Quiescence as an explanation of Gompertzian tumor growth revisited

Modeling and analysis of a multilayer solid tumour with cell physiological age and resource limitations

We study an avascular spherical solid tumour model with cell physiological age and resource constraints in vivo. We divide the tumour cells into three components: proliferating cells, quiescent cells and dead cells in necrotic core. We assume that the division rate of proliferating cells is nonlinear due to the nutritional and spatial constraints. The proportion of newborn tumour cells entering directly into quiescent state is considered, since this proportion can respond to the therapeutic effect of drug. We establish a nonlinear age-structured tumour cell population model. We investigate the existence and uniqueness of the model solution and explore the local and global stabilities of the tumour-free steady state. The existence and local stability of the tumour steady state are studied. Finally, some numerical simulations are performed to verify the theoretical results and to investigate the effects of different parameters on the model.

Stability analysis of a multiscale model of cell cycle dynamics coupled with quiescent and proliferating cell populations

In this paper, we perform a mathematical analysis of our proposed nonlinear, multiscale mathematical model of physiologically structured quiescent and proliferating cell populations at the macroscale and cell-cycle proteins at the microscale. Cell cycle dynamics (microscale) are driven by growth factors derived from the total cell population of quiescent and proliferating cells. Cell-cycle protein concentrations, on the other hand, determine the rates of transition between the two subpopulations. Our model demonstrates the underlying impact of cell cycle dynamics on the evolution of cell population in a tissue. We study the model's well-posedness, derive steady-state solutions, and find sufficient conditions for the stability of steady-state solutions using semigroup and spectral theory. Finally, we performed numerical simulations to see how the parameters affect the model's nonlinear dynamics.

Distinguishing between multiple mathematical models of neural stem cell quiescence and activation during age-related neural stem cell decline in neurogenesis

Stem cells are required for tissue maintenance and homeostasis during an organism's lifetime. Neural stem cells (NSCs) can be in an actively dividing state or in a quiescent state. The balance between stem cell quiescence and cycling activity determines the rate of neurogenesis. With age, more NSCs enter the quiescent state, while the total number of NSCs decreases. Here we reconsider an existing mathematical model of how neural stem cells switch between active and quiescent states from the point of view of control theory by considering the activation rate, self-renewal probability, and division rate as control parameters rather than as pre-defined functions. Our goal is to test whether those modifications to the basic model could explain the observed decline of neural stem cells with age better than Gomerzian time-dependent parameters, and compare the output from different model variants to experimental data from mice using AIC. We find that time-dependent activation rate provides the best fit to the activated cell fraction (ACF) of NSCs over time, but that other model variants with constant parameter values can better fit the total number of NSCs over time. We also consider an alternate model for NSCs with nonlinear feedback from progenitor cells that affect NSC parameters, and compare all models to experimental stem cell and progenitor data. However, all of the feedback models considered provide a worse fit to the experimental data. This suggests that when switching between active and quiescent stem cells is considered, a time-dependent linear model outperforms the integral feedback mechanism considered by other models of stem cell lineages. Fitting progenitor data for both the time varying models and feedback models indicates that four or five intermediate transit amplifying progenitor states are necessary. Our modeling suggests that in order to determine whether an increase in age-related neural stem cell quiescence is determined by by a decreasing stem cell activation rate or an increased stem cell depletion rate, additional experiments should be designed to explore whether or not depletion of the stem cell pool is occurring, and that a higher resolution time series for activated cell fraction (ACF) would be best to resolve this issue.

Modeling and dynamics of physiological and behavioral resistance of Asian citrus psyllid

The Asian citrus psyllid (ACP) survival in the presence of contact insecticides may be through physiological adaptations or by behaviorally avoiding. Curiously, although the first alternative is the object of frequent attention, the second was often neglected, but both may lead to insecticide resistance. In this paper, we characterize the growth dynamics of ACP population using a novel impulsive differential equation model to account for the effect of physiological and behavioral resistance, and investigate the threshold conditions for the extinction of ACP population. Furthermore, we discuss the optimal switching methods for insecticides based on two different criteria. Our numerical result suggests that ignoring both resistances or behavioral resistance would underestimate the transmission risk of Huanglongbing, whereas only considering behavioral resistance leads to an overestimation.

Elements and evolutionary determinants of genomic divergence between paired primary and metastatic tumors

Can metastatic-primary (M-P) genomic divergence measured from next generation sequencing reveal the natural history of metastatic dissemination? This remains an open question of utmost importance in facilitating a deeper understanding of metastatic progression, and thereby, improving its prevention. Here, we utilize mathematical and computational modeling to tackle this question as well as to provide a framework that illuminates the fundamental elements and evolutionary determinants of M-P divergence. Our framework facilitates the integration of sequencing detectability of somatic variants, and hence, paves the way towards bridging the measurable between-tumor heterogeneity with analytical modeling and interpretability. We show that the number of somatic variants of the metastatic seeding cell that are experimentally undetectable in the primary tumor, can be characterized as the path of the phylogenetic tree from the last appearing variant of the seeding cell back to the most recent detectable variant. We find that the expected length of this path is principally determined by the decay in detectability of the variants along the seeding cell's lineage; and thus, exhibits a significant dependence on the underlying tumor growth dynamics. A striking implication of this fact, is that dissemination from an advanced detectable subclone of the primary tumor can lead to an abrupt drop in the expected measurable M-P divergence, thereby breaking the previously assumed monotonic relation between seeding time and M-P divergence. This is emphatically verified by our single cell-based spatial tumor growth simulation, where we find that M-P divergence exhibits a non-monotonic relationship with seeding time when the primary tumor grows under branched and linear evolution. On the other hand, a monotonic relationship holds when we condition on the dynamics of progressive diversification, or by restricting the seeding cells to always originate from undetectable subclones. Our results highlight the fact that a precise understanding of tumor growth dynamics is the sine qua non for exploiting M-P divergence to reconstruct the chronology of metastatic dissemination. The quantitative models presented here enable further careful evaluation of M-P divergence in association with crucial evolutionary and sequencing parameters.

Close Encounters of the Cell Kind: The Impact of Contact Inhibition on Tumour Growth and Cancer Models

Cancer is a complex phenomenon, and the sheer variation in behaviour across different types renders it difficult to ascertain underlying biological mechanisms. Experimental approaches frequently yield conflicting results for myriad reasons, and mathematical modelling of cancer is a vital tool to explore what we cannot readily measure, and ultimately improve treatment and prognosis. Like experiments, models are underpinned by certain biological assumptions, variation of which can lead to divergent predictions. An outstanding and important question concerns contact inhibition of proliferation (CIP), the observation that proliferation ceases when cells are spatially confined by their neighbours. CIP is a characteristic of many healthy adult tissues, but it remains unclear to which extent it holds in solid tumours, which exhibit regions of hyper-proliferation, and apparent breakdown of CIP. What precisely occurs in tumour tissue remains an open question, which mathematical modelling can help shed light on. In this perspective piece, we explore the implications of different hypotheses and available experimental evidence to elucidate the implications of these scenarios. We also outline how erroneous conclusions about the nature of tumour growth may be arrived at by looking selectively at biological data in isolation, and how this might be circumvented.

Metabolism is the tie: The Bertalanffy-type cancer growth model as common denominator of various modelling approaches

Cancer or tumour growth has been addressed from a variety of mathematical modelling perspectives in the past. Examples are single variable growth models, reaction diffusion models, compartment models, individual cell-based models, clonal competition models, to name only a few. In this paper, we show that the so called Bertalanffy-type growth model is a macroscopic model variant that can be conceived as an optimal condensed modelling approach that to a high degree preserves complexity with respect to the aforementioned more complex modelling variants. The derivation of the Bertalanffy-type model is crucially based on features of metabolism. Therefore, this model contains a shape parameter that can be interpreted as a resource utilisation efficiency. This shape parameter reflects features that are usually captured in much more complex models. To be specific, the shape parameter is related to morphological structures of tumours, which in turn depend on metabolic conditions. We, furthermore, show that a single variable variant of the Bertalanffy-type model can straightforwardly be extended to a multiclonal competition model. Since competition is crucially based on available shared or clone-specific resources, the metabolism-based approach is an obvious candidate to capture clonal competition. Depending on the specific context, metabolic reprogramming or other oncogene driven changes either lead to a suppression of cancer cells or to an improved competition resulting in outgrowth of tumours. The parametrisation of the Bertalanffy-type growth model allows to account for this observed variety of cancer characteristics. The shape parameter, conceived as a classifier for healthy and oncogenic phenotypes, supplies a link to survival and evolutionary stability concepts discussed in demographic studies, such as opportunistic versus equilibrium strategies.

A Simple PSA-Based Computational Approach Predicts the Timing of Cancer Relapse in Prostatectomized Patients

Recurrences of prostate cancer affect approximately one quarter of patients who have undergone radical prostatectomy. Reliable factors to predict time to relapse in specific individuals are lacking. Here, we present a mathematical model that evaluates a biologically sensible parameter (α) that can be estimated by the available follow-up data, in particular by the PSA series. This parameter is robust and highly predictive for the time to relapse, also after administration of adjuvant androgen deprivation therapies. We present a practical computational method based on the collection of only four postsurgical PSA values. This study offers a simple tool to predict prostate cancer relapse. Cancer Res; 76(17); 4941-7. ©2016 AACR.

Correlation Factors Analysis of Breast Cancer Tumor Volume Doubling Time Measured by 3D-Ultrasound

BACKGROUND Tumor volume doubling time (TVDT) is relatively important for breast cancer diagnosis and prognosis evaluation. This study aimed to analyze the related factors that may affect the TVDT of breast cancer by three-dimensional ultrasound (3D-US). MATERIAL AND METHODS A total of 69 breast cancer patients were selected. 3D-US was applied to measure the volume of breast lumps diagnosed as BI-RADS-US 4A by conventional ultrasound. TVDT was calculated according to the formula TVDT=DT×log2/log(V2/V1). Multiple linear regression analysis was performed to analyze the factors influencing breast cancer TVDT. RESULTS The mean and median TVDT were 185±126 (range 66-521) and 164 days, respectively. TVDT showed no statistical significance according to regular shape, coarse margin, spicule sign, peripheral hyperechoic halo, microcalcification, and different posterior echo characteristics (P>0.05). Patients grouped by age, axillary lymphatic metastasis, histological differentiation, and Nottingham prognostic index (NPI) score exhibited significantly different TVDT (P<0.05). On the contrary, patients with different menstrual conditions, breast cancer family history, or pathological types presented similar TVDT (P>0.05). TVDT was obviously different in breast cancer with different ER, PR, Ki-67, and molecular subtyping but not HER2 expression. Multivariate analysis revealed that NPI score, axillary lymphatic metastasis, Ki-67, and molecular subtyping were risk factors of TVDT in breast cancer (P<0.05). CONCLUSIONS Breast cancer TVDT was significantly correlated with NPI score, axillary lymphatic metastasis, Ki-67, and molecular subtyping. Triple-negative breast cancer exhibited the most rapid growth.

Comparison between model-predicted tumor oxygenation dynamics and vascular-/flow-related Doppler indices

PURPOSE

Mathematical modeling is a powerful and flexible method to investigate complex phenomena. It discloses the possibility of reproducing expensive as well as invasive experiments in a safe environment with limited costs. This makes it suitable to mimic tumor evolution and response to radiotherapy although the reliability of the results remains an issue. Complexity reduction is therefore a critical aspect in order to be able to compare model outcomes to clinical data. Among the factors affecting treatment efficacy, tumor oxygenation is known to play a key role in radiotherapy response. In this work, we aim at relating the oxygenation dynamics, predicted by a macroscale model trained on tumor volumetric data of uterine cervical cancer patients, to vascularization and blood flux indices assessed on Ultrasound Doppler images.

METHODS

We propose a macroscale model of tumor evolution based on three dynamics, namely active portion, necrotic portion, and oxygenation. The model parameters were assessed on the volume size of seven cervical cancer patients administered with 28 fractions of intensity modulated radiation therapy (IMRT) (1.8 Gy/fraction). For each patient, five Doppler ultrasound tests were acquired before, during, and after the treatment. The lesion was manually contoured by an expert physician using 4D View^® (General Electric Company - Fairfield, Connecticut, United States), which automatically provided the overall tumor volume size along with three vascularization and/or blood flow indices. Volume data only were fed to the model for training purpose, while the predicted oxygenation was compared a posteriori to the measured Doppler indices.

RESULTS

The model was able to fit the tumor volume evolution within 8% error (range: 3-8%). A strong correlation between the intrapatient longitudinal indices from Doppler measurements and oxygen predicted by the model (about 90% or above) was found in three cases. Two patients showed an average correlation value (50-70%) and the remaining two presented poor correlations. The latter patients were the ones featuring the smallest tumor reduction throughout the treatment, typical of hypoxic conditions. Moreover, the average oxygenation value predicted by the model was close to the average vascularization-flow index (average difference: 7%).

CONCLUSIONS

The results suggest that the modeled relation between tumor evolution and oxygen dynamics was reasonable enough to provide realistic oxygenation curves in five cases (correlation greater than 50%) out of seven. In case of nonresponsive tumors, the model failed in predicting the oxygenation trend while succeeded in reproducing the average oxygenation value according to the mean vascularization-flow index. Despite the need for deeper investigations, the outcomes of the present work support the hypothesis that a simple macroscale model of tumor response to radiotherapy is able to predict the tumor oxygenation. The possibility of an objective and quantitative validation on imaging data discloses the possibility to translate them as decision support tools in clinical practice and to move a step forward in the treatment personalization.