Mathematical Aspects of a New Synergy Theory Applicable to Malstressor-Dominated Mixtures which Include Damage-Ameliorating Countermeasures.

In radiobiology, and throughout translational biology, synergy theories for multi-component agent mixtures use 1-agent dose-effect relations (DERs) to calculate baseline neither synergy nor antagonism mixture DERs. The most used synergy theory, simple effect additivity, is not self-consistent when curvilinear 1-agent DERs are involved, and many alternatives have been suggested. In this paper we present the mathematical aspects of a new alternative, generalized Loewe additivity (GLA). To the best of our knowledge, generalized Loewe additivity is the only synergy theory that can systematically handle mixtures of agents that are malstressors (tend to produce disease) with countermeasures - agents that oppose malstressors and ameliorate malstressor damage. In practice countermeasures are often very important, so generalized Loewe additivity is potentially far-reaching. Our paper is a proof-of-principle preliminary study. Unfortunately, generalized Loewe additivity's scope is restricted, in various unwelcome but perhaps unavoidable ways. Our results illustrate its strengths and its weaknesses. One area where our methodology has potentially important applications is analyzing counter-measure mitigation of galactic cosmic ray damage to astronauts during interplanetary travel.

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.

Multivariate distributions of clinical covariates at the time of cancer detection.

Many screening trials conducted in the past have generated a wealth of interesting data. These data represent an invaluable source of information for furthering our knowledge about the natural history of the disease. The traditional approach to modeling cancer screening tends to describe the process of tumor development in only one dimension, that is, the time natural history. A broader methodological idea is to construct a stochastic model of cancer development and detection that yields the multivariate distribution of observable variables at the time of diagnosis. By focusing on such multivariate observations, rather than just on the age of patients at diagnosis, this idea seeks to invoke an additional source of information (available only at the time of detection) in order to improve an estimation of unobservable quantitative parameters of cancer latency. In this article, we discuss modeling techniques that make the above-mentioned problems approachable. A special focus is placed on analytical tools for deriving joint distributions of clinical covariates at the time of cancer detection under an arbitrary screening protocol. In addition, some future research avenues and public health implications of the proposed approach are discussed.

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.

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.

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.

Combinatorics of giant hexagonal bilayer hemoglobins.

The paper discusses combinatorial and probabilistic models allowing to characterize various aspects of spacial symmetry and structural heterogeneity of the giant hexagonal bilayer hemoglobins (HBL Hb). Linker-dodecamer configurations of HBL are described for two and four linker types (occurring in the two most studied HBL Hb of Arenicola and Lumbricus, respectively), and the most probable configurations are found. It is shown that, for HBL with marked dodecamers, the number of 'normal-marked' pairs of dodecamers in homological position follows a binomial distribution. The group of symmetries of the dodecamer substructure of HBL is identified with the dihedral group D6. Under natural symmetry assumptions, the total dipole moment of the dodecamer substructure of HBL is shown to be zero. Biological implications of the mathematical findings are discussed.

Electrospray ionization mass spectrometric determination of the molecular mass of the approximately 200-kDa globin dodecamer subassemblies in hexagonal bilayer hemoglobins.

Hexagonal bilayer hemoglobins (Hbs) are approximately 3.6-MDa complexes of approximately 17-kDa globin chains and 24-32-kDa, nonglobin linker chains in a approximately 2:1 mass ratio found in annelids and related species. Studies of the dissociation and reassembly of Lumbricus terrestris Hb have provided ample evidence for the presence of a approximately 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 and Arenicola 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 the L. terrestris subassembly decreased linearly with an increase in de-clustering voltage from approximately 215,400 Da at 60 V to approximately 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 approximately 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 M(3)T(3). Because there are three isoforms of M and four isoforms of T in Lumbricus and two isoforms of M and 5 isoforms of T in Arenicola, the masses of the M(3)T(3) subassemblies 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 Arenicola Hb, in good agreement with the experimental values.

Identifiability of parameters in the Yakovlev-Polig model of carcinogenesis.

The paper discusses the problem of identifiability for two versions of a two-stage model of carcinogenesis recently introduced by Yakovlev and Polig. In this model, cell killing is allowed to compete with tumor promotion. In the first version of the Yakovlev-Polig model, which is referred to as Model 1, cell killing starts immediately after a carcinogen is administered. In the second version, called Model 2, it is assumed that a cell may be killed only after the process of initiation has been completed. The two versions of the Yakovlev-Polig model suggest explicit formulas for the distribution of time to tumor onset (that is, appearance of the first malignant clonogenic cell) counted from the initial moment of the exposure to a carcinogen. A model of carcinogenesis is identifiable if the set of all model parameters is uniquely determined by the distribution of time to tumor onset. It is shown that, under a natural necessary condition of overlap of supports of the dose-rate function h and the promotion time distributions from a family F, Model 1 is identifiable in the family F for many practically important functions h. In particular, this is the case for a simple model of spontaneous carcinogenesis (h = 1) and for a class of piecewise constant dose-rate functions h with arbitrary family F. Also, this holds for the family of gamma distributions and h supported on an interval and non-vanishing in the interior of this interval. More restrictions need to be imposed on the dose-rate function and the family of promotion time distributions for Model 2 to be identifiable. In particular, for h = 1, Model 2 turns out to be non-identifiable even in the family of gamma distributions.

A nonidentifiability aspect of the two-stage model of carcinogenesis.

This paper discusses identifiability of the two-stage birth-death-mutation model of carcinogenesis. It is shown that the homogeneous version of the model is nonidentifiable; the same is all the more evident for its nonhomogeneous versions. This result implies that the model parameters cannot be uniquely estimated from time-to-tumor observations.

A distribution of tumor size at detection and its limiting form.

A distribution of tumor size at detection is derived within the framework of a mechanistic model of carcinogenesis with the object of estimating biologically meaningful parameters of tumor latency. Its limiting form appears to be a generalization of the distribution that arises in the length-biased sampling from stationary point processes. The model renders the associated estimation problems tractable. The usefulness of the proposed approach is illustrated with an application to clinical data on premenopausal breast cancer.