A mathematical model of the treatment and survival of patients with high-grade brain tumours

More years of life per patient are lost as the result of primary brain tumours than any other form of cancer. The most aggressive of these is known as glioblastoma (GBM). The median survival time of patients with GBM is under 10 months and the outlook has hardly improved over the past 20 years. Generally, these tumours are remarkably resistant to radiotherapy and yet about 2-3% of all GBMs appear to be cured.The objectives of this study were to formulate a mathematical and phenomenological model of tumour growth in a population of patients with GBM to predict survival, and to use the model to extract biological information from clinical data.The model describes the growth of the tumour and the resulting damage to the normal brain using simple concepts borrowed from chemical reaction engineering. Death is assumed to result when the amount of surviving normal brain falls to a critical level. Radiotherapy is assumed to destroy tumour but not healthy brain. Simple rules are included to represent approximately the clinician's decisions about what type of treatment to offer each patient. A population of patients is constructed by assuming that key parameters can be sampled from statistical distributions. Following Monte Carlo simulation, the model can be fitted to data from clinical trials.The model reproduces clinical data extremely accurately. This suggests that the long-term survivors are not a separate sub-population but are the ‘lucky tail’ of a unimodal distribution. The estimated values of radiation sensitivity (represented as SF2, the survival fraction after 2 Gy) suggest the presence of severe hypoxia, which renders cells less sensitive to radiation. The model can predict the probable age distribution of tumours at presentation. The model shows the complicated effects of waiting times for treatment on the survival outcomes, and is used to predict the effects of escalation of radiotherapy dose.The model may aid the design of clinical trials using radiotherapy for patients with GBM, especially in helping to estimate the size of trial required. It is also designed in a generic form, and might be applicable to other tumour types.     

A mathematical model of the impact of novel treatments on the A beta burden in the Alzheimer's brain,CSF and plasma

With the advent of novel therapies for Alzheimer’s disease, there is a pressing need for biomarkers that are easy to monitor, such as the amyloid-beta (Aβ) levels in the cerebrospinal fluid (CSF) and plasma. To gain a better understanding of the explanatory power of these biomarkers, we formulate and analyze a compartmental mathematical model for the Aβ accumulation in the brain, CSF and plasma, throughout the course of Alzheimer’s treatment. Our analysis reveals that the total Aβ burden in the brain is dictated by a unitless quantity called the polymerization ratio, which is the product of the production and elongation rates divided by the product of the fragmentation and loss rates. In this ratio, the production rate and loss rate include a source and sink term, respectively, related to the intercompartmental transport. Our results suggest that production inhibitors are likely to reduce the Aβ levels in all three compartments. In contrast, agents that ingest monomers off of polymers, or that increase fragmentation or block elongation, may also reduce Aβ burden in the brain, but may produce little change in—or even transiently increase—CSF and plasma Aβ levels. Hence, great care must be taken when interpreting these biomarkers.     

Lipoprotein nanoplatform for targeted delivery of diagnostic and therapeutic agents

Low-density lipoprotein (LDL) provides a highly versatile natural nanoplatform for delivery of visible or near-infrared fluorescent optical and magnetic resonance imaging (MRI) contrast agents and photodynamic therapy and chemotherapeutic agents to normal and neoplastic cells that overexpress low-density lipoprotein receptors (LDLRs). Extension to other lipoproteins ranging in diameter from about 10 nm (high-density lipoprotein [HDL]) to over a micron (chylomicrons) is feasible. Loading of contrast or therapeutic agents onto or into these particles has been achieved by protein loading (covalent attachment to protein side chains), surface loading (intercalation into the phospholipid monolayer), and core loading (extraction and reconstitution of the triglyceride/cholesterol ester core). Core and surface loading of LDL have been used for delivery of optical imaging agents to tumor cells in vivo and in culture. Surface loading was used for delivery of gadolinium-bis-stearylamide contrast agents for in vivo MRI detection in tumor-bearing mice. Chlorin and phthalocyanine near-infrared photodynamic therapy agents (相似文献   

A mathematical model of blood,cerebrospinal fluid and brain dynamics

Using first principles of fluid and solid mechanics a comprehensive model of human intracranial dynamics is proposed. Blood, cerebrospinal fluid (CSF) and brain parenchyma as well as the spinal canal are included. The compartmental model predicts intracranial pressure gradients, blood and CSF flows and displacements in normal and pathological conditions like communicating hydrocephalus. The system of differential equations of first principles conservation balances is discretized and solved numerically. Fluid–solid interactions of the brain parenchyma with cerebral blood and CSF are calculated. The model provides the transitions from normal dynamics to the diseased state during the onset of communicating hydrocephalus. Predicted results were compared with physiological data from Cine phase-contrast magnetic resonance imaging to verify the dynamic model. Bolus injections into the CSF are simulated in the model and found to agree with clinical measurements.

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Biphasic toxicodynamic features of some antimicrobial agents on microbial growth: a dynamic mathematical model and its implications on hormesis

Background  

In the present work, we describe a group of anomalous dose-response (DR) profiles and develop a dynamic model that is able to explain them. Responses were obtained from conventional assays of three antimicrobial agents (nisin, pediocin and phenol) against two microorganisms (Carnobacterium piscicola and Leuconostoc mesenteroides).     

A mathematical model for the design of fibrin microcapsules with skin cells

The use of fibrin in tissue engineering has greatly increased over the last 10 years. The aim of this research was to develop a mathematical model to relate the microcapsule-size and cell-load to growth and oxygen depletion. Keratinocytes were isolated from rat skins and microencapsulated dropping fibrinogen and thrombin solutions. The cell growth was measured with MTT-assay and confirmed using histochemical technique. The oxygen was evaluated using a Clark sensor. It was found that Fick–Monod model explained the cell growth for the first 48 h, but overestimated the same thereafter. It was necessary to add a logistic equation to reach valid results. In relation to the preferred implant alternative, when considering large initial cell loads, the possibility to implant small loads of fast-growing cells arises from the simulations. In relation to the microcapsule size, it was found that a critical diameter could be established from which cell growth velocity is about the same.     

A mathematical model of intracranial pressure dynamics for brain hypothermia treatment

Brain hypothermia treatment is used as a neuroprotectant to decompress the elevated intracranial pressure (ICP) in acute neuropatients. However, a quantitative relationship between decompression and brain hypothermia is still unclear, this makes medical treatment difficult and ineffective. The objective of this paper is to develop a general mathematical model integrating hemodynamics and biothermal dynamics to enable a quantitative prediction of transient responses of elevated ICP to ambient cooling temperature. The model consists of a lumped-parameter compartmental representation of the body, and is based on two mechanisms of temperature dependence encountered in hypothermia, i.e. the van't Hoff's effect of metabolism and the Arrhenius' effect of capillary filtration. Model parameters are taken from the literature. The model is verified by comparing the simulation results to population-averaged data and clinical evidence of brain hypothermia treatment. It is possible to assign special model inputs to mimic clinical maneuvers, and to adjust model parameters to simulate pathophysiological states of intracranial hypertension. Characteristics of elevated ICP are quantitatively estimated by using linear approximation of step response with respect to ambient cooling temperature. Gain of about 4.9 mmHg degrees C(-1), dead time of about 1.0 h and a time constant of about 9.8h are estimated for the hypothermic decompression. Based on the estimated characteristics, a feedback control of elevated ICP is introduced in a simulated intracranial hypertension of vasogenic brain edema. Simulation results suggest the possibility of an automatic control of the elevated ICP in brain hypothermia treatment.     

A mathematical model for the evolution of cell populations under the action of mutagenic agents

The evolution of a homogeneous cell population under the action of mutagenic agents is investigated and account taken of the reversible character of genetic damage and the cell replication cycle. A mathematical model is obtained and consists of a system of first order partial differential equations.     

NG2 precursor cells in neoplasia: Functional, histogenesis and therapeutic implications for malignant brain tumours

Diffusely infiltrating astrocytic tumours of the central nervous system (CNS) are the most frequent intracranial neoplasms and account for more than 60% of all primary brain tumours in man. Until recently, it was generally accepted that the glial component of the mature CNS, consisted of differentiated astrocytes, ependymal cells, oligodendrocytes and the non-neuro-ectodermal microglial cells. There exists a recently recognised population of glial cells that express the NG2 proteoglycan (NG2 cells). NG2 cells are dynamic and undergo rapid morphological changes in response to a variety of CNS pathologies. They are highly motile cells, which interact with various extracellular matrix (ECM) in association with the integrin receptors. During angiogenesis and response to tissue injury, NG2 precursor cells are recruited to sites where vessel growth and repair are occurring. NG2 is over-expressed by both tumour cells and pericytes on the blood vessels of malignant brain tumours. The function of NG2 cells in the CNS, and the notion of them as a source of and/or lineage marker for some gliomas are discussed. In addition, their possible role in glioma angiogenesis, proliferation and invasion will be considered as will their value in provision of targets for clinical and pre-clinical therapeutic strategies in brain tumours.     

A mathematical model of compartmentalized neurotransmitter metabolism in the human brain

After administration of enriched [1-13C]glucose, the rate of 13C label incorporation into glutamate C4, C3, and C2, glutamine C4, C3, and C2, and aspartate C2 and C3 was simultaneously measured in six normal subjects by 13C NMR at 4 Tesla in 45-ml volumes encompassing the visual cortex. The resulting eight time courses were simultaneously fitted to a mathematical model. The rate of (neuronal) tricarboxylic acid cycle flux (V(PDH)), 0.57 +/- 0.06 micromol. g(-1). min(-1), was comparable to the exchange rate between (mitochondrial) 2-oxoglutarate and (cytosolic) glutamate (Vx), 0.57 +/- 0.19 micromol. g(-1). min(-1)), which may reflect to a large extent malate-aspartate shuttle activity. At rest, oxidative glucose consumption [CMR(Glc(ox))] was 0.41 +/- 0.03 miccromol. g(-1). min(-1), and (glial) pyruvate carboxylation (VPC) was 0.09 +/- 0.02 micromol. g(-1). min(-1). The flux through glutamine synthetase (Vsyn) was 0.26 +/- 0.06 micromol. g(-1). min(-1). A fraction of Vsyn was attributed to be from (neuronal) glutamate, and the corresponding rate of apparent glutamatergic neurotransmission (VNT) was 0.17 +/- 0.05 micromol. g(-1). min(-1). The ratio [VNT/CMR(Glcox)] was 0.41 +/- 0.14 and thus clearly different from a 1:1 stoichiometry, consistent with a significant fraction (approximately 90%) of ATP generated in astrocytes being oxidative. The study underlines the importance of assumptions made in modeling 13C labeling data in brain.     

A high density assay format for the detection of novel cytotoxic agents in large chemical libraries

In response to the need for inexpensive high throughput assays for anti-cancer drug screening, a 1536-well microtiter plate based assay utilizing the Alamar Blue fluorescent dye as a measure of cellular growth was validated in 10 μL assay volume. Its robustness was assessed in a screen against a library of 2000 known bioactives; with an overall Z′ value of 0.89 for assay robustness, several known cytotoxic agents were identified including and not limited to anthracyclines, cardiac glycosides, gamboges, and quinones. To further test the sensitivity of the assay, IC₅₀ determinations were performed in both 384-well and 1536-well formats and the obtained results show a very good correlation between the two density formats. These findings demonstrate that this newly developed assay is simple to set up, robust, highly sensitive and inexpensive. It could potentially provide a rapid way to screen established and primary tumor cell lines against large chemical libraries.     

A high density assay format for the detection of novel cytotoxic agents in large chemical libraries

In response to the need for inexpensive high throughput assays for anti-cancer drug screening, a 1536-well microtiter plate based assay utilizing the Alamar Blue fluorescent dye as a measure of cellular growth was validated in 10 microL assay volume. Its robustness was assessed in a screen against a library of 2000 known bioactives; with an overall Z' value of 0.89 for assay robustness, several known cytotoxic agents were identified including and not limited to anthracyclines, cardiac glycosides, gamboges, and quinones. To further test the sensitivity of the assay, IC50 determinations were performed in both 384-well and 1536-well formats and the obtained results show a very good correlation between the two density formats. These findings demonstrate that this newly developed assay is simple to set up, robust, highly sensitive and inexpensive. It could potentially provide a rapid way to screen established and primary tumor cell lines against large chemical libraries.     

Overcoming the phage replication threshold: a mathematical model with implications for phage therapy

Prior observations of phage-host systems in vitro have led to the conclusion that susceptible host cell populations must reach a critical density before phage replication can occur. Such a replication threshold density would have broad implications for the therapeutic use of phage. In this report, we demonstrate experimentally that no such replication threshold exists and explain the previous data used to support the existence of the threshold in terms of a classical model of the kinetics of colloidal particle interactions in solution. This result leads us to conclude that the frequently used measure of multiplicity of infection (MOI), computed as the ratio of the number of phage to the number of cells, is generally inappropriate for situations in which cell concentrations are less than 10(7)/ml. In its place, we propose an alternative measure, MOI(actual), that takes into account the cell concentration and adsorption time. Properties of this function are elucidated that explain the demonstrated usefulness of MOI at high cell densities, as well as some unexpected consequences at low concentrations. In addition, the concept of MOI(actual) allows us to write simple formulas for computing practical quantities, such as the number of phage sufficient to infect 99.99% of host cells at arbitrary concentrations.     

A systems-based mathematical modelling framework for investigating the effect of drugs on solid tumours

Background and Purpose

Most information on the dose-response of radiation-induced cancer is derived from data on the A-bomb survivors. Since, for radiation protection purposes, the dose span of main interest is between zero and one Gy, the analysis of the A-bomb survivors is usually focused on this range. However, estimates of cancer risk for doses larger than one Gy are becoming more important for radiotherapy patients. Therefore in this work, emphasis is placed on doses relevant for radiotherapy with respect to radiation induced solid cancer.

Materials and methods

For various organs and tissues the analysis of cancer induction was extended by an attempted combination of the linear-no-threshold model from the A-bomb survivors in the low dose range and the cancer risk data of patients receiving radiotherapy for Hodgkin's disease in the high dose range. The data were fitted using organ equivalent dose (OED) calculated for a group of different dose-response models including a linear model, a model including fractionation, a bell-shaped model and a plateau-dose-response relationship.

Results

The quality of the applied fits shows that the linear model fits best colon, cervix and skin. All other organs are best fitted by the model including fractionation indicating that the repopulation/repair ability of tissue is neither 0 nor 100% but somewhere in between. Bone and soft tissue sarcoma were fitted well by all the models. In the low dose range beyond 1 Gy sarcoma risk is negligible. For increasing dose, sarcoma risk increases rapidly and reaches a plateau at around 30 Gy.

Conclusions

In this work OED for various organs was calculated for a linear, a bell-shaped, a plateau and a mixture between a bell-shaped and plateau dose-response relationship for typical treatment plans of Hodgkin's disease patients. The model parameters (α and R) were obtained by a fit of the dose-response relationships to these OED data and to the A-bomb survivors. For any three-dimensional inhomogenous dose distribution, cancer risk can be compared by computing OED using the coefficients obtained in this work.     

A mathematical model of phytoremediation for petroleum-contaminated soil: model development

We present a simple model for root length density that combines the generally accepted spatial (exponential decrease with depth) and temporal (sinusoidal) variability of root length. Parameters in this model for root length density can be determined from assumed or measured information regarding the annual biomass turnover, maximum standing biomass, and maximum depth of root penetration. The root length density model, coupled with information regarding the average root lifespan, gives specific root growth and senescence functions that are the forcing functions for the phytoremediation model. We present a screening level mathematical model for phytoremediation that accounts for the growth and senescence of roots in the system. This is an important factor for recalcitrant, immobile compounds found in weathered crude oil contaminated soils. The phytoremediation model is based on variable volume compartments that have individual first-order degradation rate constants; as the roots move through the soil, the soil cycles through the rhizosphere zone, decaying root zone and bulk soil zone. Thus, although the oil is immobile, as the roots penetrate through the soil the oil is brought into contact with the rhizosphere.     

Immune-based mechanisms of cytotoxic chemotherapy: implications for the design of novel and rationale-based combined treatments against cancer

Conventional anticancer chemotherapy has been historically thought to act through direct killing of tumor cells. This concept stems from the fact that cytotoxic drugs interfere with DNA synthesis and replication. Accumulating evidence, however, indicates that the antitumor activities of chemotherapy also rely on several off-target effects, especially directed to the host immune system, that cooperate for successful tumor eradication. Chemotherapeutic agents stimulate both the innate and adaptive arms of the immune system through several modalities: (i) by promoting specific rearrangements on dying tumor cells, which render them visible to the immune system; (ii) by influencing the homeostasis of the hematopoietic compartment through transient lymphodepletion followed by rebound replenishment of immune cell pools; (iii) by subverting tumor-induced immunosuppressive mechanisms and (iv) by exerting direct or indirect stimulatory effects on immune effectors. Among the indirect ways of immune cell stimulation, some cytotoxic drugs have been shown to induce an immunogenic type of cell death in tumor cells, resulting in the emission of specific signals that trigger phagocytosis of cell debris and promote the maturation of dendritic cells, ultimately resulting in the induction of potent antitumor responses. Here, we provide an extensive overview of the multiple immune-based mechanisms exploited by the most commonly employed cytotoxic drugs, with the final aim of identifying prerequisites for optimal combination with immunotherapy strategies for the development of more effective treatments against cancer.     

A tumor vasculature targeted liposome delivery system for combretastatin A4: Design, characterization, and in vitro evaluation

The objective of this study was to develop an efficient tumor vasculature targeted liposome delivery system for combretastatin A4, a novel antivascular agent. Liposomes composed of hydrogenated soybean phosphatidylcholine (HSPC), cholesterol, distearoyl phosphoethanolamine-polyethylene-glycol-2000 conjugate (DSPE-PEG), and DSPE-PEG-maleimide were prepared by the lipid film hydration and extrusion process. Cyclic RGD (Arg-Gly-Asp) peptides with affinity for αvβ3-integrins expressed on tumor vascular endothelial cells were coupled to the distal end of PEG on the liposomes sterically stabilized with PEG (long circulating liposomes, LCL). The liposome delivery system was characterized in terms of size, lamellarity, ligand density, drug loading, and leakage properties. Targeting nature of the delivery system was evaluated in vitro using cultured human umbilical vein endothelial cells (HUVEC). Electron microscopic observations of the formulations revealed presence of small unilamellar liposomes of ∼120 nm in diameter. High performance liquid chromatography determination of ligand coupling to the liposome surface indicated that more than 99% of the RGD peptides were reacted with maleimide groups on the liposome surface. Up to 3 mg/mL of stable liposomal combretastatin A4 loading was achieved with ∼80% of this being entrapped within the liposomes. In the in vitro cell culture studies, targeted liposomes showed significantly higher binding to their target cells than non-targeted liposomes, presumably through specific interaction of the RGD with its receptors on the cell surface. It was concluded that the targeting properties of the prepared delivery system would potentially improve the therapeutic benefits of combretastatin A4 compared with nontargeted liposomes or solution dosage forms.     

Mesenchymal stromal cells in wound healing applications: role of the secretome,targeted delivery and impact on recessive dystrophic epidermolysis bullosa treatment

Mesenchymal stromal cells (MSCs) are multi-potent stromal-derived cells capable of self-renewal that possess several advantageous properties for wound healing, making them of interest to the field of dermatology. Research has focused on characterizing the unique properties of MSCs, which broadly revolve around their regenerative and more recently discovered immunomodulatory capacities. Because of ease of harvesting and expansion, differentiation potential and low immunogenicity, MSCs have been leading candidates for tissue engineering and regenerative medicine applications for wound healing, yet results from clinical studies have been variable, and promising pre-clinical work has been difficult to reproduce. Therefore, the specific mechanisms of how MSCs influence the local microenvironment in distinct wound etiologies warrant further research. Of specific interest in MSC-mediated healing is harnessing the secretome, which is composed of components known to positively influence wound healing. Molecules released by the MSC secretome can promote re-epithelialization and angiogenesis while inhibiting fibrosis and microbial invasion. This review focuses on the therapeutic interest in MSCs with regard to wound healing applications, including burns and diabetic ulcers, with specific attention to the genetic skin disease recessive dystrophic epidermolysis bullosa. This review also compares various delivery methods to support skin regeneration in the hopes of combating the poor engraftment of MSCs after delivery, which is one of the major pitfalls in clinical studies utilizing MSCs.