Exemplary Discussion Draft 3

Term Paper Model Discussion

A reduced mathematical model of the acute inflammatory response: I. Derivation of model and analysis of anti-inflammation.

Angela Reynolds et. al.

Discussion

Our model is a reduced model of the acute inflammatory response of the body to a pathogen. By Reynold's own admission, this model as been immensely simplified, and therefore is considered reduced. We observed the 4 variables of $P$ (the amount of pathogen), $N^*$ (the amount of phagocytes), $D$ (the amount of tissue damage), and $C_{A}$ (the amount of anti-inflammatory mediator). Recall that the hypothesis that we tested by recreating Reynold's model, was that a time-dependent anti-inflammatory response results in a healthy immune response, compared to a static anti-inflammatory response, and the time-dependent anti-inflammatory response is characterized by a more stable equilibrium between aseptic and septic death which defines the healthy state. In two of our figures, we observed that our model did indeed support this hypothesis.

Hypothesis Support

In Figure 7, we observed that the model of a dynamic anti-inflammatory mediator ($C_{A}$) which is responsive to time and the fluctuations of the variables for activated phagocytes ($N^*$) and tissue damage ($D$), showed a higher boundary between life and death. When compared to constant values of the $C_{A}$, the maximum amount of initial pathogen tolerated in the body, and still allowing a healthy resolution, was greatest when $C_{A}$ was able to change and fluctuate. This evidence strongly supports our hypothesis, perhaps not in regard to the stability of the health and death boundary, but it does promote the importance of modeling the acute immune response with a time-dependent anti-inflammatory response. It is also important to note that Figure 7 depicts the boundary between life and death over a range of pathogen growth rate ($k_{pg}$) values and at the smallest values of $k_{pg}$, the boundaries are undefined because the only outcome is health.

In Figure 8, we depicted the relationship between the initial amount of anti-inflammatory mediator ($C_{A}$) and the initial amount of pathogen ($P$) and found the equilibrium points which defines the boundary between life and death. At smaller growth rates $k_{pg}$, of the pathogen, there is a greater chance of a healthy resolution to the infection. The faster the growth rate of the pathogen, the smaller chance of survival for the patient. This figure was not specifically supportive of our hypothesis, except to show that at lower growth rates, the initial value of $C_{A}$ is critically linked to the intital value of pathogen. In the figure, we see that if $k_{pg}$ is at a low level, by increasing the initial value of $C_{A}$, we can allow the system to handle a larger initial value of $P$.

Reynolds et. al. Conclusions

As our results and the original figures had minimal discrepancies, we draw similar conclusions to those of Reynolds et. al., though they discuss some additional observations. First, they claim that the anti-inflammation expands the basin of attraction of health compared to that present in models lacking anti-inflammation, which is a desirable feature, because we wish to avoid sepsis or septic shock in patients. Second, Reynolds et. al. assert that the figures demonstrate an advantage, in terms of healthy resolution of infection, conferred by the dynamic nature of the anti-inflammatory response, in comparison to a tonic response. This advantage holds in all situations except for the mildest of infections, which, in any case, do not present a vital threat. This is illustrated in current clinical practice, where distressing symptoms associated with mild infections are alleviated by the co-administration of antibiotics and anti-inflammatory mediators. The reduced model also underlines the importance of the different response rates of substances promoting inflammation, represented in the model by N*, and of the anti-inflammatory mediator that limit this response. They suggest that these rates are fairly well tuned to optimize healthy outcomes to pathogenic infection.

Limitations of Recreations

We have found the limitations of this reduced model to be difficult when attempting to translate our results into applicable conclusions. The lack of unit for each of the state variables is particularly frustrating because we cannot correlate any of the figures to a defined situation that may be experienced in a clinical setting. If the initial amount of pathogen and anti-inflammatory mediator could be determined in a patient and figure 8 could be recreated with pertinent units, this figure could serve as a basis for determining how aggressive the treatment methods should be.

This model could provide a way to diagnose the possible resolution of a patient's infection, if certain parameters could be measured. Currently, when a doctor is diagnosing a patient with sepsis or septic shock, it is not based on the measurement of pathogen, tissue damage or the anti-inflammatory mediator explicitly. Below are several indicators, according to the American College of Chest Physicians, which should be used to diagnose sepsis:

1. a body temperature greater than 38C or less than 36C
2. a heart rate greater than 90 beats per minute
3. tachypnea, manifested by a respiratory rate greater than 20 breaths per minute, or hyperventilation, as indicated by a PaCO2 of less than 32 mm Hg
4. an alteration in the white blood cell count, such as a count greater than 12,000/cu mm, a count less than 4,000/cu mm, or the presence of more than 10 percent immature neutrophils (“bands”)

In the list, it can be observed that only the amount of phagocytes in the body may lead the clinician to a diagnosis of sepsis or septic shock. Suspicion of infection is involved in the diagnosis, but with no need to quantify the amount of pathogen. The inclusion of the relationship between the 4 state variables involved in Reynolds model to the diagnostic variables of patient temperature, heart rate, and respiratory rate would allow this model to become a diagnostic tool.

Recreation Discrepancies

We were able to reproduce figure 5 perfectly, with no observable deviations for the original figure. We were also able to reproduce figure 6 with slight formatting difference from the original model. We saw an extra branch to the bifurcation diagram, which was not included by Reynold's because the branch represented non-physiological conditions.

In Figure 7, our figure deviates a little bit in the shape of the curves from the original figure, but I believe this is only due to our method of defining the equilibrium points. The integrity of the figure is maintained because the same conclusions can be drawn.

We saw some discrepancies between our recreation of Figure 8 and the original figure from Reynold's paper. The specific curvature of each boundary curve is not observed in our recreation. A deviation of our reproduction is the smooth nature of the lines. I suspect that this is due to the method of recreation we chose. We may not have had the time or resources available to get the fine detail as shown in the original model. As a result of meshing our results, we lose some of the local variation of each boundary line. I do not believe this detracts from the significance of the figure, nor the conclusions we can draw from the figure.

Recreation Applied

In their model, Reynolds et. al. continue their analysis with a figure 9, which depicts the effect that manipulating the anti-inflammatory variable can have. In many studies, it has been observed that a large dose of anti-inflammation drug can be disastrous for patients who have proceeded far enough into sepsis . The high amount of anti-inflammatory mediator does not halt the progression of the infection, but rather drives it to result in further damage, and typically death. The complications associated with the use of corticosteroids (anti-inflammatory mediators) are dependent on the dose, the dosing strategy, and the duration of therapy. Much research, including Reynold's, discusses the potential applications for a low dose of anti-inflammation drug applied repeatedly, though the results remain controversial. In 1995, two analyses found no benefit for high dose corticosteroids in sepsis and septic shock and in 2004 another two analyses found benefit for long courses of low dose corticosteroids. This redundant application method is purported to be better for stabilizing the patient and slowly reduce the infection. Shocking the system with a high dose of corticosteroids can be ineffective and even harmful, and there are no studies documenting that stress doses of steroids improve the outcome of sepsis in the absence of shock unless the patient requires stress dose replacement due to a prior history of steroid therapy or adrenal dysfunction. By acknowledging that a single stressful dose, or even a static amount of anti-inflammatory mediator in the body is ineffective for creating a healthy resolution, we confirm that a time-dependent response of the anti-inflammatory mediator is necessary, both in a clinical setting and to produce an accurate model. We can expand the model to include the repeated small dosing treatment that has become common as a treatment for sepsis or septic shock.