Comparison with a brass comb model of burn: 2D simulations
), Singer et al. designed an experimental porcine model to study burn damage. Their model used a 20 mm × 20 mm × 55 mm brass comb with four 10 mm × 20 mm prongs separated by three 5-mm notches that produces four distinctive burn sites separated by three interspaces of unburned skin (refer to Fig. 1 in (31
)). The brass comb was preheated in boiling water (100°C) for five minutes and applied without pressure on one side of the back of the pig for a period of 30 seconds, resulting in four full-thickness burns separated by three unburned interspaces (refer to Fig. 2 in (31
)). The interspaces are not directly injured, but within hours, they undergo progressive ischemia, and at 24 to 48 hours, they become necrotic.
We compare our model with the experimental results of Singer et al., by comparing the burn propagation along the surface of the skin. In order to establish “proof of concept”’ of our model, we first assumed that all the chemicals does not vary much over the depth of the wound, and therefore simulated the model for geometry of the brass comb in the 2-D case. This assumption can also be thought as averaging all the variables over the depth of the wound, therefore we used the same diffusion rates as in the three dimensional case.
depicts the 2-D geometry of the brass comb; the initial burn is confined to the four rectangular regions. shows the numerical solution of the concentrations of lipid and lipid peroxide 12 hours after the initial burn. The parameters used in our simulations are based on and , and the simulated physical time is 12 hours. The simulations are performed in a large rectangle (7 cm in the x - direction and 3 cm in the y - direction) which contains all the four small rectangles. We use no-flux boundary conditions for all the species on four boundaries of the rectangular computational domain (−3.5 ≤ x ≤ 3.5, −1.5 ≤ y ≤ 1.5). We assume that initially, in the burn area, [LH] =0, [TocOH] =0, [LOOH] =0, and in the healthy area, [LH]=L0, [TocOH]=A0, [LOOH] =0, with all the other initial species solved by assuming the quasi-steady-state assumptions. The tissue in the interspace between rectangles clearly suffered excessive vascular damage by the burn. We account for this by decreasing the replenish rate of the natural vitamin E, ka1, in the interspace between the rectangles by a factor of 10. Although it is not possible to exactly compare the profiles in to tissue damage shown in the experiments of Singer et al., the progression of the burn in the simulation and the experiments shows the same pattern.
Figure 3 Two-dimensional simulation for brass comb model of a burn at 12 hour after the initial heat shock. (A) 2D surface view of the geometry of brass comb burn. (B) Contours of concentrations of lipid (L) and lipid peroxide (P) (zoomed in for viewing the interspace (more ...)
The effect of vitamin E intervention: 3D simulations
We simulated the burn wound in three-dimensional space, without intervention and with intervention by vitamin E, i.e., ka2 = 0 and ka2 = 5ka1. The goal was to test, by the mathematical model, the efficacy of vitamin E in slowing down the lipid peroxide propagation. For simplicity we assumed that the epidermal layer and the dermal layer are both homogeneous and parallel to the horizontal plane. Although the diffusion rate of the chemical species in the epidermal layer may be smaller than that of the dermal layer, for simplicity, we take them to be the same. We also assume, for simplicity, that the initial wound is a restricted within the pink cube with −0.5 ≤ x ≤ 0.5, −0.5 ≤ y ≤ 0.5, −1 ≤ z ≤ 0, and take the computational domain to be −1 ≤ x ≤ 1, −1 ≤ y ≤ 1, −2 ≤ z ≤ 0, as illustrated in .
Illustration of the three-dimensional computational domain.
The initial conditions were taken by assuming that the initial damage is proportional to the temperature of the tissue after the burn. More specifically, we assumed that at the initial instant, U = 0 in the burned area, and U = 1 elsewhere, and solved the equation
in the pink region in during the burning, with DU
, and the boundary conditions
We further define U = 1 outside the burned area, and then take 1-U to represent the temperature distribution immediately after the burn. To incorporate the temperature effect of the initial burn, which has contact only on the surface of the skin z = 0, we take the following initial conditions for the species to be
Where U is the temperature at the end of the burning, i.e., t = 30 sec.
Under the above described three-dimensional scenario, the dynamics of the lipid and lipid peroxide are shown in respectively, with both top view and side view at time 1, 6, and 12 hours. The parameters are taken as in and unless otherwise noted. In order to quantitatively analyze the propagation of the burn wound, we tracked the location where the lipid concentration is 90% of its maximum, and use it to quantify the size of the damage in . We note that this criterion gives similar result by using the peak of the lipid peroxide concentration. We found that the progression of damage without vitamin E intervention is rapid, having increased 20 % in the xy-direction and 20 % in xz-direction in 12 hours. However, with vitamin E treatment, the resulting propagation of the burn wound at time 1, 6, 12 hours is reduced to 10%. shows that the size of the wound increases approximately linearly in these 12 hours, with or without vitamin E, and the rate of propagation of the burn wound with vitamin E treatment is about 50% of that without vitamin E treatment. Since we did not take into account different compositions of the skin layers, the propagation speed in horizontal direction and in the vertical direction does not vary. The propagation speed predicted from our simulation is probably more rapid than what has been observed in clinics, because that we did not consider the effect of the heterogeneous complex composition of the skin, which can potentially slow down the diffusion of the chemicals. The propagation may slow down afterwards, as the reparative processes of wound healing may take over.
Figure 5 3D simulations of the burn wound at 1, 6 and 12 hour after the initial heat shock. (A) Without vitamin E treatment (ka2= 0), distributions lipid (L) and lipid peroxide (P); the upper panels (top view) displays the contours of distributions of those species (more ...)
We compared the numerical results with those performed in a larger computational domain (not shown here) and found that the solutions near the initial burn are virtually indistinguishable. This confirms that there is no boundary effect for the computational domain used here.
We have performed PRCC sensitivity analysis (35
) for the burn propagation at T
= 12 in two-dimensional simulations on 19 parameters. The two-dimensional simulations are based on the assumption of homogeneity in the z
-direction of the three-dimensional model.
We estimated the expansion of wound along x
-axis. The region of the wound is defined as the region where the lipid concentration is less than 90% of that of the healthy tissue. Latin hypercube sampling method was used to sample the parameters. The range of each parameter is chosen as 0.5 to 1.5 times the values listed in and . Each range is divided into 1000 intervals of uniform length. Each interval for each parameter is sampled exactly once (without replacement), so that the entire range of each parameter is explored. summarizes the PRCC values and p-values for these 19 parameters (see Fig. S1
). The significant parameters are those parameters that have p-value less than 0.01. We found that the following parameters DLOOH
(diffusion rate of lipid peroxide), μa
(degradation rate for Vitamin E), λ4
(reaction rate of lipid peroxyl radial to lipid) are highly positively correlated while the ka1
(growth rate for vitamin E from diet), λ9
(degradation rate for lipid peroxyl radical) are highly negatively correlated. The results are consistent with the intuition that the faster the lipid peroxide diffuses, the faster the burn propagates; and similarly the faster vitamin E degrades, or the slower vitamin E from diet is absorbed, or the slower lipid peroxyl radical degrades, the faster the burn is expected to propagate.
We further investigated the effectiveness of vitamin E treatment against the uncertainties of the parameters (). The results show that five times topical vitamin E treatment can slow down the propagation of burn on average by at least 50%, and this prediction is quite robust for a large range of randomly chosen parameters.
Figure 6 Statistics of 1000 simulations with randomly selected parameters, for burn wound propagation, with and without vitamin E treatment. The parameter ka1 is a random variable in an interval which contains the natural production rate of vitamin E, 1.4×10 (more ...)