A mathematical model for calcium homeostasis

In vivo control of calcium is analysed under the assumption that hormonal influences via plasma levels of parathormone and calcitonin are of prime (but not absolutely dominating) importance. A brief review concerning the physiological significance of body calcium and the mode of action of these two hormones is presented as an introduction to the basic philosophy of the study. A theoretical quasi-linear lumped-parameter model is developed to describe variations in ionic calcium, parathormone and calcitonin plasma concentrations to specific input stimuli. Formal evaluation of the system response requires the determination of ten constants, together with quantitation of ingested calcium entry into the plasma compartment which isindependent of hormonal influences. Values for various parameters are deduced from published data and experimental procedures are outlined to facilitate determination of the remaining unknowns. It is suggested that the proposed model should prove useful for investigations concerning general hormonal actions on calcium homeostatic mechanisms in both normal and diseased states, with particular reference to calcitonin.     

A mathematical model for vasogenic brain edema

Animals reproducing sexually are faced with a dilemma. They can mate in the local area where they were born, but must then take the risk that they will mate with a close relative and produce offspring suffering from inbreeding depression. Or they can migrate to a new area and mate with an unrelated animal, but then there is a chance that they will die or lose valuable reproductive time during the migration.This dilemma is here described by some simple models. The condition for spreading of a new reproductive strategy in a population is given. The results are discussed using data from the Great Tit, Man and the Japanese Quail on the effect of inbreeding.     

A mathematical model of the folate cycle: new insights into folate homeostasis

A mathematical model is developed for the folate cycle based on standard biochemical kinetics. We use the model to provide new insights into several different mechanisms of folate homeostasis. The model reproduces the known pool sizes of folate substrates and the fluxes through each of the loops of the folate cycle and has the qualitative behavior observed in a variety of experimental studies. Vitamin B(12) deficiency, modeled as a reduction in the V(max) of the methionine synthase reaction, results in a secondary folate deficiency via the accumulation of folate as 5-methyltetrahydrofolate (the "methyl trap"). One form of homeostasis is revealed by the fact that a 100-fold up-regulation of thymidylate synthase and dihydrofolate reductase (known to occur at the G(1)/S transition) dramatically increases pyrimidine production without affecting the other reactions of the folate cycle. The model also predicts that an almost total inhibition of dihydrofolate reductase is required to significantly inhibit the thymidylate synthase reaction, consistent with experimental and clinical studies on the effects of methotrexate. Sensitivity to variation in enzymatic parameters tends to be local in the cycle and inversely proportional to the number of reactions that interconvert two folate substrates. Another form of homeostasis is a consequence of the nonenzymatic binding of folate substrates to folate enzymes. Without folate binding, the velocities of the reactions decrease approximately linearly as total folate is decreased. In the presence of folate binding and allosteric inhibition, the velocities show a remarkable constancy as total folate is decreased.     

A mathematical model for human brain cooling during cold-water near-drowning

A two-dimensionalmathematical model was developed to estimate the contributions ofdifferent mechanisms of brain cooling during cold-water near-drowning.Mechanisms include 1) conductive heat loss through tissue to the water at the head surface and in theupper airway and 2) circulatorycooling to aspirated water via the lung and via venous return from thescalp. The model accounts for changes in boundary conditions, bloodcirculation, respiratory ventilation of water, and head size. Resultsindicate that conductive heat loss through the skull surface or theupper airways is minimal, although a small child-sized head willconductively cool faster than a large adult-sized head. However,ventilation of cold water may provide substantial brain cooling throughcirculatory cooling. Although it seems that water breathing is requiredfor rapid "whole" brain cooling, it is possible that conductivecooling may provide some advantage by cooling the brain cortexperipherally and the brain stem centrally via the upper airway.

     

A mathematical model for brain tumor response to radiation therapy

Gliomas are highly invasive primary brain tumors, accounting for nearly 50% of all brain tumors (Alvord and Shaw in The pathology of the aging human nervous system. Lea & Febiger, Philadelphia, pp 210-281, 1991). Their aggressive growth leads to short life expectancies, as well as a fairly algorithmic approach to treatment: diagnostic magnetic resonance image (MRI) followed by biopsy or surgical resection with accompanying second MRI, external beam radiation therapy concurrent with and followed by chemotherapy, with MRIs conducted at various times during treatment as prescribed by the physician. Swanson et al. (Harpold et al. in J Neuropathol Exp Neurol 66:1-9, 2007) have shown that the defining and essential characteristics of gliomas in terms of net rates of proliferation (rho) and invasion (D) can be determined from serial MRIs of individual patients. We present an extension to Swanson's reaction-diffusion model to include the effects of radiation therapy using the classic linear-quadratic radiobiological model (Hall in Radiobiology for the radiologist. Lippincott, Philadelphia, pp 478-480, 1994) for radiation efficacy, along with an investigation of response to various therapy schedules and dose distributions on a virtual tumor (Swanson et al. in AACR annual meeting, Los Angeles, 2007).     

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 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.

---

     

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 brain-liver circuit regulates glucose homeostasis

Increased glucose production (GP) is the major determinant of fasting hyperglycemia in diabetes mellitus. Previous studies suggested that lipid metabolism within specific hypothalamic nuclei is a biochemical sensor for nutrient availability that exerts negative feedback on GP. Here we show that central inhibition of fat oxidation leads to selective activation of brainstem neurons within the nucleus of the solitary tract and the dorsal motor nucleus of the vagus and markedly decreases liver gluconeogenesis, expression of gluconeogenic enzymes, and GP. These effects require central activation of ATP-dependent potassium channels (K(ATP)) and descending fibers within the hepatic branch of the vagus nerve. Thus, hypothalamic lipid sensing potently modulates glucose metabolism via neural circuitry that requires the activation of K(ATP) and selective brainstem neurons and intact vagal input to the liver. This crosstalk between brain and liver couples central nutrient sensing to peripheral nutrient production and its disruption may lead to hyperglycemia.     

A mathematical model and computer simulation study of insulin sensitive glucose transporter regulation

A mathematical model of insulin sensitive glucose transporter regulation is developed. Model structure is based on experimental evidence from adipocytes and myocytes. Model parameters correspond with known cellular processes. As an example, computer simulation results are compared with data from rat adipocytes. Cellular processes explicitly represented in the model include state-dependent glucose transporter synthesis and degradation rates, insulin sensitive glucose transporter translocation rates, and a glucose transporter endocytosis rate. Most of these processes are represented as first-order events. Using more complex representations of the model structure (e.g. higher order rate constants or saturable pathways) or alternative structures did not result in qualitatively better results. The model is able to accurately simulate the insulin sensitive, insulin concentration dependent, reversible translocation of glucose transporters observed in normal adipocytes. The model is also able to accurately simulate the changes in regulation of glucose transporter translocation observed with increases in cell surface area. Finally, the model can simulate pathogenic states which induce impairment of glucose transporter regulation (e.g. altered glucose transporter regulation in adipocytes from rats on high fat diets, rats with streptozotocin induced diabetes, and fasted rats). Since the structure of our model is sufficient to explain glucose transporter regulation in both normal and pathological states, it may aid in understanding the post-receptor components of insulin resistance (decreased sensitivity or responsiveness to insulin) seen in pathological states such as obesity and diabetes mellitus.     

Regulation of Ca2+ homeostasis by glucose metabolism in rat brain

In a previous communication we reported that glucose deprivation from KHRB medium resulted in a marked stimulation of Ca²⁺ uptake by brain tissue, suggesting a relationship between glucose and Ca²⁺ homeostasis in brain tissue [17]. Experiments were carried out to investigate the significance of glucose in Ca²⁺ transport in brain cells. The replacement of glucose with either D-methylglucoside or 2-deoxyglucose, non-metabolizable analogues of glucose, resulted in stimulation of Ca²⁺ uptake just as by glucose deprivation. These data show that glucose metabolism rather than glucose transfer was necessary to stimulate Ca²⁺ uptake in brain tissue. Inhibition of glucose metabolism with either NaF, NaCN, or iodoacetate resulted in stimulation of Ca²⁺ uptake similar to that produced by glucose deprivation. These results lend further support for the concept that glucose metabolism is essential for Ca²⁺ homeostasis in brain. Anoxia promotes glucose metabolism through glycolytic pathway to keep up with the demand for ATP by cellular processes (the Pasteur effect). Incubation of brain slices under nitrogen gas did not alter Ca²⁺ uptake by brain tissue, as did glucose deprivation and the inhibitors of glucose metabolism. We conclude that glucose metabolism resulting in the synthesis of ATP is essential for Ca²⁺ homeostasis in brain. Verapamil and nifedipine which block voltage-gated Ca²⁺ channels, did not alter Ca²⁺ uptake stimulated by glucose deprivation, indicating that glucose deprivation-enhanced Ca²⁺ uptake was not mediated by Ca²⁺ channels. Tetrodotoxin which specifically blocks Na⁺ channels, abolished Ca²⁺ uptake enhanced by glucose deprivation, but had no effect on Ca²⁺ uptake in presence of glucose (controls). These results suggest that stimulation of Ca²⁺ uptake by glucose deprivation may be related to Na⁺ transfer via Na-Ca exchange in brain.     

A mathematical model of cleavage

In the present paper, we propose a mathematical model of cleavage. Cleavage is a process during the early stages of development in which the fertile egg undergoes repeated division keeping the cluster size almost constant. During the cleavage process individual cells repeat cell division in an orderly manner to form a blastula, however, the mechanism which achieves such a coordination is still not very clear. In the present research, we took sea urchin as an example and focused on the diffusion of chemical substances from the animal and vegetal pole. By considering chemotactic motion of the centrosomes, we constructed a mathematical model that describes the processes in the early stages of cleavage.     

Ghrelin and glucose homeostasis

Ghrelin plays an important physiological role in modulating GH secretion, insulin secretion and glucose metabolism. Ghrelin has direct effects on pancreatic islet function. Also, ghrelin is part of a mechanism that integrates the physiological response to fasting. However, pharmacologic studies indicate the important obesogenic/diabetogenic properties of ghrelin. This is very likely of physiological relevance, deriving from a requirement to protect against seasonal periods of food scarcity by building energy reserves, predominantly in the form of fat. Available data indicate the potential of ghrelin blockade as a means to prevent its diabetogenic effects. Several studies indicate a negative correlation between ghrelin levels and the incidence of type 2 diabetes and insulin resistance. However, it is unclear if low ghrelin levels are a risk factor or a compensatory response. Direct antagonism of the receptor does not always have the desired effects, however, since it can cause increased body weight gain. Pharmacological suppression of the ghrelin/des-acyl ghrelin ratio by treatment with des-acyl ghrelin may also be a viable alternative approach which appears to improve insulin sensitivity. A promising recently developed approach appears to be through the blockade of GOAT activity, although the longer term effects of this treatment remain to be investigated.     

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 adult GnRH neurons in mouse brain and its bifurcation analysis

GnRH neurons are hypothalamic neurons that secrete gonadotropin-releasing hormone (GnRH) which stimulates the release of gonadotropins, one of the crucial hormones for sexual development, fertility and maturation. A mathematical model was built to help elucidate the mechanisms underlying electrical bursting and synchronous [Ca²⁺] transients in GnRH neurons (Lee et al., 2010). The model predicted that bursting in GnRH neurons (at least of the short-bursting type) requires the existence of a [Ca²⁺]-dependent slow after-hyperpolarisation current (sI_AHP-UCL), and this predicted current was found experimentally. GnRH behaviour under a wide range of conditions (inhibition of Na⁺ channels, IP₃ receptors, [Ca²⁺]-dependent K⁺ channels, or Ca²⁺ pumps, or in the presence of zero extracellular [Ca²⁺]) is successfully reproduced by the model. In this paper, a simplified version of the previous model, with the same qualitative behaviour, is constructed and studied using timescale separation techniques and bifurcation analysis.