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.     

Nonparametric estimation of the size-metastasis relationship in solid cancers.

This paper is concerned with the relationship between the occurrence of metastases and the size of primary cancers. We consider two probabilistic characterizations of this relationship. First is the distribution function of tumor sizes at the point of metastatic transition; second is the probability that detectable metastases are present when the cancer comes to medical attention. The equation relating these two functions is developed and conditions for their being identical are explored. Since the tumor size at the point of metastasis is not usually observable, estimation of the first distribution requires the use of the EM algorithm. Nonparametric methods of estimating both functions are explored, with attention to the fact that tumors often fail to be measured, particularly those that are known to be metastatic. The methods are applied to the estimation of primary tumor size at the point of distant metastasis in lung cancer (epidermoid and adenocarcinoma) and colorectal cancer and at the point of nodal metastasis in breast cancer. Monte Carlo experiments confirm that the bias inherent in the methodology is acceptably small.     

The size distribution of human hematogenous metastases.

The development of primary cancers and their subsequent metastases occur through a complex sequence of discrete steps. A hypothesis is proposed here whereby the time available for the growth of metastases is normally distributed, presumably as a consequence of the summation of multiple independently distributed time intervals from each of the steps and of the Central Limit Theorem. For exponentially growing metastases, the corresponding size distribution would be lognormal; Gompertzian growth would imply a modified (Gompertz-normal) distribution, where larger metastases would occur less frequently as a consequence of a decreased growth rate. These two size distributions were evaluated against 18 human autopsy cases where precise size measurements had been collected from over 3900 macroscopic hematogenous organ metastases. The lognormal distribution provided an approximate agreement. Its main deficiency was a tendency to over-represent metastases greater than 10 mm diameter. The Gompertz-normal distribution provided more stringent agreement, correcting for this over-representation. These observations supported the hypothesis of normally distributed growth times, and qualified the utility of the lognormal and Gompertz-normal distributions for the size distribution of metastases.     

Mathematical and numerical analysis for a model of growing metastatic tumors

In cancer diseases, the appearance of metastases is a very pejorative forecast. Chemotherapies are systemic treatments which aim at the elimination of the micrometastases produced by a primitive tumour. The efficiency of chemotherapies closely depends on the protocols of administration. Mathematical modeling is an invaluable tool to help in evaluating the best treatment strategy. Iwata et al. [K. Iwata, K. Kawasaki, N. Shigesad, A dynamical model for the growth and size distribution of multiple metastatic tumors, J. Theor. Biol. 203 (2000) 177.] proposed a partial differential equation (PDE) that describes the metastatic evolution of an untreated tumour. In this article, we conducted a thorough mathematical analysis of this model. Particularly, we provide an explicit formula for the growth rate parameter, as well as a numerical resolution of this PDE. By increasing our understanding of the existing model, this work is crucial for further extension and refinement of the model. It settles down the framework necessary for the consideration of drugs administration effects on tumour development.     

Computation of the Full Likelihood Function for Estimating Variance at a Quantitative Trait Locus

The proportion of alleles identical by descent (IBD) determines the genetic covariance between relatives, and thus is crucial in estimating genetic variances of quantitative trait loci (QTL). However, IBD proportions at QTL are unobservable and must be inferred from marker information. The conventional method of QTL variance analysis maximizes the likelihood function by replacing the missing IBDs by their conditional expectations (the expectation method), while in fact the full likelihood function should take into account the conditional distribution of IBDs (the distribution method). The distribution method for families of more than two sibs has not been obvious because there are n(n - 1)/2 IBD variables in a family of size n, forming an n X n symmetrical matrix. In this paper, I use four binary variables, where each indicates the event that an allele from one of the four grandparents has passed to the individual. The IBD proportion between any two sibs is then expressed as a function of the indicators. Subsequently, the joint distribution of the IBD matrix is derived from the distribution of the indicator variables. Given the joint distribution of the unknown IBDs, a method to compute the full likelihood function is developed for families of arbitrary sizes.     

A finite locus effect diffusion model for the evolution of a quantitative trait

A diffusion model is constructed for the joint distribution of absolute locus effect sizes and allele frequencies for loci contributing to an additive quantitative trait under selection in a haploid, panmictic population. The model is designed to approximate a discrete model exactly in the limit as both population size and the number of loci affecting the trait tend to infinity. For the case when all loci have the same absolute effect size, formal multiple-timescale asymptotics are used to predict the long-time response of the population trait mean to selection. For the case where loci can take on either of two distinct effect sizes, not necessarily with equal probability, numerical solutions of the system indicate that response to selection of a quantitative trait is insensitive to the variability of the distribution of effect sizes when mutation is negligible.     

Plasma fibronectin in mammary and uterine carcinomas

Plasma fibronectin was determined in cancer patients and in age- and sex-matched controls and analyzed as a function of age, size of tumor, receptor content of the tumor, metastases and treatment. In the control population, plasma fibronectin increased with age exponentially. The age-dependent increase in plasma fibronectin was strongly attenuated in the cancer population. As normal and cancer curves intersect at about 40-46 years, below this age cancer plasmas have slightly higher values than normal, above this age the inverse is true. No correlation was found between estrogen or progesterone receptor levels and plasma fibronectin values, nor with plasma albumin. Tumor patients with distant metastases gave slightly but significantly higher values than those with local or no metastases. No significant difference was found between tumors when Bloom grading was taken as the second parameter instead of age. The size of the tumor or the type of treatment had no influence. Increased proteolytic activity, increased trapping of plasma fibronectin in tissues and especially in the stromal (desmoplastic) reaction and/or modifications in plasma fibronectin biosynthesis may well be responsible for these results.     

Combination of the c-Met Inhibitor Tivantinib and Zoledronic Acid Prevents Tumor Bone Engraftment and Inhibits Progression of Established Bone Metastases in a Breast Xenograft Model

Bone is the most common metastatic site for breast cancer. There is a significant need to understand the molecular mechanisms controlling the engraftment and growth of tumor cells in bone and to discover novel effective therapeutic strategies. The aim of this study was to assess the effects of tivantinib and Zoledronic Acid (ZA) in combination in a breast xenograft model of bone metastases. Cancer cells were intracardially implanted into immunodeficient mice and the effects of drugs alone or in combination on bone metastasis were evaluated by in vivo non-invasive optical and micro-CT imaging technologies. Drugs were administered either before (preventive regimen) or after (therapeutic regimen) bone metastases were detectable. In the preventive regimen, the combination of tivantinib plus ZA was much more effective than single agents in delaying bone metastatic tumor growth. When administered in the therapeutic schedule, the combination delayed metastatic progression and was effective in improving survival. These effects were not ascribed to a direct cytotoxic effect of the combined therapy on breast cancer cells in vitro. The results of this study provide the rationale for the design of new combinatorial strategies with tivantinib and ZA for the treatment of breast cancer bone metastases.     

A mathematical algorithm that computes breast cancer sizes and doubling times detected by screening

This paper presents a mathematical algorithm that computes the sizes and growth rates of breast cancer detected in a hypothetical population that is screened for the disease. The algorithm works by simulating the outcomes of the hypothetical population twice, first without screening and then with screening. The simulation without screening relies on an underlying model of the natural history of the disease. The simulation with screening uses this natural history model to track the growth of breast tumors backwards in the time starting from the time they would have been detected without screening. The method of tracking tumor growth backward in time is different from methods that track tumor growth forward in time by starting from an estimated time of tumor onset. The screening algorithm combines the natural history model, the method tracking of tumor growth backward in time, the age group, the interval between screening exams, and the detection threshold of the screening exam to compute the joint distribution of tumor size and growth rate among screen-detected and interval patients. The algorithm also computes the sensitivity and leadtime distribution. It allows for arbitrary age groups, detection thresholds and screening intervals and may contribute to the design of future screening trials.     

Generality of the Final Size Formula for an Epidemic of a Newly Invading Infectious Disease

The well-known formula for the final size of an epidemic was published by Kermack and McKendrick in 1927. Their analysis was based on a simple susceptible-infected-recovered (SIR) model that assumes exponentially distributed infectious periods. More recent analyses have established that the standard final size formula is valid regardless of the distribution of infectious periods, but that it fails to be correct in the presence of certain kinds of heterogeneous mixing (e.g., if there is a core group, as for sexually transmitted diseases). We review previous work and establish more general conditions under which Kermack and McKendrick's formula is valid. We show that the final size formula is unchanged if there is a latent stage, any number of distinct infectious stages and/or a stage during which infectives are isolated (the durations of each stage can be drawn from any integrable distribution). We also consider the possibility that the transmission rates of infectious individuals are arbitrarily distributed—allowing, in particular, for the existence of super-spreaders—and prove that this potential complexity has no impact on the final size formula. Finally, we show that the final size formula is unchanged even for a general class of spatial contact structures. We conclude that whenever a new respiratory pathogen emerges, an estimate of the expected magnitude of the epidemic can be made as soon the basic reproduction number ℝ₀ can be approximated, and this estimate is likely to be improved only by more accurate estimates of ℝ₀, not by knowledge of any other epidemiological details.     

Feasibility and limits of an orthotopic human colon cancer model in nude mice

We sought to develop an accurate colorectal cancer model in nude mice with stable local growth, tumor cell dissemination, and reproducible metastatic capacity. To this end, we orthotopically transplanted histologically intact human colorectal cancer tissue from 10 human patients into nude mice. After successful local tumor growth, tumor tissues were retransplanted as many as 9 times in serial passage. All specimens were transplanted using microsurgical techniques. Histologic, immunohistochemical, and polymerase chain reaction techniques were used to determine tumor growth rates and kinetics, development of regional lymph node and distant hepatic metastases, and the induction of minimal residual disease (MRD). Stable local tumor growth rates with variable growth kinetics were detected in 73.4% of all mice. The lymph node and hepatic metastasis rates were low, at 18.4% and 4.9%, respectively. MRD, as reflected by CK20 positivity of the bone marrow in animals with lymph node and hepatic metastases, was present in 22.2%. The orthotopic colorectal cancer model described here is feasible for the induction of reproducible local tumor growth but is limited by variable growth kinetics and the low rate of lymph node and hepatic metastases. Cytokeratin-positive cells indicative of MRD could be detected in the bone marrow of approximately 25% of the nude mice with metastases. The observed induction of MRD after orthotopic implantation of intact human colon cancer in animals with lymph node and hepatic metastases might be improved if established colon cancer cell lines were used.     

Predictive biochemical-markers for the development of brain metastases from lung cancer: Clinical evidence and future directions

BackgroundBrain metastases are a common complication of patients with lung cancer and lung cancer is one of the most common causes of brain metastases. The occurrence of brain metastases is associated with poor prognosis and high morbidity, even after intensive multimodal therapy. Therefore, identifying lung cancer patients with who are at high risk of developing brain metastases and applying effect intervention is important to reduce or delay the incidence of brain metastases. Biochemical-markers may meet an unmet need for following patients’ mechanisms of brain metastases.MethodsData for this review were identified by searches of Pubmed and Cochrane databases, and references from relevant articles using the search terms “lung cancer” and “brain metastasis”. Meeting abstracts, unpublished reports and review articles were not considered.ResultsClinical results for pathological and circulating markers including cancer molecular subtypes, miRNA, single nucleotide polymorphisms, and other markers are presented. However, these biochemical-markers are not yet established surrogate assessments for prediction of brain metastases.ConclusionsBiochemical-markers reported allowed physicians to identify which patients with lung cancer are at high risk for brain metastases. Prospective randomized clinical studies are needed to further assess the utility of these biochemical-markers.     

Computational modeling of pancreatic cancer reveals kinetics of metastasis suggesting optimum treatment strategies

Pancreatic cancer is a leading cause of cancer-related death, largely due to metastatic dissemination. We investigated pancreatic cancer progression by utilizing a mathematical framework of metastasis formation together with comprehensive data of 228 patients, 101 of whom had autopsies. We found that pancreatic cancer growth is initially exponential. After estimating the rates of pancreatic cancer growth and dissemination, we determined that patients likely harbor metastases at diagnosis and predicted the number and size distribution of metastases as well as patient survival. These findings were validated in an independent database. Finally, we analyzed the effects of different treatment modalities, finding that therapies that efficiently reduce the growth rate of cells earlier in the course of treatment appear to be superior to upfront tumor resection. These predictions can be validated in the clinic. Our interdisciplinary approach provides insights into the dynamics of pancreatic cancer metastasis and identifies optimum therapeutic interventions.     

Mitochondrial metabolism in cancer metastasis

We have recently proposed a new two-compartment model for understanding the Warburg effect in tumor metabolism. In this model, glycolytic stromal cells produce mitochondrial fuels (L-lactate and ketone bodies) that are then transferred to oxidative epithelial cancer cells, driving OXPHOS and mitochondrial metabolism. Thus, stromal catabolism fuels anabolic tumor growth via energy transfer. We have termed this new cancer paradigm the “reverse Warburg effect,” because stromal cells undergo aerobic glycolysis, rather than tumor cells. To assess whether this mechanism also applies during cancer cell metastasis, we analyzed the bioenergetic status of breast cancer lymph node metastases, by employing a series of metabolic protein markers. For this purpose, we used MCT4 to identify glycolytic cells. Similarly, we used TO MM20 and COX staining as markers of mitochondrial mass and OXPHOS activity, respectively. Consistent with the “reverse Warburg effect,” our results indicate that metastatic breast cancer cells amplify oxidative mitochondrial metabolism (OXPHOS) and that adjacent stromal cells are glycolytic and lack detectable mitochondria. Glycolytic stromal cells included cancer-associated fibroblasts, adipocytes and inflammatory cells. Double labeling experiments with glycolytic (MCT4) and oxidative (TO MM20 or COX) markers directly shows that at least two different metabolic compartments co-exist, side-by-side, within primary tumors and their metastases. Since cancer-associated immune cells appeared glycolytic, this observation may also explain how inflammation literally “fuels” tumor progression and metastatic dissemination, by “feeding” mitochondrial metabolism in cancer cells. Finally, MCT4(+) and TO MM20(-) “glycolytic” cancer cells were rarely observed, indicating that the conventional “Warburg effect” does not frequently occur in cancer-positive lymph node metastases.     

Persistence of Tandem Arrays: Implications for Satellite and Simple-Sequence Dnas

Recombination processes acting on tandem arrays are suggested here to have probable intrinsic biases, producing an expected net decrease in array size following each event, in contrast to previous models which assume no net change in array size. We examine the implications of this by modeling copy number dynamics in a tandem array under the joint interactions of sister-strand unequal crossing over (rate gamma per generation per copy) and intrastrand recombination resulting in deletion (rate epsilon per generation per copy). Assuming no gene amplification or selection, the expected mean persistence time of an array starting with z excess copies (i.e., array size z + 1) is z(1 + gamma/epsilon) recombinational events. Nontrivial equilibrium distributions of array sizes exist when gene amplification or certain forms of selection are considered. We characterize the equilibrium distribution for both a simple model of gene amplification and under the assumption that selection imposes a minimal array size, n. For the latter case, n + 1/alpha is an upper bound for mean array size under fairly general conditions, where alpha(= 2 epsilon/gamma) is the scaled deletion rate. Further, the distribution of excess copies over n is bounded above by a geometric distribution with parameter alpha/(1 + alpha). Tandem arrays are unlikely to be greatly expanded by unequal crossing over unless alpha much less than 1, implying that other mechanisms, such as gene amplification, are likely important in the evolution of large arrays. Thus unequal crossing over, by itself, is likely insufficient to account for satellite DNA.     

Power calculation for cross-sectional stepped wedge cluster randomized trials with variable cluster sizes

Standard sample size calculation formulas for stepped wedge cluster randomized trials (SW-CRTs) assume that cluster sizes are equal. When cluster sizes vary substantially, ignoring this variation may lead to an under-powered study. We investigate the relative efficiency of a SW-CRT with varying cluster sizes to equal cluster sizes, and derive variance estimators for the intervention effect that account for this variation under a mixed effects model—a commonly used approach for analyzing data from cluster randomized trials. When cluster sizes vary, the power of a SW-CRT depends on the order in which clusters receive the intervention, which is determined through randomization. We first derive a variance formula that corresponds to any particular realization of the randomized sequence and propose efficient algorithms to identify upper and lower bounds of the power. We then obtain an “expected” power based on a first-order approximation to the variance formula, where the expectation is taken with respect to all possible randomization sequences. Finally, we provide a variance formula for more general settings where only the cluster size arithmetic mean and coefficient of variation, instead of exact cluster sizes, are known in the design stage. We evaluate our methods through simulations and illustrate that the average power of a SW-CRT decreases as the variation in cluster sizes increases, and the impact is largest when the number of clusters is small.     

Parathyroid hormone related protein and hypercalcaemia in breast cancer.

OBJECTIVE--To see whether parathyroid hormone related protein has a humoral role in breast cancer. DESIGN--Plasma concentrations and tumour expression of parathyroid hormone related protein were determined (by two site immunoradiometric assay and immunohistochemistry respectively) in women with breast cancer and related to the presence of bone metastases and serum calcium concentrations. SUBJECTS--Plasma concentrations of parathyroid hormone related protein were measured in 57 women with early breast cancer without apparent bone metastases, 28 women with bone metastases, and 13 women with bone metastases and hypercalcaemia. Tissue positivity for parathyroid hormone related protein was determined retrospectively in 106 primary breast tumours from women without apparent bone metastases and 72 tumours from women with bone metastases, 25 of whom subsequently developed hypercalcaemia. RESULTS--Plasma parathyroid hormone related protein concentrations were detectable (greater than 0.23 pmol/l) in 12 (92%) of the 13 hypercalcaemic patients with bone metastases compared with 10 (36%) of the 28 normocalcaemic patients with bone metastases and five (9%) of the 57 normocalcaemic patients without bone metastases. Parathyroid hormone related protein concentrations were significantly higher in hypercalcaemic than normocalcaemic patients with bone metastases. Tumour staining was positive for parathyroid hormone related protein in 22 (88%) of the 25 primary breast cancers from patients with bone metastases. Tumour staining was positive for parathyroid hormone related protein in 22 (88%) of the 25 primary breast cancers from patients with bone metastases who later developed hypercalcaemia compared with 25 (53%) of the 47 from women in this group who remained normocalcaemic and 55 (52%) of the 106 early breast cancers from women without known metastases. CONCLUSION--Tumour derived parathyroid hormone related protein may have an important humoral role in hypercalcaemia associated with metastatic breast cancer.