Leonid Hanin

Modeling cancer detection: tumor size as a source of information on unobservable stages of carcinogenesis

This paper is concerned with modern approaches to mechanistic modeling of the process of cancer detection. Measurements of tumor size at diagnosis represent a valuable source of information to enrich statistical inference on the processes underlying tumor latency. One possible way of utilizing this information is to model cancer detection as a quantal response variable. In doing so, one relates the chance of detecting a tumor to its current size. We present various theoretical results emerging from this approach and illustrate their usefulness with numerical examples and analyses of epidemiological data. An alternative approach based on a threshold type mechanism of tumor detection is briefly described.

Electrospray Ionization Mass Spectrometric Determination of the Molecular Mass of the ∼200-kDa Globin Dodecamer Subassemblies in Hexagonal Bilayer Hemoglobins

Hexagonal bilayer hemoglobins (Hbs) are ∼3.6-MDa complexes of ∼17-kDa globin chains and 24–32-kDa, nonglobin linker chains in a ∼2:1 mass ratio found in annelids and related species. Studies of the dissociation and reassembly ofLumbricus terrestris Hb have provided ample evidence for the presence of a ∼200-kDa linker-free subassembly consisting of monomer (M) and disulfide-bonded trimer (T) subunits. Electrospray ionization mass spectrometry (ESI-MS) of the subassemblies obtained by gel filtration of partially dissociated L. terrestris andArenicola marina Hbs showed the presence of noncovalent complexes of M and T subunits with masses in the 213.3–215.4 and 204.6–205.6 kDa ranges, respectively. The observed mass of theL. terrestris subassembly decreased linearly with an increase in de-clustering voltage from ∼215,400 Da at 60 V to ∼213,300 Da at 200 V. In contrast, the mass of the A. marina complex decreased linearly from 60 to 120 V and reached an asymptote at ∼204,600 Da (180–200 V). The decrease in mass was probably due to the progressive removal of complexed water and alkali metal cations. ESI-MS at an acidic pH showed both subassemblies to consist of only M and T subunits, and the experimental masses demonstrated them to have the composition M3T3. Because there are three isoforms of M and four isoforms of T inLumbricus and two isoforms of M and 5 isoforms of T inArenicola, the masses of the M3T3subassemblies are not unique. A random assembly model was used to calculate the mass distributions of the subassemblies, using the known ESI-MS masses and relative intensities of the M and T subunit isforms. The expected mass of randomly assembled subassemblies was 213,436 Da for Lumbricus Hb and 204,342 Da for ArenicolaHb, in good agreement with the experimental values.

Distribution of the number of clonogens surviving fractionated radiotherapy: a long-standing problem revisited

Purpose : A long-standing problem is addressed: what form of the probability distribution for the number of clonogenic tumor cells remaining after fractionated radiotherapy should be used in the analysis aimed at evaluating the efficacy of cancer treatment? Over a period of years, a lack of theoretical results leading to a closed-form analytic expression for this distribution, even under very simplistic models of cell kinetics in the course of fractionated radiotherapy, was the most critical deterrent to the development of relevant methods of data analysis. Materials and methods : Rigorous mathematical results associated with a model of fractionated irradiation of tumors based on the iterated birth and death stochastic process are discussed. Results : A formula is presented for the exact distribution of the number of clonogenic tumor cells at the end of treatment. It is shown that, under certain conditions, this distribution can be approximated by a Poisson distribution. An explicit formula for the parameter of the limiting Poisson distribution is given and sample computations aimed at evaluation of the convergence rate are reported. Another useful limit that retains a dose-response relationship in the distribution of the number of clonogens has been found. Practical implications of the key theoretical findings are discussed in the context of survival data analysis. Conclusions : This study answers some challenging theoretical questions that have been under discussion over a number of years. The results presented in this work provide mechanistic motivation for parametric regression models designed to analyze data on the efficacy of radiation therapy.

A new minimum volume straight cooling fin taking into account the length of arc

Abstract The problem of determining the shape of a straight cooling fin of minimum volume without the “length of arc” assumption is addressed. Proceeding from the conventional assumptions of one-dimensionality of the temperature distribution and its linearity for the minimum volume fin we found the profile of the optimum fin to be a circular arc and computed its geometric parameters. The volume of the optimum circular fin found in this paper is 6.21–8 times smaller than the volume of the corresponding Schmidt’s parabolic optimum fin. The optimum circular fin tends to be shorter and to have a larger base height than Schmidt’s fin.

Iterated birth and death process as a model of radiation cell survival

The iterated birth and death process is defined as an n-fold iteration of a stochastic process consisting of the combination of instantaneous random killing of individuals in a certain population with a given survival probability s with a Markov birth and death process describing subsequent population dynamics. A long standing problem of computing the distribution of the number of clonogenic tumor cells surviving a fractionated radiation schedule consisting of n equal doses separated by equal time intervals tau is solved within the framework of iterated birth and death processes. For any initial tumor size i, an explicit formula for the distribution of the number M of surviving clonogens at moment tau after the end of treatment is found. It is shown that if i-->infinity and s-->0 so that is(n) tends to a finite positive limit, the distribution of random variable M converges to a probability distribution, and a formula for the latter is obtained. This result generalizes the classical theorem about the Poisson limit of a sequence of binomial distributions. The exact and limiting distributions are also found for the number of surviving clonogens immediately after the nth exposure. In this case, the limiting distribution turns out to be a Poisson distribution.

Cell-survival probability at large doses: an alternative to the linear-quadratic model

A model of irradiated cell survival based on rigorous accounting of microdosimetric effects is developed. The model does not assume that the distribution of lesions is Poisson and is applicable to low, intermediate and high acute doses of low or high LET radiation. For small doses, the model produces the linear-quadratic (LQ) model. However, for high doses the best-fitting LQ model grossly underestimates cell survival. The same is also true for the conventional LQ model, only more so. It is shown that for high doses, the microdosimetric distribution can be approximated by a Gaussian distribution, and the corresponding cell survival probabilities are compared.

A survival model for fractionated radiotherapy with an application to prostate cancer

This paper explores the applicability of a mechanistic survival model, based on the distribution of clonogens surviving a course of fractionated radiation therapy, to clinical data on patients with prostate cancer. The study was carried out using data on 1,100 patients with clinically localized prostate cancer who were treated with three-dimensional conformal radiation therapy. The patients were stratified by radiation dose (group 1: <67.5 Gy; group 2: 67.5-72.5 Gy; group 3: 72.5-77.5 Gy; group 4: 77.5-87.5 Gy) and prognosis category (favourable, intermediate and unfavourable as defined by pre-treatment PSA and Gleason score). A relapse was recorded when tumour recurrence was diagnosed or when three successive prostate specific antigen (PSA) elevations were observed from a post-treatment nadir PSA level. PSA relapse-free survival was used as the primary end point. The model, which is based on an iterated Yule process, is specified in terms of three parameters: the mean number of tumour clonogens that survive the treatment, the mean of the progression time of post-treatment tumour development and its standard deviation. The model parameters were estimated by the maximum likelihood method. The fact that the proposed model provides an excellent description both of the survivor function and of the hazard rate is prima facie evidence of the validity of the model because closeness of the two survivor functions (empirical and model-based) does not generally imply closeness of the corresponding hazard rates. The estimated cure probabilities for the favourable group are 0.80, 0.74 and 0.87 (for dose groups 1-3, respectively); for the intermediate group: 0.25, 0.51, 0.58 and 0.78 (for dose groups 1-4, respectively) and for the unfavourable group: 0.0, 0.27, 0.33 and 0.64 (for dose groups 1-4, respectively). The distribution of progression time to tumour relapse was found to be independent of prognosis group but dependent on dose. As the dose increases the mean progression time decreases (41, 28.5, 26.2 and 14.7 months for dose groups 1-4, respectively). This analysis confirms that, in terms of cure rate, dose escalation has a significant positive effect only in the intermediate and unfavourable groups. It was found that progression time is inversely proportional to dose, which means that patients recurring in higher dose groups have shorter recurrence times, yet these groups have better survival, particularly long-term. The explanation for this seemingly illogical observation lies in the fact that less aggressive tumours, potentially recurring after a long period of time, are cured by higher doses and do not contribute to the recurrence pattern. As a result, patients in higher dose groups are less likely to recur; however, if they do, they tend to recur earlier. The estimated hazard rates for prostate cancer pass through a clear-cut maximum, thus revealing a time period with especially high values of instantaneous cancer-specific risk; the estimates appear to be nonproportional across dose strata.

DOES EXTIRPATION OF THE PRIMARY BREAST TUMOR GIVE BOOST TO GROWTH OF METASTASES? EVIDENCE REVEALED BY MATHEMATICAL MODELING

A comprehensive mechanistic model of cancer natural history was utilized to obtain an explicit formula for the distribution of volumes of detectable metastases in a given secondary site at any time post-diagnosis. This model provided an excellent fit to the volumes of n=31, 20 and 15 bone metastases observed in three breast cancer patients 8 years, 5.5 years and 9 months after primary diagnosis, respectively. The model with optimal parameters allowed us to reconstruct the individual natural history of cancer for the first patient. This gave definitive answers, for the patient in question, to the following three questions of major importance in clinical oncology: (1) How early an event is metastatic dissemination of breast cancer? (2) How long is the metastasis latency time? and (3) Does extirpation of the primary breast tumor accelerate the growth of metastases? Specifically, according to the model applied to the first patient, (1) inception of the first metastasis occurred 29.5 years prior to the primary diagnosis; (2) the expected metastasis latency time was about 79.5 years; and (3) resection of the primary tumor was followed by a 32-fold increase in the rate of metastasis growth. The model and our conclusions were validated by the results for the two other patients.

Biologically-equivalent dose and long-term survival time in radiation treatments.

Within the linear-quadratic model the biologically-effective dose (BED)-taken to represent treatments with an equal tumor control probability (TCP)-is commonly (and plausibly) calculated according to BED(D) = -log[S(D)]/alpha. We ask whether in the presence of cellular proliferation this claim is justified and examine, as a related question, the extent to which BED approximates an isoeffective dose (IED) defined, more sensibly, in terms of an equal long-term survival probability, rather than TCP. We derive, under the assumption that cellular birth and death rates are time homogeneous, exact equations for the isoeffective dose, IED. As well, we give a rigorous definition of effective long-term survival time, T(eff). By using several sets of radiobiological parameters, we illustrate potential differences between BED and IED on the one hand and, on the other, between T(eff) calculated as suggested here or by an earlier recipe. In summary: (a) the equations currently in use for calculating the effective treatment time may underestimate the isoeffective dose and should be avoided. The same is the case for the tumor control probability (TCP), only more so; (b) for permanent implants BED may be a poor substitute for IED; (c) for a fractionated treatment schedule, interpreting the observed probability of cure in terms of a TCP formalism that refers to the end of the treatment (rather than T(eff)) may result in a miscalculation (underestimation) of the initial number of clonogens.

Why Victory in the War on Cancer Remains Elusive: Biomedical Hypotheses and Mathematical Models

We discuss philosophical, methodological, and biomedical grounds for the traditional paradigm of cancer and some of its critical flaws. We also review some potentially fruitful approaches to understanding cancer and its treatment. This includes the new paradigm of cancer that was developed over the last 15 years by Michael Retsky, Michael Baum, Romano Demicheli, Isaac Gukas, William Hrushesky and their colleagues on the basis of earlier pioneering work of Bernard Fisher and Judah Folkman. Next, we highlight the unique and pivotal role of mathematical modeling in testing biomedical hypotheses about the natural history of cancer and the effects of its treatment, elaborate on model selection criteria, and mention some methodological pitfalls. Finally, we describe a specific mathematical model of cancer progression that supports all the main postulates of the new paradigm of cancer when applied to the natural history of a particular breast cancer patient and fit to the observables.