Mathematical modeling of metabolism and hemodynamics.

We provide a mathematical study of a model of energy metabolism and hemodynamics of glioma allowing a better understanding of metabolic modifications leading to anaplastic transformation from low grade glioma. This mathematical analysis allows ultimately to unveil the solution to a viability problem which seems quite pertinent for applications to medecine.

Predicting the outcome of grade II glioma treated with temozolomide using proton magnetic resonance spectroscopy.

BACKGROUND: This study was designed to evaluate proton magnetic resonance spectroscopy ((1)H-MRS) for monitoring the WHO grade II glioma (low-grade glioma (LGG)) treated with temozolomide (TMZ). METHODS: This prospective study included adult patients with progressive LGG that was confirmed by magnetic resonance imaging (MRI). Temozolomide was administered at every 28 days. Response to TMZ was evaluated by monthly MRI examinations that included MRI with volumetric calculations and (1)H-MRS for assessing Cho/Cr and Cho/NAA ratios. Univariate, multivariate and receiver-operating characteristic statistical analyses were performed on the results. RESULTS: A total of 21 LGGs from 31 patients were included in the study, and followed for at least n=14 months during treatment. A total of 18 (86%) patients experienced a decrease in tumour volume with a greater decrease of metabolic ratios. Subsequently, five (28%) of these tumours resumed growth despite the continuation of TMZ administration with an earlier increase of metabolic ratios of 2 months. Three (14%) patients did not show any volume or metabolic change. The evolutions of the metabolic ratios, mean(Cho/Cr)(n) and mean(Cho/NAA)(n), were significantly correlated over time (Spearman ρ=+0.95) and followed a logarithmic regression (P>0.001). The evolutions over time of metabolic ratios, mean(Cho/Cr)(n) and mean(Cho/NAA)(n), were significantly correlated with the evolution of the mean relative decrease of tumour volume, mean(ΔV(n)/V(o)), according to a linear regression (P<0.001) in the 'response/no relapse' patient group, and with the evolution of the mean tumour volume (meanV(n)), according to an exponential regression (P<0.001) in the 'response/relapse' patient group. The mean relative decrease of metabolic ratio, mean(Δ(Cho/Cr)(n)/(Cho/Cr)(o)), at n=3 months was predictive of tumour response over the 14 months of follow-up. The mean relative change between metabolic ratios, mean((Cho/NAA)(n)-(Cho/Cr)(n))/(Cho/NAA)(n), at n=4 months was predictive of tumour relapse with a significant cutoff of 0.046, a sensitivity of 60% and a specificity of 100% (P=0.004). CONCLUSIONS: The (1)H-MRS profile changes more widely and rapidly than tumour volume during the response and relapse phases, and represents an early predictive factor of outcome over 14 months of follow-up. Thus, (1)H-MRS may be a promising, non-invasive tool for predicting and monitoring the clinical response to TMZ.

[Phosphorus magnetic resonance spectroscopy: Brain pathologies applications]. / Spectroscopie du phosphore 31 par résonance magnétique : applications en pathologies cérébrales.

Until recent years, brain applications of (31)P magnetic resonance spectroscopy were poor. Arising of clinical high field strength magnets (three Tesla) as well as dedicated brain coils (eg: bird cage), using specific and useful sequences providing appropriate spatial localisation and signal to noise ratio brought highlights on multinuclear spectroscopy. Better understanding of brain metabolism emphasizes the role of phosphoenergetic compounds and its potential issues in tumoral, metabolic and degenerative diseases. In the present paper, we report 1 year of experience and preliminary results for 40 patients as well as review of the literature. By successive in vivo determination and quantitation of numerous metabolites it allows, multinuclear spectroscopy may provide additional information to biomathematical models of brain metabolism.

The influence of plasma membrane electrostatic properties on the stability of cell ionic composition.

An electro-osmotic model is developed to examine the influence of plasma membrane superficial charges on the regulation of cell ionic composition. Assuming membrane osmotic equilibrium, the ion distribution predicted by Gouy-Chapman-Grahame (GCG) theory is introduced into ion transport equations, which include a kinetic model of the Na/K-ATPase based on the stimulation of this ion pump by internal Na(+) ions. The algebro-differential equation system describing dynamics of the cell model has a unique resting state, stable with respect to finite-sized perturbations of various types. Negative charges on the membrane are found to greatly enhance relaxation toward steady state following these perturbations. We show that this heightened stability stems from electrostatic interactions at the inner membrane side that shift resting state coordinates along the sigmoidal activation curve of the sodium pump, thereby increasing the pump sensitivity to internal Na(+) fluctuations. The accuracy of electrostatic potential description with GCG theory is proved using an alternate formalism, based on irreversible thermodynamics, which shows that pressure contribution to ion potential energy is negligible in electrostatic double layers formed at the surfaces of biological membranes. We discuss implications of the results regarding a reliable operation of ionic process coupled to the transmembrane electrochemical gradient of Na(+) ions.

Modelling of the coupling between brain electrical activity and metabolism.

In order to make an attempt at grouping the various aspects of brain functional imaging (fMRI, MRS, EEG-MEG, ...) within a coherent frame, we implemented a model consisting of a system of differential equations, that includes: (1) sodium membrane transport, (2) Na/K ATPase, (3) neuronal energy metabolism (i.e. glycolysis, buffering effect of phosphocreatine, and mitochondrial respiration), (4) blood-brain barrier exchanges and (5) brain hemodynamics, all the processes which are involved in the activation of brain areas. We assumed that the correlation between brain activation and metabolism could be due to either changes in the concentrations of ATP and ADP following activation of Na/K ATPase that result from the changes in ion concentrations, or the involvement, in different phases of metabolism, of a second messenger such as calcium. In this article, we show how this type of model enables interpretation of MRS and fMRI published data that were obtained during prolonged stimulations.

A model study of cellular short-term memory produced by slowly inactivating potassium conductances.

We analyzed the cellular short-term memory effects induced by a slowly inactivating potassium (Ks) conductance using a biophysical model of a neuron. We first described latency-to-first-spike and temporal changes in firing frequency as a function of parameters of the model, injected current and prior history of the neuron (deinactivation level) under current clamp. This provided a complete set of properties describing the Ks conductance in a neuron. We then showed that the action of the Ks conductance is not generally appropriate for controlling latency-to-first-spike under random synaptic stimulation. However, reliable latencies were found when neuronal population computation was used. Ks inactivation was found to control the rate of convergence to steady-state discharge behavior and to allow frequency to increase at variable rates in sets of synaptically connected neurons. These results suggest that inactivation of the Ks conductance can have a reliable influence on the behavior of neuronal populations under real physiological conditions.

A few comments on electrostatic interactions in cell physiology.

The role of fixed charges present at the surface of biological membranes is usually described by the Gouy-Chapman-Grahame theory of the electric double-layer where the Grahame equation is applied independently on each side of the membrane and where the capacitive charges (linked to the transmembrane ionic currents) are disregarded. In this article, we generalize the Gouy-Chapman-Grahame theory by taking into account both intrinsic charges (resulting from the dissociation of membrane constituents) and capacitive charges, in the density value of the membrane surface charges. In the first part, we show that capacitive charges couple electrostatic potentials present on both sides of the membrane. The intensity of this coupling depends both on the value of the membrane specific capacitance and the transmembrane electric potential difference. In the second part, we suggest some physiological implications of membrane electric double-layers.

Bistable behaviour in a neocortical neurone model.

Intracellular recordings have shown that neocortical pyramidal neurones have an intrinsic capacity for regenerative firing. The cellular mechanism of this firing was investigated by computer simulations of a model neurone endowed with standard action potential and persistent sodium (gNaP) conductances. The firing mode of the neurone was determined as a function of leakage and NaP maximal conductances (gl and gNaP). The neurone had two stable states of activity (bistable) over wide range of gl and gNaP, one at the resting potential and the other in a regenerative firing mode, that could be triggered by a transient input. This model points to a cellular mechanism that may contribute to the generation and maintenance of long-lasting sustained neuronal discharges in the cerebral cortex.

Effect of enzyme organization on the stability of Yates-Pardee pathways.

Yates-Pardee-type metabolic pathways in a heterogeneous cell milieu are modeled as a system of coupled non-linear partial differential equations. A numerical solution to this system is described and some properties of such a physiological system are studied. Confinement with and without a membrane is considered and it is shown how confinement results in an increase in the stability of the metabolite concentrations. These results suggest that the enzyme organization may contribute to the stability of the cellular metabolism.

On the functional organization in a biological structure: the example of enzyme organization.

In this paper, we have considered how the spatial localization of enzymatic reactions, ranging from the elementary type (one step) to that of a metabolic pathway in 2 different phases, may affect the stability of metabolite concentrations. The spatial localization of molecules in the reactions involves: (1) the confinement of some enzymes to cellular substructures (organelles, membranes, cytoskeleton, multienzyme complexes); (2) exchanges of metabolites between cellular substructures (local phase) and cytosol. This organization may be called as structural. Under these conditions, we have studied the dynamical behaviour of the metabolic pathway investigating the velocity of convergence towards the reference steady-state after perturbation of metabolite concentrations. This type of stability may be called as functional stability. We show that an increase in exchanges by diffusion of metabolites between the local phase and cytosol from one hand, or a decrease in the local phase volume on the other hand, result in an increase of the functional stability around the steady-state. This is verified for one step of the pathway as well as for the entire pathway or when the pathway is present in the local phase and in the cytosol.

Formation of cyclobutane thymine dimers photosensitized by pyridopsoralens: a triplet-triplet energy transfer mechanism.

The 365 nm irradiation of thymine thin films in the presence of pyridopsoralens is shown to induce the formation of cyclobutane thymine dimers, in contrast to other compounds such as 8- and 5-methoxypsoralen. In order to elucidate the mechanism of such a photosensitized reaction, we have determined the energy of the lowest triplet state (T1) of these compounds, using phosphorescence spectroscopy and CNDO/S quantum chemistry calculations. The T1 energy values were found to be significantly higher for pyridopsoralens--up to 0.3 eV--than for 8- and 5-methoxypsoralen (approximately 2.8 eV), which are not able to photoinduce cyclobutane thymine dimers. The determination of the relative efficiency of cyclobutane thymine dimer formation was performed using chromatographic analysis. A good correlation was found between the energy of the T1 state of the psoralen derivatives and the related cyclobutane thymine dimer formation. Moreover, the photosensitized cyclobutane thymine dimer formation appeared to be temperature-dependent. Our results are consistent with a mechanism involving a triplet energy transfer from the pyridopsoralen to thymine.