*2.2.2 Leaky aquifer*

Aquifers that are fully unconfined or confined appear less often than aquifers that are leaky, or semi-confined. This is a common occurrence of plains, alluvial valleys, or former lake basins where a semi-pervious aquitard or semi-confining bed is underlain or overlain by a permeable layer. Pumping water from a well into a leaky aquifer eliminates water in two directions such as the vertical flow into the aquifer through the aquitard and the horizontal flow in the aquifer.

## *2.2.3 Fractured aquifer*

The fractured rock aquifers vary from the subsurface water systems that are stored in the geological formation. Although sedimentary aquifers hold and move a significant amount of water between specific sedimentary granules through pore spaces, however, fractured rock aquifers hold and move water in an otherwise impermeable rock mass through as cracks, joints, and fractures (**Figure 2**).

Therefore, fractured rock aquifers have hydraulic characteristics that vary from those found in sedimentary aquifers with accessible water (common in terms to be described as bore yield) and are typically defined by nature (opening, size, and extent) and degree of interconnection between discontinuities in the rock mass. The longterm yield from well in fractured rock aquifers depends on the location of the degree of discontinuity and the relationship of discontinuities in the total mass of the rock instead of on the permeability of the geological substances near the extraction phase. The aquifers in fractured rock typically depend on the amount of precipitation that caused the surface water runoff of which considerably greater than in flat regions. Moreover, permeability fractured rock aquifers can also be dramatically decreased by the weight of the overlying rock mass as open spaces progressively decrease between fractures and cracks.


**Leelyn Chong, Pengcheng Guo and Yingfang Zhu \***

State Key Laboratory of Crop Stress Adaptation and Improvement, School of Life Sciences, Henan University, Kaifeng 475001, China<EMAIL_ADDRESS>(L.C<EMAIL_ADDRESS>(P.G.)

**\*** Correspondence<EMAIL_ADDRESS>
Received: 25 September 2020; Accepted: 19 October 2020; Published: 20 October 2020

**Abstract:** As an evolutionarily conserved multi-protein complex, the Mediator complex modulates the association between transcription factors and RNA polymerase II to precisely regulate gene transcription. Although numerous studies have shown the diverse functions of Mediator complex in plant development, flowering, hormone signaling, and biotic stress response, its roles in the Abscisic acid (ABA) signaling pathway and abiotic stress response remain largely unclear. It has been recognized that the phytohormone, ABA, plays a predominant role in regulating plant adaption to various abiotic stresses as ABA can trigger extensive changes in the transcriptome to help the plants respond to environmental stimuli. Over the past decade, the Mediator complex has been revealed to play key roles in not only regulating the ABA signaling transduction but also in the abiotic stress responses. In this review, we will summarize current knowledge of the Mediator complex in regulating the plants' response to ABA as well as to the abiotic stresses of cold, drought and high salinity. We will particularly emphasize the involvement of multi-functional subunits of MED25, MED18, MED16, and CDK8 in response to ABA and environmental perturbation. Additionally, we will discuss potential research directions available for further deciphering the role of Mediator complex in regulating ABA and other abiotic stress responses.

**Keywords:** Mediator complex; transcription; ABA signaling; abiotic stress response


**4. The practicality of computer-aided mathematical model generation**

I will now turn to the question of whether model generation is practical. Numerous examples of my work in brain research provide evidence that formal generation of mathematical models has an attractive place in knowledge discovery. I will highlight oxygen exchange and blood flow in the aging brain to show how mathematical models can serve as an instrument of knowledge discovery. The mathematical modeling paradigm we propose for the brain is based on model generation from medical images, and we term it anatomical model generation. I demonstrate this in an example of the generation of the mathematical model for the mouse as well as the synthesis of vascular trees in humans. The guiding principle of generating mathematical models shown in **Figure 5** is the convergence of medical images into mathematical model representations. The schematic outlines this process. For instance, in this case study of the mouse brain, two-photon microscopy was used to acquire the anatomy of the brain cells and blood vessels in a large section of the primary somatosensory cortex of the mouse. My lab then used image segmentation to create vectorized image data to create an inventory of all the blood vessels and the cells capturing their precise location, diameter, and connectivity. This vectorized image data were encoded into a network graph using adjacency matrices to store connections

**29**

**Figure 6.**

*representation of vectorized data using graphs (fight).*

*Mathematical Modeling: The Art of Translating between Minds and Machines…*

of nodes with arcs and property vectors for Cartesian dimensions, diameters, and sizes. Based on this image-derived domain representation, we have generated an anatomically concise network topology of the primary somatosensory cortex of the mouse. **Figure 4** shows examples of four different sample sections, which are digital representations of the same cortical region of the brain of four different animals. Once we have generated the topological representation, in this case the anatomy of the cerebral cortex, we can automatically generate the equations from this network. The network representation of the mouse brain enables the computer to perform the

task of synthesizing transport equations using a set of rules automatically.

oxygen extraction at an unprecedented level of detail as is shown in **Figure 8** for four samples of the primary somatosensory cortex. The diagram shows with

*Information flow for anatomical model generation from medical image. Two photon image acquisition of the mouse brain in an open cranial window (left), vectorization of multiscale data (middle) and network*

**Figure 6** depicts the phenomenological description of the model generation methodology. Biphasic blood flow equations for mass conservation as well as a simplified momentum equation can be generated automatically for each node of the vascular network.  It is also very well known that red blood cells in the microcirculation do not travel uniformly between the branches of the microcirculation but essentially tend to concentrate in vessels with higher flow and larger diameter. Therefore, we need a biphasic representation of blood flow which was implemented with a very simple drift flux model of biphasic blood flow [16] depicted in **Figure 6**. This kinematic model is equivalent to a mixture with species of different volatilities; thus, red blood cells tend to divide into the thinner or thicker branch of a bifurcation as a function of their relative kinematic affinity (volatility) described by an empirical drift flux parameter, m. This simple hematocrit split rule allows us to predict the uneven distribution of red blood cells; the descriptive equations of the biphasic drift flux models are again automatically synthesized. Additional modeling details instantiate equations for oxygen transport to brain tissues, expressed by molar flux balances for red blood cells and oxygen unbinding from hemoglobin into plasma according to dissociation kinetics. The entire set of these complex model equations fit on a single piece of paper as shown in **Figure 7**. The anatomical modeling approach enables the automatic generation of system equations from information encoded in the network topology. Once these network equations are automatically generated, the numerical solution of these equations on the computer yields predictions of microcirculatory blood flow patterns and

*DOI: http://dx.doi.org/10.5772/intechopen.83691*

#### *Mathematical Modeling: The Art of Translating between Minds and Machines… DOI: http://dx.doi.org/10.5772/intechopen.83691*

of nodes with arcs and property vectors for Cartesian dimensions, diameters, and sizes. Based on this image-derived domain representation, we have generated an anatomically concise network topology of the primary somatosensory cortex of the mouse. **Figure 4** shows examples of four different sample sections, which are digital representations of the same cortical region of the brain of four different animals. Once we have generated the topological representation, in this case the anatomy of the cerebral cortex, we can automatically generate the equations from this network. The network representation of the mouse brain enables the computer to perform the task of synthesizing transport equations using a set of rules automatically.

**Figure 6** depicts the phenomenological description of the model generation methodology. Biphasic blood flow equations for mass conservation as well as a simplified momentum equation can be generated automatically for each node of the vascular network.  It is also very well known that red blood cells in the microcirculation do not travel uniformly between the branches of the microcirculation but essentially tend to concentrate in vessels with higher flow and larger diameter. Therefore, we need a biphasic representation of blood flow which was implemented with a very simple drift flux model of biphasic blood flow [16] depicted in **Figure 6**. This kinematic model is equivalent to a mixture with species of different volatilities; thus, red blood cells tend to divide into the thinner or thicker branch of a bifurcation as a function of their relative kinematic affinity (volatility) described by an empirical drift flux parameter, m. This simple hematocrit split rule allows us to predict the uneven distribution of red blood cells; the descriptive equations of the biphasic drift flux models are again automatically synthesized. Additional modeling details instantiate equations for oxygen transport to brain tissues, expressed by molar flux balances for red blood cells and oxygen unbinding from hemoglobin into plasma according to dissociation kinetics.

The entire set of these complex model equations fit on a single piece of paper as shown in **Figure 7**. The anatomical modeling approach enables the automatic generation of system equations from information encoded in the network topology. Once these network equations are automatically generated, the numerical solution of these equations on the computer yields predictions of microcirculatory blood flow patterns and oxygen extraction at an unprecedented level of detail as is shown in **Figure 8** for four samples of the primary somatosensory cortex. The diagram shows with

#### **Figure 6.**

*Information flow for anatomical model generation from medical image. Two photon image acquisition of the mouse brain in an open cranial window (left), vectorization of multiscale data (middle) and network representation of vectorized data using graphs (fight).*

*Technology, Science and Culture - A Global Vision*

*batch operations and stream table for a batch recipe.*

BDK system was a model generation framework that used natural language input, synthesized equations, and created a virtual representation of materials and eventually solved these equations to predict the physiochemical state changes resulting from material transformations, phase separations, and reactions. The idea of using formal algorithms to support mathematical model generations has received attention in the community and continues to be an interesting research task in system science.

*List of operational tasks in the BDK batch for natural language input of chemical recipes (right). Flowsheet of*

**4. The practicality of computer-aided mathematical model generation**

I will now turn to the question of whether model generation is practical. Numerous examples of my work in brain research provide evidence that formal generation of mathematical models has an attractive place in knowledge discovery. I will highlight oxygen exchange and blood flow in the aging brain to show how mathematical models can serve as an instrument of knowledge discovery. The mathematical modeling paradigm we propose for the brain is based on model generation from medical images, and we term it anatomical model generation. I demonstrate this in an example of the generation of the mathematical model for the mouse as well as the synthesis of vascular trees in humans. The guiding principle of generating mathematical models shown in **Figure 5** is the convergence of medical images into mathematical model representations. The schematic outlines this process. For instance, in this case study of the mouse brain, two-photon microscopy was used to acquire the anatomy of the brain cells and blood vessels in a large section of the primary somatosensory cortex of the mouse. My lab then used image segmentation to create vectorized image data to create an inventory of all the blood vessels and the cells capturing their precise location, diameter, and connectivity. This vectorized image data were encoded into a network graph using adjacency matrices to store connections

**28**

**Figure 5.**

$$0 = \nabla \cdot D\_{\mu \wr \rhd\_1} \cdot \nabla C\_{\mu \wr \rhd\_1 \smile \mu\_{\rho \wr}} \cdot \omega\_{\rho \wr} \cdot \nabla C\_{\mu \wr \rhd\_1} + \dot{R}\_{\rho \smile \nu^p} - S\_b \frac{U}{\varkappa \sigma\_b} \left(C\_{\mu \wr \rhd\_1} - C\_{B \boxtimes \heartsuit \rhd\_1}\right)$$

**31**

**Figure 9.**

*(capillaries).*

*Mathematical Modeling: The Art of Translating between Minds and Machines…*

in the anatomical vessel hierarchy. Our research shows that the hematocrit distribution, the hydrostatic pressure, and red blood cell saturation all experience large variability as a function of the different pathways red blood cells can take to traverse the network. These findings are nonintuitive and have been revealed with the help of the anatomical mathematical model. Key findings include that hemodynamic states in the microcirculation are not uniform, but that the tissue is evenly oxygenated, and that pressure drop occurs mainly in the capillary evenly [17]. None to these findings were previously predicted by prior models that did not offer the fine-grained level of anatomical detail

**5. Growing circulatory trees to explore cortical oxygen transport in the**

I attempt to address how oxygen is supplied to the human brain. In humans, it is not possible to access microcirculation data through open cranial windows as was shown for rodents; rather, a noninvasive approach was needed. My lab successfully deployed a model generation methodology to overcome this limitation. In humans, we used a modified constructive constrained optimization algorithm [18] originally developed by Wolfgang Schreiner for the synthesis of coronal arterial networks [19]. Schreiner randomly added segments to a main coronal arterial tree and determined the optimal segment location in the tree hierarchy, its coordinates, and segment diameters by minimizing the vascular tree volume subjected to flow conditions. Remarkably, when sequentially repeating the process of random segment addition followed by deterministic optimization, a tree emerges whose topology resembles natural vasculature. This discovery suggests that in nature, vascular trees grow in a manner that perfuses capillaries evenly, while at the same time, the segment diameters as well as locations of bifurcations are chosen so that the total required blood filling volume is at a minimum. We have modified this original algorithm to generate vascular structures for very complicated organs such as the brain. Our modified algorithm is versatile and is capable of delineating vascular structures in quite complicated domains. The example in **Figure 9** shows the initials of my laboratory (lppd-laboratory for product and process design) literally painted in blood. Each letter constitutes a physiological vascular tree that discharges the exact same amount of flow through its terminal nodes (capillaries). We have successfully used vascular synthesis to generate cerebrovascular models for rodents as well as for humans that are virtually indistinguishable from real vascular structures. Specifically, we made a computer-generated anatomical model of human microcirculation. **Figure 10** depicts a comparison between the synthetic vasculature structure and a real sample. Using an artificially generated human cortical structure, we were able to predict oxygen exchange in humans at a length scale that has not been acquired

*Demonstration of vascular growth in a complex domain. The example shows a synthetic blood vessel networks delineated the initials of the laboratory for product and process design-lppd. Each letter constitutes a physiological vascular tree that discharges the exact same amount of flow through its terminal nodes*

*DOI: http://dx.doi.org/10.5772/intechopen.83691*

shown in this research.

**human brain**

#### **Figure 7.**

*Overview of the equation generation mechanism for cortical blood flow and oxygen exchange mechanism.*

#### **Figure 8.**

*Predictions of hemodynamic states in microcirculation of mouse, blood pressure, hematocrit, and red blood cell (RBC) oxygen saturation.*

microscopic detail the distribution of red blood cells, blood oxygen saturation, the uneven distribution of hematocrit, and the patterns of blood pressure for any capillary and its surrounding tissue in the mouse cortex. The computer-generated mathematical model allows analysis at an unprecedented scale, down to the detail of individual cells or capillaries. The model predicted that the blood pressure is not uniform, but there are large deviations of hemodynamic states along different paths traversing the microcirculatory network. Previously, it was believed that in the capillary bed there are representative average conditions of pressures or oxygen saturation as a function of level *Mathematical Modeling: The Art of Translating between Minds and Machines… DOI: http://dx.doi.org/10.5772/intechopen.83691*

in the anatomical vessel hierarchy. Our research shows that the hematocrit distribution, the hydrostatic pressure, and red blood cell saturation all experience large variability as a function of the different pathways red blood cells can take to traverse the network. These findings are nonintuitive and have been revealed with the help of the anatomical mathematical model. Key findings include that hemodynamic states in the microcirculation are not uniform, but that the tissue is evenly oxygenated, and that pressure drop occurs mainly in the capillary evenly [17]. None to these findings were previously predicted by prior models that did not offer the fine-grained level of anatomical detail shown in this research.


**7. Filtration**

*New Advances on Fermentation Processes*

production [16].

fermentation [2].

yeast [17].

for changes in beer quality [2].

**6. Maturation**

Continuous fermentation systems, based on immobilised cells, were condemned to failure for several reasons. Engineering problems like excess biomass, problems with CO2 removal, optimisation of operating conditions, clogging and channelling of the reactor, unbalanced beer flavour, altered cell physiology and cell ageing lead to unrealised cost disadvantages such as high carrier prices at complex and unstable operations [16]. Pilot-plant and full industrial-scale processes showed engineering problems. The carrier material, the reactor design, together with the effect of immobilisation on yeast physiology, and the risk of contamination end up in a hardly predictable flavour profile of the beer produced. Therefore, despite the economic advantages expected, the continuous process has so far been industrially applied only in beer maturation and alcohol-free beer

The crucial step forward in continuous technology was certainly the development of commercial immobilised yeast reactors. This approach was of sufficient interest to form the subject of an entire European Brewing Convention Symposium "Immobilised Yeast Applications in the Brewing Industry" held in Finland in 1985 [17, 18]. The advantage of immobilised reactors is that very high yeast concentrations are achievable. This allows a very rapid process throughput which is of particular benefit when applied to rapid beer maturation. A single immobilised yeast reactor can eliminate the time-consuming warm conditioning step for diacetyl reduction at the end of a lager beer

The application of gel occlusion systems in the brewing process, even if associated with many advantages over conventional fermentation technology, has some important drawbacks, particularly diffusional limitations which impact negatively on yeast growth, metabolic activity and beer flavour, Masschelein et al. concluded at EBC Congress in 1984 [18]. Nakanishi et al. recognised that fermentation activity in continuous working fermenters fell gradually during continuous operation of the system. It could be maintained for 2 months by periodic aeration in which 290 mg/g-yeast (dry matter) of oxygen was supplied to the immobilised

Continuous fermentation suits best in breweries making only one style of beer, because its time and capacity consume to stop the process and start up again with a new beer [14]. Immobilised yeast reactors have also found use in new fermentation processes, for example, in the production of low-alcohol or alcohol-free beers [2], where yeast has more clarification tasks than fermenting and propagation. The major strength of the batch system, using several vessels, is that it is able to cope with seasonal or shorter-term fluctuations in demand. It can easily be adapted to vary the spectrum of production of several different beer varieties and qualities. On the other hand, benefits of continuous fermentation are realised when the systems are operating at a stable status for a long period of time with minimum downtime

Some of these yeast metabolism byproducts (vicinal diketones, acetaldehyde, dimethyl sulphide) impart undesirable flavours to the green beer. The main aim of maturation is to reduce the concentration of such unfavourable flavour compounds in the green beer, to saturate the final beer with CO2 and to remove the haze-forming components from beer within 7–30 days [16]. Fumigation with CO2 under counter-pressure to avoid too much foam may strip the unwanted flavour. The flavour can be removed from the CO2 with active carbon so that the CO2 may be

**74**

Filtration and stabilisation of the beer are carried out in order to achieve microbial, colloidal and flavour stability so that no visible changes occur for a long time and the beer looks and tastes the same as when it was made [16]. Particular for higher amounts of yeast cells, tangential flow or crossflow filters are good prefilters before flash pasteurisation or membrane filtration. Although batch flushes can extend the continuous filtration of a crossflow filter, fouling layers will clog the membranes. A chemical recovery of the filter modules is necessary; the continuous process has an end. Usually bright beer tanks collect a batch for the final quality control, and they are the buffers for the following filling of the beer in bottles, cans, kegs or road tankers.