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Mathematical models of the acute inflammatory response.

PURPOSE OF REVIEW: Trauma and infection elicit an acute inflammatory response. In certain circumstances the degree of the acute inflammatory response may result in pathologic manifestations, namely, sepsis and multiple organ failure. Despite an extensive series of clinical trials designed to modulate inflammation in sepsis, only one compound, activated protein C, has emerged from more than 250 failed trials. There is a growing recognition that the complexity of the acute inflammatory response precludes the efficient development of therapies for sepsis and multiple organ failure until systems approaches are brought to bear on this problem. RECENT FINDINGS: Work carried out by the authors' groups suggests that mathematical modeling can provide a means by which in vitro and in vivo data can be synthesized into system-level analytic models of the acute inflammatory response. The authors have focused on agent-based modeling and modeling with ordinary differential equations. Some of the advantages and disadvantages of these modeling approaches are presented, and methods for calibration and validation of these models are discussed. Finally, the usefulness of mathematical models to evaluate the prospective therapeutic strategies in clinical trials of sepsis and trauma is examined. SUMMARY: Simulations using various methods can shed insight into the pathophysiology of the acute inflammatory response and may lead to better design of clinical trials in sepsis and trauma.

Acute Disease↗

The evolution of mathematical modeling of glioma proliferation and invasion.

Gliomas are well known for their potential for aggressive proliferation as well as their diffuse invasion of the normal-appearing parenchyma peripheral to the bulk lesion. This review presents a history of the use of mathematical modeling in the study of the proliferative-invasive growth of gliomas, illustrating the progress made in understanding the in vivo dynamics of invasion and proliferation of tumor cells. Mathematical modeling is based on a sequence of observation, speculation, development of hypotheses to be tested, and comparisons between theory and reality. These mathematical investigations, iteratively compared with experimental and clinical work, demonstrate the essential relationship between experimental and theoretical approaches. Together, these efforts have extended our knowledge and insight into in vivo brain tumor growth dynamics that should enhance current diagnoses and treatments.

Animals↗

Analysis of the incidence of variola minor in an outbreak by use of a mathematical model.

The incidence of variola minor during an outbreak was analyzed by use of mathematical model B proposed by Chelsky and Angulo. Several parameters were estimated. Among these, the propagating rate (ratio of receptor cases to source cases) seems to predict the outbreak decline better than inspection of the incidence curve. The estimated mean generation interval (18.6 days) supports the thesis that variola minor is not transmitted at onset of illness but, on the average, about 4 days later. The ratio of clinical to subclinical infections (about 1 to 1) approximates those ratios found in serological surveys of variola minor and variola major outbreaks.

Brazil↗

Cell growth and division. I. A mathematical model with applications to cell volume distributions in mammalian suspension cultures.

A mathematical model is formulated for the development of a population of cells in which the individual members may grow and divide or die. A given cell is characterized by its age and volume, and these parameters are assumed to determine the rate of volume growth and the probability per unit time of division or death. The initial value problem is formulated, and it is shown that if cell growth rate is proportional to cell volume, then the volume distribution will not converge to a time-invariant shape without an added dispersive mechanism. Mathematical simplications which are possible for the special case of populations in the exponential phase or in the steady state are considered in some detail. Experimental volume distributions of mammalian cells in exponentially growing suspension cultures are analyzed, and growth rates and division probabilities are deduced. It is concluded that the cell volume growth rate is approximately proportional to cell volume and that the division probability increases with volume above a critical threshold. The effects on volume distribution of division into daughter cells of unequal volumes are examined in computer models.

Animals↗

A mathematical model of drug resistance applied to treatment for small-cell lung cancer.

A mathematical model has been applied to patients with small-cell lung cancer (SCLC) in order to estimate the proportions of resistant and sensitive tumor at presentation, and the efficacy of the treatment, measured in terms of proportions of tumor killed with each cycle of therapy. The model uses estimates of tumor volume obtained from computed tomographic (CT) scans of the chest before each course of chemotherapy. Application of the model to a trial using single-agent high-dose cyclophosphamide (HDC) showed that HDC killed approximately 94% of the sensitive tumor on each application, but that the proportion of tumor resistant to HDC rose from an average of 1% to an average of 15% after the first cycle, assuming a 30-day tumor doubling time. These estimates proved fairly insensitive to different assumptions about tumor doubling time and inaccuracies in volume measurement and may thus provide a useful additional evaluation technique for some clinical trials.

Carcinoma, Small Cell↗

Mathematical modelling for intra-specific brood-parasitism: coexistence between parasite and non-parasite.

In this paper, we consider an aspect of the intra-specific brood-parasitism with a mathematical modelling. As in case of moorhen Gallinula chloropus, the case dealt with in this paper in such that just a part of the whole population has the parasitising behaviour against the individuals belonging to another part of non-parasite subpopulation that does not have such behaviour. Analysing the expected fitness gain from the brood-parasitism, we consider the condition in order that parasite individuals coexist with non-parasite ones within a population. From the mathematical modelling analysis, it is shown that the stable equilibrium frequency of parasite individuals within a population, if exists, depends on the difference among individuals in terms of the individual quality reflected to the survival probability of bred offsprings.

Animals↗

A mathematical model to calculate temperature distributions in human and rabbit eyes during hyperthermic treatment.

A mathematical model based on the finite difference method has been developed to calculate transient and steady state temperature distributions in normal unexposed human and rabbit eyes, and human and rabbit eyes heated by various heating techniques. The normal steady state temperature distributions in human and rabbit eyes are given. The model has been experimentally fitted to data obtained from measurements on rabbit eyes. The heat transfer from the choroid to the body core temperature of the rabbit is described by the heat transfer coefficient hs = 65 W m-2 degrees C-1, and from the cornea to the surrounding air temperature by hc = 20 W m-2 degrees C-1. The thermal conductivity and the specific heat of the lens of the rabbit eye were determined empirically to be 0.40 W m-1 degree C-1 and 3.0 J g-1 degree C-1 respectively. The thermal properties of the vitreous humour were taken to be equal to the thermal properties of water.

Animals↗

[Mathematical model of the zone formation in polyene antibiotic diffusion in an inoculated gel].

A mathematical model of inhibition growth zone formation on diffusion of polyenic antibiotics in gel is described. Dynamic equations for the zone formation describing the changes in time of the microbial population density (test microbe) in the presence of the diffusing antibiotics, as well as equations for diffusion of the antibiotics with an account of their inactivation were developed. Extreme cases of no inactivation of the drug by the test microbe are discussed. For the cases of spontaneous inactivation of the drugs due to instability of their molecules there were developed expressions showing correlation between the zone size and antibiotic initial concentration in the hole and other parameters characterizing the antibiotic and test microbe.

Antifungal Agents↗

Mathematical model of regulation of oxidative phosphorylation in intact mitochondria.

1. A mathematical model of the regulation of mitochondrial ATP synthesis by the extramitochondrial ATP/ADP ratio is presented taking into account the transport processes of phosphate and of adenine nucleotides by their specific translocators. 2. In agreement with known experimental data the model describes the control of respiration by the extramitochondrial ATP/ADP ratio as well as the distribution of adenine nucleotides and of inorganic phosphate between the extramitochondrial and the intramitochondrial compartment. 3. In the extramitochondrial compartment the phosphorylation potential is predicted by the model to be higher than in the matrix space. 4. Despite the differences in the phosphorylation potentials no particular translocation energy is necessary. This has been achieved by postulating a charge compensation between the movement of adenine nucleotides and the uptake of phosphate during ATP synthesis. 5. The proton stoichiometry of the proton pump must be higher than stated by the chemiosmotic coupling hypothesis in its present form, otherwise sufficient results could not be obtained. 6. With increasing activities of non-phosphorylating energy requiring side reactions (as uncoupling) at first the difference of the respiration rates between the phosphorylating and the non-phosphorylating state disappears, at higher activities the ADP phosphorylation stops, but the membrane potential collapses at very high activities only.

Adenosine Diphosphate↗

A simple approximate mathematical model to predict the number of severe acute respiratory syndrome cases and deaths.

BACKGROUND: Severe acute respiratory syndrome (SARS) is currently spreading in many countries. This paper proposes a simple approximate mathematical model for public health practitioners to predict the number of SARS cases and deaths. METHODS: The model is based on four parameters: R(o) (basic reproductive number), F (case-fatality rate), i (incubation period), and d (duration of disease). The calculations can be done by hand or by using a computer spreadsheet. RESULTS: The best parameters to fit Canadian data as of 6 April 2003 (before infection controls took effect) are R(o) = 1.5, F = 30%, i = 5 days, d = 14 days. On 6 April (day 40) there were 74 cases and 7 deaths. If this trend continues, SARS numbers in Canada are predicted to be as follows: 387 cases and 34 deaths by 26 April (day 60), 4432 cases and 394 deaths by 26 May (day 90), and 50 500 cases and 4489 deaths by 25 June (day 120). By comparison, the best parameters to fit Hong Kong data as of 10 April 2003 are R(o) = 2.0, F = 20%, i = 5 days, d = 14 days. CONCLUSIONS: Using the proposed mathematical model, it was estimated that about 1.5 to 2 new infectious cases were produced per infectious case every five days. Also, about 20% to 30% of the cases die within 14 days. The case-fatality may therefore be considerably higher than initially thought. The model indicates that SARS can spread very fast when there are no interventions.

Canada↗

A mathematical model of hematopoiesis: II. Cyclical neutropenia.

Cyclical neutropenia is a dynamical disease of the hematopoietic system marked by an oscillation in circulating leukocyte (e.g. neutrophil) numbers to near zero levels and then back to normal. This oscillation is also mirrored in the platelets and reticulocytes which oscillate with the same period. Cyclical neutropenia has an animal counterpart in the grey collie. Using the mathematical model of the hematopoietic system of Colijn and Mackey [A mathematical model of hematopoiesis: I. Periodic chronic myelogenous leukemia. Companion paper to the present paper.] we have determined what parameters are necessary to mimic laboratory and clinical data on untreated grey collies and humans, and also what changes in these parameters are necessary to fit data during treatment with granulocyte colony stimulating factor (G-CSF). Compared to the normal steady-state values, we found that the major parameter changes that mimic untreated cyclical neutropenia correspond to a decreased amplification (increased apoptosis) within the proliferating neutrophil precursor compartment, and a decrease in the maximal rate of re-entry into the proliferative phase of the stem cell compartment. For the data obtained during G-CSF treatment, good fits were obtained only when parameters were altered that would imply that G-CSF led to higher amplification (lower rate of apoptosis) in the proliferating neutrophil precursors, and a elevated rate of differentiation into the neutrophil line.

Adult↗

A mathematical model of B lymphocyte differentiation: control by antigen.

A mathematical model of B lymphocyte differentiation, based on experimental results, has been developed. The model focuses on the role of antigen in initiating and regulating B cell differentiation while other mechanisms, acting in concert with antigen but the functioning of which can be circumvented under appropriate conditions, are not considered. The importance of presence of antigen at individual stages of B cell differentiation was studied in experiments with an easily metabolizable antigen. Immunocompetent cells (ICC), arising by antigen-independent differentiation of stem cells, are activated by antigen (they become immunologically activated cells--IAC). Excess of antigen drives IAC into the terminal stage (antibody-forming cells--AFC) thereby restricting proliferation. Exhaustive terminal differentiation results in tolerance. A low primary dose permits IAC to escape antigen; IAC proliferate and later give rise to resting memory cells (MC) which are amenable to reactivation. MC have higher avidity for antigen (due to higher affinity, number and density of receptors) and the effect of different doses of antigen on MC is diverse. A very low secondary dose induces tolerance, a medium dose secondary response, and the administration of a high dose of antigen also brings about tolerance. The model suggests that the fate of memory cells is controlled by the ratio R:Ag, of the number of immunoglobulin receptors on B cells (R) to the number of available antigenic molecules (Ag), low values R:Ag favouring stimulation to differentiation while high values of R:Ag favouring inactivation. A nonlinear system of ordinary differential equations, describing the development of the populations involved in antigen-driven B cell differentiation, was used to simulate experiments and good qualitative agreement was achieved.

Animals↗

[Evaluation of the applicability of a mathematical model in the x-ray cephalometric diagnosis of sagittal bite anomalies. A clinical test].

The mathematical model (Järvinen) for measuring the sagittal difference between the maxillary and mandibular apical bases has been clinically tested. The material for this study consisted of 30 lateral skull radiographs of orthodontically untreated children (aged seven to 15 years) with different types of skeletal and/or dento-alveolar malocclusions. A comparison of the model with two conventional and two more developed methods to measure the sagittal apical base difference showed that the correlation between the new and the older methods increased as the errors of the older methods were eliminated. The results seemed to indicate that sagittal malocclusions could be accurately diagnosed by means of the model.

Adolescent↗

Mathematical modeling of the human fetal arterial blood circulation.

A mathematical model of the human fetal arterial circulation based on mass and momentum conservation for one-dimensional flow is presented. We simplified the fetal arterial vascular system from the heart to the placenta, defined 16 anatomical segments and studied the characteristics of the vascular system in relation to changes in morphology and hemodynamics. The two-step Lax-Wendroff finite difference scheme was used to solve the system of equations, after introducing the rheological constants, the diameter and length of the segments measured by two-dimensional imaging and the mean arterial velocity at the inlet segments obtained by pulsed Doppler. The model was validated by comparing the numerical results to our non-invasive ultrasound direct measurements and to previous published data.

Arteries↗

Mathematical model for contact inhibited cell division.

We present a mathematical model for cell growth, which takes into account cell-cell interactions and leads to non-exponential inhibited growth of number of cells. The resulting difference equation is solved and extended to a differential equation which turns out to be of a non-linear diffusion type.

Cell Communication↗

A mathematical model for assessing risk of postoperative wound infection.

A mathematical model for predicting the risk of acquisition of postoperative wound infection in individuals or groups of patients is described. It is based on data from prevalence surveys of 41 hospitals and includes 1980 wounds. The factors included in the model, i.e., age, sex, length of pre-operative stay, type of operation, wound drainage, number of occupied beds in ward, and special factors, e.g., diabetes, steroid therapy, were obtained by stepwise regression analysis of the original data. Most of the ward facilities and practices were excluded as non-significant. The model has been modified for use with incidence studies and its accuracy confirmed in a further 1331 patients by comparing predicted and recorded infection rates.

Age Factors↗

Mathematical model for transient isoelectric focusing of simple ampholytes.

A mathematical model describing transient processes in isoelectric focusing (IEF) of L biprotic ampholytes is presented. The model is a generalization of our previous research on steady slate in IEF and consists of L nonlinear partial differential equations coupled with 2L+2 algebraic equations. Constraints imposed by the mode of operation, viz., constant current. voltage or power, are described. Due to the nonlinearity of the equations, analysis of the model requires computer simulation. Model equations suitable for computer implementation are derived.

Journal Article↗

Analysis of two-component signal transduction by mathematical modeling using the KdpD/KdpE system of Escherichia coli.

A mathematical model for the KdpD/KdpE two-component system is presented and its dynamical behavior is analyzed. KdpD and KdpE regulate expression of the kdpFABC operon encoding the high affinity K+ uptake system KdpFABC of Escherichia coli. The model is validated in a two step procedure: (i) the elements of the signal transduction part are reconstructed in vitro. Experiments with the purified sensor kinase and response regulator in presence or absence of DNA fragments comprising the response regulator binding-site are performed. (ii) The mRNA and molecule number of KdpFABC are determined in vivo at various extracellular K+ concentrations. Based on the identified parameters for the in vitro system it is shown, that different time hierarchies appear which are used for model reduction. Then the model is transformed in such a way that a singular perturbation problem is formulated. The analysis of the in vivo system shows that the model can be separated into two parts (submodels which are called functional units) that are connected only in a unidirectional way. Hereby one submodel represents signal transduction while the second submodel describes the gene expression.

Base Sequence↗