**4. Case study presentation**

**i.** *Chemical compounds' uptake by plants*: Chemicals' uptake by plant organisms is a system of complex and multi-step processes. These processes could be classified firstly as chemical uptake and transportation between different anatomical compartments (e.g., root to any other anatomical compartment) and secondly as chemical uptake from different environmental compartments (route of exposure) and plant anatomical compartments (particle deposition, vapor uptake from the atmosphere, and so on). The amplitude of these processes is determined by physicochemical properties of the chemical that is under uptake [5]. Current literature presents clear-cut evidence that the availability of most organic chemical compounds is governed on the one hand by their lipophilicity and on the other hand depends on the organic matter (OM) content of the soil under consideration [6]. Some compounds form "bound" residues with organic matter (OM) or humus particles in the soil. Besides, the nature and rooting pattern of the vegetation will have greater influence on the solubility of chemicals. Exuding up to 25% of the net carbon fixed during photosynthesis into the rhizosphere, plants modify given soil-chemical interactions in multiple ways. Secondary plant products (phenolic) and soil bioactive compounds (carbohydrates, organic acids, etc.) could also impact soil micro-biodiversity that could influence in a positive way transformation of organic pollutants to reactive metabolites [7]. For example, it has been demonstrated that isoproturon is metabolized to available plant and reactive compounds in rhizosphere soil [8], while the bacterial conversion of arochlors to reactive metabolites has been one of the early results of bioremediation studies [9].

**ii.** Probably, one of the most effective ways to study chemical behavior and fate is to *use mathematical fate models*. Mechanistic environmental models use mathematical equations which describe the parameters of an environment (e.g. data on flows, depths, pH, temperature, etc.) interconnected with the physicochemical properties of the chemical compounds under various conditions with the final aim of predicting their fate in the environment. According to [10], this can be an inexpensive and suitable approach for setting the limits for discharges in the environment of certain chemical compounds, and since the initial parameter description has been set up and validated by in situ and laboratory data, it can be studied with a minimum set of analysis (e.g., only the quantification of

The ability of numerical models to accurately predict concentrations of target chemical compounds in any living organism depends on the model's ability to mimic the processes involved in their uptake, and this must be assessed before they can be confidently applied [10]. After that it is necessary to consider all of these processes in order to include them in the

*Soil-root transport*: The uptake of chemicals by the root from the soil is mediated in high percentage by soil water content through the plant transpiration process [12]. A large number of organic chemicals also can be sorbed or bound to the components in soil (clay, iron oxides, organic matter),

chemical compound inputs to the ecosystem) [10].

**environment and living organisms**

200 Numerical Simulations in Engineering and Science

numerical model that wants to be developed [11].

**3. Framework for chemical compound interaction with the** 

The properties of wild growing mushrooms make them valuable resources both in culinary practices and in pharmaceutical practices. They are recognized as healthy food with low

contents of calories and fats but high in vitamins, minerals, and vegetable proteins. Their suitability for use by the pharmaceutical industry is given by their rich antioxidant chemical constituents that are capable of preventing the human body from oxidative damage [22]. It is also known that mushrooms could be considered as good bioindicators for the evaluation of environmental pollution, since they are known to accumulate a broad range of chemical compounds [23]. The aim of this study was to propose a numerical procedure which estimates the highest accumulation rate (*R*) of a chemical compound on the entire anatomical compartments of a mushroom body. Such data could lead to improvement in both food quality assurance and environment safety assessment. Analytical assessments on mushroom samples have shown that the accumulation potential of chemical compounds varies with mushroom species and varieties and also varies between the same mushroom anatomical compartments

Numerical Modeling of Chemical Compounds' Fate and Kinetics in Living Organisms: An Inverse…

http://dx.doi.org/10.5772/intechopen.76611

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(see **Figure 2**) as well between mushroom development stages ("age").

**numerical method for rate estimation from concentration**

*dz*[*p*(*z*)(*WC* <sup>−</sup> *ET*¯

the diameter [mm], *p* is the porosity, *χ* is the hydration coefficient, *Wc*

factor, *ET*¯ is the evaporation-transpiration coefficient, *Upf*

concentration data of the target chemical compound [ng⋅g−1].

into three concentric subintervals with regard to diameter.

using smoothing spline functions [27, 28].

on the following differential equation:

*dz* <sup>−</sup> \_\_\_*<sup>d</sup>*

*<sup>d</sup>*(*p*(*z*)*C*(*z*)) \_\_\_\_\_\_\_\_\_\_\_

**5. Solving modality path selection and motivation: the inverse** 

Inverse problems are extremely frequent in interdisciplinary science subjects. A large scale of mathematical and numerical techniques for solving scattering problems as well as other inverse problems usually exist [24]. These methods are often very different from the methods used for solving direct problems due to the differences in mathematical structure and input data [25].

In our study, the estimation process of the chemical compound accumulation rate was built

) *dC*(*z*) \_\_\_\_\_

where the elements which may affect the rates are given as follows: *z* is the height [mm], Φ is

cumulation factor, *R* is the rate of accumulation for the chemical compound of interest, *C* is the

The rate estimation model had as a starting point the one-dimensional transport-reaction equation for dissolved compounds presented by Lettmann et al. [26]. In this chapter the same type of equation was used but this time the equation is based on the main factors that can influence in some way the assimilation rate of a chemical compound in a vegetal organism specifically in a mushroom body. Our model was supported by concentration data (*C*) of the target compounds, which were measured in the laboratory from cross-sections taken at every 2 mm over the whole body of the studied mushroom species. Also, the concentration measurements correspond to cross-sections taken at every 2 mm, and each section was divided

The goal of the first step is to approximate *R* using the left-hand side of Eq. (1). The approximation of differential operators from the left side of the proposed equation has been solved

*dz* ] <sup>+</sup> *<sup>p</sup>*(*z*) *Upf*(*C*(*z*) <sup>−</sup> *Baf*) <sup>=</sup> *<sup>R</sup>* (1)

is the uptake factor, *Baf*

the is saturated hydration

is the bioac-

**Figure 2.** Chemical compound concentration variations in different anatomical compartments**:** (*a*) concentration variation in the first anatomical compartment (basal bulb) of the mushroom; (*b*) concentration variation in the second anatomical compartment (stipe) of the mushroom; (*c*) concentration variation in the third anatomical compartment (cap) of the mushroom.

contents of calories and fats but high in vitamins, minerals, and vegetable proteins. Their suitability for use by the pharmaceutical industry is given by their rich antioxidant chemical constituents that are capable of preventing the human body from oxidative damage [22]. It is also known that mushrooms could be considered as good bioindicators for the evaluation of environmental pollution, since they are known to accumulate a broad range of chemical compounds [23]. The aim of this study was to propose a numerical procedure which estimates the highest accumulation rate (*R*) of a chemical compound on the entire anatomical compartments of a mushroom body. Such data could lead to improvement in both food quality assurance and environment safety assessment. Analytical assessments on mushroom samples have shown that the accumulation potential of chemical compounds varies with mushroom species and varieties and also varies between the same mushroom anatomical compartments (see **Figure 2**) as well between mushroom development stages ("age").
