Predictive modeling of molds effective elimination by external inactivation sources

Presented paper deals with a novel application of the (nonlinear) logistic equation to model an elimination of microscopic ﬁlaments types of fungi-molds from affected materials via different external inactivation techniques. It is shown that if the inactivation rate of the external source is greater than the maximum natural growth rate of mycelium, the mold colony becomes destroyed after a ﬁnite time. Otherwise, the mycelium may survive the external attack only at a sufﬁciently large initial concentration of the inoculum. Theoretically determined growth curves are compared with the experimental data for Aspergillus brasiliensis mold inactivated by using both cold atmospheric plasma (CAP) and UV-germicidal lamp. Model presented in the article may be applied also to other classes of microorganisms (e.g. bacteria).


Introduction
The formation and presence of molds on the surfaces (or in the bulks) of various materials (foods, historical artifacts, building materials, etc. [1][2][3][4][5] ) represent a serious problem not only from the aesthetic standpoint, but they are potentially responsible for serious human/animal health problems as a consequence of produced mycotoxins (e.g.aflatoxin, collagenase, coagulase, hemolysis, kinases [6][7][8] ).Furthermore, molds may gradually degrade the surface of material during their reproduction cycles resulting in considerable cultural and economical losses.To reduce the proliferation of the molds through the invaded system, various external inactivation techniques are applied in practice.Traditionally used biocidal chemical agents (aldehydes, peroxo and iodine compounds, active chlorine, etc.) exhibit a relatively high index of killing effect in suppression of the molds, but, on the other hand, due to their carcinogenic and toxic effects their application is restricted in more than 30 countries 9 .Among broadly applied methods (in particular, in the food industry) lacking the above disadvantages is heating the damaged system up to sufficiently large temperatures resulting usually in destroying undesirable pathogens in the foods.However, so-called heat-resistant molds may survive this operation [10][11][12] to continue spoilage of the heat-processed food.Further increasing of temperature may lead to a decrease of nutritional forces of the sterilized food.
Another innovative treatment of the system damaged by the molds is application of either cold atmospheric plasma (CAP)     or the UV-germicidal lamp.While the former one utilizes the synergistic effect of the reactive species produced by CAP (e.g.ozone, single oxygen, hydrogen peroxide, hydroxyl radicals, etc. 13,14 ) responsible for the lethal effects on the mold and its spores 15 , latter one works with the interaction of radiation of given wavelength with the cells of mold.(It has to be stressed out, however, that the efficacy of UV-lamps is much higher when applied to the bacterial systems).
Laboratory tests of mold inactivation belong rather to long-term experiments (consuming more than 300 hours).Consequently, the modeling of this process is welcomed not only due to time-saving reasons, but, in particular, for the possibility to determine straightforwardly the influence of parameters adjustable from the outside on the elimination process.
Regardless of the source of lethal agents (chemical-biocides, fungicides, physical-plasma, UV radiation, heat 10 ), in predictive microbiology it is usually assumed that the deactivation process follows first-order kinetics [16][17][18] .This approach results in exponential decreasing of the number of the microorganisms in the affected sample with time.Or, alternatively, in linear decreasing of survival curves with time 11 .Since one of the important shortcomings of such a linear model is its inability to correctly describe convex or concave survival curves, further models have been developed to cover this gap.In particular, the Weibull distribution and its variations (see, e.g., 17,19 ), resp.the Gompertz (sigmoid) model in its alternative forms (e.g. 12,20,21 )belong to those frequently used in microbiology.Novelty of the model presented in this article consists in application of nonlinear logistic equation with the additional term (representing inactivation rate of external source within finite time interval) to describe the extinction of mold mycelium in the sample.This approach also extends number of similar models applied in variety of unrelated fields (e.g.predator-prey interaction with harvesting 22,23 , fishery 24,25 , infectious dynamics 26 and population models 27 ).

Results
Due to the topologically complex texture of the network-like structure of mycelium on the substrate, the measure of its proliferation has to be defined rather in terms of more continuous characteristics as compared with growth of bacteria.9][30] ).

Model and calculations
Assume that the surface of substrate in the Petri dish is inoculated with the spores of an initial coverage θ (t = 0) = θ 0 .Further proliferation of the inoculate depends on the interplay between the ability of the mold to gradually cover the surface of the substrate and the inactivation rate I of the external source.Since the inactivation mechanism is poorly understood (e.g. 20), it is assumed (within context of presented phenomenological model) that the mycelium elimination occurs via two-step process: I. External inactivation source is activated at t = t 0 , when the part of the substrate is already covered by the growing mycelium and the source is switched off at t = t 1 .
II.From this moment the mycelium destruction process starts.The damaged colony defends against the external attack through adaptive reparation mechanisms trying to absorb newly established unfavorable environmental conditions.Assume further that this revitalization process terminates at t = t 2 and the surviving part of the mycelium (if any) is prepared from this moment for the further proliferation over the substrate.(This adaptation mechanism can be related to the so-called lag-phase of microorganisms evolution 16 which can span between several hours to several weeks 11 . However, it has to be stressed out that understanding of this complex phenomenon requires deeper microbiological analysis which is out of scope of this article.) Resulting mathematical picture of this modeled situation in such a way results in (nonlinear) logistic equation with initial condition Above, θ (t) is the coverage of substrate by mycelium, H(•) stands for the Heaviside generalized function, r represents the proportional increase of the surface coverage (due to intrinsic metabolism of mold combined with complex interplay of mold with its environment), K is the carrying capacity of the habitat corresponding to the maximum coverage that may be sustained by available resources inside the Petri dish (nutrition, water, living space, etc.) and I expresses the inactivation rate of the external source.(Here, the simplest case I = const.within interval ⟨t 0 ,t 2 ⟩ will be assumed).
This equation can be solved in quadratures or within the context of qualitative analysis allowing to analyze the properties and behavior of the exact solution.
A. 0 ≤ t < t 0 .In this case, the exact solution of the equation with the known initial condition θ (t = 0) = θ 0 can be expressed as B. t 0 ≤ t < t 2 .The logistic equation within this time interval reads with the initial condition 2][33][34] ), the first step consists of determination of equilibrium (stationary) points of eq. ( 5).These points (if exist) correspond to situation when there are no changes in surface coverage by mycelium, i.e., when the following quadratic equation is fulfilled Since the equilibrium points are real numbers, the condition I ≤ rK/4 has to be satisfied.
When the inactivation rate I of the external source is relatively moderate and less than the maximum natural growth rate of mycelium (i.e.I < rK/4), then the equilibrium points exist, viz. and where D = 1 − 4I/rK and, successively, If the power of the external inactivation source gradually increases (in limiting case to the critical value I c = rK/4) then only one equilibrium point (double root of eq. ( 7)) exists: Finally, for the supercritical values of I (i.e., I > rK/4) there are no equilibrium points, because the roots of quadratic equation ( 7) are complex numbers without any connection to the microbiological reality.In this sense I c = rK/4 represents a threshold value of the external source inactivation rate associated with a dramatic change of the solution of eq. ( 5).(5) undergoes a change in its behavior: the saddle-node bifurcation (for details, see e.g. 32,35 ; B = I/I c then plays the role of bifurcation parameter. , with the exact analytical solution 3/10 satisfying the initial condition where Furthermore, as it can be shown with help of Figure 1 (through identification of regions where dθ /dt > 0, resp.dθ /dt < 0) for subcritical B < 1, the equilibrium point θ 1 represents a source, whereas θ 2 is a sink (attractor).It means that θ 2 is an asymptotically stable and θ 1 is an unstable point.Indeed, as it follows from solution (11), when α > θ 1 , then lim t→∞ θ (t) = θ 2 for r, D > 0. On the other hand, in the case α < θ 1 , then there exists such a time t = τ > 0, for which denominator of solution ( 11) is zero.Thus, lim t→∞ θ (t) = −∞, since θ 2 > θ 1 and α < θ 1 (as it is assumed).In practice, it means that if the surface coverage by mycelium at the moment of the external source switch-on is smaller than the unstable equilibrium point, θ (t = t 0 ) = α < θ 1 , then the mycelium tends to go extinct in finite time.Otherwise, when α > θ 1 , the damaged mycelium is able to survive the external source intervention and to continue in the proliferation process after a sufficiently large revitalization period.
C. t > t 2 .Logistic equation has the form (the external source is in the switched-off regime) where r 2 is the intrinsic (natural) growth rate of mycelium.(It is assumed, in general, that r 2 may differ from r due to possible modification of the substrate by the impact of the external source).
Analytical solution of eq. ( 14) has the form satisfying initial condition and with α given by (12).

Comparison with experimental data
It is noteworthy that the model presented in this paper depends, in fact, on six parameters.Four of them (initial surface coverage θ 0 , the external source switch on/off times and carrying capacity K) are arbitrarily adjustable from the outside.(In our case, K can be related to the maximum coverage of substrate inside the Petri dish, i.e., K = 100%).Remaining inputs are uniquely determined by choice of the mold genus (via intrinsic growth rate r depending both on metabolism of given mold and its interaction with the environment) and by the parameters of external source (via I).
The solution of appropriate logistic equations are tested with a view to validating the model formulation using datasets resulting from experiments with Aspergillus brasiliensis inactivated by two different external sources (cold atmospheric plasma (CAP) and UV-germicidal lamp, respectively).
The probability of the mold survival after plasma intervention was studied in two operating modes.In the first experiment, the plasma was activated at t 0 = 72 hours after initial inoculation of the substrate and with operating time ∆t = t 1 − t 0 = 10 minutes.In this case (as it was proved by a set of numerical experiments), the optimization procedure operates in the regime very sensitive to slight perturbations in the values of optimized fit.Consequently, the optimization of a growth curve under such conditions is a very complex problem and has to be solved via alternative methods.Inserting experimentally obtained points A = [48 h, 1.4%] and B = [72 h, 5.6%] lying on the exponential part of the growth curve (see Figure 2) to the eqn.( 4), the trivial calculation gives θ 0 = 0.0813% and r = 0.0596 h −1 .
On the other hand, the growth curve remains unchanged for t > t 0 (see Figure 2) as it follows from relationships (11) and (12).
Furthermore, for D = 0 the bifurcation parameter B = 1 and inactivation rate , the plasma attack leads to extinction of the mold on the substrate.Presented model was also applied for a reproduction of A. brasiliensis growth when plasma begins to act after t 0 = 117 hours after MEA inoculation (see Figure 3).Using both optimal fit values and also relations (8a), resp.( 12), one obtains a set of key parameters: critical inactivation rate I c = 0.68 % • h −1 , bifurcation parameter B = 0.96, unstable equilibrium point θ 1 = 39.52% and the surface coverage at t 0 = 117 hours, θ (t = t 0 ) = 58.89%.The inequality θ 1 = 39.52% < 58.89% = θ (t = t 0 ) indicates that the mycelium has a clear chance to survive the intervention of lethal agents.
Finally, the mold resistance to extinction caused by other external sources like plasma was studied by the use of a UV-germicidal lamp (operating at the wavelength 253.7 nm where it provides the maximum germicidal effect).
Partially grown mold was irradiated (in operating time 10 minutes) by the germicidal UV lamp after 24 hours after inoculation.Then the sample was kept in the unchanged environmental conditions up to 360 hours and its further development was regularly recorded.The evolution of the irradiated mold is represented in Figure 4.

Discussion
In many sectors of real life (civil engineering, public health, food-processing industry), the dominant interest is focused on the elimination of the molds proliferation on the surfaces of material applying environmentally acceptable techniques.We herein address, for the first time, a general model of the mold inactivation by use of various external sources exhibiting inactivation rate I. (This approach has been applied to specific cases of the inactivation of A. brasiliensis using both cold atmospheric Optimal fit: 0 = 5.54% r = 2.73×10 2 h 1 r 2 = 3.79×10 2 h 1 I = 6.53×10 1 % h 1 Experimental data plasma and also UV-germicidal lamp).The resulting mathematical picture is a nonlinear logistic balance equation involving inactivation term (operating during a certain time interval) describing the molds growth on the substrate of given carrying capacity.As all calculations are performed by analytical closed form, the presented model is simpler and more effective as compared with other purely numerical approaches.(Although Einstein never said that "Everything should be made as simple as possible, but not simpler," this statement played a key role in the build-up of the model.) The principal conclusions drawn from the results presented in this study are as follows: A) In the supercritical region B > 1 (i.e., when the inactivation rate is greater than the critical one), the mycelium always tends to extinction in finite time.B) In the subcritical case (when B < 1) the probability of mycelium survival after plasma intervention depends on the value of its surface coverage θ (t 0 ) just at a moment t = t 0 of plasma switch-on.When θ (t 0 ) is greater than the asymptotically unstable point θ 1 , then the mycelium has a good chance to survive the plasma attack.Otherwise, if θ (t 0 ) < θ 1 , mycelium tends to be destroyed.
C) It is shown that the application of CAP is much more efficient than the use of a UV-germicidal lamp.D) Presented model is not necessarily limited only to CAP application for mold inactivation, but also other various sources of lethal agents (heat, UV-radiation, certain chemicals) can be applied.E) In this study, the constant inactivation rate I was considered in order to simplify appropriate analytical calculations.
Consequently, the future extension to overcome this limitation includes the precise modeling of inactivation term related  both to real experimental arrangement (e.g.distance of electrode from the mycelium surface, duration of the plasma intervention, power of plasma device, etc.) and also following microbiological characterization of inactivation process.
F) It has to be pointed out one of the advantages of the presented model.While the real experiments are highly timeconsuming (in order of weeks), this approach allows, at least, to obtain related information in order of minutes.

Methods
In order to compare experimental data with the theoretical results, the mold Aspergillus brasiliensis CCM8222 (ATCC 16404) was chosen as a suitable candidate for inactivation procedure (in particular, due to high resolution in coloration between surviving and destroyed parts of mycelium after plasma treatment, see Figure 5).
A. brasiliensis suspension was inoculated in known concentration onto the surface of melt extract agar (MEA) using three 0.1 ml droplets.Inoculated plates (Petri dishes with diameter of 90 mm) were placed in an inoculator set to room temperature 23 • C. The inoculated plates were then exposed to cold atmospheric plasma (CAP) produced by applying voltage to accelerate electrons, resulting in partial ionization of the surrounding air between the electrode and the surface of mycelium.This process leads to the generation of reactive species (charged and neutral particles, photons) destructively propagating within the mold.The experiments were performed by a Diffuse Coplanar Surface Barrier Discharge (DCSBD) device by Roplass, CZ with a processing power 300 W. (In the experiments, the samples were treated by CAP during a period of 10 minutes after chosen time after inoculation).
The percentage of mold coverage θ was determined using θ = θ m /θ A × 100% (where θ m is the area covered by mold and θ A represents the total area).The evaluation of surface mold growth coverage was performed using image analysis.Samples  were withdrawn from the climate chamber on a daily basis to assess mold growth.The surface area was detected by capturing high-resolution photographs of the samples using a camera Canon G1X.Subsequently, the images were analyzed using the Fiji software offering several advantages (simplicity, objective and reproducibility) making it a valuable and widely used tool in research.
Within the comparative test, the mold was also treated by use of germicidal lamp LB 301.2 with operational power 2 × 30 W and wavelength 253.7 nm.
Curve fitting was done numerically using SciPy python package function scipy.optimize.curve_fitwhich implements the Levenberg-Marquardt algorithm (LMA) to solve the nonlinear least squares problem.The plots were created using the Matplotlib package.

Figure 2 .
Figure 2. Growth curve of A. brasiliensis after plasma intervention at t 0 = 72 hours after inoculation.

Figure 3 .
Figure 3. Modeling of the growth curve of A. brasiliensis.

Figure 4 .
Figure 4. Growth curve of A. brasiliensis irradiated by UV-germicidal lamp after 24 hours after inoculation.

Figure 5 .
Figure 5.b: Plasma treated A. brasiliensis on MEA using diffuse coplanar surface barrier discharge (DCSBD) in ambient air.