*Article* **Black Sorghum Phenolic Extract Modulates Platelet Activation and Platelet Microparticle Release**

### **Borkwei Ed Nignpense 1, Kenneth A Chinkwo 1,2, Christopher L Blanchard 1,2 and Abishek B Santhakumar 1,2,\***

Received: 20 May 2020; Accepted: 10 June 2020; Published: 12 June 2020

**Abstract:** Platelet hyper-activation and platelet microparticles (PMPs) play a key role in the pathogenesis of cardiovascular diseases. Dietary polyphenols are believed to mimic antiplatelet agents by blunting platelet activation receptors via its antioxidant phenomenon. However, there is limited information on the anti-platelet activity of grain-derived polyphenols. The aim of the study is to evaluate the effects of sorghum extract (Shawaya short black 1 variety), an extract previously characterised for its high antioxidant activity and reduction of oxidative stress-related endothelial dysfunction, on platelet aggregation, platelet activation and PMP release. Whole blood samples collected from 18 healthy volunteers were treated with varying non-cytotoxic concentrations of polyphenol-rich black sorghum extract (BSE). Platelet aggregation study utilised 5 μg/mL collagen to target the GPVI pathway of thrombus formation whereas adenine phosphate (ADP) was used to stimulate the P2Y1/P2Y12 pathway of platelet activation assessed by flow cytometry. Procaspase-activating compound 1 (PAC-1) and P-selectin/CD62P were used to evaluate platelet activation- related conformational changes and degranulation respectively. PMPs were isolated from unstimulated platelets and quantified by size distribution and binding to CD42b. BSE treatment significantly reduced both collagen-induced platelet aggregation and circulatory PMP release at 40 μg/mL (*p* < 0.001) when compared to control. However, there was no significant impact of BSE on ADP-induced activation-dependent conformational change and degranulation of platelets. Results of this study suggest that phenolic rich BSE may confer cardio-protection by modulating specific signalling pathways involved in platelet activation and PMP release.

**Keywords:** black sorghum; polyphenols; platelets; platelet microparticles; atherosclerosis

### **1. Introduction**

According to a World Health Organisation report, cardiovascular diseases accounted for an estimated 31% of deaths globally with majority being a result of stroke or heart attack [1,2]. In clinical settings, treatment involves blunting the activity of platelets using antiplatelet drugs. These drugs interfere with the thrombotic pathophysiology—wherein a rupture of an atherosclerotic plaque triggers platelet hyper-activation resulting in unwanted clot formation and occlusion of the blood vessel. Macrovesicles referred to as platelet microparticles (PMPs) are released following platelet activation and can contribute to the thrombotic situation [3,4].

The several signalling pathways involved in platelet activation and thrombus formation include receptor-agonist pathways such as P2Y1/P2Y12-ADP, GPVI-collagen, PAR1-thrombin and the COX-1-thromboxane [5]. An agonist such as collagen when exposed by atherosclerotic plaque may activate nearby platelets by binding to their GPVI receptor resulting in complex intracellular signalling

that produce a conformational change (indicated by GPIIb/IIIa receptor expression), degranulation (indicated by P-selectin secretion) and subsequent formation of platelet aggregates [5]. In addition, PMP released upon activation possess adhesive and pro-coagulant platelet-derived receptors that further enhance thrombus formation, thereby acting as biomarkers of platelet activation [3]. The common antiplatelet agents, clopidogrel and aspirin, used in clinical treatments inhibit platelet activation and its circulating biomarkers by selectively targeting P2Y1/P2Y12-ADP and COX-1-thromboxane respectively [6]. Unfortunately, because of the unresponsiveness and side effects associated with administration there have been considerable research in dietary bioactive agents known as polyphenols [7].

One such example of a polyphenol-rich functional food is sorghum whole grain. Although mainly used as animal feed, studies have demonstrated that it possesses anti-inflammatory, anti-cancer and antioxidant properties which add value to it as a food for human consumption [8]. Sorghum of different types exist that are classified based on the pigmentation of the pericarp and vary in their phenolic content [9]. The polyphenols found in sorghum that contribute to its bioactivity include flavonoids, hydroxybenzoic acids and hydroxycinnamic acid [8]. Furthermore Francis et al. [10] recently demonstrated that black sorghum rich in catechins and their derivatives may confer cardioprotective properties. The treatment of human umbilical vein cells with flavonoid-rich extract was found to prevent oxidative stress-related endothelial dysfunction through the modulation of gene expression.

Furthermore, these cardio-protective benefits of polyphenols apply in the context of platelet function. Several studies have demonstrated that polyphenols may inhibit platelet activation, adhesion, degranulation and aggregation by targeting specific thrombogenic pathways for example P2Y1/P2Y12-ADP, GPVI-collagen, PAR1-thrombin and the COX-1-thromboxane. As reviewed by Ed Nignpense et al. [11] many of the studies that investigate the polyphenol impact on platelet function and PMP generation utilise aggregometry and flow cytometry. However there is limited research on sorghum-derived polyphenols in modulating biomarkers of platelet activation. This study aims to investigate the impact of black sorghum derived polyphenol extracts on collagen-induced platelet aggregation, ADP-induced platelet activation and PMP generation.

### **2. Materials and Methods**

### *2.1. Research Ethics*

The study protocol was approved by the Charles Sturt University Human Research Ethics Committee (HR17012) and the Institutional Biosafety Committee (19HB02). The study was performed in compliance with relevant laws and institutional guidelines.

### *2.2. Volunteer Recruitment*

Eighteen healthy volunteers between 18–65 years of age (9 males and 9 females) were recruited from Charles Sturt University and the local community. Informed consent was obtained from all participants prior to commencement of the study. The criteria for recruitment involved normal health status with no history of conditions such as cardiovascular, metabolic, liver or lung disease. Other parameters that could affect the integrity of the analysis such as alcohol consumption, smoking, pregnancy, allergies or venepuncture difficulty were considered during the recruitment process. A health screening questionnaire was used to assess the already mentioned parameters. A dietary questionnaire (adapted from WINTEC and NZ academy of sport) was used to assess the usual dietary intake of volunteers and to avoid recruitment of participants on a high antioxidant diet. The cut-off figure for each type of food listed in the questionnaire was based on nutrient reference ranges for Australia and New Zealand—recommended daily nutrient intake values.

### *2.3. Blood Collection and Processing*

After fasting for at least 8 h, whole blood was collected from each participant by a trained phlebotomist into a tri-potassium ethylene diamine tetra-acetic acid (EDTA-1.8 mg/mL concentration) anticoagulant tube (Vacuette Greiner Bio-one, Interpath Services, Heidelberg West, VIC, Australia) and a tri-sodium citrate (28.12 g/L concentration) anticoagulant tube (Becton, Dickson and Company, North Ryde, NSW, Australia). A 20-mL syringe (Becton, Dickson and Company, North Ryde, NSW, Australia) and 21-gauge 1.5-inch needle (Terumo Medical Corporation, Macquarie Park, Australia) were used to draw blood from the median cubital vein. The purpose of choosing a larger gauge was to avoid the activation of platelets while drawing or dispensing blood. Utmost care was taken to ensure samples were not obtained through a traumatic collection and that none contained obvious clots. In addition, care was taken to ensure minimal specimen handling and agitation in order to prevent artefactual platelet activation. The first 2 mL of blood was discarded before drawing into the tri-sodium citrate tubes in order to avoid the risk of collecting platelets activated by venepuncture. Tri-sodium tubes were used for aggregometry and flow cytometry assays whereas the EDTA was used to perform full blood examinations.

### *2.4. Full Blood Examination*

Using an Abbott CELL-DYN Emerald 22 Haematology Analyser (Abbott Diagnostics, Illinois, USA), a full blood examination (FBE) was performed on all samples. The FBE results of volunteers indicated that the blood cell parameters were within normal reference ranges as determined by the Royal College of pathologists of Australia. Individuals with cell counts, especially platelet counts, outside of the reference range were excluded from the study. Quality control validation and maintenance were all performed according to the Abbott CELL-DYN Emerald 22 Haematology Analyser manual.

### *2.5. Extraction of Black Sorghum Polyphenols*

Sorghum (*Sorghum bicolor*) samples of six different pericarp varieties were obtained from glasshouse trials conducted by Curtin University, Perth, Western Australia. Six pigmented varieties of sorghum were cultivated under the same conditions, grown in a glasshouse equipped with Lumisol Clear AF cover (200 μm thick, at a transparency of ca UV-A 94%, UV-B 84% and photosynthetic active radiation (PAR, 400-700 nm) 93% [8]. Extraction and analysis of phenolic composition and antioxidant activity were performed previously using methods described by Rao et al. [8]. Among the different sorghum varieties, the black pericarp variety (Shawaya short black 1) was selected for this study because of its relatively high antioxidant activity when analysed with ferric reducing antioxidant power (FRAP; 20.19 ± 2.69 mg/g TE) and 2,2-dipheny-1-picrylhydrazyl (DPPH; 18.04 ± 3.53 mg/g TE) antioxidant assays (Supplementary Figure S1, Tables S1 and S2). The highest level of polyphenols found in the BSE included catechin derivatives, catechins and pentahydroxyflavanone-(3->4)-catechin-7-O- glucoside (Supplementary Table S2). Stock concentrations of BSE (20 mg/mL in 50% DMSO) were diluted in phosphate buffered saline (PBS) to achieve desired concentrations (5 μg/mL, 20 μg/mL and 40 μg/mL) in whole blood. Desired concentrations were selected based on viability studies done by Francis et al. [10].

### *2.6. Whole Blood Platelet Aggregometry*

Platelets in whole blood were stimulated for aggregation using 5 μg/mL collagen exogenous platelet agonists (DSKH Australia Pty. Ltd., Hallam, VIC, Australia) to investigate the effect of BSE treatment on the platelet aggregation. Five hundred microliters of citrated whole blood were added to 100 μL of 0.1% DMSO control (Sigma-Aldrich, Castle Hill, NSW, Australia) or BSE stock concentrations (5 μg/mL, 20 μg/mL and 40 μg/mL) and mixed with 400 μL of saline. The sample was then incubated at 37 ◦C for 20 min. Using a Chrono-log model 700 lumi-aggregometer (DKSH Australia Pty. Ltd., Hallam, VIC, Australia) the samples were analysed by means of electrical impedance (ohms) to determine the amount of platelet aggregation occurring in the sample over a 6-min time period (Supplementary Figure S2).

### *2.7. Flow Cytometry*

### 2.7.1. Standardisation

Flow-check fluorospheres were run as quality control for optimal laser alignments. Antibody capture beads (Anti-Mouse Ig, K CompBeads, BD Biosciences, North Ryde, NSW, Australia) were used for single colour compensation controls in order to achieve optimal compensation. Megamix beads (0.1 μm, 0.3 μm, 0.5 μm, 1 μm) from Biocytex, Marseille, France were used as per manufacturer's instructions to set up an appropriate gating to detecting microparticles. They were run before each PMP analysis.

### 2.7.2. Measurement of Platelet Activation-Dependent Conformational Change and Degranulation

The effects of BSE on ADP-induced platelet activation were analysed and interpreted using a Gallios flow cytometer (Beckman Coulter, Inc., Lane Cove NSW, Australia). The protocols were adopted from the method described by Santhakumar et al. [12] with some modifications. Platelet activation and thrombogenic indicators were assessed via activation-dependent platelet monoclonal antibodies (mAbs) purchased from Becton, Dickinson and Company, North Ryde, NSW, Australia. Procaspase activating compound-1 (PAC-1)-fluorescein isothiocyanate-fluorescein isothiocyanate was used to detect platelet activation-related conformational change and P-selectin/CD62P-allophycocyanin highlighted activation dependent degranulation. CD42b-phycoerythrin identified the GPIb-IX-V receptor, a common receptor found on the surface of all platelets, activated and resting included. A decreased expression of mAb exhibits alleviation of thrombogenesis. Within 5 min of collection tri-sodium citrated whole blood was used for assay preparation to avoid artefactual activation of platelets. A volume of 40 μL of blood was incubated with DMSO control or the various BSE concentration for 20 min at 37 ◦C in the dark. A 10-μL mixture of all three monoclonal antibodies (3.33 μL each of PAC-1, CD62P and CD42b) was added to blood samples and incubated for 20 min at room temperature in the dark. To induce platelet activation, 10 μM ADP (Helena laboratories Pty. Ltd., Mt Waverly, VIC, Australia) was added, and samples were incubated for a further 15 min in the dark at room temperature, after which erythrocytes were lysed (575 μL of 10 % lysing solution). Samples were thoroughly vortexed to ensure homogeneity and incubated in the dark at room temperature for a further 15 min and then analysed. In all, 10,000 platelet events were acquired, gated based on light scatter and CD42b mAb expression and activated platelets were articulated as mean fluorescence intensity (MFI) (Supplementary Figure S3).

### 2.7.3. Measurement of Circulatory PMPs

Using the microparticle gating established with Megamix beads, PMPs were identified and quantified by size distribution and binding to CD42b (Supplementary Figure S4). The protocol for circulatory PMP analysis was adapted from Lu et al. [13]. A volume of 1 mL whole blood was added to micro-centrifuge tubes in the presence of PGE1 (120 nmol/L; Sigma-Aldrich, Castle Hill, NSW, Australia). PGE1 was added to prevent artefactual activation during centrifugation. The blood samples were incubated with the respective 100 μL BSE concentrations and DMSO control for 20 min at 37 ◦C in the dark. Each sample was then centrifuged for 15 min at a 1000 rpm and the resultant platelet rich plasma (PRP) was discarded. The remaining blood was spun a further 15 min at 3000 rpm. The supernatant rich in PMPs (40 μL) was collected into flow tubes and incubated with 4 μL of CD42b and 6 μL of stain buffer (Becton, Dickson and Company, North Ryde, NSW, Australia) in dark room for 15 min. Four percent formaldehyde was used to fix any activation of platelets left in the supernatant. After a 10-min incubation period the samples were run for PMP analysis on the flow cytometer.

### *2.8. Statistical Analysis*

A two-way ANOVA following Tukey's post comparison test was performed using GraphPad Prism version 8.0 for Windows (GraphPad Software, La Jolla, California, USA). A minimum sample size of 14 participants in total was required for 80% power to detect a 5% variation in the laboratory parameters measured where a 3–5% standard deviation exists in the population, assuming an alpha error of 0.05. All the data were expressed as mean ± standard deviation (SD). Differences between the groups were significant when *p* < 0.05. Any significant statistical interactions were included in the analysis where applicable.

### **3. Results**

The baseline parameters including full blood counts for all 18 participants were within normal reference ranges set by the Royal College of Pathologists of Australasia (Table 1) [14].

**Table 1.** Baseline full blood counts of participants.

Values are represented as mean ± Standard deviation (SD). RBC, red blood cell, PCV, packed cell volume, MCV, mean cell volume, MCH, mean cell haemoglobin, MCHC, mean cell haemoglobin concentration, RDW, red cell distribution width, MPV, mean platelet volume.

### *3.1. E*ff*ect of BSE on Whole Blood Platelet Aggregation and Platelet Activation*

BSE at 40 μg/mL concentration significantly reduced platelet aggregation stimulated by collagen by 19 % (*p* = 0.0004) (Figure 1). BSE at lower concentrations did not exhibit any significant reduction in aggregation.

It was observed that whole blood treatment with the varying concentrations of BSE did not significantly affect ADP-induced platelet conformational change and degranulation indicated by PAC-1 and P-selectin expression respectively (Supplementary Figures S5 and S6).

### *3.2. E*ff*ect of BSE on Circulatory PMPs*

BSE at a concentration of 40 μg/mL significantly reduced the amount of circulatory PMPs in whole blood by 47% (*p* = 0.0008). Lower concentrations of BSE did not exhibit any significant reduction to the amount circulatory PMPs (Figure 2).

**Figure 1.** The effect of varying concentrations of BSE on collagen-induced aggregation. BSE at 40 μg/mL significantly reduced platelet aggregation (5.3 ± 1.3; *p* value = 0.0004). BSE at 5 μg/mL and 20 μg/mL did not reduce platelet aggregation when compared to control (*p* value > 0.1). N = 18 and data is represented in aggregation (Ohms) versus BSE concentrations. \*\*\* signifies statistical significance *p* < 0.001 compared to control. Error bars expressed as mean ± SD.

**Figure 2.** The effect of varying concentrations of BSE on circulatory PMP production in vitro. BSE at 40 μg/mL significantly reduced the amount of circulatory PMPs (<24190 ± 4935, *p* = 0.0008). BSE at 5 μg/mL and 20 μg/mL did not reduce platelet aggregation when compared to control (*p* value > 0.1). N = 14 and data are represented in number of PMP events versus BSE concentrations. \*\*\* signifies statistical significance *p* < 0.001 compared to control. Error bars expressed as mean ± SD.

### **4. Discussion**

There is growing interest in understanding the therapeutic benefits of functional foods. Sorghum for example is one of the functional foods that is showing promise in this area. With sorghum-derived polyphenols already having demonstrated anti-inflammatory, anti-cancer and antioxidant properties, the current study aimed to evaluate the effects of polyphenol-rich BSE on platelet function in terms of aggregation, conformational change, degranulation and circulatory PMP production [8–10]. It was observed that BSE significantly inhibited collagen-induced platelet aggregation and decreased the release of circulatory PMPs but did not have a significant effect on ADP-induced platelet conformational change or degranulation. Although these results do not reflect a typical dose-dependent inhibition, they suggest a potential role of BSE polyphenols at optimum concentrations to interfere with pathways in the GPVI-collagen signalling and the release of circulatory PMPs but little or no effect on P2Y1/ P2Y12-ADP pathway.

To the best of our knowledge only a few studies have investigated the antiplatelet effects of sorghum extracts. Li, Yu and Fan et al. [15] extracted alditols and monosaccharides from sorghum vinegar to evaluate their anti-aggregation activity using the turbidimetric method. Results from their study indicated a significant dose-dependent inhibition of aggregation via multiple agonists, arachidonic acid, collagen, ADP and thrombin. Furthermore, a different study by Fan et al. [16] reported in vitro inhibition of ADP- and thrombin- induced rabbit platelet aggregation by methanolic extracts of aged sorghum vinegar with the half maximal inhibitory concentrations (IC50) of 1.7 ± 0.3 and 8.9 ± 1.9 mg/mL respectively. When rats were orally administered the extracts (>100 mg/kg), both collagen- and epinephrine-induced pulmonary thrombosis were inhibited. In comparison with the present study it is to be noted that these studies employed platelet-rich plasma rather than whole blood hence not accounting for the possible involvement of other blood cells and extracellular mediators involved in thrombus formation. In addition, sorghum vinegar extracts were used at higher concentrations; milligrams compared to micrograms used in this study. This raises the question of bioavailability and the importance of employing physiological concentrations of extracts.

Although the BSE concentration of 40 μg/mL at which antiplatelet effect were observed is relatively lesser in concentration than used in the other studies, the most bioactive compounds with respect to antioxidant activity were catechins and other flavonoids which are usually considered to have low bioavailability [8]. It has been suggested that the total plasma polyphenol concentration rarely exceeds 1 μM and that their antiplatelet effects are only found at high non-physiological concentrations (greater than 50 μM) [17,18]. However, it is likely that these plasma concentrations are underestimations due to the ability of polyphenols to bind to the surface of red blood cells and thereby exert their bioactivity [19]. Furthermore polyphenols (structurally related to catechin) and their metabolites have been shown to inhibit platelet function in vitro [20]. This highlights the possibility of sorghum catechins and their metabolites having antiplatelet effects in vivo despite bioavailability concern. Interestingly, an in vivo human dietary intervention trial compared consumption of red and white wholegrain sorghum-based pasta to a control pasta in order to investigate its acute effect on the total phenol content and antioxidant activity in the plasma of healthy subjects [21]. Results showed that when compared to the control pasta, the red sorghum pasta showed significantly increased net plasma phenolic content and antioxidant activity post consumption (from 216.90 ± 2.62 at baseline to 269.40 ± 2.33 at 2 h; *p* < 0.001), thus demonstrating a plausible correlation between antioxidant activity and sorghum polyphenol consumption—which in turn may contribute to antiplatelet effects.

The antiplatelet effects was observed with BSE-included inhibition of collagen-induced aggregation and circulatory PMP production but no effect on ADP-induced platelet activation. The absence of antiplatelet effects on the P2Y1/ P2Y12-ADP activation pathway suggests that BSE polyphenols are not mimicking the action of drugs such as clopidogrel that antagonise P2Y12 receptor activation [22]. However, the inhibition of collagen-induced aggregation suggest that BSE polyphenols interfere with GPVI-collagen signalling pathways by either blunting the GPVI receptor directly or by other mechanisms [5]. Previous studies have demonstrated that flavonoids, specifically quercetin and catechin, can act synergistically to inhibit collagen-induced aggregation by blunting the associated burst of H2O2 and subsequent PLC activation [23,24]. Thus, a possible mechanism of inhibition BSE flavonoids may be a synergistic antagonism of the positive feedback activation of intracellular signals triggered by H2O2. Moreover, it has been shown that the phosphorylation cascade initiated by collagen can be inhibited by flavonoids [25,26]. Flavones, especially apigenin and luteolin, by virtue of a double bond in the C2-C3 and the keto group in C4 can also inhibit collagen-induced activation by antagonizing the TxA2 receptor activation which is also involved in the positive feedback loop [27]. Besides inhibition of the GPVI-collagen signalling, BSE polyphenols showed inhibition of the circulatory PMP production.

To the best of our knowledge, this is the first study investigating the effect of sorghum-derived polyphenols on PMP production. In contrast to this study, other PMP studies have employed the use of Annexin V as well as the platelet specific antibody CD42b, to identify pro-coagulant PMPs by their phosphatidylserine expression and to limit background noise [28]. However, because of the heterogeneity of PMPs, not all PMPs express phosphatidylserine [29]. Moreover, the measurement of CD42b-positive PMPs alone is significant as its increase has been associated with an increased risk of coronary heart disease [30]. From the current study, the significant inhibition of CD42b-positive circulatory PMPs observed in vitro may be attributed to the antioxidant properties of BSE polyphenols. It is believed that the inhibition of PMP generation may be the result of neutralisation of H2O2, scavenging of other free radicals or interaction with intracellular signalling leading to PMP release.

The juxtaposition of both the present study and that of an earlier study by Francis et al. [10] highlights the multifaceted role of BSE polyphenols in cardio-protection. The group investigated the effects of BSE polyphenols on the expression of antioxidant- and inflammatory-linked genes involved in endothelial dysfunction under oxidative stress. Results indicated that BSE polyphenols alleviate oxidative stress–induced damage to endothelial cells. Since vascular dysfunction is a precursor to cardiovascular diseases, the current study builds upon earlier findings by exhibiting the antiplatelet effects of BSE. In the context of endothelial dysfunction, platelet activation and circulatory PMPs play central roles in the pathogenesis of atherothrombosis. The disruption of the plaque exposes collagen that binds to the GPVI receptor resulting in platelet activation and subsequent thrombus formation [31]. Circulatory PMPs may contribute to thrombosis via GPIb-IX-V receptor binding and have a pro-inflammatory effect to promote the development of the plaque [32]. Therefore, by reducing collagen-induced platelet aggregation and circulatory PMP generation, BSE polyphenols may be displaying the potential to augment thrombosis.

### **5. Conclusion and Future Considerations**

In summary, the present study contributes to the growing body of literature on bioactivity of sorghum polyphenols and highlights possible mechanisms of antiplatelet action that may result in cardiovascular health benefits. Because of the ability to reduce collagen-induced platelet activation and circulatory PMP generation, BSE polyphenols demonstrate the potential to interfere with pathological processes involved in vascular disorders and thrombotic complications. However, a larger panel of agonist for the flow cytometry and aggregometry studies will aid to further elucidate antiplatelet mechanisms. Because of the bioavailability concerns, well-controlled dietary intervention trials using larger sample sizes to evaluate the antiplatelet effects of sorghum consumption in healthy and pro-thrombotic populations are warranted to justify our findings. Because of the varied phenolic profiles of the different sorghum varieties, further research comparing the antiplatelet therapeutic potential of different grains is also warranted. Furthermore, this study attest to the measurement of circulatory PMPs as a biomarker of platelet activation to assess the bioactivity of functional foods.

**Supplementary Materials:** The following are available online at http://www.mdpi.com/2072-6643/12/6/1760/s1, Figure S1: Characterisation of phenolic compounds in BSE. Ultra-high-performance liquid chromatography (UHPLC) was employed to quantify the different phenolic compounds identified by the peak (on top). An online 2,2 -azino-bis(3-ethylbenzothiazoline-6-sulfonic acid) (ABTS) was coupled with UHPLC to quantify the relative antioxidant activity (peaks below) of each compound identified, Figure S2: A report derived from the collagen induced platelet aggregation study using the Chrono-log model 700 lumi-aggregometer (DKSH Australia Pty. Ltd, Hallam, VIC, Australia). The blue tracing represents the control (whole blood with no BSE) and the black tracing represents the whole blood pre-treated with 5 μg/mL BSE. The addition of BSE reduced the maximum platelet aggregation expressed in Ohms from 15 ohms to 11 ohms, Figure S3: A report from the ADP-induced platelet activation analysis using Kaluza Flow Cytometry Software (Beckman Coulter, Brea, CA, USA). Results indicate the gating of whole platelet population (CD42b positive events) and the proportion of activated platelets indicated by PAC-1 and P-selectin expression, Figure S4: A report from the PMP analysis using Kaluza Flow Cytometry Software (Beckman Coulter, Brea, CA, USA). Microparticle gating was established using Megamix beads of standard sizes. PMPs were distinguished from other microparticles by size (0.5 μm - 0.9 μm) and expression of CD42b. The number of CD42b positive events in the microparticle gate was used to quantify the PMPs, Figure S5: The effect of varying concentrations of BSE on PAC-1 expression. BSE did not significantly reduce ADP-induced platelet conformational change detected by PAC-1 expression (*p* values > 0.1 compared to control) N=14 and data is represented in mean fluorescence intensity (MFI) versus BSE concentrations. Error bars expressed as mean ± SD, Figure S6: The effect of varying concentrations of BSE on P-selectin expression. BSE did not significantly reduce ADP-induced platelet degranulation detected by P-selectin expression (*p* values > 0.1 compared to control ) N=14

and data is represented in mean fluorescence intensity (MFI) versus BSE concentrations Error bars expressed as mean ± SD, Table S1: Phenolic composition and antioxidant activity of sorghum varieties on as is basis, Table S2: List of top ten phenolic compounds identified in the black sorghum phenolic rich extracts by Q-TOF LC/MS and quantified using UHPLC-Online ABTS system (Adapted from Rao et al., 2018).

**Author Contributions:** B.E.N. conducted the experiments outlined in this study and drafted the manuscript. K.A.C., C.L.B. and A.B.S. were involved in the designing and critical reviewing of the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This study was funded by the Australian Research Council Industrial Transformations Training Centre for Functional Grains (Project ID 100737).

**Acknowledgments:** The authors would like to acknowledge Graham Centre for Agricultural Innovation for providing funding towards open access publication of this article. Borkwei Ed Nignpense is a recipient of a PhD Scholarship from the Australian Government Research Training Program and also a recipient of a top-up scholarship by the Graham Centre.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **Abbreviations**

### **References**

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

*Article*


*2.1. In Rice*

Rice is one of the most important food crops in the world. Investigating the genes controlling rice yield, quality, and stress resistance using genomics leads to significant improvement. The grain length of rice seed is a crucial trait determining rice yield. In contrast, japonica rice in the Heilongjiang region of China is significantly shorter than the indica rice varieties in southern China. Two japonica rice varieties, Pin20 and Songjing 15, with significant differences in grain shape, were used to map a new QTL *qGL9.1* controlling the grain length, which promotes the study of the molecular mechanism controlling the grain length of japonica rice and lays the theoretical foundation for japonica rice improvement in northern China [4]. The salt tolerance of rice is important to improve the adaptation and yield of rice in saline–alkali soil. Low phosphate root 5 (OsLPR5) and ferroxidase activity levels were expressed during salt stress. Then, the physiological role of *OsLPR5* in salt stress was demonstrated via overexpression and using CRISPR/Cas9-mediated mutation techniques. This study identified a novel molecular mechanism in which *OsLPR5* positively regulates salt tolerance in rice [5]. Perennial rice plays a vital role in protecting the ecological environment and alleviating the labor shortage of farmers. The development of axillary buds is a crucial trait of perennial rice (regenerative rice). It was reported that sucrose promotes the growth of axillary buds and rhizomes, which might be related to fructose and glucose. The increased osmotic pressure of the cells results in the absorption of water, which encourages cell growth and eventually leads to the development of axillary buds and rhizomes [6]. The domestication of wild rice is caused by genomic variation, including the synonymy codon usage bias (SCUB) due to synonymy nucleotide substitution. SCUB was affected in the same way during the process of domestication of Asian and African rice. The analysis of cytosine to thymine transformation mediated via SCUB and DNA methylation showed the close relationship between genetic variation and epigenetic variation in domestication [7]. *LEAF TIP RUMPLED 1* (*LTR1*) positively regulates the rice yield and salt tolerance. This study showed that *LTR1* maintained the salt tolerance of rice by mediating the activities of aquaporins and ion transporters. The research on *LTR1*

**Citation:** Huang, J.; Li, Z.; Zhang, J. Research on Plant Genomics and Breeding. *Int. J. Mol. Sci.* **2023**, *24*, 15298. https://doi.org/10.3390/ ijms242015298

Received: 22 September 2023 Revised: 7 October 2023 Accepted: 8 October 2023 Published: 18 October 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

provides a theoretical basis for studying salt tolerance and high rice yields [8]. NF-YCs family transcription factors, including seed development, play a key role in plant growth and development. *NF-YC8*, *9*, *10*, *11*, and *12* are homologous genes that are expressed at highly similarity levels in seeds. Systematic genetic experiments suggested that these five genes synergistically regulate GA and ABA responses, thus affecting the rice seed quality and germination [9].

#### *2.2. In Wheat*

Wheat is the world's largest cultivated and most widely distributed food crop. Unlike rice, wheat is a dryland crop. Therefore, it is important to improve wheat's drought tolerance. Two previously reported KASP markers, TaDreb-B1 and 1-FEH w3, were employed for the marker-assisted selection (MAS) of drought tolerance. These two molecular markers were used to detect drought tolerance in winter and spring wheat populations, and the correlation of TaDreb-B1 with drought tolerance was better than that of marker 1-FEH w3 [10]. Cytokinin is an important hormone that determines the wheat yield. Members of the *TaCKX* gene family (GFMs) encode the cytokinin oxygenase/dehydrogenase enzyme (CKX), which causes the irreversible degradation of cytokinin. Thus, these genes might regulate wheat yield. It has been speculated that it is feasible to select high-yielding wheat families using the *TaCKX* GFMs cross-generation expression pattern [11].

#### *2.3. In Maize*

Maize is the world's most productive food crop per unit area, reaching up to 30,000 kg per hectare. While abiotic stresses such as drought and salinization severely restrict maize yield, it is important to resolve the mechanism of maize flowering time and abiotic stress to improve maize yield and resistance. It has been reported that protein arginine methyltransferase (PRMTs) is mainly responsible for the histone methylation of specific plant arginine residues. Ling et al. found that the overexpression of the maize *ZmPRMT* gene in *Arabidopsis* can promote early flowering and heat tolerance. First, it was reported that *ZmPRMT1* gene regulates the flowering time and resistance to heat stress in plants, which will provide an essential theoretical basis to further reveal the functions of *ZmPRMT* gene and epigenetic regulation mechanism of maize growth and development and response to abiotic stress [12]. Corn stalks can be used as an animal feed, and sugar in the stalk is also a carbon source for corn seeds. Increasing the sugar content of corn stalks can not only promote the development of animal husbandry, but also increase the yield. Chen et al. investigated 188 waxy, sweet, and hybrid corn resources with different stalk sugar contents for GWAS analysis. The results showed that the expression levels of six candidate genes were significantly different among the other stalk sugar-containing materials, which provides a significant insight into the genomic footprints of stalk sugar content in fresh corn and facilitates the breeding of corn cultivars with a higher stalk sugar content [13].

### **3. Genome Research in Cash Crops**

The production of cash crops (like cotton, soybean, rapeseed, *Brassica oleracea,* etc.) plays a decisive role in industry development, especially the light industry, which is also the main source of exports, foreign exchange earnings, and increasing national economic income.

#### *3.1. In Oil Crops*

Soybean sterile line SXCMS5A is an H3A cytoplasmic male sterile line (CMS) developed from JY20. Bai et al. conducted the transcriptome analysis of the sterile line SXCMS5A and the maintainer line SXCMS5B to find the differentially expressed genes (DEGs) and the metabolic pathways related to pollen sterilization. These findings might provide useful information to facilitate soybean hybrid breeding [14]. Excellent root development is an important factor for plants' high nitrogen-use efficiency (NUE). Ahmad et al. revealed 16 genes involved in rapeseed root development under low nitrogen (LN) stress through

the integrated analysis of GWAS, a weighted gene co-expression network, and DEGs. Seven of these genes have been previously reported to be associated with root development and NUE. This study provides genetic/SNPs resources for studying low nitrogen tolerance in rapeseed [15].

#### *3.2. In Horticultural Crops*

Khusnutdinov et al. found that the DNA binding structure of transcription factor *MYBL2-1* (a negative regulator of anthocyanin synthase *Dihydroflavonol-4-reductase* (*DFR*)) of purple varieties of *Brassica oleracea* L. had two SNP mutations, leading to an increase in *DFRs* expression. This is the reason for the high anthocyanin content of purple cabbage [16]. *Ipomoea aquatica* is a leafy vegetable rich in essential amino acids, flavonoids, and various mineral elements (calcium, potassium, phosphorus, etc.). The role of *IabHLHs* family transcription factors in the biosynthesis of anthocyanins in *I. aquatica* was analyzed using bioinformatics combined with molecular experiments. This work provides valuable clues to further explore *IabHLH*'s function and facilitating the breeding of anthocyanin-rich varieties of *I. aquatica* [17]. Improving the cold tolerance of herbaceous peonies (*Paeonia lactiflora*) is essential to plant them at low latitudes. Wang et al. modified the previously reported multi-criteria decision-making (MCDM) model. This model was implemented to analyze 15 peony varieties at different latitudes. As a result, 'Meiju', 'Hang Baishao', 'Hongpan Tuojin', and 'Bo Baishao' were excellent varieties with strong environmental adaptability. This research can provide a reference for the breeding and cultivating of perennial herbs [18]. NAC transcription factors are crucial in plant growth, development, and stress responses. Zhao et al. found that *AeNAC83* positively regulates the salt tolerance of okra (*Abelmoschus esculentus*). Exactly as the expression of *AeNAC83* gene is up-regulated in okra under salt stress, the down-regulation of *AeNAC83* is caused by virus-induced gene silencing enhanced plant sensitivity to salt stress [19]. Lotus (*Nelumbo nucifera*), which is in the Nelumbonaceae family, is a relict plant with crucial scientific research and economic values. Such as the sequencing of the lotus genome and the assembly of several high-quality genomes, the investigation of lotus functional genome has been extensively promoted. The resequencing of natural and genetic populations and different levels of genomic studies benefit the classification of different lotus germplasm resources and the identification of genes controlling various traits [20].

#### *3.3. In Other Crops*

Liu et al. analyzed the genetic basis of senescence in cotton using 355 upland cotton accessions planted in multiple environments for GWAS. From the candidate genes that have been linked, *GhMKK9* silencing improves the drought resistance of cotton, whereas *GhMKK9* overexpression accelerates senescence in *Arabidopsis* [21]. *Areca catechu* is a commercially important medicinal plant widely cultivated in tropical regions. Zhou et al. investigated 12 *natural resistance-associated macrophage protein* (*NRAMP*) genes from the whole genome of *A. catechu* with Fe and Zn deficiency, and their sequence characteristics, gene structure, phylogeny, promoter sequence, and collinearity were analyzed. This research revealed the regulatory response of *NRAMP* family genes in *A. catechu* under Fe and Zn deficiency stress [22]. Yao et al. treated tartary buckwheat (*Fagopyrum tataricum* L.) leaves with different concentrations of H2O2, and then analyzed the growth, photosynthesis, antioxidant enzyme activity, and related gene expression under salt stress. This study explored the mechanism of H2O2 enhancing salt tolerance of Tartary buckwheat from a physiological point of view [23].

#### **4. Conclusions and Perspectives**

In summary, we reviewed 20 plant genomics articles on different plant species. These functional gene studies consider the yield, quality, fertility, abiotic stress, and evolution of rice, wheat, maize, and cash crops. These researchers have made outstanding contributions to plant genomics and provided valuable genetic resources for crop breeding. Based on these genetic resources controlling different traits, gene-editing methods, such as CRISPR-cas9 technology, can alter gene expression to improve the crop yield, quality, and resistance. Finally, strengthening the interface between the application and industrialization of functional genomics research would help to accurately quantify traits, such as the crop yield, stress resistance, quality, and nutrient content. Meanwhile, promoting more efficient and precise crop breeding is expected to bring about new transformations in the global agricultural field.

**Author Contributions:** All the authors participated in the editing of this Research Topic. J.H. wrote the draft, and all the other authors provided suggestive comments on the Editorial. All authors have read and agreed to the published version of the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**

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**2.5 Bronchus**

172 Front Lines of Thoracic Surgery

Fig. 2. Our studies on 101 angiograms showed the wider the main to branch PA angle, the better PA size. Narrow L/R PA angle resulted in bilateral PA hypoplasia in the upper panel. Thus wider angle facilitates bilateral PA growth before ASO. Any maneuver that did not correct this inborn error of ventriculoarterial discordance or make even narrower angle like

The aortic arch is kept wide open by the presence of the MPA bifurcation below it (Chiu et al., 2000b; Chiu et al., 2002b). Neo-aortic kinking is the narrowing of the aortic arch window and may become slit-like, can also be called aortic neocoarctation (Figure 3, Chiu et al., 2010; Muster et al., 1987). One group reported an incidence of 0.54% (Serraf et al., 1995). A coarctation is not present and the arch is wide open before conventional ASO with Lecompte maneuver. A pressure gradient across it is rare, less seen than is supravalvular PS

Fig. 3. After conventional arterial switch with Lecompte maneuver, a slit like arch window

can be seen in the right panel, which was wide open before switch.

Lecompte maneuver will compromise PA growth.

because of systemic-pressure in aorta.

**2.4 Aorta**

The aorta is not the only structure behind the MPA bifurcation after Lecompte maneuver, the left main bronchus (airway-pressure with cartilage) is also present. Bronchial compression or even atelectasis has been reported after Lecompte maneuver (Robotin et al., 1996; Toker et al., 2000), although in the majority of cases the bcronchial patency was not so severely compromised.

Fig. 4. After conventional ASO with Lecompte maneuver, not only supra-aortic stenosis occurs at the site posterior to the anteriorly mobilized pulmonary bifurcation (\*), but also the left main brobchus, that is cartilage stented, is ccompreesed; its lumen becomes pin hole like on brochoscope, both virtual and real. In the right lower panel, the aortic arch window becomes slit-like at lesser curvature site.

Restoration of Transposed Great Arteries With or Without Subpulmonary Obstruction to Nature 175

Fig. 5. (continued) anastomosis site bleeder can be drained into PAs. Nonfacing sinus was cut open for exposure during coronary transfer. (C) The transparent ascending aorta, with a left aortic lip (procured by oblique amputation of aorta, Chiu et al., 2000b), was rotated counterclockwise to sit on the neoaortic stump. The posterior cut-edge of MPA could be attached to the posterior neoaorta at this stage as leftward as possible.(D) The RPA orifice must be visible behind the aorta to ensure patency of RPA, not just probed with a Hegar dilator. A big enough MPA flap could be reattached to the neoaorta without pericardial patch. (Ao = Aorta, L or R = Left or Right portion of nonfacing sinus wall after cut back, LCA = Left Coronary Artery, LPA =Left Pulmonary Artery, RCA = Right Coronary Artery,

No-fault transfer of the coronary artery is the cornerstone of a successful ASO. Various techniques (Aubert et al., 1978; Yacoub and Randly Smith, 1978; Kurosawa et al., 1986; Quaegebeur et al., 1986; Brawn and Mee, 1988; Idriss et al., 1988; Bove et al., 1989; Takeuchi and Katogi, 1990; Mee, 1994; de Leval et al., 1994; Murthy & Cherian, 1996; Chiu et al., 1996a & 1997) have been proposed to achieve this goal: de Leval et al. (1994) pointed out that the key point is to take the aorta away from the coronary arteries and the MPA is brought to them, rather than moving the coronaries from the aorta and transferring the coronary scallops to the MPA or neoaorta. To implement this concept, *in situ* transfer technique, proposed by Aubert et al. (1978) and Takeuchi and Katogi (1990), was developed by Murthy and Cherian (1996). This principle is the key (Figure 5) we adopted to redirect the neoaortic outflow tract to the coronary arteries as *in situ* as possible by a common wall concept. In this way the coronary redirection can be achieved more securely than the conventional

*In situ* transfer and the common wall concept not only for the coronary arteries, but also for both great arteries are our guiding principle. We have designed two pedicled grafts, aortic lip and MPA flap, to achieve our goal. The purpose of these two flaps is described below. The aortic lip (Figures 2, 3A and 3B) is initially called left lip (Chiu et al., 2002b). In our first spiral arterial switch, we used a free flap taken from MPA to cover the left-sided portion of the neoaortic stump (Chiu et al., 2000b), to solve the size discrepancy when connecting small aorta to a very large, original MPA stump; the other effect is to act as the floor of the neopulmonic pathway. Since the nonfacing aortic sinus was cut back (Figure 5B and 5C) for exposure during coronary transfer, later we used an aortic pedicled graft including the nonfacing aortic sinus wall (Figure 6A and 6B) to achieve the above 2 purposes of the aortic lip. In addition, using the aortic lip taken from **anterior** sinus wall to cover the **left-sided** portion (Figure 6B and 6C, light blue arrow), we can achieve the effect of counterclockwise rotation of the ascending aorta. This, in turn, would give way to the RPA so that it can sling around the ascending aorta (Figure 6C, Hegar dilator in RPA). The fourth effect of the aortic lip might lessen the chance of coronary kinking by avoiding take-up the larger stump on neoaortic anastomosis, as noted by Quaegebeur et

The purposes of the MPA flap include: (1) accommodating the ascending aorta, (2) achieving a more leftward shift of RPA, and (3) attaching to the ascending aorta without pericardium. Thus, the spiral relationship of the great arteries is resumed by using the

autologous pedicled flaps tailored from both arterial trunks, respectively.

RPA = Right Pulmonary Artery)

"coronary transfer".

al. (1986).


**1. Introduction**

Darcy's law [1] establishes that the water flow in porous media is proportional to the hydraulic gradient; the proportionality coefficient is denoted hydraulic conductivity (K). The law, discovered in the context of water flow in saturated soils, has since been generalized to flow in unsaturated soil [2]. In saturated soils, conductivity is independent of water pressure, whereas in unsaturated soils it is a highly nonlinear function of pressure (ψ), or volumetric water content (θ) [3–8].

The saturated hydraulic conductivity, denoted KS, is at most a function of spatial coordinates. In unsaturated soils the hydraulic conductivity is a function of water pressure K(ψ) as well as spatial coordinates. In such cases the soil-water retention curve θ(ψ) is needed to relate volumetric water content to soil-water pressure. The two curves θ(ψ) and K(ψ) are known as the soil hydrodynamic characteristics and are important to the study of mass and transfers such as infiltration, drainage and evaporation, and groundwater recharge [2,9–12].

The aim of the present work is to establish relationships between the soil-water retention curve and hydraulic conductivity curve, using concepts of probability theory and fractal geometry in order to reduce the amount of unknown functions in the unsaturated soil zone. Soil here is considered as a set of Lebesgue measure different than zero. This does not consider sets which porosity is a unity, such as Menger sponge, which may not represent natural soil.

#### **2. Materials and Methods**

#### *2.1. A General Model of Hydraulic Conductivity*

A conceptual model for the hydraulic conductivity based on Poiseuille law of water flow in capillary tubes has been proposed at the literature [13–18]. The model has the general form:

$$\mathbf{K} = \mathbf{f} \mathbf{C}\_{\mathbf{f}} \int\_{\Omega} (\mathbf{R} / \mathbf{T})^2 \mathbf{d}w \tag{1}$$

where Ω represents the water flow area divided by total area of the exposed face [13]; f = ρw g/η is the fluidity, ρw is water density, η is the dynamical viscosity coefficient, g is the gravitational acceleration; R is the pore radius; the non-dimensional coefficient (Cf) take in account the irregular shape of the pore perimeter, for a circular pore Cf = 1/8; if R is taken as the hydraulic radius then Cf is called Koseny coefficient [19] with Cf = 1/2 for a circle; T is the tortuosity factor defined as T = dzf/dz ≥ 1, where z is the rectilinear path of water particles following macroscopic direction of the movement and zf is the actual path of water particles [20]; ω is the water flow effective area relative to total area of soil or partial effective areal porosity.

The effective area definition is established by Fuentes et al. [14–18] from the probabilistic idea of Childs and Collis–George [21] and fractal geometry concepts. After a perpendicular cut to macroscopic trajectory of water we obtain two faces, which are located at z and z + dz positions (Figure 1); the radii of pores of the z-face are denoted by r and those of the z + dz-face are denoted by ρ. A water particle in a pore of z-face can continue its trajectory by the same pore or by other pore of equal or different radius. The introduction of the joint probability of the two faces at intermediate point z + 1/2 dz allows the modeling of these possibilities.

**Figure 1.** A cross section in the perpendicular macroscopic direction of the water flow. In the z-face the different pore radius are represented by r and in the z + dz-face the different pore radius are represented by ρ, where dz is the pore size order.

We consider a completely saturated soil. Childs and Collis–George [21] assume that the pore size distribution f(r) is the same at both faces and that dθ(r) = f(r)dr is the water content in the pore interval containing r, (r − 1/2 dr, r + 1/2 dr), and dθ(ρ) = f(ρ)dρ is the water content in the pore interval containing ρ, (ρ − 1/2 dρ, ρ + 1/2 dρ), at the other face. The completely random joint probability of all the pores represented by these intervals is equal to the product of these probabilities, provided that dz is of the order of a pore size and less than a characteristic particle size. The product of dθ(r) and dθ(ρ) represents the flow effective area dω(r,ρ) = dθ(r)dθ(ρ) = f(r)dr f(ρ)dρ, which integration over whole pore domain gives the total effective flow area μ = φφ = φ2, where μ represents also a total effective areal porosity and φ the total volumetric porosity. In a parallel capillary system, a water

particle moves always in the same capillary and the effective area is equal to volumetric porosity dω(r) = dθ(r) = f(r)dr, which integration gives μ = φ [22].

In probabilistic terms, the Purcell model represents a complete correlation between the two faces, whereas the Childs and Collis–George model represents a complete decorrelation, these models represent the possible extreme behaviors. Mualem and Dagan [13] show that the Purcell model can be formally deduced from the Childs and Collis–George model if the flow effective area is defined as dω(r, ρ) = f(r)dr δ(ρ − r) dρ, where δ indicates the Dirac delta.

An intermediate approach between the Purcell and Childs and Collis–George models is proposed by Millington and Quirk [23]. These authors suppose that, if the areal porosity at each face is φs, then the effective areal porosity at intermediate point is μ = φsφs where s = 1/2 represent the Purcell model and s = 1 the Childs and Collis–George model. Further, since φ2s ≤ φ, Millington and Quirk proposed to add the larger solid area (1 − φ) s to φ2s, in order to obtain the total area, that is (1 − φ) s + φ2s = 1. In most studies about soil structure, it is studied as a fractal object [16,17,24–28]. In this context, we'll offer an interpretation of the relationship (1 − φ) s + φ2s = 1.

In Euclidean geometry we have V ∝ L3 and A ∝ L2, where V, A, and L are the volume, area and length in an object, respectively; for example in a sphere V = 4/3πr3, A = 4πr2, L = r; since L ∝ V1/3 we have A ∝ V2/3. Using the Mandelbrot [29] area–volume relationship in fractal geometry we have A ∝ VD/E, where D is the particle surface fractal dimension or the particle-pore interface fractal dimension and E = 3 is the Euclidean dimension of the space where the object is embedded.

In a cross section, the total area is the sum of the solid cross-sectional area and the pore cross-sectional area. If μs = 1 − μ is the solid cross-sectional area relative to the total area (call it the areal solidity), and φs = 1 − φ is the solid volume relative to soil volume (call it volumetric solidity), then μs = φs s, or 1 − μ = (1 − φ) s, where s = D/E is the fractal dimension relative to Euclidean dimension. Following the probabilistic idea, the areal and volumetric porosities relationship resulting is μ = φsφs = φ2s; s = 1/2 corresponds to Purcell model and s = 1 to Childs and Collis–George model. Notice that the solid area takes the exponent s rather than 1-s; this is because μs = φs s is Hausdorff measure of the solid phase. The exponent 1-s is important in the calculus of the parallel body volume of a fractal [30].

The r-parallel body of a set F is defined by Pr(F) = x ∈E : x − y ≤ r, y ∈ F where E is the E-cartesian product of real numbers and represents soil (both pores and solids), F is the solids and E − F the pores. Clearly, F is contained in Pr(F). The volume of a parallel body is obtained as the product of the cover set's volume, crE where c is a form coefficient (c = 1 if all covers are parallelepipeds), and the number of covers; therefore: volE(Pr) = NrcrE. Considering that Nr ∝ r−D when r → 0, we obtain volE(Pr) ∝ rE−D. It must be noted that the body parallel volume of the solids is not the same as the porous volume; it would, however, be the same if porosity tends to unity and if F were dense in E.

Since μs + μ = 1, the relationship between the relative fractal dimension and the total volumetric porosity is defined implicitly by the equation:

$$(1 - \phi)^{s} + \phi^{2s} = 1\tag{2}$$

where s = D/E. It can be shown that μ ≤φ, s→1/2 when φ→0, and s→1 when φ→1, in other words 1/2 < s < 1 when 0 < φ < 1.

From mass additive property we deduce the relationship ρt = ρs φs + ρv φ, where ρt is the total density of soil, ρs the density of solids, and ρv the density of pores; if this last one is considered null, we get the classic φ = 1 − ρt/ρs formula for estimating porosity, where ρt becomes the total density of dry soil. The comparison with Equation (2) allows us to deduce that ρs/ρt = φs−1 s and ρv/ρt = φ2s−1. The first could also be written as ρs/ρt = φs−1 s /φs, which is the quotient of Hausdorff measure and Lebesgue measure of solids. The second one can be interpreted as the quotient of the pores Hausdorff measure and the pores parallel body volume if we write ρv/ρt = φs/φ1−s. It can be shown that φs ≤ φ1−s s / and φ ≤ φ1−s.

Clearly the relation s(φ) defined by Equation (2) cannot by applied, for instance, to Menger sponge, where φ= 1 and s = log20/3log3 - 0.9089; in other words, this relationship cannot be applied to any abstract sets where Lebesgue measure is zero (φs = 0) and s < 1. Because relation s(φ) established a one-to-one relationship between porosity and the fractal dimension of solid pore interphase, it may not work for certain types of soils. However, their implications in the modeling of hydraulic conductivity are being investigated.

For the modeling of the hydraulic conductivity of unsaturated soils we accept the classic hypothesis that the water is contained in saturated pores with radius r, where 0 < r < R, for a given water content θ(R). Consequently, the effective area of flux, or partial area porosity, is the generalization of μ = φs φs:

$$\mathbf{d}\,\omega(\mathbf{r},\boldsymbol{\uprho}) = \mathbf{d}\theta^{\mathbf{s}}(\mathbf{r})\mathbf{d}\theta^{\mathbf{s}}(\boldsymbol{\uprho})\tag{3}$$

As concerns the tortuosity factor, it was demonstrated in Fuentes et al. [17] that R is the pore radius measured perpendicularly to actual trajectory of the water particles (zf) and Rs is its projection in the macroscopic direction (z), consequently Rs/R = dz/dzf = 1/T. According to the probabilistic idea we have Rs ∝ Rs Rs = R2s, which we can write as an equality Rs/Rso = (R/Ro) 2s with Rso = Ro/To, where Ro is a reference radius and To the associated tortuosity factor. The pore radius-tortuosity factor relationship resulting is:

$$\text{T(R)} = \text{T}\_{\text{o}} \text{(R}\_{\text{o}}/\text{R)}^{\text{\textdegree}} \tag{4}$$

where 0 < δ = 2s − 1 < 1.

The general hydraulic conductivity model results of the introduction of the Equations (2) and (4) in the Equation (1):

$$\mathbf{K} = \mathbf{f} \frac{\mathbf{C}\_{\mathbf{f}}}{\mathbf{T}\_{\mathbf{o}}^{2} \mathbf{R}\_{\mathbf{o}}^{2(2s-1)}} \int\_{\Omega} \mathbf{R}^{4s}(\mathbf{r}, \boldsymbol{\rho}) \mathbf{d}\theta^{s}(\mathbf{r}) \mathbf{d}\theta^{s}(\boldsymbol{\rho}) \tag{5}$$

The saturated hydraulic conductivity (Ks) is obtained from Equation (5), replace Ω by the total pore domain ΩT.

#### *2.2. Classical Models of Hydraulic Conductivity*

Classic models reported in literature may be deduced from the proposed general model, provided that certain hypotheses are introduced. From Equation (3) we can deduce dω(r, ρ) = s2[θ(r)]s−1 [θ(ρ)]s−1 dθ(r)dθ(ρ), with 0 < r < R and 0 < ρ < R. Assuming that the multiplicative function of the θ(r) and θ(ρ) differentials can be replaced by a medium value, which clearly depends on a superior limit, we have:

$$\mathbf{d}d\boldsymbol{\omega}(\mathbf{r},\boldsymbol{\rho};\mathbf{R}) = \left[\boldsymbol{\theta}(\mathbf{R})\right]^{2s-2} \mathbf{d}\boldsymbol{\theta}(\mathbf{r})\mathbf{d}\boldsymbol{\theta}(\boldsymbol{\rho})\tag{6}$$

where the s2 term has been eliminated to satisfy:

$$\int\_{0}^{\mathbb{R}} \int\_{0}^{\mathbb{R}} \text{d}\omega(\mathbf{r}, \mathfrak{p}; \mathbb{R}) = \omega = \mathfrak{e}^{2s} \tag{7}$$

Likewise, tortuosity will depend only on the major radius, that of θ(R). We know that for a small R θ(R) = φ(R/Ro) λ, where Ro is the radius of a reference pore and λ > 0 is an index of pores (Brooks and Corey, 1964). From Equation (4) we obtain that:

$$\mathrm{T}(\mathbb{R}) = \mathrm{T}\_{\mathrm{o}} \Big[ \frac{\Phi}{\theta(\mathbb{R})} \Big]^{\mathrm{y}} \text{ with } \mathrm{y} = \frac{\delta}{\lambda} \tag{8}$$

In several articles [24,25] has been suggested that partial volumetric porosity is proportional to the volume of the parallel body, which means that θ(R)∝RE-D, where the fractal dimension D is estimated from soil-water retention curve, a result that's valid when porosity tends to unity. If that's the case, we get:

$$
\lambda = \mathcal{E} - \mathcal{D} \tag{9}
$$

Considering Equations (6) and (8), Equation (1) becomes:

$$\mathbf{K} = \mathbf{f} \frac{\mathbf{C}\_{\mathbf{f}}}{\mathbf{T}\_{\mathbf{o}}^{2}} \boldsymbol{\Phi}^{2s-2} \begin{bmatrix} \boldsymbol{\Phi} \\ \boldsymbol{\Phi} \end{bmatrix}^{\mathbb{P}} \int \left[ \mathbf{R}(\mathbf{r}, \boldsymbol{\rho}) \right]^{2} \mathbf{d}\boldsymbol{\theta}(\mathbf{r}) \mathbf{d}\boldsymbol{\theta}(\boldsymbol{\rho}) \text{ with } \mathbf{p} = (2s - 2) + (2\chi) \tag{10}$$

In the power p, the first addend represents the global effects of the correlation among pores, whereas the second represents the global effects caused by the tortuosity of flow trajectories.

For each relation between the effective radius R(r,ρ) and the radii r and ρ we can deduce a special model of the hydraulic conductivity. We will obtain four models corresponding to four relations R(r,ρ) utilized in models of the hydraulic conductivity reported at the literature: (i) 'small pore' model R(r,ρ) = min(r,ρ) used by Childs and Collis–George [21]; (ii) 'geometric pore' model R(r, ρ) = √rρ used by Mualem [31]; (iii) 'neutral pore' model R(r,ρ) = either r or ρ corresponding to Burdine (1953) model; and iv) 'large pore' model R(r,ρ) = max(r,ρ) used by Fuentes [14]. Following the general indications of Brutsaert [32] for the integration of the Equation (10), the resulting special models are:

Small pore model: R(r, ρ) = min(r, ρ).

$$\mathbf{K}(\boldsymbol{\theta}) = \mathbf{f} \frac{2\mathbf{C}\_{\mathbf{f}}}{\mathbf{T}\_{\mathbf{o}}^{2}} \boldsymbol{\Phi}^{2s-2} \left[\frac{\boldsymbol{\Phi}}{\boldsymbol{\Phi}}\right]^{\mathbf{P}} \int\_{0}^{\boldsymbol{\Phi}} [\boldsymbol{\Phi} - \boldsymbol{\Phi}] \mathbf{r}^{2} \mathbf{d}\boldsymbol{\Phi} \tag{11}$$

This is the model of Childs and Collis–George [21], with a correction factor [θ/φ] p. Geometric pore model: R(r, ρ) = √rρ.

$$\mathbb{K}(\boldsymbol{\theta}) = \mathbf{f} \frac{\mathbf{C}\_{\text{f}}}{\mathbf{T}\_{\text{o}}^{2}} \boldsymbol{\Phi}^{2s-2} \left[\frac{\boldsymbol{\Theta}}{\boldsymbol{\Phi}}\right]^{\mathbb{P}} \left[\int\_{0}^{\boldsymbol{\Theta}} \mathbf{r} \,\mathrm{d}\,\boldsymbol{\theta}\right]^{2} \tag{12}$$

This model presents the structure of Mualem's model [31] with p = 1/2. Neutral pore model: R(r, ρ) = either r or ρ.

$$\mathbf{K}(\boldsymbol{\theta}) = \mathbf{f} \frac{\mathbf{C}\_{\mathbf{f}}}{\mathbf{T}\_{\mathbf{o}}^{2}} \boldsymbol{\Phi}^{2\mathbf{s}-1} \left[\frac{\boldsymbol{\Theta}}{\boldsymbol{\Phi}}\right]^{\mathbb{P}+1} \int\_{0}^{\boldsymbol{\Theta}} \mathbf{r}^{2} \mathbf{d} \,\mathbf{s} \tag{13}$$

This model presents the structure of Burdine's model [33] with p = 1. Large pore model: R(r, ρ) = max(r, ρ).

$$\mathbf{K}(\boldsymbol{\theta}) = \mathbf{f} \frac{2\mathbf{C}\_{\mathbf{f}}}{\mathbf{T}\_{\mathbf{o}}^{2}} \boldsymbol{\Phi}^{2s-2} \left[\frac{\boldsymbol{\Theta}}{\boldsymbol{\Phi}}\right] \int\_{0}^{\boldsymbol{\Theta}} \mathbf{r}^{2} \boldsymbol{\Phi} \,\mathrm{d}\boldsymbol{\Phi} \tag{14}$$

Childs and Collis–George [21] suppose that the resistance to the flow is determined by the small pore. Whereas Fuentes [14] proposes that, to deduce the opposite behavior, the conductance is determined by the pore of greater size [17]. Mualem [31] gives a little more weight to the large pore by proposing the geometric mean. The Burdine model is deduced when the pores sizes have the same weight.

The classic models are reported for relative hydraulic conductivity [Kr(θ)] and they are in function of the retention curve ψ(θ). These are obtained from Equations (11)–(14) with the rule Kr(θ) = K(θ)/K(θs), and the introduction of the Young–Laplace–Jurin law for the capillary rise phenomena:

$$
\psi = -\frac{\ell\_\mathrm{L}^2}{\mathrm{R}} \cos(\alpha\_\mathrm{c}) \tag{15}
$$

where the scale or capillary number (L) is defined by L = 2σ/ρwg [34], σ is the interfacial tension, L - 0.386 cm at 20 ◦C; αc is the contact angle formed between the air–water interface and the solid particles, assumed generally constant and equal to zero.

Classic models also consider the residual water content θr, defined by Brooks and Corey [35] as K(θr) = 0. This can be incorporated in the precedent models replacing θ by the effective water content θef = θ − θr, and φ by the effective volumetric porosity φef = φ − θr. The exponent s must be calculated by replacing φ with effective porosity (φef) in Equation (3); θr is added to solid particles, i.e., φsef = 1 − φef = φs + θr. In the classical conductivity models, the porosity φ is replaced by the volumetric water content to natural saturation θs, when entrapped air is considered.

The power p that appears in models (11)–(14) has been considered as an empiric parameter. This power (p = p1 + p2) is the result of the effects of correlation between pores, [θ(R)] p1 with p1 = 2s − 2, and tortuosity T2(R) = T2 o[φ/θ(R)] p2 with p2 = 2(2s − 1)/λ. To know the order of magnitude of the power p2 we assume that the particle surface fractal dimension is roughly equal to the fractal dimension estimated from the soil-water retention curve, i.e., from Equation (9) λ - 3(1 − s), hence p2 - 2(2s − 1)/3(1 − s). In consequence, the value of p may be estimated from porosity, some of which are shown in Table 1.

**Table 1.** Predicted values of the exponent p of the classical hydraulic conductivity models, which results from the effect of the pore correlation (p1) and the tortuosity factor (p2), for some values of the total volumetric porosity.

The approximate value of p ≈ 1/2 was obtained by Mualem [31] from the calibration of Equation (12) over the experimental data of 45 soils reported in different works, with total volumetric porosity in the range of 0.4 < φ < 0.7. According to Table 1, these soils may be represented by a soil with an average porosity of roughly 0.6.

#### **3. Results and Discussion**

#### *3.1. Some New Models of Hydraulic Conductivity*

We may obtain new hydraulic conductivity models from the general model established by Equation (5), without the introductions of those hypotheses established in Equations (6) and (8) to deduce the classic conductivity models, which may be restricted.

Each of the R(r,ρ) relationships mentioned above corresponds to a specific model of hydraulic conductivity. And again, following Brutsaert [32] for the integration of the Equation (5), the resulting special models are:

Small pore model: R(r, ρ) = min(r, ρ)

$$\mathbf{K}(\mathbf{R}) = \mathbf{f} \frac{2\mathbf{C}\_{\mathbf{f}}}{\mathbf{T}\_{\mathbf{o}}^{2}\mathbf{R}\_{\mathbf{o}}^{2(2s-1)}} \int\_{0}^{\mathbf{R}} [\boldsymbol{\Theta}^{s}(\mathbf{R}) - \boldsymbol{\Theta}^{s}(\mathbf{r})] \cdot \mathbf{r}^{4s} d\boldsymbol{\Theta}^{s}(\mathbf{r}) \tag{16}$$

Geometric pore model: R(r, ρ) = √rρ.

$$\mathbf{K}(\mathbf{R}) = \mathbf{f} \frac{\mathbf{C}\_{\mathbf{f}}}{\mathbf{T}\_{\mathbf{o}}^{2} \mathbf{R}\_{\mathbf{o}}^{2(2s-1)}} \left[ \int\_{0}^{\mathbf{R}} \mathbf{r}^{2s} \mathbf{d}\theta^{s}(\mathbf{r}) \right]^{2} \tag{17}$$

Neutral pore model: R(r, ρ) = r or R(r, ρ) = ρ.

$$\mathbf{K}(\mathbf{R}) = \mathbf{f} \frac{\mathbf{C}\_{\mathbf{f}}}{\mathbf{T}\_{\mathbf{o}}^{2} \mathbf{R}\_{\mathbf{o}}^{2(2s-1)}} \boldsymbol{\Theta}^{s}(\mathbf{R}) \left[ \int\_{0}^{\mathbb{R}} \mathbf{r}^{4s} \mathbf{d} \boldsymbol{\Theta}^{s}(\mathbf{r}) \right] \tag{18}$$

Large pore model: R(r, ρ) = max(r, ρ).

$$\mathbf{K}(\mathbf{R}) = \mathbf{f} \frac{\mathbf{2C\_f}}{\mathbf{T}\_o^2 \mathbf{R}\_o^{2(2s-1)}} \int\_0^\mathbf{R} \mathbf{r}^{4s} \theta^s(\mathbf{r}) \mathbf{d}\theta^s(\mathbf{r}) \tag{19}$$

Note that the Equations (17) and (18) can be generalized assuming R(r, ρ) = rαρ1−α, where 0 ≤ α ≤ 1; the Equation (17) follow with α = 1/2 and the Equation (18) with α = 0 or α = 1.

To obtain specific functions from new special models is necessary to provide the function θ(R). This can be obtained from the soil-water retention curve θ(ψ), relating the soil-water content to soil-water pressure, and the Laplace law, defined by the Equation (15).

From Equations (16)–(19) we can obtain the corresponding models to calculate the relative hydraulic conductivity from the retention curve:

$$\frac{\mathcal{K}(\Theta)}{\mathcal{K}\_{\mathsf{s}}} = \int\_{0}^{\Theta} (\Theta^{s} - \mathfrak{P}^{s}) \frac{\mathfrak{P}^{s-1}}{\left| \psi(\mathfrak{P}) \right|^{4s}} d\mathfrak{P} \Big/ \int\_{0}^{1} (1 - \mathfrak{P}^{s}) \frac{\mathfrak{P}^{s-1}}{\left| \psi(\mathfrak{P}) \right|^{4s}} d\mathfrak{P} \tag{20}$$

$$\frac{\mathcal{K}(\Theta)}{\mathcal{K}\_{\mathfrak{s}}} = \left[ \int\_0^{\Theta} \frac{\mathcal{S}^{s-1}}{\left| \psi(\mathfrak{s}) \right|^{2s}} d\mathfrak{s} \Big| \int\_0^1 \frac{\mathcal{S}^{s-1}}{\left| \psi(\mathfrak{s}) \right|^{2s}} d\mathfrak{s} \right]^2 \tag{21}$$

$$\frac{\mathcal{K}(\Theta)}{\mathcal{K}\_{8}} = \Theta^{8} \left[ \int\_{0}^{\Theta} \frac{\mathfrak{P}^{s-1}}{\left| \psi(\mathfrak{d}) \right|^{4s}} \mathrm{d}\mathfrak{d} \bigg/ \int\_{0}^{1} \frac{\mathfrak{P}^{s-1}}{\left| \psi(\mathfrak{d}) \right|^{4s}} \mathrm{d}\mathfrak{d} \right] \tag{22}$$

$$\frac{\mathcal{K}(\Theta)}{\mathcal{K}\_{\mathsf{K}}} = \int\_{0}^{\Theta} \frac{\mathfrak{s}^{2s-1}}{\left| \psi(\mathfrak{s}) \right|^{4s}} d\mathfrak{s} \Big| \int\_{0}^{1} \frac{\mathfrak{s}^{2s-1}}{\left| \psi(\mathfrak{s}) \right|^{4s}} d\mathfrak{s} \tag{23}$$

where Θ = (θ − θr)/(θs − θr) is an effective degree of saturation.

#### *3.2. Applications*

#### 3.2.1. Brooks and Corey Equation

The equation proposed by Brooks and Corey [35] to represent the soil-water retention curve is:

$$
\Theta = (\psi\_{\rm cr}/\psi)^{\lambda} \tag{24}
$$

if ψ < ψcr, and Θ = 1 if ψcr ≤ ψ; where ψcr is a critical pressure and λ > 0 is an index of the pore distribution. The assumption λ = E − D [24,25] is not used here.

The introduction of the Equation (24) in the Equations (20)–(23) gives the same expression for the relative hydraulic conductivity:

$$\mathbb{K}(\Theta)/\mathbb{K}\_{\sf s} = \Theta^{2\sf s(2/\lambda + 1)}\tag{25}$$

The saturated hydraulic conductivity is given by:

$$\mathbf{K}\_{\mathfrak{s}} = \mathbf{f} \mathbf{C}\_{\mathfrak{f}} (\boldsymbol{\theta}\_{\mathfrak{s}} - \boldsymbol{\theta}\_{\mathfrak{r}})^{2s} (\mathbf{R}\_{\mathfrak{o}} / \mathbf{T}\_{\mathfrak{o}})^{2} \boldsymbol{\Lambda} \tag{26}$$

where we have defined Ro = λL λL/ ψcr . The factor Λ is different for each model: small pore Λs = 1/[2(2/λ + 1/2)(2/λ + 1)];geometric poreΛg = 1/(2/λ + 1) 2 ; neutral poreΛN = 1/[2(2/λ + 1/2)]; and large pore ΛL = 1/(2/λ + 1). The following inequalities are satisfied Λs < Λg < ΛN < ΛL, the equalities are given at extremes (λ → 0, ∞). When λ → 0 we have: Λs = λ2/8 < Λg = λ2/4 < ΛN = λ/4 < ΛL = λ/2.

The corresponding saturated hydraulic conductivity value satisfies the inequalities: Kss < Ksg < KsN < KsL.

#### 3.2.2. Generalized Power Function

One of the larger groups of models used to represent the soil–water retention curve is the following power function [36]:

$$
\psi = \psi\_{\rm d} \Theta^{-1/\lambda} \big( 1 - \Theta^{1/\rm m} \big)^{1/n} \tag{27}
$$

where ψd is a pressure scale; m > 0, n > 0 and λ > 0 are three form parameters.

In Equation (27), we can note when Θ → 0 we obtain the Brooks and Corey [35] equation, and when λ = mn we obtain the van Genuchten equation [37]:

$$\Theta(\psi) = \left[1 + \left(\psi/\psi\_{\rm d}\right)^{n}\right]^{-m} \tag{28}$$

Introducing the Equation (27) in the Equations (20)–(23) we obtain, respectively:

$$\frac{\mathbf{K}(\Theta)}{\mathbf{K}\_{\mathbf{s}}} = \frac{\Theta^{\mathbf{s}} \mathbf{B}\_{\mathbf{l}} \big( \Theta^{1/\mathbf{m}}; 4\mathbf{s}\mathbf{m}/\lambda + \mathbf{s}\mathbf{m}, \ 1 - 4\mathbf{s}/\mathbf{n} \big) - \mathbf{B}\_{\mathbf{l}} \big( \Theta^{1/\mathbf{m}}; 4\mathbf{s}\mathbf{m}/\lambda + 2\mathbf{s}\mathbf{m}, \ 1 - 4\mathbf{s}/\mathbf{n} \big)}{\mathbf{B}(4\mathbf{s}\mathbf{m}/\lambda + \mathbf{s}\mathbf{m}, \ 1 - 4\mathbf{s}/\mathbf{n}) - \mathbf{B}(4\mathbf{s}\mathbf{m}/\lambda + 2\mathbf{s}\mathbf{m}, \ 1 - 4\mathbf{s}/\mathbf{n})} \tag{29}$$

$$\mathbf{K}(\Theta)/\mathbf{K}\_{\mathbf{s}} = \left[ \beta\_{\mathrm{l}} \big( \Theta^{1/\mathrm{m}}; 2\mathrm{s}\mathbf{m}/\lambda + \mathrm{s}\mathbf{m}, \ 1 - 2\mathrm{s}/\mathbf{n} \big) \right]^{2} \tag{30}$$

$$\mathcal{K}(\Theta)/\mathcal{K}\_{\sf s} = \Theta^{\sf s} \beta\_{\sf l} \Big( \Theta^{1/\sf m}; 4\sf{sm}/\lambda + \sf{sm}, \, 1 - 4\sf{s}/\bf n\right) \tag{31}$$

$$\mathbf{K}(\Theta)/\mathbf{K}\_{\mathbf{s}} = \beta\_{\mathrm{I}} \Big( \Theta^{1/\mathrm{m}}; 4\mathrm{s}\mathbf{m}/\lambda + 2\mathrm{s}\mathbf{m}, \,\mathbf{1} - 4\mathrm{s}/\mathbf{n} \Big) \tag{32}$$

where βI(x; p, q) = BI(x; p, q)/B(p, q), BI(x; p, q) is the incomplete beta function of variable x and parameters p > 0 and q > 0 and B(p,q) = BI(1:p,q) is the complete beta function.

We can obtain closed-form equations accepting the van Genuchten [37] idea consisting in to assign integral values to parameter p of the beta function and specially p = 1; this conduces to impose relationships between the form parameters of the soil–water retention curve. This idea is only applicable to models defined by Equations (30)–(32) because the unicity of these relationships. From Equation (29) we can obtain only closed-form equations of the first or second integral of the numerator, the result is an incomplete closed-form formula (semi closed-form) of the conductivity.

Small pore model:

$$\frac{\mathbf{K}(\Theta)}{\mathbf{K}\_{\mathbf{s}}} = \frac{\Theta^{\mathbf{s}} \mathbf{B}\_{\mathbf{l}} \Big(\Theta^{1/\mathbf{m}}; 1, 1 - 4\mathbf{s}/\mathbf{n}\Big) - \mathbf{B}\_{\mathbf{l}} \Big(\Theta^{1/\mathbf{m}}; 1 + \text{sm, } 1 - 4\mathbf{s}/\mathbf{n}\Big)}{\mathbf{B}(1, 1 - 4\mathbf{s}/\mathbf{n}) - \mathbf{B}(1 + \text{sm, } 1 - 4\mathbf{s}/\mathbf{n})}, \\ \boldsymbol{\lambda} = \frac{4\mathbf{s}\mathbf{m}}{1 - \text{sm}} \tag{33}$$

$$\frac{\mathbf{K}(\Theta)}{\mathbf{K}\_{\mathbf{s}}} = \frac{\Theta^{\mathbf{s}} \mathbf{B}\_{\mathbf{l}}^{\mathbf{s}} \mathbf{B}\_{\mathbf{l}}^{\mathbf{l}} \Theta^{\mathbf{l}/\mathbf{m}} ; 1 - \text{sm, } 1 - 4\mathbf{s}/\mathbf{n} \text{)} - \mathbf{B}\_{\mathbf{l}} \big( \Theta^{\mathbf{l}/\mathbf{m}} ; 1, \, 1 - 4\mathbf{s}/\mathbf{n} \big)}{\mathbf{B} (1 - \text{sm, } 1 - 4\mathbf{s}/\mathbf{n}) - \mathbf{B} (1, \, 1 - 4\mathbf{s}/\mathbf{n})}, \, \lambda = \frac{4\mathbf{s} \text{m}}{1 - 2\mathbf{s} \text{m}} \tag{34}$$

with n > 4s and BI Θ1/m; 1, 1 − 4s/n = (1 − 4s/n) −1 1 − 1 − Θ1/m 1−4s/n Geometric pore model:

$$\mathbf{K}(\Theta)/\mathbf{K}\_{\mathbf{s}} = \left[ 1 - \left( 1 - \Theta^{1/\mathbf{m}} \right)^{1 - 2\mathbf{s}/\mathbf{n}} \right]^2, \lambda = \frac{2\mathbf{s}\mathbf{m}}{1 - \mathbf{s}\mathbf{m}} \tag{35}$$

.

with n > 2s.

Neutral pore model:

$$\mathcal{K}(\Theta)/\mathcal{K}\_{\mathbb{S}} = \Theta^{\mathbb{S}} \Big[ 1 - \left( 1 - \Theta^{1/\text{m}} \right)^{1 - 4\kappa/\text{n}} \Big], \ \lambda = \frac{4\text{sm}}{1 - \text{sm}} \tag{36}$$

with n > 4s.

Large pore model:

$$\mathbf{K}(\Theta)/\mathbf{K}\_{\mathbf{s}} = 1 - \left(1 - \Theta^{1/\mathbf{m}}\right)^{1 - 4\kappa/\mathbf{n}}, \ \lambda = \frac{4\mathbf{s}\mathbf{m}}{1 - 2\mathbf{s}\mathbf{m}}\tag{37}$$

with n > 4s.

We can note that soil–water retention curves induced by Equations (33) and (34) are equals to those induces by Equations (36) and (37), respectively.

The use of models (33)–(37) reduces the form parameters number of the soil–water retention curve defined by Equation (27): the three independent parameters {m, n, λ} are reduced to two parameters {m, n}.

The form parameters can even be reduced to one. If we assume λ = mn in the Equation (27) we obtain van Genuchten equation, Equation (28), which makes the function Θ(ψ) explicit where the form parameters {m, n} are still independent. If we accept the relationships between λ and m used to obtain Equations (33)–(37), the Equation (27) will have only one form parameter (m). The corresponding models of the conductivity associated to van Genuchten equation with a form parameter are the following:

Small pore model:

$$\frac{\mathcal{K}(\Theta)}{\mathcal{K}\_{\mathsf{s}}} = \frac{\Theta^{\mathsf{s}} \mathsf{B} \Big( \Theta^{1/\mathsf{m}}; 1, \text{ sm} \Big) - \mathsf{B} \Big( \Theta^{1/\mathsf{m}}; 1 + \text{sm}, \text{ sm} \Big)}{\mathsf{B} (1, \text{ sm}) - \mathsf{B} (1 + \text{sm}, \text{ sm})}, \\ 0 < \text{sm} = 1 - 4\mathsf{s}/\mathsf{n} < 1 \tag{38}$$

$$\frac{\mathbf{K}(\Theta)}{\mathbf{K}\_{\mathbf{s}}} = \frac{\Theta^{\mathbf{s}} \mathbf{B}\_{\mathbf{l}} \Big( \Theta^{1/\mathbf{m}}; 1-\text{sm, sm} \Big) - \mathbf{B}\_{\mathbf{l}} \Big( \Theta^{1/\mathbf{m}}; 1,\text{sm} \Big)}{\mathbf{B}(1-\text{sm, sm}) - \mathbf{B}(1,\text{sm})}, \ 0 < 2\text{sm} = 1 - 4\text{s}/\text{m} < 1\tag{39}$$

Geometric pore model:

$$\mathrm{K}(\Theta)/\mathrm{K}\_{\mathrm{s}} = \left[1 - \left(1 - \Theta^{1/\mathrm{m}}\right)^{\mathrm{sm}}\right]^2, \ 0 < \mathrm{sm} = 1 - 2\mathrm{s}/\mathrm{n} < 1\tag{40}$$

Neutral pore model:

$$\mathcal{K}(\Theta)/\mathcal{K}\_{\sf s} = \Theta^{\sf s} \Big[ 1 - \left( 1 - \Theta^{1/\sf m} \right)^{\rm sm} \Big], \ 0 < \text{sm} = 1 - 4\sf s /\bf n < 1 \tag{41}$$

Large pore model:

$$\mathcal{K}(\Theta)/\mathcal{K}\_{\mathbb{S}} = 1 - \left(1 - \Theta^{1/\text{m}}\right)^{2\text{sm}}, \ 0 < 2\text{sm} = 1 - 4\text{s/n} < 1\tag{42}$$

A first evaluation of the predictive capacity of the relative hydraulic conductivity models defined by Equations (38)–(42), on fifty soils of the GRIZZLY database reported by Haverkamp et al. [38], was presented by Fuentes et al. [18] with acceptable results.

The classical version of the models defined by Equations (21) and (22) corresponds to the Mualem [31] and Burdine [33] models that are defined in Equations (12) and (13). The use of these models has been proved by different authors [37]. We will show the capacity of prediction of the model that corresponds to the hypothesis of the large pore defined in Equation (23) and in particular the close-form equation of the relative hydraulic conductivity defined in Equation (42).

We use three of the five soils that van Genuchten [37] analyzes, holding the values θr and θs that the author use and the parameter s is estimated from θs. In Table 2 we present some properties of the three soils. The parameters ψd and m obtained by least squares method corresponding to three different models are presented in Table 3.

**Table 2.** Some physical properties of the three analyzed soils.

**Table 3.** Parameters values of the soil water retention curve of the three analyzed soils corresponding to three different models, Equations (40)–(42).

Figures 2–4 present the fitted soil water retention curves, and the predicted relative hydraulic conductivity curves by the geometric pore, neutral pore and large pore models for the three studied soils. We can observe that the predictions are good enough in these three soils. In addition, in Figure 5 we illustrate the prediction capability of the small pore model using the relationships between m and n provides by the neutral pore and large pore models.

The analysis performed on the classical models and the comparison between experimental and predicted relative hydraulic conductivity with the four special new models allows us to show that the different models may be used to estimate relative hydraulic conductivity.

**Figure 2.** Observed and calculated curves of the soil hydrodynamic characteristics of hygiene sandstone, Equations (40)–(42).

**Figure 3.** Observed and calculated curves of the soil hydrodynamic characteristics of Touchet silt loam G.E.3, Equations (40)–(42).

**Figure 4.** Observed and calculated curves of the soil hydrodynamic characteristics of silt loam G.E.3, Equations (40)–(42).

**Figure 5.** Observed and calculated curves of the relative hydraulic conductivity with small pore model, Equations (38) and (39).

#### **4. Conclusions**

The proposed hydraulic conductivity model has been deduced using classic probabilistic theory and fractal geometry concepts in order to approach the flow effective area and the tortuosity of flow trajectories for each pore radius. The introduction of four definitions used at the literature of the pore effective radius that is involved in the general model has permitted to establish four new models to predict the relative hydraulic conductivity.

Some additional considerations related to the definitions of flow effective area and the tortuosity factor have allow us to deduce four classical models that are extensively used in different studies. In particular, we have given some interpretations of its empirical parameters in the fractal geometry context. The power function proposed by Brooks and Corey (1964) to represent the water retention curve leads to a power function to represent the relative hydraulic conductivity. The difference between the new models consists in the prediction of the relative hydraulic conductivity.

We have studied a general power function with three form parameters to represent the water retention curve reported by Braddock et al. [36], which presents as a special case the equation of Brooks and Corey [35] with a form parameter and the van Genuchten equation [37] with two form parameters. With this function we have gotten close-form equations of hydraulic conductivity with two and one form parameter. The general power function of the soil water retention curve and the close-form equations of hydraulic conductivity, obtained through special fractal models, may be used for the study of mass and energy transferences through soil, as infiltration, drainage, evaporation and ground water recharge.

**Author Contributions:** Formal analysis, C.F., C.C. and F.B.; investigation, C.F., C.C. and F.B.; software, C.F., C.C. and F.B.; C.F., C.C. and F.B.; writing—original draft, C.F. and F.B.; writing—review and editing, C.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.


**Yuan-Sen Yang 1,\*, Chia-Hao Chang 1 and Chiun-lin Wu 2**

Received: 14 September 2019; Accepted: 3 October 2019; Published: 4 October 2019

**Abstract:** Image analysis techniques have been applied to measure the displacements, strain field, and crack distribution of structures in the laboratory environment, and present strong potential for use in structural health monitoring applications. Compared with accelerometers, image analysis is good at monitoring area-based responses, such as crack patterns at critical regions of reinforced concrete (RC) structures. While the quantitative relationship between cracks and structural damage depends on many factors, cracks need to be detected and quantified in an automatic manner for further investigation into structural health monitoring. This work proposes a damage-indexing method by integrating an image-based crack measurement method and a crack quantification method. The image-based crack measurement method identifies cracks locations, opening widths, and orientations. Fractal dimension analysis gives the flexural cracks and shear cracks an overall damage index ranging between 0 and 1. According to the orientations of the cracks analyzed by image analysis, the cracks can be classified as either shear or flexural, and the overall damage index can be separated into shear and flexural damage indices. These damage indices not only quantify the damage of an RC structure, but also the contents of shear and flexural failures. While the engineering significance of the damage indices is structure dependent, when the damage indexing method is used for structural health monitoring, the damage indices safety thresholds can further be defined based on the structure type under consideration. Finally, this paper demonstrates this method by using the results of two experiments on RC tubular containment vessel structures.

**Keywords:** image-based measurement; crack measurement; shear cracks; flexural cracks; damage index


*1.1. Literature Review*

Numerous studies have attempted to address the drawbacks of the traditional SCOPF model.

To the best of the authors' knowledge, there are currently two strategies to reduce the calculation burden of traditional SCOPF and make it easier to solve. One strategy uses a contingency filtering (CF) [8,9] technique to reduce the number of contingencies. Usually, an index that ranks the severity of a contingency is used to filter contingencies; thus, only the contingency that exceeds the severity threshold is included in the contingency set. However, choosing the severity threshold itself is a challenge, for example, a very severe contingency may have a very low probability of occurring, and controlling it through SCOPF may result in excessive costs. The second strategy is to use Benders decomposition (BD) [10–12] to decompose the original SCOPF problem into a master problem and several subproblems. In this way, parallel computing technology can be used to improve the computing efficiency; however, BD requires convexity of the feasible region, which is not guaranteed in an SCOPF problem [12].

The concept of risk-based SCOPF [13–15] has been proposed as a method that comprehensively considers the probability and severity of contingencies. The risk of a contingency is defined as the product of the probability and severity of a contingency. Risk-based SCOPF uses risk as constraints to achieve a tradeoff between economic and security. This method relaxes the constraints of a single contingency [14] but controls the total risk of a contingency set to a certain level. Although the security and economy of power system operations are enhanced by risk-based SCOPF, the uncertainty of RES and load are not taken into consideration because measuring system risk under uncertainty is a challenging task. Moreover, the optimization formulation of risk-based SCOPF is complicated, and the calculation time is 4–7 times that of traditional SCOPF [14], which makes it difficult to apply in a real-time dispatch.

Chance-constrained optimization (CCO) [16–26] is a promising method to handle the uncertainty in power systems and it has been successfully applied to many problems. Instead of rigid constraint, CCO ensures a certain level of probability that the constraint is satisfied. The work of Bienstock [19] provides a solid foundation for incorporating CCO with OPF. This model was further extended in [22] to incorporate corrective SCOPF. Li et al. [23] provided a novel transmission expansion planning approach based on CCO and BD. Liu et al. [24], proposed solutions based on CCO for peak power shaving and frequency regulation in microgrids. Based on CCO, a day-ahead scheduling approach is proposed in [25] and a volt/var control approach is provided in [26]. Although CCO has been successfully applied to a variety of problems, the probability distribution of the uncertainty source is usually assumed to follow a Gaussian distribution. Studies [27–29] have indicated that the distribution of wind power forecast error and photovoltaic power is very different from a Gaussian distribution; therefore, the existing models should be improved so that they are able to handle arbitrary distributions. Moreover, there are few CCO models that consider contingency probability.


**2.3 Drone system to help exercise of the visually impaired for the University of Nevada**

The visually impaired people have limitations in some exercise, such as running outside without a guide, but it has been confirmed that the visually impaired have higher spatial localization skills than the general public. This study became the basis for the ability of the visually impaired people to follow drones in a running track environment. In fact, the University of Nevada, Reno (UNR), developed a drone system (**Figure 5**) that helps exercise of the visually impaired people through low-cost flying drones [15]. Equipped with a total of two cameras, a downward-facing camera that follows the track's line and a separate camera that focuses on the marker on the runner's shirt, the drone flies about 10 feet ahead of the runner running at eye level and provides sound guidance. As the runner speeds up or slows down, the drone adjusts its own speed to guide the movement of the visually impaired. The study was conducted with two visually impaired persons, and the results of the study confirmed that the visually impaired could accurately identify and follow the drone, and the qualitative results showed that the participants were accustomed to following the drone, and that the drone system had high efficiency when following and locating the drone.


**1. Introduction**

The past few decades have been detrimental for understanding the mechanism of appearance and evolution of many myeloid and lymphoid diseases. Previous immunobiology and molecular biology techniques along with most current sequencing technologies and their

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

implementation in molecular diagnostics altogether contributed toward characterizing such conditions.

Novel pharmaceutical approaches, such as targeted therapies, optimized chemotherapy regimens, radiotherapy, and others have been developed. Yet, many of these diseases still present poor survival outcomes. In such cases, hematopoietic stem cell transplantation (HSCT) is considered as final resort therapeutic approach, whenever all other options have failed [1].

The success of HSCT depends on various factors that should be taken into consideration in advance. Reduced immunological reaction is such a major factor. This is only accomplished when donor and recipient of the graft are immunologically compatible. During the previous decades, immunology and molecular biology techniques have been moving toward delineating the biological mechanism of this compatibility.

During the past few years, the advent of high throughput sequencing technologies has helped move toward this direction with a much faster pace. In this chapter, we will review the past, present, and future of these technologies in this particular area of research.