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MBW:Dynamical System Analysis Of Staphylococcus Epidermidis Bloodstream Infection

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By Gerald Feichtinger

Dynamical System Analysis Of Staphylococcus Epidermidis Bloodstream Infection

Article

This articles subject is a review of the paper Dynamical System Analysis Of Staphylococcus Epidermidis Bloodstream Infection.

Authors: Hangyul M. Chung, Megan M. Cartwright, David M. Bortz, Trachette L. Jackson and John G Younger.

Background

Staphylococcus Epidermidis (SE), one of thirty three known species of the Staphylococcus bacteria, is one of the most used species in laboratory testing. Usually, the bacteria is non-pathogenic, but nevertheless part of the most frequently identified causes of bloodstream infections in the United States.Recent studies showed that these bacterias cause between 11% and 33% of nosocomial bloodstream infections, whereas patients with a weak immune system are most often affected by an infection.

Using new improved methods of mathematical thinking about interactions during bloodstream infections provide us a bunch of new possibilities in physiologically based pharamacokinetik modeling. The authors of the paper provide a well-functioning mathematical model covering bacterial growth, clearance and intercompartmental transfer between the blood, liver, lung and spleen, whose details will now be covered below.

For testing bacteria was injected into five mice, which where studied under identical experimental conditions. Because luminescent SE was not available during testing, therefore Klebsiella pneumoniae Xen 39 has been chosen.

Model

In figure 1 a schematic diagram illustrates the interactions between the four compartments blood, lung, liver and spleen. The authors assumed that the bacteria exists in one of four compartments. Within one compartment the bacteria can increase at the rate a and decrease at rate d. The transfer between the compartments is a function of blood per volume (mL/h) defined as q/v. Additionally a compartment specific partitioning coefficient p (dimensionless) is necessary.

The basic model is defined as follows:

Definition

The growth rate (a >= 0) and death rate (d >= 0) are defined as the proportion to the number of bacteria within the compartment. The term (a – d) is defined as the net-growth-rate of the bacteria, whereas the upper bound of (a – d) was established at 0.92 experimentally. The parameter q is defined as the rate of blood flow to each compartment and v as the volume of each compartment. The partitioning coefficient p is defined with a value of greater than 1, if bacteria will be filtered from the blood and between 0 and 1 if bacteria will be released into the blood.

The mathematical model is based on four differential equations (one for each compartment) as listed below.

(1) Lung: Equation #1: Lung

(2) Spleen: Equation #2: Spleen

(3) Liver: Equation #3: Liver

(4) Blood: Equation #1: Blood

Within the equations l, s, h and b describe the bacterial concentrations in the lung, spleen, liver and blood. As illustrated in figure 1 above, the circulating bood can transfer bacteria into the liver, the lung and the spleen. The liver and the lung can transfer bacteria back into the blood, whereas the spleen can only transfer bacteria into the liver.

The initial conditions of the parameters have to be defined for two scenarios: the normodynamic and the hyperdynamic blood flow. In the former case the blood flow parameter q for the liver is set to 1.5 mL/min (qh), for the spleen to 1.2 (qs) and for the lung to 6.84 (ql). In the latter case these values change to qh = 2.4, qs = 1.9 and ql = 10.05. The organ mass volume v for the liver is set to 1.99g (vh), the spleen to 0.15g (vs), the lung to 2.1g (vl) and the blood to 1.8mL (vb). The net-growth-rate of the bacteria for the liver is set to -0.24, for the lung to -4.7 and for the spleen and blood to 0.92. Finally the partitioning coefficients p are set to 79 for the liver (ph), 3 for the lung (pl) and 28 for the spleen (ps).

Result

After finally calibrating the model with all given parameters, the population of the SE bacteria will initially just exist within the lung (set to 1) and won't exist within the other compartments (set to zero). The results of the two scenarios are more or less similar. In the first case, the normodynamic, the population of the bacteria within the lung will decrease immediately. After a few ups and downs the bacteria population within the liver and blood will stabilize at a very low level, but will increase slightly within the spleen.

File:Model Result hyper.jpg
Hyperdynamic case

In the second case, the hyperdynamic, the situation is almost the same. In the beginning the bacteria decreases rapidly within the lung, stabilizes within the blood and liver at a certain leven and increases within the spleen.

Matlab Code

The implementation of the model in Matlab is divided into three sections: the implementation of the differential equations for the normodynamic case LBHSrhs_nor.m, the hyperdynamic case LBHSrhs_hyp.m and the main body bacteria.m.