes related to both normal and cancerous cells. It is important to study how these various hypotheses are interrelated and attempt to integrate all different facets of cancer cells into a plausible mechanism of carcinogenesis. It would also be beneficial for future research efforts to study the synergistic effects of both epigenetic factors and genetic alterations on normal cells.

In this chapter, radiation quantities and units and the induction and repair of radiation-induced DNA damage are briefly reviewed. Results of the authors' theoretical studies of DNA damage induction and outcomes from the excision repair of clustered DNA lesions are reported for selected types of low- and high-LET (linear energy transfer) radiation. Models for the conversion of clusters into small-scale mutations and aberrations are presented and used to illustrate trends in mutagenesis as a function of dose and particle LET.

Recent results indicate that cells not directly targeted with radiation exhibit responses directly attributed to their exposure to the irradiated cells or some product of those cells. Many responses, called “bystander effects,” have been measured in unirradiated cells, ranging from transient gene expression changes to permanent, heritable changes to the genetic material. In this review, current literature regarding the transmission and nature of the molecule responsible for the bystander effect and the variables surrounding both production of and receipt of these signals is presented. Finally, consideration of the risks posed and the evolutionary implications of this phenomenon are discussed.

Based on recent biological studies, in this chapter we have developed a stochastic model for human colon cancer involving five different pathways. These pathways are: the sporadic LOH pathway (about 70–75%), the familial LOH pathway (about 10–15%), the FAP pathway (about 1%), the sporadic MSI pathway (about 10%) and the HNPCC pathway (about 5%). For this model, we have combined the data augmentation method (equivalent to the EM algorithm in sampling theory framework) with the genetic algorithm (GA algorithm) and the state space model to estimate the genetic parameters of these pathways. We use the Bayesian approach to estimate the parameters through the posterior modes of the parameters by combining the genetic algorithm with the mean numbers of state variables. We have applied this model to fit and analyze the SEER data of human colon cancers from NCI/NIH. Our results indicate that the model not only provides a logical avenue to incorporate biological information but also fits the data much better than other models including the four-stage single pathway model. This model not only would provide more insights into human colon cancer but also would provide useful guidance for its prevention and control and for prediction of future cancer cases.

This article illustrates how to use stochastic models and state space models to assess risk of environmental agents. In this state space model, the stochastic system model is the general stochastic multi-stage model of carcinogenesis whereas the observation model is a statistical model based on cancer incidence data. To analyze the stochastic system model, in this chapter we introduce the stochastic equations for the state variables and derive the probability distributions of these variables. In this chapter we also introduce the genetic algorithm, the multi-level Gibbs sampling procedure and the predicted inference procedure to estimate the unknown genetic parameters and to predict the state variables. As an application, the model and the method are used to illustrate how the arsenic in drinking water induces the bladder cancer in human beings using cancer incidence data given in Morale et al. (2000). Our analysis clearly indicated that in the induction of human bladder cancer the arsenic in drinking water was both an initiator and a weak promoter. These results are not possible by purely statistical methods.

Two stochastic models, the multistage model and a geometric model, are presented to describe formation and growth of cancer precursor lesions. Both models are applied to data from animal experiments by maximum likelihood methods. The models are used to test biological hypotheses about the process of formation and growth of preneoplastic lesions and to describe the dose-response relationship for model parameters.

Two factors constitute major impediments to the successful chemotherapy: toxicity of chemotherapeutic agents affecting healthy cells and drug resistance. This chapter reviews how drug resistance can be incorporated into models describing dynamics of population of cancer cells. It begins with a simple example illustrating necessity of including the mechanisms of drug resistance in any dynamical model of chemotherapy. Afterwards, more complex compartmental models are introduced, including infinite-dimensional ones. The problem of optimization of chemotherapy is also stated, followed by its solution for several models.

Bladder cancer is the fifth leading cause of cancer-related deaths in the United States. A common test for the cancer is urine dipsticks or standard urinalysis. It is safe, but has a very poor specificity, as low as 70%. The finding is usually at a late stage and not able to provide accurate location and staging of the lesion. As the main method of investigating bladder abnormalities, fiberoptic cystoscopy is more accurate. But it is invasive, time-consuming, expensive, uncomfortable, and has risk of urinary track infection. Recently, virtual cystoscopy (VC) has been developed as an alternative means for bladder cancer detection and evaluation. In this chapter, we present a novel mixture-based computer-aided detection (CAD) system for VC using multi-spectral (T₁- and T₂-weighted) magnetic resonance (MR) images, where a better tissue contrast can be achieved as compared to computer tomography (CT) images. As multi-spectral images are spatially registered over three-dimensional space, information extracted from multi-spectral MR images is obviously more valuable than that extracted from each (CT or MR) image individually. In addition, the urine has significantly different T₁ and T₂ relaxations compared to the bladder wall, the MR-based VC procedure can be completely non-invasive. Because bladder tumors tend to develop gradually and migrate slowly from the mucosa into the bladder wall/muscle, our focus is on the mucosa layer by mixture image segmentation. In order to obtain both geometry and texture information, we scan the bladder at two states of nearly empty and nearly full. Our CAD system utilizes fully the multi-scan and dual-state MR images for tumor detection and evaluation. Experimental results show its feasibility towards screening of bladder tumors and follow-up of recurrences.

Cancer is a proteomic disease. Though MALDI-TOF mass spectrometry allows direct measurement of the protein signature of tissue, blood, or their biological samples, and holds tremendous potential for disease diagnosis and treatment, key challenges remain in the processing of proteomic data. In this chapter, we will introduce a wavelet based mathematical framework and computational tools for proteomic data processing, feature selection, and statistical analysis in cancer study.

Xenograft model is a common in vivo model in cancer research, where human cancer (e.g., sliced tumor tissue blocks, or tumor cells) are grafted and grown in severe combined immunodeficient (scid) nude mice. In cancer drug development, demonstrated anti-tumor activity in this model is an important step to bring a promising experimental treatment to human. These experiments provide important data on the mechanism of action of the drug and for the design of future clinical trials. For therapy with single agent, the experimental design and sample size formulae are quite well established. However, cancer therapy typically involves combination of multiple agents. Such studies should be optimally designed, so that with moderate sample size, the joint action of two drugs can be estimated and the best combinations identified. A typical outcome variable in these experiments is tumor volume measured over a period of time. The resulting data have several unique features. Since a mouse may die during the experiment or may be sacrificed when its tumor volume quadruples, then incomplete repeated measurements arise. The incompleteness or missingness is also caused by drastic tumor shrinkage (<0.01 cm³) or random truncation. In addition, if no treatment were given to the tumor-bearing mice, the tumors would keep growing until the mice die or are sacrificed. This intrinsic growth of tumor in the absence of treatment constrains the parameters in the regression and causes further difficulties in statistical analysis. This chapter reviews the current methods of experimental design and data analysis for xenograft experiments. We describe the optimal experimental design for combination studies in xenograft models and likelihood-based methods for estimating the dose-response relationship while accounting for the special features of data such as informative censoring and model parameter constraints.

The primary motivation for this work is drawn from problems arising in the analysis of incomplete data. Although data can be incomplete in many ways, we are most interested in those situations where an intermediate event, which is of prime importance, cannot be observed. Such types of data are also referred as interval censored data. For example, in the analysis of data from occult tumor trials, the time of tumor onset is not known. Due to incompleteness of the data analyses become very complex and many assumptions are often required to develop a basis for inference concerning the tumor incidence. We briefly discuss the assumptions made that lead to many different analyses of occult tumor trial data in past two decades. From the prospect of reducing the sample size in occult tumor trials, some models have been proposed. These semi-parametric models assume relationships in tumor-bearing and tumor-free animals and have impact on tumor onset rates and potency measures of carcinogenic substances that we have explored further.

The following sections are included:

• Index