Background

Data assimilation refers to methods for updating the state vector (initial condition) of a complex spatiotemporal model (such as a numerical weather model) by combining new observations with one or more prior forecasts. We consider the potential feasibility of this approach for making short-term (60-day) forecasts of the growth and spread of a malignant brain cancer (glioblastoma multiforme) in individual patient cases, where the observations are synthetic magnetic resonance images of a hypothetical tumor.

Results

We apply a modern state estimation algorithm (the Local Ensemble Transform Kalman Filter), previously developed for numerical weather prediction, to two different mathematical models of glioblastoma, taking into account likely errors in model parameters and measurement uncertainties in magnetic resonance imaging. The filter can accurately shadow the growth of a representative synthetic tumor for 360 days (six 60-day forecast/update cycles) in the presence of a moderate degree of systematic model error and measurement noise.

Conclusions

The mathematical methodology described here may prove useful for other modeling efforts in biology and oncology. An accurate forecast system for glioblastoma may prove useful in clinical settings for treatment planning and patient counseling.

Reviewers

This article was reviewed by Anthony Almudevar, Tomas Radivoyevitch, and Kristin Swanson (nominated by Georg Luebeck).

The characteristics of aristolochic acid nephropathy (AAN) are interstitial fibrosis and atrophy of the proximal tubules, but with no change in glomeruli. To investigate the effects of AA on renal functions and the pharmacokinetics (PKs) of p-aminohippuric acid (PAH) and inulin, New Zealand white rabbits were used in this study. The plasma concentrations of PAH and inulin were determined by validated HPLC methods. After a single intravenous administration of 0.5 mg/kg aristolochic acid sodium (AANa), rabbits exhibited mild to moderate nephrotoxicity on the 7th day. Significant tubulointerstitial damage to kidney specimens was found, but there were no remarkable glomerular changes. Clearance rates of PAH and inulin both significantly decreased in AANa-treated rabbits. In addition, there was a significant correlation among the degree of tubulointerstitial changes and PK parameters of PAH after AANa administration, but no correlation was noted with the PKs of inulin. With mild to moderate AAN in rabbits, the renal plasma flow significantly decreased by 55%, and the glomerular filtration rate also significantly decreased by 85%. In conclusion, major renal lesions were found on proximal tubules after AANa administration. The PKs of PAH and inulin significantly changed, and kidney functions, including the RPF and GFR, were reduced.

It is well known that Michaelis–Menten kinetics is suitable for the response function in chemical reaction, when the reaction rate does not increase indefinitely when an excess of resource is available. However, the existing models for insulin therapies assume that the response function of insulin clearance is proportional to the insulin concentration. In this paper, we propose a new model for insulin therapy for both type 1 and type 2 diabetes mellitus, in which the insulin degradation rate assumes Michaelis–Menten kinetics. Our analysis shows that it is possible to mimic pancreatic insulin secretion by exogenous insulin infusions, and our numerical simulations provide clinical strategies for insulin–administration practices.

Type 1 diabetics must inject exogenous insulin or insulin analogues one or more times daily. The timing and dosage of insulin administration have been a critical research area since the invention of insulin analogues. Several pharmacokinetical models have been proposed, and some are applied clinically in modeling various insulin therapies. However, their plasma insulin concentration must be computed separately from the models’ output. Furthermore, minimal analytical study was performed in these existing models. We propose two systemic and simplified ordinary differential equation models to model the subcutaneous injection of rapid-acting insulin analogues and long-acting insulin analogues, respectively. Our models explicitly model the plasma insulin and hence have the advantage of computing the plasma insulin directly. The profiles of plasma insulin concentrations obtained from these two models are in good agreement with the experimental data. We also study the dynamics of insulin analogues, plasma insulin concentrations, and, in particular, the shape of the dynamics of plasma insulin concentrations.

Purpose

In this study, we investigated the biochemical pharmacology of pirenoxine (PRX) and catalin under in vitro selenite/calcium- and ultraviolet (UV)-induced lens protein turbidity challenges. The systemic effects of catalin were determined using a selenite-induced cataractogenesis rat model.

Methods

In vitro cataractogenesis assay systems (including UVB/C photo-oxidation of lens crystallins, calpain-induced proteolysis, and selenite/calcium-induced turbidity of lens crystallin solutions) were used to screen the activity of PRX and catalin eye drop solutions. Turbidity was identified as the optical density measured using spectroscopy at 405 nm. We also determined the in vivo effects of catalin on cataract severity in a selenite-induced cataract rat model. Sodium dodecyl sulfate polyacrylamide gel electrophoresis (SDS–PAGE) was applied to analyze the integrity of crystallin samples.

Results

PRX at 1,000 μM significantly delayed UVC-induced turbidity formation compared to controls after 4 h of UVC exposure (p<0.05), but not in groups incubated with PRX concentrations of <1,000 μM. Results were further confirmed by SDS–PAGE. The absolute γ-crystallin turbidity induced by 4 h of UVC exposure was ameliorated in the presence of catalin equivalent to 1~100 μM PRX in a concentration-dependent manner. Samples with catalin-formulated vehicle only (CataV) and those containing PRX equivalent to 100 μM had a similar protective effect after 4 h of UVC exposure compared to the controls (p<0.05). PRX at 0.03, 0.1, and 0.3 μM significantly delayed 10 mM selenite- and calcium-induced turbidity formation compared to controls on days 0~4 (p<0.05). Catalin (equivalent to 32, 80, and 100 μM PRX) had an initial protective effect against selenite-induced lens protein turbidity on day 1 (p<0.05). Subcutaneous pretreatment with catalin (5 mg/kg) also statistically decreased the mean cataract scores in selenite-induced cataract rats on post-induction day 3 compared to the controls (1.3±0.2 versus 2.4±0.4; p<0.05). However, catalin (equivalent to up to 100 μM PRX) did not inhibit calpain-induced proteolysis activated by calcium, and neither did 100 μM PRX.

Conclusions

PRX at micromolar levels ameliorated selenite- and calcium-induced lens protein turbidity but required millimolar levels to protect against UVC irradiation. The observed inhibition of UVC-induced turbidity of lens crystallins by catalin at micromolar concentrations may have been a result of the catalin-formulated vehicle. Transient protection by catalin against selenite-induced turbidity of crystallin solutions in vitro was supported by the ameliorated cataract scores in the early stage of cataractogenesis in vivo by subcutaneously administered catalin. PRX could not inhibit calpain-induced proteolysis activated by calcium or catalin itself, and may be detrimental to crystallins under UVB exposure. Further studies on formulation modifications of catalin and recommended doses of PRX to optimize clinical efficacy by cataract type are warranted.

Background

Androgens bind to the androgen receptor (AR) in prostate cells and are essential survival factors for healthy prostate epithelium. Most untreated prostate cancers retain some dependence upon the AR and respond, at least transiently, to androgen ablation therapy. However, the relationship between endogenous androgen levels and cancer etiology is unclear. High levels of androgens have traditionally been viewed as driving abnormal proliferation leading to cancer, but it has also been suggested that low levels of androgen could induce selective pressure for abnormal cells. We formulate a mathematical model of androgen regulated prostate growth to study the effects of abnormal androgen levels on selection for pre-malignant phenotypes in early prostate cancer development.

Results

We find that cell turnover rate increases with decreasing androgen levels, which may increase the rate of mutation and malignant evolution. We model the evolution of a heterogeneous prostate cell population using a continuous state-transition model. Using this model we study selection for AR expression under different androgen levels and find that low androgen environments, caused either by low serum testosterone or by reduced 5α-reductase activity, select more strongly for elevated AR expression than do normal environments. High androgen actually slightly reduces selective pressure for AR upregulation. Moreover, our results suggest that an aberrant androgen environment may delay progression to a malignant phenotype, but result in a more dangerous cancer should one arise.

Conclusions

The model represents a useful initial framework for understanding the role of androgens in prostate cancer etiology, and it suggests that low androgen levels can increase selection for phenotypes resistant to hormonal therapy that may also be more aggressive. Moreover, clinical treatment with 5α-reductase inhibitors such as finasteride may increase the incidence of therapy resistant cancers.

Reviewers

This article was reviewed by Ariosto S. Silva (nominated by Marek Kimmel) and Marek Kimmel.

Malignant melanoma is a cancer of the skin arising in the melanocytes. We present a mathematical model of melanoma invasion into healthy tissue with an immune response. We use this model as a framework with which to investigate primary tumor invasion and treatment by surgical excision. We observe that the presence of immune cells can destroy tumors, hold them to minimal expansion, or, through the production of angiogenic factors, induce tumorigenic expansion. We also find that the tumor–immune system dynamic is critically important in determining the likelihood and extent of tumor regrowth following resection. We find that small metastatic lesions distal to the primary tumor mass can be held to a minimal size via the immune interaction with the larger primary tumor. Numerical experiments further suggest that metastatic disease is optimally suppressed by immune activation when the primary tumor is moderately, rather than minimally, metastatic. Furthermore, satellite lesions can become aggressively tumorigenic upon removal of the primary tumor and its associated immune tissue. This can lead to recurrence where total cancer mass increases more quickly than in primary tumor invasion, representing a clinically more dangerous disease state. These results are in line with clinical case studies involving resection of a primary melanoma followed by recurrence in local metastases.

Author Summary

Melanoma is a deadly skin cancer that invades into the dermis and metastasizes into the surrounding tissue. In clinical cases, surgical excision of the primary tumor has led to widespread and accelerated growth in metastases. We develop a mathematical model describing the basic process of melanoma invasion, metastatic spread, and the anti-tumor immune response. This model is formulated using partial differential equations that describe the spatial and temporal evolution of a number of different cellular populations, and it uses a realistic skin geometry. Using simulations, we examine the importance of the immune response when a primary tumor is spawning satellite metastases. We find that local metastases can be suppressed by the immune response directed against the primary tumor, but grow aggressively following surgical treatment. We also find that moderately metastatic tumors optimally activate the local immune response against disseminated disease, and in this case tumor excision may have profound effects on metastatic growth. We conclude that surgical perturbation of the immune response controlling local metastases is one mechanism by which cancer can recur. This could have implications as to the appropriate clinical management of melanomas and other solid tumors.