**8. Discussions and main conclusions**

Physiological events modeling, as uptake, bioaccumulation, or metabolism, and so on, in living organisms are extremely difficult both due to the complex nature of physiological processes and due to the complexity of the biological system that is modeled. For this reason the existence of implemented models in this area is very scarce, if not almost inexistent. Most existing models from the literature refer to models which are applied to a micro-scale fragment from a biological system (intercellular models) and less for globalized macro-scales that integrate multiple events [29, 39].

In this chapter we tried to overcome this challenge, trying to model the accumulation rate for a chemical compound based on experimental data obtained in the laboratory after analysis conducted on numerous mushroom specimen analyses of *Macrolepiota procera*. Our first approach was on the anatomical compartments of the mushroom body, but the results were not satisfactory.

Initially, the independent variable *z* was the height of the same compartment. Because compartments as cap and lamellae had an insufficient number of data (since the length of cap and lamellae is 1.2 cm and the minimal width for sample collection was from the section taken from 2 and 2 mm), we obtained large deviations between measured concentrations and computed concentrations at inter-compartment boundaries (see **Figure 4**).

Thus, because in chemistry in a given volume the concentration of a chemical compounds is the same in any point of these volume, we considered in the next that we have several points of concentration data of same value in horizontal sections of the cap (**Figure 3**).

These drawbacks lead us to modify the approach mentioned in Section 6 on Analysis and modeling: The estimation of R. Due to symmetry we considered a half of an axial section and a medial axis of the section. Now *z* is the length of the path on medial axis. To apply the classical theory on ordinary differential equations to Eq. (1), we need to have a function of class *C*<sup>2</sup> on the domain of *z*, while the parameters defining the rate are piecewise constant. For this reason we considered the weighted average of these parameters on the whole length of the medial axis. The diameter *Φ*(*z*) was approximated by the piecewise Hermite cubic spline of measured diameters. Our choice is motivated by the fact that these splines are shape-preserving. The

and ρ is the smoothing factor. The spaps function uses the algorithm described in Reinsch's work [32]. For additional details on smoothing and interpolation splines, see deBoor's book

**Figure 3.** Schematic representation of target chemical compound concentration distribution around mushroom anatomical

This step has a double purpose: to prove that the approximation of rate, given by Eq. (2), is sufficiently accurate and to compute the concentration from the rate, without performing any measurement. We want to solve the two-point boundary value problems Eq. (1) and bound-

> *C*(*zinitial*) = *C*0, *<sup>C</sup>*(*zfinal*) <sup>=</sup> *Cn*

Our solution uses the collocation method [38]. The independent variable *z* in Eq. (1) means the

Physiological events modeling, as uptake, bioaccumulation, or metabolism, and so on, in living organisms are extremely difficult both due to the complex nature of physiological processes

. (5)

[33] and the MATLAB curve fitting toolbox user's guide [36, 37].

**7. Validation and solution of the differential equation**

ary conditions:

compartment.

206 Numerical Simulations in Engineering and Science

{

length of the path along the medial axis.

**8. Discussions and main conclusions**

**Figure 4.** The averaging of parameters (ODE solution, smoothing spline) and target chemical concentrations on the whole mushroom.

next step was the application of Eq. (2) to compute the rate *R*. The rate *R*, computed along the medial axis, as described in the previous section, is plotted in **Figure 5**.

In order to assess the accuracy of our model we plot the initial and the computed concentra-

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There is good matching, as **Figure 8** shows. The least square deviation for the concentration is 5.1384e-005, and it was computed at points corresponding to the measured z and

The model proposed for the rate estimation of a chemical compounds in living organisms, specifically a mushroom, is new. Based on our knowledge, up to date, there is no paper on the

tion data (**Figure 7**).

**Figure 7.** The graph of initial and computed concentrations.

**Figure 8.** Differences between the initial and the computed concentration.

concentrations.

Analyzing data obtained for monitored chemical compound rate, it was possible to observe that larger fluctuations are present in the mushroom stipe while in the caps part (cap and lamellae) a decreasing tendency is registered. These data are in correlation both with infield experimental measurements and with the computed concentration obtained from our model—see **Figure 6** where the concentration *C* is presented, after the solution of the differential equation (1) with boundary conditions (5).

**Figure 5.** The graph of rate *R.*

**Figure 6.** The graph of concentration *C*, obtained by the solution of a two-point boundary value problem.

In order to assess the accuracy of our model we plot the initial and the computed concentration data (**Figure 7**).

There is good matching, as **Figure 8** shows. The least square deviation for the concentration is 5.1384e-005, and it was computed at points corresponding to the measured z and concentrations.

The model proposed for the rate estimation of a chemical compounds in living organisms, specifically a mushroom, is new. Based on our knowledge, up to date, there is no paper on the

**Figure 7.** The graph of initial and computed concentrations.

next step was the application of Eq. (2) to compute the rate *R*. The rate *R*, computed along the

Analyzing data obtained for monitored chemical compound rate, it was possible to observe that larger fluctuations are present in the mushroom stipe while in the caps part (cap and lamellae) a decreasing tendency is registered. These data are in correlation both with infield experimental measurements and with the computed concentration obtained from our model—see **Figure 6** where the concentration *C* is presented, after the solution of the differen-

**Figure 6.** The graph of concentration *C*, obtained by the solution of a two-point boundary value problem.

medial axis, as described in the previous section, is plotted in **Figure 5**.

tial equation (1) with boundary conditions (5).

208 Numerical Simulations in Engineering and Science

**Figure 5.** The graph of rate *R.*

**Figure 8.** Differences between the initial and the computed concentration.

rate estimation for a specific chemical compound in a vegetal system. The model presented by Lettmann et al. [26] has completely different premises—there is no volume limitation in their case study while in our case we were limited to the smaller dimension of the studied living organism, the mushroom species *Macrolepiota procera*. Once have the rate for a species, we can compute concentrations via the solution of Eqs. (1) + (5) without doing any additional laboratory measurement.

in Foods: A Volume in Series in Food Science, Technology and Nutrition. Woodhead Publishing Limited; Oxford, 2013: pp. 129-144. DOI: https://doi-org.am.e-nformation.

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