*3.2.2. Reproductive management and performance, and production results*

Estrous synchronisation was only carried out in the 3.20% of farms held. This practice was only observed in conventional farms, since it is not permitted in organic farming. Accordingly, only 4.80% of farms opted for artificial insemination, with all of them also carrying out natural mating, such that the use of either one or another technique was not exclusive. This scarce use of these reproductive techniques is typical in low-input beef cattle farms.

Calves weaned in All Organic had lower weights than those belonging to the conventional group, and Organic 2 farms sold less weaned calves per cow in total, thus showing a lower productivity in this regard. However, Organic 2 and All Organic sold more fattened calves per cow and also sold a higher proportion of fattened calves/total calves sold. These differences were due to the fact that the Organic 2 group was composed entirely of fattening farms, while all Organic 1 farms solely marketed calves at the age of weaning. Similarly to the Organic 1 group, 83.33% of the farms belonging to the conventional group did not carry out the fattening of any of the calves that they produced. These facts about the composition of the groups also influenced the differences between these indicators for yearlings sold per cow and calf weight at weaning.

#### *3.2.3. Breeds*

**3.2. Farm structure and management**

Organic 2 vs. 80.18 in Organic 1, *p* < 0.05).

**Parameters Conv.**

Calves sold at weaning age/cow/year (N°)

adult serviced cows in the farm. 4

1 and Organic 2. 7

and level of significance.

1

The average size of farms was 268.44 ha utilized agricultural area (UAA) (**Table 2**). Organic 2 farms were larger than 'All Organic', but the high variability within the sample did not allow identifying significant differences between this group and Organic 1. With regard to herd size, All Organic farms were also similar to Conventional farms, and an important variation within farms was identified in relation to the mean cattle herd size (140.95 livestock units (LU) in

> **Org. 1 (***n* **= 22)**

Replacement rate (%)1 11.98 11.92 13.29 0.922 12.19 1.23 12.38 0.875 Cows/bull (N°) 31.01 30.67 28.29 0.844 30.42 1.68 29.88 0.740 Estrous synchronisation (%)2 6.70 0.00 0.00 0.321 3.20 – 0.00 0.132 Artificial insemination (%)3 6.70 4.50 0.00 0.592 4.80 – 3.00 0.658 Length of mating period (months) 10.40 10.70 10.14 0.922 10.46 0.29 10.52 0.846 Fertility rate (%)4 85.15 77.70 81.49 0.187 81.91 1.82 78.97 0.091 Age at first calving (month) 30.68 33.45 33.68 0.197 32.17 0.79 33.53 0.074 Calving interval (days)5 346.50 33500 343.64 0.165 341.98 2.74 337.88 0.117 Calves born/cow/year (N°) 0.85 0.78 0.81 0.187 0.82 0.02 0.78 0.091 Weaned calves/cow/year (N°) 0.81 0.71 0.65 0.061 0.75 0.03 0.69 0.025\* Age at weaning (months) 5.86 5.82 6.00 0.886 5.87 0.13 5.88 0.944 Live weight at calving (kg) 202.33a 190.91b 193.18ab 0.037\* 196.75 2.12 191.67 0.011\*

Fattened calves sold/cow/year (N°) 0.07a 0.00a 0.45b 0.000\*\*\* 0.11 0.03 0.15 0.000\*\*\* Fattened calves/total calves sold 0.09a 0.00a 0.64b 0.000\*\*\* 0.15 0.30 0.21 0.119

Annual average proportion of: live births/serviced cows. 5

**Table 2.** Reproductive management and performance, and productive orientation. Mean values, standard deviation

In relation to the various land uses and the type of ownership, it was seen that 59% of land was in property (owned area/UAA in percentages). Fifty-two percent of UAAs had tree presence.

a, b, c Mean values with different letters in the same row are significantly different. \*

proportion of cows synchronized/total adult cows in the farm. 3

days from calving to calving in the adult cows of the farms. 6

Moreover, crop areas were almost inexistent.

Calculated as the annual average proportion of heifers bred for reproduction/number adult. 2

Analysis of Variance of the groups Conventional vs. All Organic.

**Org. 2 (***n* **= 11)** **Sig. 16 Sample (***n* **= 63)**

0.63a 0.66a 0.27b 0.000\*\*\* 0.58 0.03 0.53 0.000\*\*\*

**SD All**

p<0.05, \*\* p<0.01, \*\*\* p<0.001.

Average annual proportion of inseminated cows/total

Analysis of Variance of the groups Conventional, Organic

Annual average

Annual average number of

**Organic (***n* **= 33)**

**Sig. 27**

**(***n* **= 30)**

*3.2.1. Farm and herd characteristics*

24 Livestock Science

The breed distribution of organic farms is also an important issue, as autochthonous breeds are preferable for this production model, as indicated by Regulation 834/2007 [28]. **Table 3** shows the composition by breed of the farms.

a, b Mean values with different letters in the same row are significantly different. \* p<0.05. 1 Analysis of Variance of the groups Conventional, Organic 1 and Organic 2. 2 Analysis of Variance of the groups Conventional vs. All Organic.

**Table 3.** Farm breed structure. Mean percentage values, standard deviation and level of significance.

The percentage of purebred autochthonous cows reached 41.83%, with this percentage being higher in All Organic and Organic 2 than in the conventional group. Also, All Organic and Organic 2 showed a higher presence of these purebred cows; either autochthonous or foreign ones. The main reason for this is that Organic 2 farmers were market oriented (they had contracts with supermarkets) so that they knew that more productive breeds that allow them to produce carcass of better conformation, mainly Limousine.

Note: all these indicators were measured in terms of €/ha. 1 Value of the land at market prices. This depended on the quality of the plots (grazing resources, location and tree density, among other parameters). 2 Value of infrastructure at market prices. Years of use and level of conservation/maintenance were taken into account. 3 Value of machinery (cars, trucks, etc.) at market prices. Years of use and level of conservation/maintenance were taken into account. 4 Value of all livestock present at market prices. 5 Analysis of Variance of the groups Conventional, Organic 1 and Organic 2. 6 Analysis of Variance of the groups Conventional vs. All Organic.

**Table 4.** Fixed capital according to farm groups.

a, b Mean values with different letters in the same row are significantly different. \* p<0.05, \*\* p<0.01. 1 Expenditures (purchases) made in external feedstuffs/hectare of UAA (€/ha). 2 Expenditures (purchases) made in external feedstuffs/LU (€/LU). 3 Expenditure in seeds and fertilisers/hectare of UAA (€/ha). 4 Expenditures made in veterinary and medicines/ha UAA (€/ha). 5 Expenditures made in veterinary and medicines /LU (€/LU). 6 Expenditures made in maintenance of fixed capital/ha of UAA (€/ha). 7 Expenditures made in energy/ha UAA (petrol and electricity) (€/ha). 8 Expenditures/ha UAA made in farmers' associations, lawyers, etc. (€/ha). 9 Sum of the following costs (€/ha of UAA: External feedstuffs + Veterinary services and medicines + Energy + Maintenance of machinery and infrastructure + Other goods and Services (lawyers, farmers' associations, etc.). 10Expenditures made in salaries/ha of UAA (€/ha). 11Amortization of machinery and infrastructure = Sum of (((1/20 years amortization) × Value of infrastructures) + ((1/10 years) × Value of machinery)). 12Cost of the land rented (€/ha). 13Analysis of Variance of the groups Conventional, Organic 1 and Organic 2. 14Analysis of Variance of the groups Conventional vs. All Organic.

**Table 5.** Intermediate consumption and other costs.

#### **3.3. Economic parameters**

contracts with supermarkets) so that they knew that more productive breeds that allow them

**Org. 2 (***n* **= 11)**

Land fixed capital1 5,630.07 5,194.52 5,695.62 0.788 5,489.42 310.43 5,361.56 0.669 Buildings fixed capital2 660.75 606.48 546.10 0.935 621.78 114.32 586.35 0.748 Machinery fixed capital3 215.51 138.79 107.77 0.449 169.91 35.00 128.45 0.217 Livestock fixed capital4 624.81 416.66 329.10 0.217 500.49 69.69 387.47 0.089 Total fixed capital 7,131.14 6,356.45 6,678.59 0.443 6,781.59 430.59 6,463.83 0.443

**Sig. 15 Sample (***n* **= 63)**

Analysis of Variance of the groups Conventional, Organic 1 and Organic 2.

**Sample (***n* **= 63)**

**SD All**

Value of the land at market prices. This depended on the

**SD Sig. 113 All**

p<0.05, \*\* p<0.01. 1

Expenditures (purchases) made in external

Expenditures made in energy/ha UAA (petrol and electricity) (€/ha).

**Organic (***n* **= 33)**

Value of infrastructure at

Value of machinery (cars,

**Organic (***n* **= 33)**

Expenditures

Expenditures made in

Expenditures made in veterinary

Sum of the following costs (€/ha of UAA:

Value of all

**Sig. 214**

**Sig. 26**

to produce carcass of better conformation, mainly Limousine.

**(***n* **= 30)**

Note: all these indicators were measured in terms of €/ha. 1

Analysis of Variance of the groups Conventional vs. All Organic.

**Org. 1 (***n* **= 22)**

quality of the plots (grazing resources, location and tree density, among other parameters). 2

market prices. Years of use and level of conservation/maintenance were taken into account. 3

**(***n* **= 30)**

a, b Mean values with different letters in the same row are significantly different. \*

Organic 1 and Organic 2. 14Analysis of Variance of the groups Conventional vs. All Organic.

Expenditures/ha UAA made in farmers' associations, lawyers, etc. (€/ha). 9

Expenditure in seeds and fertilisers/hectare of UAA (€/ha). 4

External feedstuffs + Veterinary services and medicines + Energy + Maintenance of machinery and infrastructure + Other goods and Services (lawyers, farmers' associations, etc.). 10Expenditures made in salaries/ha of UAA (€/ha). 11Amortization of machinery and infrastructure = Sum of (((1/20 years amortization) × Value of infrastructures) + ((1/10 years) × Value of machinery)). 12Cost of the land rented (€/ha). 13Analysis of Variance of the groups Conventional,

Expenditures made in veterinary and medicines /LU (€/LU). 6

(purchases) made in external feedstuffs/hectare of UAA (€/ha). 2

feedstuffs/LU (€/LU). 3

8

and medicines/ha UAA (€/ha). 5

maintenance of fixed capital/ha of UAA (€/ha). 7

**Table 5.** Intermediate consumption and other costs.

trucks, etc.) at market prices. Years of use and level of conservation/maintenance were taken into account. 4

**Org. 1 (***n* **= 22)**

Feed/ha UAA1 109.69a 17.55b 96.63ab 75.24 112.18 0.009\*\* 43.91 0.019\* Feed/LU2 161.59a 38.27b 220.54a 128.82 165.70 0.003\*\* 99.03 0.136 Seeds and fertilisers3 7.51 3.10 1.00 4.84 12.61 0.252 2.40 0.108 Veterinary and medicines/ha UAA4 17.87a 4.51b 4.84b 10.93 21.16 0.043\* 4.62 0.012\* Veterinary and medicines /LU5 20.32a 7.45b 11.64ab 14.31 15.14 0.006\*\* 8.84 0.002\*\* Maintenance of fixed capital6 15.74 18.95 22.60 18.06 22.68 0.681. 20.17 0.444 Energy7 24.24 22.44 18.27 22.57 22.84 0.765 21.05 0.584 Other expenditure8 24.32 20.87 21.88 22.69 33.67 0.934 21.20 0.717 Intermediate consumption9 199.38 87.42 165.22 154.32 177.86 0.077 113.36 0.054 Remuneration of employees10 60.29 42.48 61.24 54.24 100.69 0.799 48.73 0.653 Fixed capital consumption11 54.59 44.20 38.08 48.08 66.62 0.744 42.16 0.464 Land rented12 30.56 30.96 23.25 29.42 38.47 0.846 28.39 0.825

**Org. 2 (***n* **= 11)**

**Parameters Conv.**

26 Livestock Science

livestock present at market prices. 5

**Table 4.** Fixed capital according to farm groups.

**Parameters Conv.**

6

#### *3.3.1. Analysis of fixed capital*

This analysis allowed identification of similarities between organic and conventional systems (**Table 4**), with regard to infrastructure, land and animals. It is worth highlighting the high average value of land fixed capital 5489.42 €/ha that accounted for the 81% of total fixed capital (**Table 5**).

Note: Those parameters whose unit is not showed in the table are measured per ha of UAA ((€/ha). 1 Value of all the products of agricultural activities. All agricultural output was recorded except that which was solely produced by units for their own consumption. 2 It measures the value created by all agricultural output after the consumption of fixed capital. That output is valued at basic prices and intermediate consumption is valued at purchase prices. It was calculated as follows: (Gross output – Intermediate consumption – Amortisation) + (Those subsidies not related to livestock farming). 3 It measures the yield from land, capital and unpaid labour. It is the balance of the generation of income account which indicates the distribution of income between the factors of production and the general government sector. 4 Obtained by adding the interest received and then deducting rent (i.e., farm and land rents) and interest payments, measuring compensation of unpaid labour, remuneration from land belonging to units and the yield arising from the use of capital. 5 Ratio between net surplus and average capital assets, estimated from the value of total fixed capital and the value of capital. 6 Analysis of Variance of the groups Conventional, Organic 1 and Organic 2. 7 Analysis of Variance of the groups Conventional vs. All Organic.

**Table 6.** Economic and productive performance and subsidies.

**Table 6** shows the economic and productive performance of the farm groups, as well as aspects related to subsidies, where the Organic 1 group can be seen to have lower livestock sales per hectare of UAA and lower gross production.

Conventional farms proved to sell more calves per hectare and year, which is due to their shorter productive cycle and the low productivity of Organic 1. No differences were found for the remaining indicators, but some interesting results were found and therefore comments are necessary. Organic farms (especially Organic 2) revealed higher numerical values for other sales, which reflect a higher level of business diversification, something that is key in the farms' flexibility and adaptability to the changing market environment. Moreover, organic farms (especially Organic 2) tended to be more dependent on subsidies.


**2.2 Organic bulk heterojunction solar cells**

416 Solar Cells – New Aspects and Solutions

been shown to be compatible with solution-processing at room temperature, for instance, by high-throughput printing techniques. Processing on flexible substrates is possible, thus allowing for roll-to-roll manufacturing as well as influencing the properties of the finished electronic devices. The recent considerable achievements in terms of power conversion efficiency have been made possible now by more than 15 year long research and development on solution-processed organic solar cells. Nevertheless, in order to let the scientific progress be followed by a commercial success, further improvements in term of

In this chapter, we will briefly introduce the basic working principles of organic solar cells and present an overview of the most often studied PPV-type materials as applied within the

The first organic solar cells consisted of a single layer of photoactive material sandwiched between two electrodes of different work functions (Chamberlain, 1983; Wohrle & Meissner, 1991). However, due to the high binding energy of the primary photoexcitations, the separation of the photogenerated charge carriers was so inefficient that far below 1% power

The next breakthrough was achieved in 1986 by introducing the bilayer heterojunction concept, in which two organic layers with specific electron or hole transporting properties were sandwiched between the electrodes (Tang, 1986). In this organic bilayer solar cell were consisting of a light-absorbing copper phthalocyanine layer in conjunction with an electronegative perylene carboxylic derivative. The differing electron affinities between these two materials created an energy offset at their interface, thereby driving exciton

The efficiencies of the first organic solar cells reported in the 1980s were about 1% at best at that time. Primarily, this is due to the fact that absorption of light in organic materials almost always results in the production of a mobile excited state, rather than free electronhole pairs as produced in inorganic solar cells. This occurs because in organic materials the weak intermolecular forces localize the exciton on the molecules. Since the exciton diffusion lengths in organic materials are usually around 5-15 nm (Haugeneder et al., 1999), much shorter than the device thicknesses, exciton diffusion limits charge-carrier generation in these devices because most of them are lost through recombination. Photogeneration is

The discovery of ultrafast photoinduced electron transfer (Sariciftci et al., 1992) from a conjugated polymer to buckminsterfullerene (C60) and the consequent enhancement in charge photogeneration provided a molecular approach to achieving higher performances from solution-processed systems. In 1995 the first organic bulk heterojunction organic solar cell was fabricated based on a mixture of soluble *p*-phenylene-vinylene (PPV) derivative with a fullerene acceptor (Yu et al., 1995). In 2001, Shaheen et al. obtained the first truly promising results for bulk heterojunction organic solar cells when mixing the conjugated polymer poly(2-methoxy-5-(3',7'-dimethyl-octyloxy)-*p*-phenylene vinylene) (MDMO-PPV) and methanofullerene [6,6]-phenyl C61-butyric acid methyl ester (PCBM) yielding a power

therefore a function of the available mechanisms for excitons dissociation.

conversion efficiency of 2.5% (Shaheen et al., 2001).

efficiency and device lifetime have to be made.

conversion efficiency could be achieved.

photoactive layer.

**2.1 A brief history**

dissociation.

**2. Organic solar cells**

The sequential process involved in the light into electricity conversion can be summarized by the following steps: First, incident light is absorbed within the photoactive layer leading to the created of a bound electron-hole pairs (singlet excitons); the created excitons start to diffuse within the donor phase leading to charge separation; the separated charge carriers are transported to the corresponding electrodes.

Fig. 1. (a) Schematic device structure and (b) energy diagram for an organic bilayer solar cell

Figure 1 (a) shows the simplest structure of an organic bilayer solar cell appears to be the superposition of donor and acceptor materials on top of each other, providing the interface needed to ensure the charge transfer. The schematic energy diagram of such an organic bilayer solar cell is depicted in Figure 1 (b). The excitons photogenerated in the donor or in the acceptor can diffuse to the interface where they are dissociated. According to the Onsager theory (Onsager, 1938) that can be invoked as a first approximation in organic semiconductors, photoexcited electrons and holes, by virtue of the low dielectric constant intrinsic to conjugated polymers, are coulombically bound. Due to the related exciton binding energy, which with around 0.5 eV is much larger than the thermal energy, the photogenerated excitons are not easily separated. Once excitons have been generated by the

Organic Bulk Heterojunction Solar Cells Based on Poly(*p*-Phenylene-Vinylene) Derivatives 419

Fig. 3. MIM picture for a polymer diode under different operating modes. (a) open circuit

injection is hindered by the field present in the device (Dennler & Sariciftci, 2005).

When polarized in the forward direction (high work function electrode (ITO) connected to (+) and low work function electrode (Al) connected to (-)) as in Figure 3 (c), electrons can be injected from the Al electrode to ITO electrode and holes from ITO electrode to Al electrode. The effective field in the device will ensure the drift of electrons from Al electrode to ITO electrode and hole from ITO electrode to Al electrode. Finally, when the device is polarized in the reverse direction (ITO connected to (-) and Al connected to (+)) (Figure 3 (d), charge

condition, (b) short circuit condition, (c) forward bias, (d) reverse bias.

absorption photons, they can diffuse over a length of approximately 5-15 nm (Haugeneder et al., 1999). Since the exciton diffusion lengths in conjugated polymers are less than the photon absorption length, the efficiency of a bilayer cell is limited by the number of photons that can be absorbed within the effective exciton diffusion range at the polymer/electron interface. This limits drastically the photocurrent and hence the overall efficiency of the organic bilayer solar cells. To overcome this limitation, the surface area of the donor/acceptor interface needs to be increased. This can be achieved by creating a mixture of donor and acceptor materials with a nanoscale phase separation resulting in a threedimensional interpenetrating network: the "bulk heterojunction solar cells" (Figure 2).

Fig. 2. (a) Schematic device structure and (b) energy diagram for an organic bulk heterojunction solar cell

The discovery of 1-(3-methoxycarbonyl)propyl-1-phenyl[6]C61 (PCBM) (Hummelen et al., 1995), a soluble and processable derivative of fullerene C60, allowed the realization of the first organic bulk heterojunction solar cell by blending it with poly(2-methoxy-5-(2'-ethyl-hexoxy)- 1,4-phenylene-vinylene) (MEH-PPV) (Yu et. Al., 1995). Figure 2(b) demonstrates the schematic energy diagram of an organic bulk heterojunction solar cell. Contrary to Figure (1b), excitons experience dissociation wherever they are generated within the bulk. Indeed, the next interface between donor and acceptor phases is present within the exciton diffusion length everywhere in the device. After having been generated throughout the bulk, the free carriers have to diffuse and/or be driven to the respective electrodes (Dennler & Sariciftci, 2005).


*Article* **Development and Demonstration of an Endocrine-Disrupting Compound Footprint Calculator**

**Rachel Taylor 1, Kathryn Hayden 1, Marc Gluberman 1, Laura Garcia 1, Serap Gorucu 1,2, Bryan Swistock 3 and Heather Preisendanz 1,4,\***

**Abstract:** Chemicals in personal care products used in everyday lives become part of the wastewater stream. Wastewater treatment plants were not designed to remove these chemicals; therefore, these products and their metabolites persist in the effluent. Many of these chemicals are known, or suspected to be, endocrine-disrupting compounds (EDCs) and can cause adverse impacts to aquatic organisms at trace concentrations.  The calculator prompts users to input the number of products they own in each of three categories: health and beauty, laundry, and cleaning. The calculator, which is programmed with average values of EDCs in each product, outputs an estimate of the user's EDC footprint (mass) and ranks the contribution of each product to the footprint. When used by a group of 39 citizen scientists across the Susquehanna River Basin in the northeastern United States, the average household EDC footprint was ~150 g. Results of this tool aid in decision making by providing users with the information necessary to reduce the household's footprint through product selection that avoids specific ingredients or by replacing the top-ranking products with greener alternatives.

**Keywords:** chemical footprint; emerging contaminants; endocrine-disrupting compounds; footprint calculator; personal care products; water quality


**2.5 Anticancer effect**

The crude extract of *P. lenticus* leaves and fruits significantly inhibited the growth of melanoma cell line (B16F10 cells). The leaves and fruits significantly inhibited the B16F10 cells (IC50 = 56.40 and 58.04 μg/mL, respectively) [3], whereas the seed oil and phenolic compounds fraction of *P. lenticus* showed inhibitory potential against BHK21 cell line. The IC50 was 0.029 g/mL and the percent effect was 42.4 at the concentration of 0.09 g/mL [23]. The essential oils extracted from the fruits and leaves of *P. lenticus* were tested for anticancer effect. The oil of leaves exhibited interesting anticancer activity as compared to fruits [16] on RD and L20B cell lines with IC50 values of 26.43 ± 2.18 and 33.02 ± 2.84μg/mL, respectively. A protective effect of the oil of *P. lenticus* has been reported in bleomycin-induced lung fibrosis and oxidative stress in rats [24]. A significant anti- proliferation potential of *P. atlantica* against COLO205 has been noticed along with good antioxidant effect [25]. Colorectal cancer is one of the major malignant forms of cancer, which has been blocked by the essential oil from the *P. palaestina* [26]. The ethanol extract of *P. atlantica* demonstrated a significant effect against the gastric and cervical carcinoma along with antioxidant effect, which is attributed to the presence of phenolic compounds in the extract [27]. The ethyl acetate extract of *P. vera* L. also attenuated the growth of MCF-7 human breast cancer [28]. The anticancer effect of these plants against various cells line is well established. Interestingly, the antiemetic effect of these plants is very good regarding the anticancer effect because most anticancer drugs have emesis as a significant side effect. If a natural remedy has anticancer and antiemetic potential, it might be one of the best therapeutic mixtures.


• **The third challenge: Obtain better performance and compete with other technologies.**

So far, the record performance achieved by CIGSe-based thin-film photovoltaic technology is 22.6% [1], close to the highest yield of 24.9% recorded by silicon technology. The main goal of the present development, is to undertake scientific investigations to go beyond the above-mentioned values.

#### **2.4 Thin film issue**

In principle, a thin layer of a given material is an element of this material, one of its dimensions, called the thickness, has been greatly reduced so that it is expressed in nanoscale and that this small distance between the two boundary surfaces (almost two-dimensionality) leads to a disturbance of the majority of the physical properties.

The second essential characteristic of a thin film is such that whatever the procedure used for its manufacture; a thin film is almost part of the support on


Suggestions for Therapy (Conservative or Surgical) before Joint Arthroplasty

**87**

**Chapter 5**

**Abstract**

*and Abdullah M. Kaki*

erative knee osteoarthritis.

sion and restore gliding [1, 2].

**1. Introduction**

cartilage degeneration, regenerative therapy repair

The Regenerative Effect of Intra-

Articular Injection of Autologous

The study started in 2010 to find the effect of autologous fat micrograft for osteoarthritis (OA); the result was published on normal animal's model, in 10 patients, then in 80 patients with knee osteoarthritis, and the current study in 205 patients. The study was conducted at King Abdulaziz University Hospital (January 2012–October 2015); 80 adult patients were suffering from moderate to severe knee osteoarthritis. About 10–20 mL fat micrograft was prepared with liposuction and injected intra-articularly into the affected knee/s. The results revealed that pain improvement after the fat injection during rest and with activity with the visual analogue scale. The Western Ontario and McMaster Universities Osteoarthritis Index (WOMAC) indicated improvement, both in the three domains (pain, stiffness, and physical function) and in total. The use of intra-articular autologous fat micrograft is simple, safe, and effective for degen-

**Keywords:** autologous fat micrograft, intra-articular injection, knee osteoarthritis,

Fat grafting and its use in aesthetic and reconstructive surgeries are considered a state of art, but looking back at the history, it is just a revival in the techniques, which was described previously by others when **Gustav Neuber** on 1893 was the first to perform fat graft for orbital depression in human and Erich Lexer who is a skilled German orthopedic and plastic surgeon reported a variety of clinical uses of fat graft in management of knee ankylosis and fat graft wrapped around the tendon during tendon repair to prevent skin tendon adhe-

The revolution in surgical specialty directed toward minimally invasive therapeutic modalities where endoscopic surgery replaced the open surgical operations; similarly, the recent discovery of the regenerative effect of fat micrograft due to presence of adipose-derived stem cells (ADSCs), cytokines, growth factors, pre adipocytes, and mature adipocytes led to a growing interest for the

Fat Micro-Graft in Treatment of

Chronic Knee Osteoarthritis

*Sabah S. Moshref, Yasir S. Jamal, Amro M. Al-Hibshi*

### **Chapter 5**


Preface

Water holds the key to life on Earth. Every cell, every ecosystem, and every human society thrives on its availability. Yet, access to clean, safe water is far from universal. Globally, 2.7 billion people live in water-stressed countries, facing scarcity that worsens with climate change and population growth.

*Water Purification – Present and Future* presents a comprehensive review of the different types of water, the elements that negatively affect their quality, and the industrial processes for water purification, desalination, and energy production. It includes twelve chapters organized into five sections:

We would like to thank all the authors for their contributions, and we hope that this book will help save our planet from the water crisis, as without water, there will be no living organisms, including us!

## **Magdy M.M. Elnashar**

Professor, Faculty of Health Sciences, School of Medicine, Curtin University, Perth, Australia

National Research Center, Center of Excellence for Advanced Sciences, Cairo, Egypt

### **Selcan Karakuş**

Associate Professor, Faculty of Engineering, Department of Chemistry, Istanbul University-Cerrahpaşa, Istanbul, Turkey

Section 1


**1. Introduction**

The placental syncytiotrophoblast (STB) layer that covers the floating chorionic villi in direct contact with maternal blood in the intervillous space constitutes a major barrier to vertical transmission of parasites and microorganisms. The consequences of STB damage have been studied in first

trimester placental explants where the subsyncytial cytotrophoblasts (CTB) becomes infected by *Listeria monocytogenes* if the STB layer is damaged with collagenase [1]. With *Toxoplasma gondii*, the situation is more complex, as multiparasite vacuoles in chorionic villi are present not only immediately below or besides syncytial interruptions, but also at sites of no visible syncytial interruptions, suggesting the possibility of hard to detect breaks in the STB layer or of earlier breaks that had subsequently healed [2]. These observations highlight the importance of considering syncytial permeability upon studying placental vulnerability to infection.

Several viruses cross the uterine-placental interface, infecting the fetus and causing birth defects, including rubella, varicella-zoster, parvovirus B19, human cytomegalovirus, hepatitis E type 1 and Zika (ZIKV) (see [3]). ZIKV is a flavivirus which is transmitted by *Aedes* mosquitoes; it was first identified in 1947 in an African forest, and triggered epidemics in the South Pacific in 2007 and in the Americas in 2015–2017 [4]. ZIKV infection during pregnancy is associated to an array of devastating birth defects known as congenital Zika syndrome, which include microcephaly, brain calcifications, neurological impairment, and retinal damage.

We are interested in exploring the mechanisms that allow ZIKV transplacental transmission in humans. During the development of human placenta, CTB epithelial cells may be di fferentiated in two ways. Firstly, they aggregate into cellular columns that invade the uterine interstitium and colonize the spiral arterioles, allowing the anchorage of the fetus to the mother and the flow of blood to the placenta. Secondly, CTB form a bilayer in which the cells of the external sheet fuse to form the multinucleated STB that covers the chorionic villi. The STB layer is crucial for the interchange of ions, nutrients, gases, and waste between the fetus and the mother. During pregnancy, as a result of syncytial ruptures or focal degeneration of STB, lateral cell membranes subdivide segments of STB from the surrounding STB continuum. This appears to be a dynamic process where the disconnected parts of the STB eventually fuse after the disintegration of the lateral separating membranes [5].

Tight junctions (TJs) regulate transit through the paracellular pathway of epithelial cells. In the STB, these cell-cell adhesion structures located at the uppermost portion of the lateral membranes that subdivide the STB layer constitute a cornerstone of the blood-placental barrier (BPB) that protects the fetus from toxins and pathogens. TJs in the STB of human placental chorionic villi have been observed by freeze fracture, and their function as paracellular seals has been demonstrated by electron microscopy in thin sections, with the blockade of the transit of electron-dense markers through the paracellular pathway [6–9]. The apical surface of the STB of chorionic villi expresses several TJ proteins, including the integral proteins occludin, claudins -1, -3 and -16, and the adaptor protein ZO-1 [9,10], while claudin-4 is strongly expressed during all trimesters of pregnancy [11], but localizes at the basal membrane of the STB [9].

ZIKV infects cells that strongly express TIM-1 [12], a cell surface phosphatidylserine and phosphatidylethanolamine receptor [13]. ZIKV in humans replicates in the glandular epithelium of the decidua and in decidual cells and infects invasive CTB, the CTB of cell columns, as well as fetal fibroblasts and macrophages known as Hofbauer cells which are present in the parenchyma of floating chorionic villi [12,14,15]. These observations led to the proposal of a ZIKV transmission route that goes from the cells of the basal decidua in the mother to the fetal invading CTB, followed by the infection of cell columns of CTB and Hofbauer cells present in the parenchyma of chorionic villi [12]. Moreover, since the envelope (E) proteins between the dengue virus (DENV) and ZIKV are very similar structurally [16], cross-reactive antibodies are generated that may enhance ZIKV infection [17]. In this respect, it has been observed that ZIKV infection in human placental explants are enhanced by the presence of dengue virus antibodies, suggesting that ZIKV immune complexes could use the neonatal Fc receptor for IgG as a transport system to transcytose across the STB layer to infect Hofbauer cells in the chorionic villi [18]. However, since the clinical severity of maternal ZIKV infection has not being associated with the existence of prior dengue antibodies [19], and not all pregnant women infected with ZIKV have been previously infected with DENV, here, we explore another complementary route of vertical ZIKV transmission. Taking into account that STB cells are not infected by ZIKV [12], we explore

the possibility of ZIKV reaching the chorionic mesenchyme via transit through the paracellular route of the STB.

Virus passage through the paracellular route of epithelial and endothelial cells has previously been reported. Thus, after human airway epithelia infection with adenovirus, the viruses are first released to the basolateral surface and then escape to the apical surface. This process involves the binding of the fiber proteins of the adenovirus to its receptor CAR located within TJs, which triggers a disruption of junctional integrity that allows the virus to escape through the paracellular pathway between the cells to reach the apical surface [20]. Another case is that of human immunodeficiency virus (HIV), whose disruption of TJs and adherens junctions (AJs) in oral epithelial cells facilitates the paracellular spread of the herpes simplex virus 1 (HSV1), emerging as a mechanism to explain the rapid development of HSV-associated oral lesions in HIV infected individuals [21]. Likewise, sealing the blood-brain barrier (BBB) is compromised by HIV-1 induced inflammatory cytokines; in this respect, TNFα has been shown to open the paracellular route for HIV-1 invasion across the BBB [22]. With regards to *Flaviviruses*, the West Nile virus disrupts the BBB in mice inducing an increase in BBB permeability and a reduced expression of TJ and AJ proteins [23]; moreover, in vitro infection of mouse brain endothelia with DENV delocalizes TJ proteins from the membrane to the cytoplasm, reduces the transendothelial electrical resistance, and increases the macromolecule permeability and the paracellular passing of free virus particles [24].

Here, we analyzed the permeability of the paracellular pathway and the molecular composition of TJs in the STB cell layer of placentae derived from ZIKV-infected women. In addition, in the trophoblast cell line BeWo, we observed that basolateral incubation with ZIKV reduces the transepithelial electrical resistance (TER).

Our results indicate that ZIKV infection alters the composition of placental TJs and increases paracellular permeability.


*3.4. Color Tests*

Figure 6 shows the color coordinates L\*, a\* and b\* for the mixes studied in the different media tested, in the test phases after 28 and 120 days. Each pigment displays the luminosity values L\* on the left and the chromaticity diagram a\*-b\* on the right. In all cases, the black circles represent the coordinates of the pure pigment, the blue circles the binders, the white circles the pastes after 28 days and the grey circles the pastes after 120 days.

With regard to the binders, they all feature very small tonal amounts (a\*-b\*) of red and blue, except the water glass, whose tones are blue and green. The luminosities (L\*) are extremely high in all cases, with values which are very close to 100%, generating a visual impression that is very similar to white. Meanwhile, the pigments used in all of the mixes in proportions of 20% feature shades which are characteristic of the color palette designed, and they are obviously affected by their interactions with the different binders on forming the pastes.

In terms of shade, the CY pigment is composed of a significant proportion of yellow and also features a considerable amount of green (a\*-b\*). The luminosity of this chromium pigment is very close to 100%. The luminosity of the pastes hardened for 28 or 120 days tended to differ, darkening in some way by approximately 5%; the color saturation decreased more markedly—by up to 40%.

The MO pigment is largely composed of yellow and red tones (a\*-b\*), and the luminosity is low: slightly higher than 50%. The changes induced by the different binders increased the luminosity in all cases, this being greater after 120 days, with values of close to 13%. In the chromaticity diagram the saturation of the pastes drastically reduces to levels which are very close to those of the binders (≈80%). Equally noteworthy is the reduction induced by the binders in relation to the yellow shades, at the expense of the associated red shade.

**Figure 6.** CIELab1976 colour space representation of binders and their corresponding pigmentcoloured pastes (PPS, PALS, PHLS and PWGS). Plot L\*, indicates brightness values of the different samples. Plot a\*-b\*, indicates chromaticity values of the different samples.

The NS pigment has high luminosity (L\*) values, of close to 90%, and where chromaticity (a\*-b\*) is concerned, yellow stands out as the principal shade alongside small amounts of red. The pastes obtained provoke subsaturation, with values which are close to those of the binders, although the relative reduction was ≈10%. The luminosity increased ≈ 14%. It is notable how there were significant reductions in the yellow and red shades, and the red was replaced by green in the PWGS paste, after 120 days.

The GE pigment has average luminosity (≈60%) and chromaticity values (a\*-b\*), along with comparatively small amounts of yellow and green. The pastes featured increased luminosity (≈12%), particularly those measured after 120 days, contrary to what happens in the chromaticity diagram, where the pastes display relative subsaturation of ≈8% after 28 and 120 days.

The CG pigment has low luminosity (<50%) and chromaticity components (a\*-b\*) based on green and yellow shades. The variations in luminosity were significant, particularly the increase in white, which occurred in the pastes after 28 and 120 days (≈15%). The most significant variations in terms of chromaticity stemmed from subsaturation, which arose in the mixes after 28 and 120 days; these have a clear tendency towards the binder values (≈3%).

The O pigment reached luminosity values slightly higher than 50% and tonal proportions (a\*-b\*) of yellow and red. The changes were similar to those experienced by the rest of the pigments and their pastes: a clear increase in luminosity (≈6%) and a substantial loss of saturation (≈14%). The PWGS sample after 120 days provided a slight green tone compared to the reddish tones of the other samples (PPS, PALS and PHLS).

The ZY pigment has very high luminosity values (≈90%), whilst the chromaticity includes tonal components based on yellow and green. The variations experienced by the pastes on hardening with the binders were very similar to those described in all cases: increased luminosity (≈3%) and loss of saturation (≈30%).

The UB pigment has low L\* values, making it dark (≈25%). The changes induced by the inclusion of the binders studied increased luminosity by up to (≈11%) after 120 days. The chromaticity values include a blue tonal component, which dominates in relation to the red tone, which represents half the amount of the blue. The pastes provoked a loss of saturation which left it close to the binders' values (≈33%).

Table 8 shows that all of the paste samples studied experienced total color variations (ΔE) clearly detectable by the human eye ΔE ≥ 3 [76] after 28 or 120 days. These changes are determined by the chroma variation (ΔC), which entails variations in saturation (subsaturation). This doubled after 120 days in the majority of cases, compared to the amount after 28 days. Likewise, the increased luminosity (ΔL) induced by the binders used to prepare the pastes is a decisive factor in the total color variation (ΔE). The only exception of note occurred in the mixes containing pigment O, whose variations just exceeded the limit of detection by the human eye.

**Table 8.** CIELab\* 1976 values for the pigments and pastes used in this study and total color variations (ΔE), chromaticity variations (ΔC) and luminosity variations (ΔL) of the pigments used in the preparation of the pastes after 28 and 120 days.

**Table 8.** *Cont.*

All of these changes are represented in the form of a color chart in Figure 7, taking into account the average values for a\*, b\* and L\* for each group of samples. For each pigment and paste type, the results are shown for 28 days and 120 days according to their color coordinates (CIELab 1976).

Щ

Щ

**Figure 7.** Color chart for the different mixes and inorganic pigments used, with an indication of their color code for both of the test phases (28 days and 120 days).

## **4. Conclusions**

**Author Contributions:** Conceptualization, M.P.S.-P. and J.A.D.-S.; methodology, M.P.S.-P. and J.A.D.-S.; validation, M.P.S.-P. and J.A.D.-S.; formal analysis, M.P.S.-P. and J.A.D.-S.; investigation, M.P.S.-P. and J.A.D.-S.; resources, M.P.S.-P., J.A.D.-S., A.V.-V. and T.G.-L.; data curation, M.P.S.-P. and J.A.D.-S.; writing—original draft preparation, M.P.S.-P. and J.A.D.-S.; writing—review and editing, M.P.S.-P., J.A.D.-S., A.V.-V. and T.G.-L.; supervision, M.P.S.-P. and J.A.D.-S., A.V.-V. and T.G.-L. All authors have read and agreed to the published version of the manuscript.

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

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** This work was supported by the REMINE Programme for Research and Innovation Horizon 2020 Marie Skłodowska–Curie Actions and WARMEST H2020-MSCA-RISE-2020 (Marie Skłodowska–Curie Research and Innovation Staff Exchange and was carried out under the auspices of Research Groups RNM 0179 and HUM 629 of the Junta de Andalucía.

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

#### **References**


**5. Simulation Results**

Simulations using the PSIM software tool were conducted to validate the proposed method. The system configuration was similar to that shown in Figure 1. The DC link voltage (*Vdc*) was 300 V, and was distributed equally between the top capacitance voltage (*Vtop*) and the bottom capacitance voltage (*Vbottom*). The total DC link capacitance was 2200 μF and the switching frequency was 10 kHz. The value of |*Elimit*| was set empirically at 2.6 V. The simulation parameters are given in Table 2. After synthesizing the reference voltage vectors with the minimized cost function using the deadbeat DSVM-MPC system, the reference voltage vector was applied to the grid-connected 3L T-type VSI.

**Table 2.** System parameters.

Figure 7 shows the simulation results for the optimal voltage vectors in the *d–q* frame for the deadbeat DSVM-MPC, which followed the reference current using only four candidate voltage vectors [20]. Figure 8 shows the simulation results for the deadbeat DSVM-MPC using the balancing method in Reference [20] and using the proposed DPWM method. It can be seen that the current waveforms became highly distorted before implementation of the neutral-point balancing of capacitance voltage, owing to large voltage deviations. Obviously, the total harmonic distortion (THD was is significantly reduced using either one of the balancing methods. Nevertheless, the proposed balancing method performed better in terms of THD, as shown in Figure 8b.

**Figure 7.** Simulation results of the optimal voltage vectors for the deadbeat FS-MPC with DSVM.

**Figure 8.** *Cont*.

**Figure 8.** Simulation results of capacitance voltage balancing for deadbeat DSVM-MPC. (**a**) Conventional balancing method [20]. (**b**) Proposed balancing method.

Figure 9 shows the simulation results for dynamic response for the deadbeat DSVM-MPC system, using the two balancing methods. As can be seen, both methods exhibited a fast dynamic current response, because the MPC method was used. On the other hand, the proposed method exhibited smaller THD compared with the method in Reference [20].

**Figure 9.** *Cont*.

 **Figure 9.** Simulation results of current dynamic response for deadbeat DSVM-MPC with (**a**) conventional balancing method [20] and (**b**) proposed balancing method.

The robustness of the proposed method was shown by varying the impedances of the grid connected system including the inductance (L) and resistance (R) (Figure 10). The values of resistance and inductance were increased (at t = 0.1 s) by 100% and 50% of the nominal values, respectively. It can be obviously seen that there was a negligible increase in the THD. Therefore, it was confirmed that the proposed balancing method in FS-MPC with DSVM and DBC is robust to the variations of impedances.

Figure 11a,b depicts the frequency spectrum of the phase voltage for the conventional method and proposed method, respectively. Apparently, the first harmonic component (i.e., switching frequency (*fsw*)) was greatly reduced with the proposed algorithm, compared to the conventional method. This indicates the proposed method has less switching frequency owing to the use of the DPWM modulator, as mentioned previously.

**Figure 10.** Simulation results of grid impedance variation for the proposed method.

**Figure 11.** Simulation results of frequency spectrum for the phase voltage with (**a**) conventional balancing method [20] and (**b**) proposed balancing method.


*Source of the Great Recession DOI: http://dx.doi.org/10.5772/intechopen.90729*

#### **Figure 7.**

*Historical decomposition of output. (a) Constant volatility Case A. (b) Stochastic volatility Case B. (c) Constant volatility data rich Case C. (d) Stochastic volatility data rich Case D. Notes: Four Cases A, B, C and D are described in Table 1. Case A; 11 observable variables and constant volatility shocks. Case B: 11 observable variables and structural shocks with SV. Case C: 40 observable variables and constant volatility shocks. Case D: 40 observable variables and structural shocks with SVs. Eight shocks are explained in Section 2 and SV shocks are explained in Section 3.*

specialized in US mortgage debt at this moment.), the SVs of net-worth shocks of financial and nonfinancial sectors have rapidly jumped to ceil for both of Case B and D, as well as other shocks such as TFP, monetary policy, IST and labor supply shocks. And levels of these SVs (deep blue lines) exceed the red dashed flat lines indicating estimation of constant volatilities as **Figure 6**.

In this study, we would like to verify whether the data-rich approach contributes to the accuracy of the estimated SVs, compared with standard data structure. **Figures 4(b)** and **5(b)** show averages of the 90% interval (light shade area) in Cases B does not look different from those of Case D. And, although **Figure 6** reports difference in sizes of the 90% intervals (light shade area) of the SVs over the entire sample period between Cases B and D, we do not find obvious improvement of 90% band by the data-rich approach in Case D. From only the three figures, we cannot yet include the data-rich environment improve the accuracy of the SVs estimates. This inquiry will be remained until further research.

shocks, but investment-specific technology (IST) shock and the labor supply shocks, look very similar in Cases B and D. And the SVs of the preference and labor supply shocks fluctuate with large amplitude during the period of the expansion between 2001:Q4 and 2007:Q4, and it indicates that they have played an important role of boom. Meanwhile, the SVs of the remaining shocks seem to be quiet and level off between 1990:Q1 and 2007:Q3. After August 2007, when the Great Recession began with the seizure in the banking system (in fact, BNP Paribas precipitated

*Stochastic volatilities of structural shocks. (a) Stochastic volatility: Case B. (b) Stochastic volatility data rich: Case D. Notes: Case B and Case D are described in Table 1. Eight shocks in our DSGE model are explained in Section 2. Corporate Net Worth shock and Bank Net Worth are balance sheet shocks of nonfinancial and financial sectors described in Figure 1(b). TFP (total factor productivity), investment specific technology, and labor shock are belong to supply shocks, whereas preference of consumers, monetary policy, and government spending shocks belong to demand shocks. The deep blue lines and blue shaded area are posterior mean and 90% credible interval of stochastic volatility (SV) of Cases B and D, respectively. The red dashed lines denote the posterior means of constant volatilities shocks estimated in Case A and C, respectively. SV shocks are explained*

**Figure 6.**

*Financial Crises - A Selection of Readings*

*in Section 3.*

**42**

ceasing investment activity and was followed by three big hedge funds that

**Figure 8.**

*Historical decomposition of investment. (a) Constant volatility Case A. (b) Stochastic volatility Case B. (c) Constant volatility data rich Case C. (d) Stochastic volatility data rich Case D. Notes: See the notes of Figure 7.*

**4.2 Historical decompositions**

*Source of the Great Recession*

*DOI: http://dx.doi.org/10.5772/intechopen.90729*

**Figure 9.**

**45**

Here, we move to discuss about difference of historical decompositions among the four cases. In particular, as can be seen from Figures 7**(a)–(d)**–10**(a)–(d)**, we focus on the periods between 2000:Q1 and 2012:Q2 of the following four observations and model variables, say (1) the real GDP of observations matching to an output gap of model variable, similarly (2) the gross private domestic investment matching to investment, (3) Moody's bond index (corporate Baa) matching to corporate borrowing rate, (4) the commercial banks' leverage ratio matching to the bank leverage ratio. The red and black circle lines represent observations and smoothed estimation, respectively. The differences between both lines indicate measurement errors of observations. In these figures, the light blue shade represents the period of the Great Recession (2007:Q3 to 2009:Q2). In order to make more visible and to concentrate on the contributions of both net-worth shocks of banking sector (deep blue shade area) and corporate sector (green shade area) for the recession by remaining key shocks like the TFP (red shade area) and monetary

*Historical decomposition of borrowing rate. (a) Constant volatility Case A. (b) Stochastic volatility Case B. (c) Constant volatility data rich Case C. (d) Stochastic volatility data rich Case D. Notes: See the notes of Figure 7.*

Finally, we turn to analyzing the monetary policy in the Great Recession and after it including an unconventional monetary policy by FRB such as Round 1 of quantitative easing policy (QE1), between 2008:Q4 and 2010:Q2 and Round 2 of quantitative easing policy (QE2) between 2010:Q4 and 2011:Q2, although our monetary policy rule follows linearized Taylor rules. As the fourth row of **Figures 4** and **5**, we can find the estimation of monetary policy shocks (deep blue lines) have two big negative troughs in this period for all cases. The first negative trough was identified at 2007:Q4 when the global financial market was disarranged by announcement of the BNP Paribas. And the second trough was ascertained at 2008:Q3, immediately before the FRB implemented QE1. Especially, the sizes of the two big negative shocks are classified in the Cases B and D with SVs shocks, as shown in **Figures 4(b)** and **5(b)**. The fourth row of **Figure 6** also draws the rapid surge of these volatilities of monetary policy shocks between 2007:Q4 and 2008:Q3. In other words, the two unconventional monetary policy might be undertaken more boldly and without hesitation as well as the case of conventional tightening policy according to the 90% credible band of the SVs.


*4.2.1. Internal scanning*

The second modality has originated from an proposal in [41]. At the time only numerical simulation and reconstruction using the Singular Value Decomposition method were performed on a turbine blade with encouraging results. One may conceive this modality as an evolved Prettyman's scatterometer, which can rotate around an axis perpendicular to its plane, realizing what Prettyman had foreseen long ago to generate more useful data for his numerical reconstruction method. In fact, as will be shown the rotational motion is necessary for the reconstruction the electron density in a trans-axial slice.

**Figure 9.** Second CST modality: internal scanning

The apparatus is is sketched in Fig. 9. An emitting radiation point source **S** is placed at a distance 2*p* from a point detector **D**. The segment *SD* joining them rotates around its middle point **O**. At site **D** is collected the single-scattered radiation flux density from the scanned object for a given angular position of the line *SD* and at a given scattering energy *E*(*ω*), (equivalently at scattering angle *ω*). Then at fixed *φ*, a multichannel analyzer records the photon counts in each energy channel. Data acquisition is performed for every angular position *φ* of *SD*.

Thus, thanks to the physics of the Compton effect, the detected radiation flux density is proportional to the integral of the electron density *n*(**M**) on a class of circular arcs sharing a chord of fixed length 2*p*, which rotates about its fixed middle point **O**.

With a polar coordinate system centered at **O**, the equation of a circular arc lying inside a circle of center *O* and radius *p* reads (see [42])

$$r = r(\cos(\theta - \phi)) = p\left(\sqrt{1 + \tau^2 \cos^2(\theta - \phi)} - \tau \cos(\theta - \phi)\right),\tag{13}$$

which is defined by two parameters *τ* and *φ*: a) *φ* is the angle made by its symmetry axis with the reference direction *Ox* and b) *τ* is related to the scattering angle *ω* by *τ* = cot *ω*. Note that *τ* is positive for 0 *< ω < π*/2 and the range of *θ* is (*φ* − *π*/2) *< θ <* (*φ* + *π*/2).

Again to simplify the notation by assuming that attenuation and beam spreading are neglected as well as by absorbing the Compton kinematic factor into one single function, we can say that the detected photon flux density *n*�(*τ*, *φ*) is the Radon transform of the electron density *n*(*r*, *θ*) along arcs of circle given by equation 13. Thus using the auxiliary angle *γ* = (*θ* − *φ*) as in the previous subsection, we may express *n*�(*τ*, *φ*) with *ds*, the integration arc element given by

$$ds = r \, d\gamma \, \sqrt{\frac{1 + \tau^2}{1 + \tau^2 \cos^2 \gamma}} = dr \, \frac{\sqrt{1 + \tau^2}}{\tau \sin \gamma}. \tag{14}$$

Now introducing the angular Fourier components of *n*�(*τ*, *φ*) and *n*(*r*, *θ*) as given by equation 9, we see that they are related by

$$\frac{\pi \widehat{n}\_l(\tau)}{\sqrt{1+\tau^2}} = 2 \int\_{p(\sqrt{1+\tau^2}-\tau)}^p \frac{dr}{\sqrt{1-\frac{1}{4\tau^2}\left(\frac{p}{r}-\frac{r}{p}\right)^2}} \cos\left[l\cos^{-1}\left(\frac{1}{2\tau}\left(\frac{p}{r}-\frac{r}{p}\right)\right)\right] \, n\_l(r). \tag{15}$$

At first this Chebyshev transform looks a bit hopeless. However if one introduces a new variable *g* defined by

$$g = \frac{1}{2} \left( \frac{p}{r} - \frac{r}{p} \right) . \tag{16}$$

one can put it under the form

$$\frac{\tau \widehat{n}\_l(\tau)}{\sqrt{1+\tau^2}} = 2 \int\_0^\tau d\mathbf{g} \, \frac{\cos l \cos^{-1}\left(\frac{\mathbf{g}}{\tau}\right)}{\sqrt{1-\frac{\mathbf{g}^2}{\tau^2}}} \, \frac{p(\sqrt{1+\mathbf{g}^2}-\mathbf{g})}{\sqrt{1+\mathbf{g}^2}} \, n\_l(p(\sqrt{1+\mathbf{g}^2}-\mathbf{g})) . \tag{17}$$

Then defining new functions by

$$\hat{N}\_l(\tau) = \frac{\tau \hat{n}\_l(\tau)}{\sqrt{1 + \tau^2}} \quad \text{and} \quad N\_l(\mathbf{g}) = \frac{p(\sqrt{1 + \mathbf{g}^2} - \mathbf{g})}{\sqrt{1 + \mathbf{g}^2}} n\_l(p(\sqrt{1 + \mathbf{g}^2} - \mathbf{g})), \tag{18}$$

we obtain

12 Will-be-set-by-IN-TECH

There are two scanning modes in this modality. If the object is "small", or can be put inside a circle of radius *p* (a mechanical adjustable parameter), one can make an "internal" scanning inside this circle. If the object is "large" or situated far way from the observer, one can use the

The second modality has originated from an proposal in [41]. At the time only numerical simulation and reconstruction using the Singular Value Decomposition method were performed on a turbine blade with encouraging results. One may conceive this modality as an evolved Prettyman's scatterometer, which can rotate around an axis perpendicular to its plane, realizing what Prettyman had foreseen long ago to generate more useful data for his numerical reconstruction method. In fact, as will be shown the rotational motion is necessary

The apparatus is is sketched in Fig. 9. An emitting radiation point source **S** is placed at a distance 2*p* from a point detector **D**. The segment *SD* joining them rotates around its middle point **O**. At site **D** is collected the single-scattered radiation flux density from the scanned object for a given angular position of the line *SD* and at a given scattering energy *E*(*ω*), (equivalently at scattering angle *ω*). Then at fixed *φ*, a multichannel analyzer records the photon counts in each energy channel. Data acquisition is performed for every angular

Thus, thanks to the physics of the Compton effect, the detected radiation flux density is proportional to the integral of the electron density *n*(**M**) on a class of circular arcs sharing

With a polar coordinate system centered at **O**, the equation of a circular arc lying inside a circle

1 + *τ*2 cos2 (*θ* − *φ*) − *τ* cos (*θ* − *φ*)

, (13)

a chord of fixed length 2*p*, which rotates about its fixed middle point **O**.

**4.2. Second CST modality (Nguyen-Truong 2010)**

*4.2.1. Internal scanning*

"external" scanning mode. The two modes shall be discussed separately.

for the reconstruction the electron density in a trans-axial slice.

**Figure 9.** Second CST modality: internal scanning

of center *O* and radius *p* reads (see [42])

*r* = *r*(cos(*θ* − *φ*)) = *p*

position *φ* of *SD*.

$$\hat{N}\_l(\tau) = 2 \int\_0^\tau d\mathbf{g} \, \frac{\cos l \cos^{-1} \left(\frac{\mathcal{S}}{\tau}\right)}{\sqrt{1 - \frac{\mathcal{S}^2}{\tau^2}}} \, N\_l(\mathbf{g})\_\prime \tag{19}$$

which is precisely of the form of equation 10, obtained in the Radon problem on circles passing through a fixed point. Hence an inversion formula exists in the (*τ*, *g*) variables, *i.e.*

$$N\_l(\mathbf{g}) = \frac{1}{\pi \mathcal{g}} \left( \int\_0^\mathcal{g} dp \, \frac{d\widehat{\mathbf{N}}\_l(\tau)}{d\tau} \, \frac{\left( (\mathbf{g}/\tau) - \sqrt{(\mathbf{g}/\tau)^2 - 1} \right)^l}{\sqrt{(\mathbf{g}/\tau)^2 - 1}} - \int\_\mathcal{g}^\infty d\tau \, \frac{d\widehat{\mathbf{N}}\_l(\tau)}{d\tau} \, \mathcal{U}\_{l-1}(\mathbf{g}/\tau) \right). \tag{20}$$

#### 14 Will-be-set-by-IN-TECH 114 Numerical Simulation – From Theory to Industry Recent Developments on Compton Scatter Tomography: Theory and Numerical Simulations 15

A closed form of the inversion formula can be deduced from equation 8

$$N(\mathbf{g}, \theta) = \frac{1}{2\pi^2 \mathcal{g}} \int\_0^{2\pi} d\phi \int\_0^\infty d\tau \,\frac{\partial \dot{N}(\tau, \phi)}{\partial \tau} \frac{1}{\mathbf{g}/\tau - \cos(\theta - \phi)}.\tag{21}$$

on this arc of circle is now expressed by the following Chebyshev integral transform for its

This equation will take the form of equation 10, as equation 19, when the intermediate variable

) = *p*(

are used. Finally, following the same steps as in the previous scanning mode, the

In this subsection we discuss two scanning modes of a third CST modality proposed in [54]. This modality has originated from a search for curves in the plane such that the Radon transform on these curves may be reduced to a Chebyshev integral transform of the form of equation 10, for circles passing through a fixed point. It turns out that Radon transforms on arcs of circles orthogonal to a fixed circle *do have this property*. The pair source-detector still moves on a circle of radius *p* (still an adjustable parameter) and centered at **O**, the origin of a polar coordinate system. But their separation distance is no longer constant as in the second CST modality: it depends on the scattering angle *ω*. The positions of **S** and **D** are given by an

1 2*τ r p* − *p r*

*nl*(*r*) cos

Recent Developments on Compton Scatter Tomography: Theory and Numerical Simulations 115

1 + *g*�2 + *g*�

1 + *g*�2

)

1

*nl*(*p*(

− cos(*θ* − *φ*)

*τ*2 cos2(*θ* − *φ*) − 1), (28)

*l* cos−1

 1 2*τ*  *r p* − *p r*

, (25)

1 + *g*�2 + *g*�

*∂*

*∂τ τ n*(*τ*, *φ*) √ 1 + *τ*2

. (24)

)), (26)

 .

(27)

*dr*

*g*� = 1 2 *r p* − *p r*

angular components

*p*(

*p*

*Nl*(*τ*) = *τ nl*(*τ*) √ 1 + *τ*2

> 1 + 1 4 *r p* − *p r* 2

reconstruction formula for *n*(*r*, *θ*) reads

*r* 2 *r p* − *p r*

**4.3. Third CST modality (Truong-Nguyen 2011)**

√1+*τ*2+*τ*)

1 − 1 4*τ*2 *r p* − *p r* 2

and *Nl*(*g*�

 2*π* 0

opening angle *γ*0, measured from the symmetry axis of the circular arc.

*r* = *p*(*τ* cos(*θ* − *φ*) −

In this case, the Radon transform is defined on the following circular arc of equation

where *τ* = 1/ cos *γ*0 and −*γ*0 *<* (*θ* − *φ*) *< γ*0. Inspection of Fig. 11 shows that *γ*0 = (*π*/2 − *ω*) for 0 *< ω < π*/2. Hence *τ* = 1/ sin *ω >* 1. So data acquisition works as follows. For fixed *φ*, the pair source - detector must be *simultaneously* displaced on the circle of radius *p* so that the opening angle *SOD* = 2*γ*0 = (*π* − 2*ω*), before registering at **D** a photon count. The number of isogonic lines is thus equal to the number of positions of the pair (**S**, **D**).

*dφ* ∞ 0 *dτ*

1 + *τ*2 = 2

and the functions

*n*(*r*, *θ*) = 1

2*π*2

*4.3.1. Internal scanning*

*τ nl*(*τ*) √

Finally going back to the original functions *n*(*r*, *θ*) and *n*(*τ*, *φ*), via equations 18, the reconstructed electron density is

$$n(r,\theta) = \frac{1}{2\pi^2} \frac{\sqrt{1 + \frac{1}{4}\left(\frac{p}{r} - \frac{r}{p}\right)^2}}{\frac{r}{2}\left(\frac{p}{r} - \frac{r}{p}\right)} \int\_0^{2\pi} d\phi \int\_0^\infty d\tau \frac{1}{\frac{1}{2\pi}\left(\frac{p}{r} - \frac{r}{p}\right) - \cos(\theta - \phi)} \frac{\partial}{\partial\tau} \left(\frac{\tau\hat{n}(\tau,\phi)}{\sqrt{1 + \tau^2}}\right). \tag{22}$$

#### *4.2.2. External scanning*

This scanning mode is illustrated by Fig. 10. Data acquisition is the same as for internal scanning, except for an object of compact support, the rotational motion may be replaced by a back and forth radar type sweeping motion.

**Figure 10.** Second CST modality: external scanning

The working is similar to the internal scanning mode, except that one uses the scattering angle range *π*/2 *< ω < π* and the "external" arc of circle given by the equation

$$r = r(\cos(\theta - \phi)) = p\left(\sqrt{1 + \tau^2 \cos^2(\theta - \phi)} + \tau \cos(\theta - \phi)\right),\tag{23}$$

where *τ* = − cot *ω >* 0 and (*φ* − *π*/2) *< θ <* (*φ* + *π*/2). This arc of circle Radon transform has not yet been considered in [42]. Similarly the Radon transform of the electron density on this arc of circle is now expressed by the following Chebyshev integral transform for its angular components

$$\frac{2\pi\hat{n}\_l(\tau)}{\sqrt{1+\tau^2}} = 2\int\_p^{p(\sqrt{1+\tau^2}+\tau)} \frac{dr}{\sqrt{1-\frac{1}{4\tau^2}\left(\frac{r}{p}-\frac{p}{r}\right)^2}} \,\eta\_l(r)\,\cos\left[l\cos^{-1}\left(\frac{1}{2\tau}\left(\frac{r}{p}-\frac{p}{r}\right)\right)\right]. \tag{24}$$

This equation will take the form of equation 10, as equation 19, when the intermediate variable

$$g' = \frac{1}{2} \left( \frac{r}{p} - \frac{p}{r} \right),\tag{25}$$

and the functions

14 Will-be-set-by-IN-TECH

Finally going back to the original functions *n*(*r*, *θ*) and *n*(*τ*, *φ*), via equations 18, the

This scanning mode is illustrated by Fig. 10. Data acquisition is the same as for internal scanning, except for an object of compact support, the rotational motion may be replaced by

The working is similar to the internal scanning mode, except that one uses the scattering angle

where *τ* = − cot *ω >* 0 and (*φ* − *π*/2) *< θ <* (*φ* + *π*/2). This arc of circle Radon transform has not yet been considered in [42]. Similarly the Radon transform of the electron density

1 + *τ*2 cos2 (*θ* − *φ*) + *τ* cos (*θ* − *φ*)

, (23)

range *π*/2 *< ω < π* and the "external" arc of circle given by the equation

1 2*τ p r* − *r p*

*∂N*(*τ*, *φ*) *∂τ*

1 *g*/*τ* − cos(*θ* − *φ*)

− cos(*θ* − *φ*)

*∂ ∂τ*

1

. (21)

 *τ n*(*τ*, *φ*) √ 1 + *τ*2

 .

(22)

A closed form of the inversion formula can be deduced from equation 8

 2*π* 0

> 2*π* 0

*dφ* ∞ 0 *dτ*

*dφ* ∞ 0 *dτ*

*N*(*g*, *θ*) = 1

reconstructed electron density is

 1 + 1 4 *p r* − *r p* 2

> *r* 2 *p r* − *r p*

a back and forth radar type sweeping motion.

**Figure 10.** Second CST modality: external scanning

*r* = *r*(cos(*θ* − *φ*)) = *p*

2*π*2

*4.2.2. External scanning*

*n*(*r*, *θ*) = 1

2*π*2*g*

$$\hat{N}\_l(\tau) = \frac{\tau \,\hat{n}\_l(\tau)}{\sqrt{1+\tau^2}} \quad \text{and} \quad N\_l(\mathbf{g'}) = \frac{p(\sqrt{1+\mathbf{g'}^2} + \mathbf{g'})}{\sqrt{1+\mathbf{g'}^2}} \, n\_l(p(\sqrt{1+\mathbf{g'}^2} + \mathbf{g'})) , \tag{26}$$

are used. Finally, following the same steps as in the previous scanning mode, the reconstruction formula for *n*(*r*, *θ*) reads

$$m(r,\theta) = \frac{1}{2\pi^2} \frac{\sqrt{1 + \frac{1}{4}\left(\frac{r}{p} - \frac{p}{r}\right)^2}}{\frac{r}{2}\left(\frac{r}{p} - \frac{p}{r}\right)} \int\_0^{2\pi} d\phi \int\_0^\infty d\tau \frac{1}{\frac{1}{2\tau}\left(\frac{r}{p} - \frac{p}{r}\right) - \cos(\theta - \phi)} \frac{\partial}{\partial\tau} \left(\frac{\tau\hat{n}(\tau,\phi)}{\sqrt{1 + \tau^2}}\right). \tag{27}$$

#### **4.3. Third CST modality (Truong-Nguyen 2011)**

In this subsection we discuss two scanning modes of a third CST modality proposed in [54]. This modality has originated from a search for curves in the plane such that the Radon transform on these curves may be reduced to a Chebyshev integral transform of the form of equation 10, for circles passing through a fixed point. It turns out that Radon transforms on arcs of circles orthogonal to a fixed circle *do have this property*. The pair source-detector still moves on a circle of radius *p* (still an adjustable parameter) and centered at **O**, the origin of a polar coordinate system. But their separation distance is no longer constant as in the second CST modality: it depends on the scattering angle *ω*. The positions of **S** and **D** are given by an opening angle *γ*0, measured from the symmetry axis of the circular arc.

#### *4.3.1. Internal scanning*

In this case, the Radon transform is defined on the following circular arc of equation

$$r = p(\tau \cos(\theta - \phi) - \sqrt{\tau^2 \cos^2(\theta - \phi) - 1}),\tag{28}$$

where *τ* = 1/ cos *γ*0 and −*γ*0 *<* (*θ* − *φ*) *< γ*0. Inspection of Fig. 11 shows that *γ*0 = (*π*/2 − *ω*) for 0 *< ω < π*/2. Hence *τ* = 1/ sin *ω >* 1. So data acquisition works as follows. For fixed *φ*, the pair source - detector must be *simultaneously* displaced on the circle of radius *p* so that the opening angle *SOD* = 2*γ*0 = (*π* − 2*ω*), before registering at **D** a photon count. The number of isogonic lines is thus equal to the number of positions of the pair (**S**, **D**).

**Figure 11.** Third CST modality: internal scanning

The integration arc element being now

$$ds = r d\gamma \sqrt{\frac{\tau^2 - 1}{\tau^2 \cos^2 \gamma} - 1} = dr \, \frac{\sqrt{\tau^2 - 1}}{\tau \sin \gamma} \, \tag{29}$$

in [54]. Consequently we recover the electron density in closed form as

 2*π* 0

*dφ* ∞ 0 *dτ*

1 2*τ p r* + *r p*

This scanning mode is shown in Fig. 12. It has the same data acquisition procedure as for internal scanning. It is appropriate for large objects or objects situated far away from the scanner, *e.g.* buried objects underground or undersea, with a back and forth radar type of

The Radon transform is now defined on the external circular arc with respect to the reference

Here *π*/2 *< ω < π* and *γ*0 = (*ω* − *π*/2) 2. As for the case of internal scanning, we have also

The integration arc element is the same as in the previous scanning mode, see equation 29. Then the Radon transform of the electron density *n*(*τ*, *φ*) becomes, in terms of function

*nl*(*r*) cos

*l* cos−1

*r* = *p*(*τ* cos(*θ* − *φ*) +

angular components, the following Chebyshev transform

2 Note that the two arcs of equations 28 and 35 belong to the same circle.

1 − 1 4*τ*2 *p r* + *r p* 2

√*τ*2−1)

*τ* = 1/ cos *γ*0 and −*γ*0 *<* (*θ* − *φ*) *< γ*0 with the same data acquisition procedure.

*dr*

1

Recent Developments on Compton Scatter Tomography: Theory and Numerical Simulations 117

− cos(*θ* − *φ*)

*τ*2 cos2(*θ* − *φ*) − 1). (35)

 1 2*τ*  *p r* + *r p* . (36)

*∂*

*∂τ τ n*(*τ*, *φ*) √*τ*2 − 1  .

(34)

*n*(*r*, *θ*) = 1

2*π*2

*4.3.2. External scanning*

scanning motion.

 1 4 *p r* + *r p* 2 − 1

> *r* 2 *p r* + *r p*

**Figure 12.** Third CST modality: external scanning

circle of radius *p* of equation

*p*(*τ*−

*p*

*τ nl*(*τ*) √*τ*2 − 1 = 2

the Radon transform of the electron density *n*(*τ*, *φ*) becomes, in terms of the angular components *nl*(*r*) and *nl*(*τ*), a Chebyshev transform similar to equation 17, *i.e.*

$$\frac{\pi \,\hat{n}\_l(\tau)}{\sqrt{\tau^2 - 1}} = 2 \int\_{p(\tau - \sqrt{\tau^2 - 1})}^p \frac{dr}{\sqrt{1 - \frac{1}{4\tau^2} \left(\frac{p}{r} + \frac{r}{p}\right)^2}} \, n\_l(r) \, \cos\left[l \cos^{-1}\left(\frac{1}{2\tau} \left(\frac{p}{r} + \frac{r}{p}\right)\right)\right]. \tag{30}$$

The structural similarity with equations 15 and 24 suggests an intermediate variable *g*" of the form

$$g'' = \frac{1}{2} \left( \frac{r}{p} + \frac{p}{r} \right) \, \tag{31}$$

which yields the following Chebyshev transform

$$\frac{\tau \widehat{n}\_l(\tau)}{\sqrt{\tau^2 - 1}} = 2 \int\_1^\tau d\mathbf{g''} \, \frac{\cos l \cos^{-1} \left(\frac{\mathbf{g''}}{\tau}\right)}{\sqrt{1 - \frac{\mathbf{g''}^2}{\tau^2}}} \, \frac{p(\mathbf{g''} - \sqrt{\mathbf{g''}^2 - 1})}{\sqrt{\mathbf{g''}^2 - 1}} \, n\_l(p(\mathbf{g''} - \sqrt{\mathbf{g''}^2 - 1})). \tag{32}$$

Now redefining the functions by

$$
\hat{N}\_l(\tau) = \frac{\tau \hat{n}\_l(\tau)}{\sqrt{\tau^2 - 1}} \quad \text{and} \quad N\_l(\mathbf{g''}) = \frac{p(\sqrt{\mathbf{g''} - \mathbf{g''}^2 - 1})}{\sqrt{\mathbf{g''}^2 - 1}} \, n\_l(p(\mathbf{g''} - \sqrt{\mathbf{g''}^2 - 1})), \tag{33}
$$

we obtain the form of equation 10, nevertheless with a lower integration bound equal to *τ* = 1 and not zero. This does not spoil the inversion procedure based on the identity 11, as shown in [54]. Consequently we recover the electron density in closed form as

$$n(r,\theta) = \frac{1}{2\pi^2} \frac{\sqrt{\frac{1}{4}\left(\frac{p}{r} + \frac{r}{p}\right)^2 - 1}}{\frac{r}{2}\left(\frac{p}{r} + \frac{r}{p}\right)} \int\_0^{2\pi} d\phi \int\_0^\infty d\tau \frac{1}{\frac{1}{2\tau}\left(\frac{p}{r} + \frac{r}{p}\right) - \cos(\theta - \phi)} \frac{\partial}{\partial\tau}\left(\frac{\tau\hat{n}(\tau,\phi)}{\sqrt{\tau^2 - 1}}\right). \tag{34}$$

#### *4.3.2. External scanning*

16 Will-be-set-by-IN-TECH

**Figure 11.** Third CST modality: internal scanning

*ds* = *r dγ*

cos *l* cos−1

 1 − *g*"2 *τ*2

1 − 1 4*τ*2 *p r* + *r p* 2

components *nl*(*r*) and *nl*(*τ*), a Chebyshev transform similar to equation 17, *i.e.*

*dr*

*g*" = 1 2 *r p* + *p r*

 *g*" *τ*

and *Nl*(*g*") = *p*(

*τ*2 − 1

*τ*2 cos2 *γ* − 1 = *dr*

the Radon transform of the electron density *n*(*τ*, *φ*) becomes, in terms of the angular

The structural similarity with equations 15 and 24 suggests an intermediate variable *g*" of the

*nl*(*r*) cos

*p*(*g*" − *g*"2 − 1) *g*"2 − 1

*g*" − *g*"2 − 1) *g*"2 − 1

we obtain the form of equation 10, nevertheless with a lower integration bound equal to *τ* = 1 and not zero. This does not spoil the inversion procedure based on the identity 11, as shown

√*τ*2 − 1

 *l* cos−1

*τ* sin *γ* , (29)

 *p r* + *r p*

, (31)

*g*"2 − 1)). (32)

*g*"2 − 1)), (33)

. (30)

 1 2*τ*

*nl*(*p*(*g*" −

*nl*(*p*(*g*" −

The integration arc element being now

 *p p*(*τ*−

√*τ*2−1)

which yields the following Chebyshev transform

 *τ* 1 *dg*"

Now redefining the functions by

√*τ*2 − 1

*Nl*(*τ*) = *τ nl*(*τ*)

*τ nl*(*τ*) √*τ*2 − 1 = 2

> *τ nl*(*τ*) √*τ*2 − 1 = 2

form

This scanning mode is shown in Fig. 12. It has the same data acquisition procedure as for internal scanning. It is appropriate for large objects or objects situated far away from the scanner, *e.g.* buried objects underground or undersea, with a back and forth radar type of scanning motion.

**Figure 12.** Third CST modality: external scanning

The Radon transform is now defined on the external circular arc with respect to the reference circle of radius *p* of equation

$$r = p(\tau \cos(\theta - \phi) + \sqrt{\tau^2 \cos^2(\theta - \phi) - 1}). \tag{35}$$

Here *π*/2 *< ω < π* and *γ*0 = (*ω* − *π*/2) 2. As for the case of internal scanning, we have also *τ* = 1/ cos *γ*0 and −*γ*0 *<* (*θ* − *φ*) *< γ*0 with the same data acquisition procedure.

The integration arc element is the same as in the previous scanning mode, see equation 29. Then the Radon transform of the electron density *n*(*τ*, *φ*) becomes, in terms of function angular components, the following Chebyshev transform

$$\frac{2\pi\widehat{n}\_l(\tau)}{\sqrt{\tau^2 - 1}} = 2\int\_p^{p(\tau-\sqrt{\tau^2-1})} \frac{dr}{\sqrt{1 - \frac{1}{4\tau^2}\left(\frac{p}{r} + \frac{r}{p}\right)^2}} \, n\_l(r) \, \cos\left[l \cos^{-1}\left(\frac{1}{2\tau}\left(\frac{p}{r} + \frac{r}{p}\right)\right)\right]. \tag{36}$$

2 Note that the two arcs of equations 28 and 35 belong to the same circle.

#### 18 Will-be-set-by-IN-TECH 118 Numerical Simulation – From Theory to Industry Recent Developments on Compton Scatter Tomography: Theory and Numerical Simulations 19

We see that the intermediate variable *g*" of equation (31) can be used again so that

$$\frac{\tau \widehat{m}\_l(\tau)}{\sqrt{\tau^2 - 1}} = 2 \int\_1^\tau d\mathbf{g''} \, \frac{\cos l \cos^{-1} \left(\frac{\mathbf{g''}}{\tau}\right)}{\sqrt{1 - \frac{\mathbf{g''}^2}{\tau^2}}} \, \frac{p(\mathbf{g''} + \sqrt{\mathbf{g''}^2 - 1})}{\sqrt{\mathbf{g''}^2 - 1}} \, n\_l(p(\mathbf{g''} + \sqrt{\mathbf{g'}^2 - 1})). \tag{37}$$

where *K* is the maximal value taken by *k*. Now introducing the indefinite integrals

and *Jl*(*x*) = � *x*

Recent Developments on Compton Scatter Tomography: Theory and Numerical Simulations 119

(*l* − 2)*Il*(*x*) = 2 (tan *x* sin(*l* − 2)*x* − cos(*l* − 2)*x*) − *l Il*−2(*x*), (44)

2 tan−1 *x* and *J*1(*x*) = ln � *x*

*δ* = *n*��

*K*−1 ∑ *k*=*j n*�� *dz* sin *lz* cos2 *z*

*xl*(1 + *x*2 − (*l* + 2)*Jl*−1(*x*), (45)

� +

*l*(*k*)(*Jl*((*k* + 1)*δ*) − *Jl*(*kδ*))

1 2(1 + *x*2)

*l*(*k*), *i.e.*

⎞

{I*o*(*i*, *j*)} ,

⎠ . (48)

*l*(*k*). (47)

. (46)

√ 1 + *x*2 , (43)

*dz e*−*lz* cosh2 *z*

*l Jl*+1(*x*) = − 1

The derivatives of *dn*�*l*(*p*)/*dp* are replaced by a simple linear interpolation *n*��

*l*(*k*)(*Il*((*k* + 1)*δ*) − *Il*(*kδ*)) −

*dp* � *nl*(*k* + 1) − *nl*(*k*)

Then the discretized value of the reconstructed angular component of *nl*(*r*), with *r* ∼ *jδ*, is

In this subsection, we present numerical simulations on the first CST modality applied to the Shepp-Logan medical phantom, see [50]. The source **S** is placed below on the left of the image and the detector moves along the line *SD*. The relevant space is represented by 256 × 256 (length unit)2. Let *Nφ* be the number of angular steps and let *Np* be the number of radii for circles going through the origin *O*. Then the following sampling steps *δφ* = 2*π*/*Nφ* and *δ* = 4 256/*Np* are taken. Since there is no rotation around the object, the (*φ*, *p*)-space is very large in comparison to the studied image. To have a good representation of the object, we need a

To estimate the reconstruction quality, we use the normalized mean square error (*NMSE*) and

{I*o*(*i*, *j*)}2 and *NMAE* = 100

*N*2

∑ (*i*,*j*)∈[1,*N*]2

> max (*i*,*j*)∈[1,*N*]2

the normalized mean absolute error (*NMAE*) (expressed as a percentage), defined by

2

we see that they obey the following recursion relations, see [50]

1

*dn*�*l*(*p*)

Equation 48 shall be used in numerical simulations3.

large maximum value of *p*, (*e.g.* four times the image size).

3 For difficulties on the handling of the inversion formula of S. J. Norton, see [55]

*Il*(*x*) = � *x*

with *I*0(*x*) = 1/ cos *x* and *I*1(*x*) = −2 ln(cos *x*), and

2(1 + *x*2) +

with

*J*0(*x*) = *x*

*nl*(*jδ*) = 1

*NMSE* = 100

*N*2

∑ (*i*,*j*)∈[1,*N*]2

> max (*i*,*j*)∈[1,*N*]2

*π*

**5.2. Simulation results**

⎛ ⎝ *j*−1 ∑ *k*=1 *n*��

Now redefining the functions by

$$
\hat{N}\_l(\tau) = \frac{\tau \hat{n}\_l(\tau)}{\sqrt{\tau^2 - 1}} \quad \text{and} \quad N\_l(\mathbf{g''}) = \frac{p(\sqrt{\mathbf{g''} + \mathbf{g''}^2 - 1})}{\sqrt{\mathbf{g''}^2 - 1}} \ n\_l(p(\mathbf{g''} + \sqrt{\mathbf{g''}^2 - 1})), \tag{38}
$$

we obtain the form of equation 10, and surprisingly the same equation as for internal scanning. Consequently one end up with the same reconstruction formula (34), which is a remarkable advantage for this modality. This is due to the fact that the two arcs are on the same circle.