PSEUDORANDOM NUMBER GENERATORS, GOODNESS OF FIT TESTS, AND STOCHASTIC. SIMULATION OF HETEROGENEOUS TUMORS. Roger S. Day ...
Proceedings of the 160th Annual Joint Statistical Meetings, Baltimore MD, 1999 PSEUDORANDOM NUMBER GENERATORS, GOODNESS OF FIT TESTS, AND STOCHASTIC SIMULATION OF HETEROGENEOUS TUMORS Roger S. Day, Qing Shou Huang, Sailesh Ramakrishnan, University of Pittsburgh Roger S. Day, U. Pittsburgh Cancer Institute, Suite 400A, 3600 Forbes Ave, Pittburgh PA 15213
"cancer-cell machines". We have also used OncoTCap in collaboration with clinical cancer researchers to explain observations, and to develop and explore new treatment ideas. The modeling laboratory is designed in a way that makes extension to reflect new biology extremely easy. Therefore it is well suited to the "postgenome" era of cancer research.
KEYWORDS: Stochastic simulation, random number generator, cancer, probability generating functions.
1. Introduction New and anticipated discoveries in the molecular biology of cancer bring to the fore a crisis; how can the great volume of very specific information be effectively synthesized and utilized to think more clearly and comprehensively about cancer? The biomathematical modeling of cancer cells is likely to be of increasing importance in answering this challenge. As both computer technology and molecular biology research continue their remarkable expansion, the practicality of very elaborate stochastic simulations grows.
Elaborate stochastic simulations of vast populations of cells present special challenges in regard to the effective use of pseudo-random number generators (PRNGs). We highlight four of these challenges encountered in the development of the Oncology Thinking Cap program. The role of pseudo-random number generators in OncoTCap is critical because one of the two computational engines is based on stochastic simulation.
It has long been believed that tumor heterogeneity is a key ingredient in the natural history of cancer, reflecting the genomic instability that is at the heart of the disease, and also making curative treatment extraordinarily challenging. The understanding of the cancer process is currently being revolutionized by new techniques that allow us to see in remarkable detail the status of genes and their expression in tumor samples and even individual cancer cells. These techniques are exemplified by the NCI's vigorous initiative called the Cancer Genome Anatomy Project, by the newly initiated Gene Annotation Index, and by the recent call for new National Cooperative Tumor Signature Groups (NCTGS) to develop new tumor classifications to augment or replace classic clinico-pathologic staging, based on high-throughput studies of cancer cell gene mutations and gene expression. These efforts will illuminate tumor heterogeneity in tremendous detail.
2. Modeling capabilities of OncoTCap Tumor heterogeneity underlies most of the central themes of cancer biology and cancer. These include: apoptotic mechanisms, cell cycle control, repair mechanisms, and mutational processes that may disrupt normal mechanisms, as well as tumor growth kinetics, sensitivity to treatments, treatment resistance mechanisms, local spread and metastasis. All of these phenomena are described in terms of variations in properties of cancer cells, properties which are heritable in a genetic, positional, or other sense. The key to extensive and flexible modeling capabilities is a unified and economical description of tumor heterogeneity. A partial list of the current modeling capabilities is seen in Table 1. OncoTCap lets users describe biological and chemical relationships as experienced by each individual tumor cell. It then maps accurately to the macroscopic behavior of the tumor in regard to growth kinetics and response to treatment. Computational engines and graphic displays incorporate all these aspects to produce individual patient simulations or calculate cure probabilities. Thus OncoTCap provides the means to synthesize these details into a model for calculating predictions about cancer patients and the success of treatment plans. The modeling framework strives towards a hypothesis-neutral ideal, to encourage the challenging of assumptions, and to be fully responsive to new developments in cancer biology [Day 1986(a); Day 1987; Day 1986(b); Norton 1991].
In the past four years, our team has developed a powerful cancer modeling workbench called OncoTCap ("Oncology Thinking Cap") [Day et al 1998] with tumor heterogeneity as its main organizing principle. The program design supports the gathering of numerous hypotheses and/or facts about cancer cells, to construct predictions of the behavior of tumor cell populations under treatment, predictions which would be mathematically accurate if the hypotheses were correct and complete. We have gained experience in designing and evaluating training modules based on encounters with virtual
-1-
Proceedings of the 160th Annual Joint Statistical Meetings, Baltimore MD, 1999
TABLE 1: Some Current Capabilities of OncoTCap Economical Description of Heterogeneous Tumors ♦ Properties/Compartments/Hypotheses o Reduction in number of parameters o General editor for elaborate model rules ♦ Broad range of biological hypotheses o Proliferation control pathways o Signalling pathways o Apoptosis pathways o Proliferation & cell death switches o Treatment resistance · Resistance by mutation · Induction of resistance by treatment · Reversion to sensitivity · Multidrug resistance · Pre-adaptation · Resistance modulation o Metastasis modeling o Local invasion modeling o Angiogenesis modeling · Induction by cancer cells · Suppression by treatment o SN mutation, gene amplification, LOH o DNA repair
Complementary Computational Engines ♦ Stochastic Simulator o Constructs individual patient histories o Patient management rules o Clinical trial management rules o Patient encounter ♦ Analytic Simulator o Solves differential equations for process o Rapid averaging over all patient histories o Explicit support for comparing regimens o Explicit support for sensitivity analysis Basic Modeling Capabilities ♦ Exponential and Gompertz growth models o Gompertz rules nested in regions ♦ Anti-cancer treatments o Single agents/combinations/courses o Episodic versus extended administration o Cytostatic & cytotoxic effects o Hormonal and antiangiogenic treatments o Toxicities
3. The rand( ), srand( ) simulator routines
Though the lagged Fibonacci sequence has good properties, it fails the "birthday-spacings test". The introduction of the arithmetic sequence solves this problem, and leads to a greatly extended period, sufficient for any current application (Marsaglia and Zaman 1990).
A very widespread PRNG is the ISO-standard rand( ) . This is a simple and fast linear congruential routine taking a 16 or 32 bit integer as seed, maintaining the current state with a single static 32 bit integer, and returning either the full 32 bits or the upper 16 bits. The version shipped with Microsoft's C library returns 16 bits. Nevertheless its period is 231-1. Therefore, an individual value can appear multiple times in the output stream, without causing the stream to return to the earlier state and duplicate subsequent values.
A 32-bit integer seed is required to initialize this random number generator. This integer is used to initialize an array of reals of length 97, from which the Fibonacci sequence xi (1) can be initiated. The arithmetic sequence yi is initiated at 362436./16777216. Each value of the seed generates a distinct sequence.
4. The "ranmar( )" random number generator Most stochastic simulation in OncoTCap is currently achieved using a method due to Marsaglia and Zaman (1990), using a popular implementation due to James (1990). This PRNG has a very long period (2144) and was designed to be reproducibly portable among different computer architectures. The strategy, in brief, is to combine two distinct PRNG generators x1, x2,… and y1, y2,… , producing the output Ui = xi ●yi where the operator ● is defined by x●y = x -y if x ≥y , x -y +1 if x
