*Edited by Pankaj Kumar Choudhury*

*and Abdel-Baset M.A. Ibrahim*

The exotic electromagnetic properties of liquid crystals can be harnessed for many technological applications in general, and in the photonics industries in particular. In addition, the varying molecular structures of crystals make them prudent for chemical industries as well. *Liquid Crystals* presents the basics of liquid crystals as well as discusses their applications in optics and field sensing. The book also discusses the interfacing of liquid crystal displays and the programming required for these displays.

ISBN 978-1-83881-944-6

Liquid Crystals

Published in London, UK © 2022 IntechOpen © smirkdingo / iStock


**4. Mineralization of NAPL**

The most widely applied soil bioremediation approach to organic contaminants involved the biostimulation of natural microbial biodegraders. Biodegradation requires a source of carbon (organic contaminant) and nutrients, as amendment. The hydrophobic organic contaminants represent the carbon source as electron donors, while nitrogen and phosphorous are essential for microbial growth for cellular metabolism. Addition of nitrogen particularly is often necessary due to heavy demands by the biodegradation process. Phosphorous is usually amended in lower concentration. Optimizing nutrient status of a contaminated soil can have direct impact on microbial activity and contaminants biodegradation. In some instances, the negative effects of high nutrients amendment with NPK on soil biodegradation especially on aromatics have been reported [19–21].

The ultimate microbial aerobic degradation process of converting bioavailable NAPL constituents in a contaminated soil matrix:

This process is commonly referred to as mineralization. The degradation process is brought about under aerobic conditions. NAPL constituents are hydrophobic organic chemicals that exhibit limited or no solubility in contaminated soils and thermodynamically tend to partition to the soil solid phase. Sorption may account for more than 95% of the total contaminant mass. As a consequence, the hydrophobic contaminant exhibits limited dissolved mass transfer phase and bioavailability, which limits its biotic degradation in the soil. Therefore, in a contaminated soil environment, biodegradation of an organic hydrophobic compound should be envisioned as a stepwise process involving contaminants bioavailability and species of biodegraders.

The use of surfactants represents a cost-effective and promising method that can enhance bioremediation of organic hydrophobic contaminants in soils. Many studies have shown that surfactants can solubilize and mobilize hydrophobic organic contaminants sorbed onto soil matrices [22–24]. Adding surfactant to a contaminated soil matrix is expected to enhance microbial degradation through mobilization or emulsification. Mobilization takes place at concentrations below CMC and the solubilization process above the surfactant CMC, whereas emulsification allows for dispersion of one phase into the other. A certain amount of surfactant in the slurry system will inevitably be sorbed onto the soil particles. Sorbed surfactant does not contribute to the solubilization and bioavailability of contaminants during treatment. The more surfactant is sorbed, the less effective will be the surfactant. Furthermore, soil hydrophobicity may increase as more surfactant becomes sorbed onto the contaminated soil matrix.

**Table 4.** Relevant environmental and surfactant considerations for *ex situ* surfactant-enhanced bioremediation.

Surfactants can enhance metabolic degradation and thereby, contaminants mineralization in the soil by two main mechanisms [25]. One mechanism involves the increase in the contaminant bioavailability for microorganisms. The second mechanism is due to interaction with cell surface resulting in the hydrophobicity increases in the cell surface allowing hydrophobic organic chemicals to interact with bacterial cells. Environmental factors and surfactant properties affecting the metabolic capability of biodegraders in the soil vis-à-vis hydrophobic organic contaminants are summarized in **Table 4**.

The role of treatability studies for *ex situ* surfactant-enhanced bioremediation of hydrophobic organic contaminants contaminated soil is vital. It will allow to derive crucial information that will serve as blueprint to optimize field operation. Typically, a treatability study will be conducted in laboratory microcosms to inform (a) on the dosage of surfactant required to optimize contaminant mass transfer, (b) on the effect of temperature on contaminant bioavailability as temperature may affect surfactant efficiency and microbial activity, (c) on optimum biostimulation through the addition of appropriate nutrient amendments such as N, P and other elements, (d) optimum moisture level as it will vary with soil type, (e) selection of appropriate surfactant, (f) modeling rate of contaminants degradation under varying environmental factors, (g) rate of oxygen and nutrients consumption under different environmental conditions, (h) implement bioaugmentation utilization by inoculation with acclimated bacteria strains, (i) the complimentary effects of combined bioaugmentation and biostimulation, (j) determine whether targeted level of cleanup is attainable, (k) formulation of an efficient and effective monitoring program for field treatment operation, (l) the engineering design, (m) potential surfactant toxicity and means to reduce it, (n) sorption behavior of a surfactant.

The two main strategies can be highlighted for assessing a bioremediation system performance. A material balance approach consists of extracting and quantifying residual parent compounds and monitoring partitioning in the headspace phase. The other strategy involves monitoring the system for CO2 production. A direct correlation occurs between mineralization of the parent compound and CO2 production.

The biodegradation during the treatability assessment may be modeled through either a firstor zero-order power rate model [26]. A zero-order reaction indicates the biodegradation of a parent contaminant in the microcosm occurs at a constant rate and independent of concentration and time. If the parent compound C is mineralized to CO2, the rate of disappearance of C is given by Eq. (9):

$$\mathbf{d} \mathbf{C} / \mathbf{d} \mathbf{t} = -\mathbf{k} \tag{9}$$

integration yields Eq. (10):

$$\mathbf{C}\_{\text{t}} = \mathbf{C}\_{\text{o}} - \mathbf{k}\mathbf{t} \tag{10}$$

where

to the soil solid phase. Sorption may account for more than 95% of the total contaminant mass. As a consequence, the hydrophobic contaminant exhibits limited dissolved mass transfer phase and bioavailability, which limits its biotic degradation in the soil. Therefore, in a contaminated soil environment, biodegradation of an organic hydrophobic compound should be envisioned as a stepwise process involving contaminants bioavailability and species of

The use of surfactants represents a cost-effective and promising method that can enhance bioremediation of organic hydrophobic contaminants in soils. Many studies have shown that surfactants can solubilize and mobilize hydrophobic organic contaminants sorbed onto soil matrices [22–24]. Adding surfactant to a contaminated soil matrix is expected to enhance microbial degradation through mobilization or emulsification. Mobilization takes place at concentrations below CMC and the solubilization process above the surfactant CMC, whereas emulsification allows for dispersion of one phase into the other. A certain amount of surfactant in the slurry system will inevitably be sorbed onto the soil particles. Sorbed surfactant does not contribute to the solubilization and bioavailability of contaminants during treatment. The more surfactant is sorbed, the less effective will be the surfactant. Furthermore, soil hydrophobicity may increase as more surfactant becomes sorbed onto the contaminated soil matrix.

Acclimation Proper biodegraders; enzymatic adjustment for metabolic process

Nutrients Sufficient N, P not limiting biodegraders growth; C:N:P ratio of 100:50:1

Effective concentration Efficient in increasing aqueous solubility of organic compounds at

CMC Effective below CMC; partial micelle encapsulation of contaminant; low sequestration vis-à-vis target contaminant

**Table 4.** Relevant environmental and surfactant considerations for *ex situ* surfactant-enhanced bioremediation.

System slurry Optimized to promote mass transfer; 50–80% of soil water intrinsic saturation

biodegraders.

**Considerations Remarks**

Metabolites Non-toxic

Temperature Mesophiles 15–45°C Oxygen Aerobes; DO > 0.30 mg/L pH Optimum range 5–9

Redox potential Greater than 70 mV; promote aerobes

318 Soil Contamination - Current Consequences and Further Solutions

Salinity Low inhibition of CMC formation

Environmental risk Pose no risk to the environment

Substrate source Not a preferential growth substrate Sorption behavior Low sorption onto soil constituents

Toxicity No inhibitory effects; not toxic to any receptors

low concentration Recalcitrancy Non-persistent; biodegradable and mineralizable

*Environmental factors*

*Surfactant properties*

Ct = parent compound present at time t

Co = initial concentration of parent compound

k = zero-order reaction rate constant

t = corresponding sampling time.

First-order reactions have rates that depend on mass transfer of parent compound concurrent to its biodegradation, Eq. (11):

$$\text{dC/dt} = -\text{kC} \tag{11}$$

where

C = parent compound concentration

t = corresponding sampling time

k = first-order reaction rate constant integration yields Eq. (12):

$$\text{Ln}\left(\text{C}\_{\text{t}}\right) - \text{Ln}\left(\text{C}\_{\text{o}}\right) = \text{Ln}\left(\text{C}\_{\text{t}}/\text{C}\_{\text{o}}\right) = -\text{kt} \tag{12}$$

where

Ct = parent compound present at time t

Co = initial concentration of parent compound

k = first-order reaction rate constant (time−1)

t = corresponding sampling time.

Solving for concentration yields Eq. (13):

$$\mathbf{C} = \mathbf{C}\_o \mathbf{e}^{-\text{kt}} \tag{13}$$

and parameters are as defined above.

Biosurfactants may be the strategic choice for increasing contaminant bioavailability in bioreactors while minimizing toxicity to biodegraders. An examination of the literature indicates that synthetic surfactants while effective for increasing contaminant mass transfers at the recommended concentration may show inhibitorial effects on the microorganisms in the bioreactor [27, 28]. In such case, this will inhibit cell proliferation and thus the biodegradation of organic contaminants. According to empirical evidence, surfactant toxicity was found to be primarily dependent on its molecular structure, in order of toxicity, generally non-ionic <anionic < cationic [28]. Several practical approaches may be implemented to reduce surfactant cytotoxicity in a bioreactor by considering a suitable biosurfactant as an alternative to a synthetic surfactant, adding a surfactant at concentration below CMC, using a suitable nonionic surfactants, using a suitable combination of biosurfactant and synthetic surfactant, in some instances, strategically increasing the surfactant concentration to decrease contact of biodegraders with the contaminant, prescreening for a suitable additive such as Ca and Mg as they were found to stabilize the cell membrane, thereby decreasing surfactant toxicity [29].


(e) Boundary conditions

Table 2 shows boundary condition types selected for the model. A commonly employed porous medium approach was used for the monolith, with a one-directional pressure gradient. Boundary conditions values are presented in Table 3.

**Table 2.** Boundary conditions included in the model.

**Table 3.** Measured temperatures, mass flow values and outlet pressure. These values have been employed as boundary conditions for the model.

A total amount of seven species are considered in gas composition (see Table 4). Oxidations of carbon monoxide and propylene, as explained in the previous section, are accounted for.

**Table 4.** Modeled species and volumetric fraction range in gas inlet composition within the test matrix.

(f) Boundary layer sensitivity analysis.

A mesh with a special refinement (three layers) near the walls and other without it were tested. Only a part of the domain was used (DOC and a section of the exhaust pipe) in order to reduce computational requirements. As can be seen in Table 5, refinement along the boundary layer adds a considerable number of extra cells and does not provide significant variations.

**Table 5.** Boundary layer sensitivity analysis results.

Boundary layer refinement results differed from the base case by less than 1% and required five times more cells. It can be stated that the model is not boundary layer-sensitive and, hence, boundary layer refinement is omitted.

(g) Wall treatment sensitivity analysis.

Three different wall approaches were tested to study their influence in the results of the model: standard, non-equilibrium and enhancement wall functions (Figure 8).

The standard wall functions proposed by Launder and Spalding [37] have been widely used, but the assumption of logarithmic velocity distribution treatment may not be adequate for complex non-equilibrium flows. To overcome this, non-equilibrium wall functions are based on pressure-gradient sensitized Launder and Spalding's [37] log-law for mean velocity.

Enhanced wall treatment is a near-wall modelling method that combines a two-layer model with enhanced wall functions. A one-equation relationship is used to evaluate the laminar sub-layer with fine mesh and transition to log-low function for the turbulent part of the boundary layer. The restriction that the near-wall mesh must be suitably fine might impose large computational requirements.

**Figure 8.** Results of test runs with different wall functions. No great disagreement between the three approaches was obtained.

More accurate wall functions such as non-equilibrium or enhanced showed no more than a 3% discrepancy with the standard approach, while requiring eight times more mesh elements. Consequently, the standard wall functions were selected.


Preface

Over the last half a century, the rapid integration of mathematical modeling, computing technology, and real-life applications has made computational simulation a powerful tool for researchers to study complex phenomena in the nature and the society around us. Along with theory and experiment, computational simulation has now become the third pillar for the foundation of scientific exploration. In many circumstances, computational simulations enable researchers to study complex problems in greater detail with better understanding than theory or experiments, such as in the study of protein dynamics. In other circumstances, computational simulations allow researchers to explore numerous different scenarios much more quickly and cost-effectively than experiments, such as in optimal design of modern aircrafts or screening of potential drug molecules.

One of the main challenges facing researchers in the field of computational simulations is the highly interdisciplinary nature of the field, which typically involves complex real-world application problems, mathematical models describing those problems, appropriate numerical solution algorithms for solving those model equations, and necessary computer hardware and software to carry out simulations. Successful computational simulation projects often require team-efforts and knowledge in mathematical modeling, computer programming, and specific application problems being studied.

The purpose of this book is to highlight the interdisciplinary nature of computational simulations and to introduce researchers and graduate students who are interested in computational simulations to a broad range of applications, with a particular emphasis on those involving computational fluid dynamics (CFD) simulations. The book is divided into three parts:

9 – 12), fluid heat exchange (chapter 13 – 14), airborne contaminant dispersion over buildings and atmospheric flow around a re-entry capsule (chapter 15 – 16), gas-solid two phase flow in long pipes (chapter 17), free surface flow around a ship hull (chapter 18), and hydrodynamic analysis of electrochemical cells (chapter 19).

 Part III covers applications of non-CFD based computational simulations, including atmospheric optical communications (chapter 20), environmental studies involving climate system simulations, porous media flow, and combustion (chapter 21 – 23), solidification (chapter 24), and sound field simulations for optimal acoustic effects (chapter 25).

I am grateful to InTech for the opportunity to serve as the editor for this book, and I wish to sincerely thank all contributing authors around the world for their diligence in following editorial guidelines, their willingness to support open access publications, and their valuable technical contributions that made this book possible. Special thanks are due to Ms. Ana Nikolic, Mr. Zeljko Spalj, and the technical staff at InTech for their editorial efforts, detailed reviews, and professional assistance. I also would like to thank the University of Texas at Arlington for supporting my participation in this book project. Without the support from the authors, InTech staff, and the University, this book could not have become a reality.

> **Jianping Zhu, Ph.D.** Professor and Chair Department of Mathematics The University of Texas at Arlington Arlington, Texas, USA


*2.3. Model Calibration and Evaluation*

The model performance metrics considered both graphical and statistical analyses to assess the best parameterization based on the coefficient of determination (R2), Root Mean Square Error (RMSE, Equation (1)), and the percent bias (PBIAS, Equation (2)), which are widely applied in hydrologic and erosion modeling [43].


**2. Materials and methods**

growing nuclear diagnostic field [1]. It is a valuable metabolic imaging approach

Moreover, PET renders high-quality tomographs, which cannot be attained by SPECT or non-nuclear counterparts [1–3]. It has a significant clinical impact, which is evident from a variety of well-conducted case studies. Oncologists, using PET imaging services, appreciate its usefulness. PET is used in a number of critical clinical applications, including cancer diagnosis and monitoring, management of

Brain-PET scanner is an imaging device used for analysis and observation of the brain metabolic activity [1–3]. The concept is to inject a controlled volume of the radioactive tracer into the patient. Brain-PET frequently uses the tracer fluorine-18 fluorodeoxyglucose (18F-FDG) [4, 5]. After a certain period, a significant amount of this tracer is accumulated across the tumor cells. Tracer releases β + positrons, annihilated by medium electrons. As a result of each annihilation, two γ-rays of 511 keV energies are emitted. These are released at about 180° relative angles at the same time. Crystal scintillators absorb energy from emitted γ-rays [6, 7]. They are turning the energy into photons of light. In the next step, photo-detectors pick up

The shape and amplitude of electrical pulses produced by photo-detectors enable undesirable interactions to be reduced [2, 3, 11, 14]. In this context, sophisticated

The PET pulses, produced by photo-detectors, are conditioned and processed to extract the information such as energies and timestamps. The extracted parameters are onward passed to the Image Reconstruction Module (IRM). It combines all pairs of γ-rays, produced by same annihilations, to generate the Line-of-Responses (LORs) [11, 15, 16]. It also uses energies of the PET pulses to measure the Depths of Interactions (DOIs) [1–3]. Afterward, IMR treats these LORs and DOIs to produce

PET pulses readout introduces artifacts [11, 12, 16]. It degrades the exactitude of measuring the PET pulses energies and timestamps [11]. It influences device consistency in calculating DOIs and LORs, which lowers the scanner's sensitivity and resolution [11, 12, 16]. BLRs, pile-up correctors, and high-precision timestamp

The readout of photo-detectors' generated PET pulses produces a shift in the baseline of the signal. It diminishes the energy resolution and the precession of DOI calculation. BLRs are utilized in this framework. Various BLRs have been suggested, ranging from analog approaches to adaptive solutions based on digital

Another primary method in PET scanners is the measurement of annihilation timestamps. It allows the collection of adequate annihilations to create LORs and also prevents the processing of ineffective information at IRM [11, 12]. Various timestamp calculators were suggested for this purpose. These are essentially formed

The timestamp calculators based on analog discriminators do not face a problem with the temporal resolution of the input signal. This problem arises, however, in the case of timestamp calculators based on digital discriminators [11, 23]. The performance of the system relates directly to the temporal resolution of the input signal and the discriminatory algorithm. The interpolation and multithreshold discriminators are used to achieve accurate measurement of digital

The remaining chapter is set out as follows. Section 2 explains the materials and methods. The VHDL-based system implementation and synthesis on FPGA are

by using discriminators and time to digital converters (TDCS) [11, 22–27].

that is probably using the most suitable radiopharmaceutical [1].

cardiology and cardiac surgery, and neurology and psychiatry.

these photons, which transform them into electrical pulses [8–13].

calculators are being used to resolve these deficits [11, 16–21].

information [12–15].

*Integrated Circuits/Microchips*

filters [11, 12, 16–21].

timestamps [11, 28].

**174**

three-dimensional tomographs.

PET pulses readout and conditioning are used to maximally preserve this

**Figure 1** displays a block diagram of the proposed system. It illustrates that in the intended patient body "1" an appropriately controlled quantity of the radioactive tracer "2" is injected. The radioactive tracer used in this analysis is 18F-FDG. Most of this tracer is distributed through contaminated brain cells after a certain time [5]. It originates β+, which is annihilated with electrons of medium. Every annihilation emits two 511-keV energy γ-rays. These are simultaneously released at about 180° relative angle and interact with crystal scintillators of the detection sensors "4." Scintillators convert energies of γ-rays into photons of light. Afterward, photo-detectors pick up these photons and turn them into electrical pulses [8–13]. A determined number of sensors are used in a detection ring "3." The front-end electronics, located in the detection sensors "4," further prepare and process PET pulses emanating from photo-detectors. For the construction of tomographs, the selected pulses with extracted parameters like addresses of the intended crystal and the involved sensor are conveyed to the IRM "5."

#### **2.1 The detection ring**

The detection ring consists of a group of four sensors, arranged axially around an "B" axis in a circle. Most contemporary PET scanners are built from radially arranged scintillators [15, 23]. The use of scintillator axial arrangements with the

**Figure 1.** *Block diagram of the proposed system.*

appropriate treatment of lateral sides of scintillators could increase the scanner's spatial resolution compared to alternatives, based on the radial arrangement of scintillators [11, 23].

**2.2 Digital conditioning of the PET pulses**

*DOI: http://dx.doi.org/10.5772/intechopen.92193*

estimate the timestamp [11, 23].

**2.3 The BLR**

**Figure 3**).

**Figure 3.**

**Figure 4.**

**177**

*The BLR and interpolator blocks.*

*The architecture of BLR module.*

The amplification enhances the resistance against noise in the PET pulses [29]. Besides, this amplification can also inject random offsets in the processed pulses. It influences the performance of energy estimators and degrades DOI accuracy. In the same manner, the digitization of selected PET pulses degrades the temporal resolu-

*Computationally Efficient Hybrid Interpolation and Baseline Restoration of the Brain-PET Pulses*

timestamp and LORs' estimators. A rise in *FS* can enhance the temporal precision. It does, however, lead to a costly solution in terms of cost and power utilization [30]. The suggested BLR and interpolator processes concepts are illustrated by using

The BLR concept is shown in **Figure 4**. Two equivalent modules are introduced, working independently to retrieve the selected pair of PET pulses: *Ska* and *Skb* (cf.

**Figure 4** shows that a real-time offset value, *OCalc*, is first estimated from the

incoming pulse by the BLR. It is determined as a mean of the baseline of the incoming pulse. The concept can be interpreted in mathematical terms using Eq. (1), where *N* represents the count of concerned samples, belonging to the digitized pulse baseline. *Skn* represents the sampled signal and *n* indexes the considered samples. *N* belongs to the set {1, 2, … , *N*}. *N* is selected in accordance to the

tion. The reduced temporal resolution lowers the performance of the post

**Figure 3**. It indicates that the BLR mainly processes the digitized versions of selected pulses in order to recover their baselines. Outputs of BLRs are conveyed to energy estimators and an adder. The added pulses are interpolated. It is performed to ameliorate the temporal resolution. Onward, the up-sampled signal is used to

Each sensor is composed of a scintillator crystal matrix. The LYSO scintillators are used in this study [6, 7]. The γ-rays interact with these crystals and turn their energies into photons of light. These photons are then sensed by two photo-detector matrices, positioned on both sides of the scintillator matrix. Such photo-detectors produce two electrical pulses as a result of each interaction [11, 26]. Each face of a crystal scintillator is positioned with a photo-detector. Only 2x*P* photo-detectors are used for a matrix of *P* crystal scintillators (cf. **Figure 2**). It reveals a γ-ray's interaction with one scintillator crystal "6." The crystal has faces "6a" and "6b" that are paired with two "7a" and "7b" MPPCs, respectively. The PET pulses produced by MPPCs are transferred to the electronics front-end module "8." These pulses are sorted and processed and the selected pulses with extracted information are conveyed to IRM "5."

A higher count of photons is attainable by MPPCs. Moreover, their behavior is free of the magnetic field influence. It is the reason behind the frequent usage of MPPCs in PET scanners as photo-detectors. The pixel size of the utilized MPPCs is 50 50μm2 . Each MPPC contains 3600 pixels. The MPPC-generated pulses are of extremely low amplitude [20]. Hundreds of microamperes bound the highest amplitude [12]. Thus, they are noise sensitive and can be completely malformed. Such pulses are amplified to enhance their immunity. It is realized by cascading a charge resistance with an appropriate bandwidth amplifier [29]. Such amplified pulses are transferred to the blocks embedded in "8" for signal selection and address encoding. They pick pulses among a matrix for the active crystal and also ensure that only one crystal is enabled at a time. If this requirement is satisfied, then the further processing of the selected pair of pulses is realized.

An appropriate baseline for the selected pulses should be available for the correct functionality of the suggested BLR [12]. Two lumped delay lines are used for each sensor in this context [30]. They incorporate a delay of 50 ns, in the selected PET pulses, with a standard deviation of only about 100 ps.

The delayed pulses, after A/D conversion, are conveyed to the IRM. To attain appropriate accuracy, the sampling frequency, FS, and the quantizer resolution should be selected tactfully [11, 12, 29]. In this case, the selected and prepared PET pulses are acquired with two 12-bit resolution Analog-to-Digital Converters (ADCs), functioning at FS = 200 MHz. Besides the chosen pulses, addresses of the effective crystal and the sensor in question are also encoded and communicated to the IRM. These data, obtained from different sensors in the detection ring, are used by the IRM for calculating DOIs and LORs. In this way, the three-dimensional tomographs are produced.

**Figure 2.** *The sensor principle.*

*Computationally Efficient Hybrid Interpolation and Baseline Restoration of the Brain-PET Pulses DOI: http://dx.doi.org/10.5772/intechopen.92193*

### **2.2 Digital conditioning of the PET pulses**

The amplification enhances the resistance against noise in the PET pulses [29]. Besides, this amplification can also inject random offsets in the processed pulses. It influences the performance of energy estimators and degrades DOI accuracy. In the same manner, the digitization of selected PET pulses degrades the temporal resolution. The reduced temporal resolution lowers the performance of the post timestamp and LORs' estimators. A rise in *FS* can enhance the temporal precision. It does, however, lead to a costly solution in terms of cost and power utilization [30].

The suggested BLR and interpolator processes concepts are illustrated by using **Figure 3**. It indicates that the BLR mainly processes the digitized versions of selected pulses in order to recover their baselines. Outputs of BLRs are conveyed to energy estimators and an adder. The added pulses are interpolated. It is performed to ameliorate the temporal resolution. Onward, the up-sampled signal is used to estimate the timestamp [11, 23].

#### **2.3 The BLR**

appropriate treatment of lateral sides of scintillators could increase the scanner's spatial resolution compared to alternatives, based on the radial arrangement of

Each sensor is composed of a scintillator crystal matrix. The LYSO scintillators are used in this study [6, 7]. The γ-rays interact with these crystals and turn their energies into photons of light. These photons are then sensed by two photo-detector matrices, positioned on both sides of the scintillator matrix. Such photo-detectors produce two electrical pulses as a result of each interaction [11, 26]. Each face of a crystal scintillator is positioned with a photo-detector. Only 2x*P* photo-detectors are used for a matrix of *P* crystal scintillators (cf. **Figure 2**). It reveals a γ-ray's interaction with one scintillator crystal "6." The crystal has faces "6a" and "6b" that are paired with two "7a" and "7b" MPPCs, respectively. The PET pulses produced by MPPCs are transferred to the electronics front-end module "8." These pulses are sorted and processed and the selected pulses with extracted information are con-

A higher count of photons is attainable by MPPCs. Moreover, their behavior is free of the magnetic field influence. It is the reason behind the frequent usage of MPPCs in PET scanners as photo-detectors. The pixel size of the utilized MPPCs is

An appropriate baseline for the selected pulses should be available for the correct functionality of the suggested BLR [12]. Two lumped delay lines are used for each sensor in this context [30]. They incorporate a delay of 50 ns, in the selected PET

The delayed pulses, after A/D conversion, are conveyed to the IRM. To attain appropriate accuracy, the sampling frequency, FS, and the quantizer resolution should be selected tactfully [11, 12, 29]. In this case, the selected and prepared PET pulses are acquired with two 12-bit resolution Analog-to-Digital Converters (ADCs), functioning at FS = 200 MHz. Besides the chosen pulses, addresses of the effective crystal and the sensor in question are also encoded and communicated to the IRM. These data, obtained from different sensors in the detection ring, are used by the IRM for calculating DOIs and LORs. In this way, the three-dimensional

extremely low amplitude [20]. Hundreds of microamperes bound the highest amplitude [12]. Thus, they are noise sensitive and can be completely malformed. Such pulses are amplified to enhance their immunity. It is realized by cascading a charge resistance with an appropriate bandwidth amplifier [29]. Such amplified pulses are transferred to the blocks embedded in "8" for signal selection and address encoding. They pick pulses among a matrix for the active crystal and also ensure that only one crystal is enabled at a time. If this requirement is satisfied, then the

further processing of the selected pair of pulses is realized.

pulses, with a standard deviation of only about 100 ps.

. Each MPPC contains 3600 pixels. The MPPC-generated pulses are of

scintillators [11, 23].

*Integrated Circuits/Microchips*

veyed to IRM "5."

tomographs are produced.

**Figure 2.** *The sensor principle.*

**176**

50 50μm2

The BLR concept is shown in **Figure 4**. Two equivalent modules are introduced, working independently to retrieve the selected pair of PET pulses: *Ska* and *Skb* (cf. **Figure 3**).

**Figure 4** shows that a real-time offset value, *OCalc*, is first estimated from the incoming pulse by the BLR. It is determined as a mean of the baseline of the incoming pulse. The concept can be interpreted in mathematical terms using Eq. (1), where *N* represents the count of concerned samples, belonging to the digitized pulse baseline. *Skn* represents the sampled signal and *n* indexes the considered samples. *N* belongs to the set {1, 2, … , *N*}. *N* is selected in accordance to the

**Figure 3.** *The BLR and interpolator blocks.*

**Figure 4.** *The architecture of BLR module.*

employed delay line and FS [29]. *N* = 8 is selected. It enables the mean value of the incoming pulse baseline to be estimated through accumulation and the right shifting while eviting the utilization of a complex conventional divider.

$$O\_{\text{Calc}} = \frac{1}{N} \sum\_{n=1}^{N} \mathbf{x}\_n \tag{1}$$

The restored pulse, free of offset, is obtained by employing Eq. (2) where *ykn* is the nth restored signal sample and *Skn* is the *nth* input signal sample.

$$\mathcal{Y}\_{k\_n} = \mathbb{S}\_{k\_n} - \mathcal{O}\_{\text{Calc}} \tag{2}$$

This *OCalc* estimation in real-time makes the suggested BLR self-adjustable. The value of *OCalc* adapts as a function of the intended PET pulse. It allows an effective restoration of the incoming pulses with a diverse range of bipolar offsets [29].

#### **2.4 The hybrid interpolator**

The concept of the suggested hybrid interpolator is clear from the block diagram, shown in **Figure 5**.

#### *2.4.1 The leading-edge selection*

**Figure 5** shows that the intended signal *xkn* , generated as sum of the outputs of both BLRs, is preliminary conveyed to the signal leading-edge selector. The attention on the leading-edge is due to the form of post timestamper, which is formed by combining multi-thresholds leading-edge discriminators [11].

The signal selection mechanism is illustrated by using **Figure 6**, where *Vmax* is the input signal maximum amplitude, *α* is a percentage of *Vmax* and is chosen equal to 10% of *Vmax*. If the *nth* input signal sample, *xkn* , crosses the α*.Vmax* threshold then *Q+2* samples are selected. The association among the *nth* nominated signal sample, *xsn*, and the *nth* input signal sample, *xkn* , can be presented in mathematical terms with Eq (3).

$$\infty\_n = \infty\_{k\_{n+i}}, where \ i = \{-1, 0, 1, 2, \ldots, N\} \tag{3}$$

Once that threshold is exceeded by an input sample, the magnitude comparator output will be high. It is used as a logic and control unit notification that allows the circular buffer to output the *xsn*. The logic and control unit is based on a counter and a J-K latch. The comparator pilots the latch. Once set, it enables the output port of the buffer. The counter provides the address of buffer registers needed to read. Finally, the logic and control unit resets the J-K latch, after reading the *xsn*.

*Computationally Efficient Hybrid Interpolation and Baseline Restoration of the Brain-PET Pulses*

The chosen portions of pulses are up-sampled with interpolation factor, *IF* = 4, by utilizing the primary interpolator stage. It means three equally spaced samples are positioned statistically between the initial two consecutive samples. It fourfold ameliorates the first stage output temporal resolution compared to its input. The Weighted Least Square Interpolator (WLSI) up-samples at the first stage [11]. WLSI coefficients are determined using 10,000 summed pulses. The summed pulses provided by the adder "9" are used as reference ones (cf. **Figure 9**). It allows the interpolated values to be compared with the reference ones and adapts the interpolation coefficients to reduce the differences from the reference signal. In particular, these coefficients are determined by utilizing the Least Square (LS) algorithm to diminish the squares of discrepancies among the interpolated values and the actual ones. Five samples of *xsn* are passed to the first stage interpolator input (cf. Section 3). After up-sampling, it outputs 17 samples. The error functions

of the WLSI coefficients for the 12 implanted samples are estimated as:

*2.4.2 First stage interpolator*

*The architecture of signal leading-edge selector.*

*DOI: http://dx.doi.org/10.5772/intechopen.92193*

**Figure 6.**

**Figure 7.**

**179**

*Concept of signal selection.*

This process of selection prevents processing of the entire signal length and thus dramatically improves the performance of the proposed system in terms of computation and power consumption [11].

The signal selector composes of a magnitude comparator, a circular buffer, and a module for logic and control. The concept is illustrated by using **Figure 7**. It displays that each input signal sample is compared with the predefined threshold *α*.*Vmax*.

**Figure 5.** *The block diagram of hybrid interpolator block.*

*Computationally Efficient Hybrid Interpolation and Baseline Restoration of the Brain-PET Pulses DOI: http://dx.doi.org/10.5772/intechopen.92193*

**Figure 6.** *Concept of signal selection.*

employed delay line and FS [29]. *N* = 8 is selected. It enables the mean value of the incoming pulse baseline to be estimated through accumulation and the right shifting

*n*¼1

The restored pulse, free of offset, is obtained by employing Eq. (2) where *ykn* is

This *OCalc* estimation in real-time makes the suggested BLR self-adjustable. The value of *OCalc* adapts as a function of the intended PET pulse. It allows an effective restoration of the incoming pulses with a diverse range of bipolar offsets [29].

The concept of the suggested hybrid interpolator is clear from the block dia-

**Figure 5** shows that the intended signal *xkn* , generated as sum of the outputs of both BLRs, is preliminary conveyed to the signal leading-edge selector. The attention on the leading-edge is due to the form of post timestamper, which is formed by

The signal selection mechanism is illustrated by using **Figure 6**, where *Vmax* is the input signal maximum amplitude, *α* is a percentage of *Vmax* and is chosen equal to 10% of *Vmax*. If the *nth* input signal sample, *xkn* , crosses the α*.Vmax* threshold then *Q+2* samples are selected. The association among the *nth* nominated signal sample, *xsn*, and the *nth* input signal sample, *xkn* , can be presented in mathematical terms

This process of selection prevents processing of the entire signal length and thus dramatically improves the performance of the proposed system in terms of compu-

The signal selector composes of a magnitude comparator, a circular buffer, and a module for logic and control. The concept is illustrated by using **Figure 7**. It displays that each input signal sample is compared with the predefined threshold *α*.*Vmax*.

*xn* (1)

*ykn* ¼ *Skn* � *OCalc* (2)

, *where i* ¼ �f g 1, 0, 1, 2, *::*, *N* (3)

*OCalc* ¼ 1 *N* X *N*

while eviting the utilization of a complex conventional divider.

the nth restored signal sample and *Skn* is the *nth* input signal sample.

combining multi-thresholds leading-edge discriminators [11].

*xsn* ¼ *xkn*þ*i*

tation and power consumption [11].

*The block diagram of hybrid interpolator block.*

**2.4 The hybrid interpolator**

*Integrated Circuits/Microchips*

*2.4.1 The leading-edge selection*

gram, shown in **Figure 5**.

with Eq (3).

**Figure 5.**

**178**

#### **Figure 7.** *The architecture of signal leading-edge selector.*

Once that threshold is exceeded by an input sample, the magnitude comparator output will be high. It is used as a logic and control unit notification that allows the circular buffer to output the *xsn*. The logic and control unit is based on a counter and a J-K latch. The comparator pilots the latch. Once set, it enables the output port of the buffer. The counter provides the address of buffer registers needed to read. Finally, the logic and control unit resets the J-K latch, after reading the *xsn*.

#### *2.4.2 First stage interpolator*

The chosen portions of pulses are up-sampled with interpolation factor, *IF* = 4, by utilizing the primary interpolator stage. It means three equally spaced samples are positioned statistically between the initial two consecutive samples. It fourfold ameliorates the first stage output temporal resolution compared to its input. The Weighted Least Square Interpolator (WLSI) up-samples at the first stage [11].

WLSI coefficients are determined using 10,000 summed pulses. The summed pulses provided by the adder "9" are used as reference ones (cf. **Figure 9**). It allows the interpolated values to be compared with the reference ones and adapts the interpolation coefficients to reduce the differences from the reference signal. In particular, these coefficients are determined by utilizing the Least Square (LS) algorithm to diminish the squares of discrepancies among the interpolated values and the actual ones. Five samples of *xsn* are passed to the first stage interpolator input (cf. Section 3). After up-sampling, it outputs 17 samples. The error functions of the WLSI coefficients for the 12 implanted samples are estimated as:

In Eqs. (4)–(6), *K* shows the total number of summed pulses, used in the measurement. In this case, *K* = 5000 is selected*. C* is the count of samples per intended pulse*.* Eq. (7) determines the matrix of coefficients *w*, referring to the

*Computationally Efficient Hybrid Interpolation and Baseline Restoration of the Brain-PET Pulses*

*w* ¼ *XT:X* � ��1

*X* ¼

*Xref* ¼

0

B@

0

B@

where *X* is the matrix of the initial values used for the interpolation. *Xref* is the matrix of reference samples at the time instants where approximations are made.

> *xs*11 … *xs*15 ……… *xsNp*1 … *xsNp*5

*xref* 11 … *xref* 112 …… … *xrefNp*1 … *xrefNp*12

The program for setting the *X* and *Xref* matrices and carrying out the matrix computations to calculate the WLSI coefficients is specifically implemented in MATLAB [31]. The architecture utilized by the designed WLSI can be seen in

The output of first stage is further up-sampled by the second stage interpolator. It is presumed that the signal between two successive samples changes linearly. Therefore, in the second stage, the SLI with an IF = 4 is deployed. The value of *nth* approximated value, *xrn*, corresponding to the *nth* interpolating instant, *trn*, is equal

to the mean of its previous and following incoming samples (cf. Eq. (10)).

*:XT:Xref* (7)

CA (8)

CA (9)

1

1

minimum errors.

*DOI: http://dx.doi.org/10.5772/intechopen.92193*

**Figure 10**.

**Figure 10.**

**181**

*The architecture of SLI.*

*2.4.3 Second stage interpolator*

**Figure 8.** *The experimental setup.*

**Figure 9.** *The architecture of WLSI.*

$$ferr\_1(w\_{1,1}, w\_{2,1}, \dots w\_{5,1}) = \sum\_{\mathcal{N} = 1}^{K} \sum\_{n=1}^{C} \left( \infty\_1.w\_{1,1} + \infty\_2.w\_{2,1} + \dots \infty\_5.w\_{5,1} - \infty\_7 \mathbb{I} \mathcal{N}p\_n \right)^2. \tag{4}$$

$$\left(ferm\_2(w\_{1,2}, w\_{2,2}, \dots w\_{5,2}) = \sum\_{Np=1}^{K} \sum\_{n=1}^{C} \left( \infty\_1.w\_{1,2} + \infty\_2.w\_{2,2} + \dots \infty\_5.w\_{5,2} - \infty\_7 \text{fNp}\_n \right)^2. \tag{5}$$

: :

$$ferr\_{12}(w\_{1,12}, w\_{2,12}, \dots w\_{5,12}) = \sum\_{Np=1}^{K} \sum\_{n=1}^{C} \left( \mathbf{x}\_1 \boldsymbol{w}\_{1,12} + \mathbf{x}\_2 \boldsymbol{w}\_{2,12} + \dots \mathbf{x}\_5 \boldsymbol{w}\_{5,12} - \mathbf{x} \mathbf{r} \mathbf{f} \mathbf{N} p\_n \right)^2 \tag{6}$$

*Computationally Efficient Hybrid Interpolation and Baseline Restoration of the Brain-PET Pulses DOI: http://dx.doi.org/10.5772/intechopen.92193*

In Eqs. (4)–(6), *K* shows the total number of summed pulses, used in the measurement. In this case, *K* = 5000 is selected*. C* is the count of samples per intended pulse*.* Eq. (7) determines the matrix of coefficients *w*, referring to the minimum errors.

$$\boldsymbol{w} = \left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{X}\_{\text{ref}}\tag{7}$$

where *X* is the matrix of the initial values used for the interpolation. *Xref* is the matrix of reference samples at the time instants where approximations are made.

$$X = \begin{pmatrix} \varkappa \mathfrak{s} \mathbf{1}\_1 & \dots & \varkappa \mathbf{1}\_5 \\ \dots & \dots & \dots \\ \varkappa \varkappa \mathbf{N} p\_1 & \dots & \varkappa \mathbf{N} p\_5 \end{pmatrix} \tag{8}$$

$$X \text{ref}^\circ = \begin{pmatrix} \varkappa \text{ref}^\circ \mathbf{1}\_1 & \dots & \varkappa \text{ref}^\circ \mathbf{1}\_{12} \\ \dots & \dots & \dots \\ \varkappa \text{ref} \mathbf{N} p\_1 & \dots & \varkappa \text{ref} \mathbf{N} p\_{12} \end{pmatrix} \tag{9}$$

The program for setting the *X* and *Xref* matrices and carrying out the matrix computations to calculate the WLSI coefficients is specifically implemented in MATLAB [31]. The architecture utilized by the designed WLSI can be seen in **Figure 10**.

#### *2.4.3 Second stage interpolator*

The output of first stage is further up-sampled by the second stage interpolator. It is presumed that the signal between two successive samples changes linearly. Therefore, in the second stage, the SLI with an IF = 4 is deployed. The value of *nth* approximated value, *xrn*, corresponding to the *nth* interpolating instant, *trn*, is equal to the mean of its previous and following incoming samples (cf. Eq. (10)).

**Figure 10.** *The architecture of SLI.*

*ferr*1ð*w*1,1, *w*2,1, … *w*5,1Þ ¼ X

**Figure 9.**

**180**

**Figure 8.**

*The experimental setup.*

*Integrated Circuits/Microchips*

*The architecture of WLSI.*

*ferr*2ð*w*1,2, *w*2,2, … *w*5,2Þ ¼ X

*ferr*12ð*w*1,12, *w*2,12, … *w*5,12Þ ¼ X

*K*

X *C*

*xs*1*:w*1,1 þ *xs*2*:w*2,1 þ … *xs*5*:w*5,1 � *xrefNpn* � �2

*xs*1*:w*1,2 þ *xs*2*:w*2,2 þ … *xs*5*:w*5,2 � *xrefNpn* � �2

*xs*1*:w*1,12 þ *xs*2*:w*2,12 þ … *xs*5*:w*5,12 � *xrefNpn* � �2

*:*

(4)

*:*

(5)

(6)

*n*¼1

X *C*

*n*¼1

X *C*

*n*¼1

: :

*Np*¼1

*K*

*Np*¼1

*K*

*Np*¼1

$$
\omega r\_n = \frac{\varkappa s\_{n-1} + \varkappa s\_n}{2} \tag{10}
$$

In this way, the temporal resolution of the second stage interpolator output is 16 fold superior than that of the original incoming signal. The second stage interpolator collects 17 samples from the prior WLSI module and outputs 65 samples. **Figure 8** illustrates the architecture and working of SLI. It indicates that 16 identical modules are operating concurrently.

#### **2.5 The experimental setup**

The system applicability is studied by using an experimental setup, as shown in **Figure 9**. It indicates that the radioactive tracer, *22Na*, is relocated by a robotic arm with respect to the utilized crystal scintillators matrix "6." The gap among successive relocation steps remains equivalent to 1 mm. The system includes a sensor, consisting of four LYSO crystals of 3 � 3 � 60 mm3 , enclosed from both sides with 3 � 3 mm2 arrays of Hamamatsu MPPCs, "7a" and "7b." MPPC array consists of four MPPCs of 3 � 3 mm2 area, to suit the used scintillators.

The front-end electronics modules, "8a" and "8b," manage the pulses, which come from both MPPC matrices "7a" and "7b." They realize amplification, selection, and addition of delay. Signal selection and address encoding block "10" is used to process outputs of "8a" and "8b." The selected pulses are then added using the adder circuit "9." Finally, the selected pulses supplied by "10" and their sum, supplied by "9" are digitized concurrently by using an oscilloscope. The LeCroy WavePro 7300A oscilloscope is utilized in this study. The pulses, produced by "10" are digitized at FS = 200 MHz. The sum of pulses produced by "9" is digitized at the sampling rate of 3.2GHz. Later, when characterizing the hybrid interpolation module, these pulses, obtained at a frequency of 3.2 GHz, are used as reference ones. The approach is described further in Section 3.2.

#### **3. VHDL implementation and synthesis on FPGA**

The designed BLR and hybrid interpolator modules are implemented in VHDL and are synthesized on an FPGA. Preliminarily the 28-nm technology-based "xc7a200t FPGA" from the "Artix-7" family is considered [32]. The circuit VHDL implementation is synthesized on the selected FPGA by using the Xilinx Synthesis Technology (XST) [33, 34]. The system Register-Transfer-Level (RTL) schematics are shown in **Figure 11a** and **Figure 11b**. **Figure 11a** shows the top-level RTL schematic and **Figure 11b** shows the components-level RTL schematic.

The post-synthesis summary of the FPGA resource utilization, by one BLR and hybrid interpolator, is presented in **Table 1**. It shows that more than 10 proposed BLR and hybrid interpolator chains can be implemented on a single *xc7a200t* chip. The available DSP-blocks, on a single *xc7a200t* chip, mainly pose this limitation on the count of proposed conditioning chains. In case of need, this limitation can be resolved by using alternate logic-cells-based architectures and implementations, which minimize the use of DSP-blocks.

#### **4. Results**

The statistical features of the PET pulses such as rise-time, fall-time, bandwidth, and magnitude are extracted. It is carried out by analyzing the PET pulses delivered by "10." Ten thousand pulses are considered through this operation. They are collected at a sampling rate of 200 MHz by utilizing the inbuilt ADCs of the

Bandwidth (MHz) 35 4 Maximum amplitude, Vmax, after addition of both pulses(mV) 1010 55

Parameter Mean value Standard deviation

Resource Input-output pins (IOs) Slice registers Slice LUTs DSP48E1s Block RAMs Used 36 1223 818 60 8 Total 500 269,200 134,600 740 365 % utilization 7.2 0.5 0.6 8.1 2.2

*Computationally Efficient Hybrid Interpolation and Baseline Restoration of the Brain-PET Pulses*

*DOI: http://dx.doi.org/10.5772/intechopen.92193*

11 1.1

146 10

oscilloscope. A summary is presented in **Table 2**.

*Summary of amplified and conditioned pulses parameters.*

*Summary of the xc7a200t resources utilization.*

*(a) The top-level RTL schematic. (b) The components-level RTL schematic.*

Rise-time (ns) 10–90%

**Figure 11.**

**Table 1.**

Fall-time (ns) 10–90%

**Table 2.**

**183**

*Computationally Efficient Hybrid Interpolation and Baseline Restoration of the Brain-PET Pulses DOI: http://dx.doi.org/10.5772/intechopen.92193*

#### **Figure 11.**

*xrn* ¼ *xsn*�1 þ *xsn*

are operating concurrently.

*Integrated Circuits/Microchips*

**2.5 The experimental setup**

consisting of four LYSO crystals of 3 � 3 � 60 mm3

The approach is described further in Section 3.2.

which minimize the use of DSP-blocks.

**4. Results**

**182**

**3. VHDL implementation and synthesis on FPGA**

four MPPCs of 3 � 3 mm2 area, to suit the used scintillators.

In this way, the temporal resolution of the second stage interpolator output is 16 fold superior than that of the original incoming signal. The second stage interpolator collects 17 samples from the prior WLSI module and outputs 65 samples. **Figure 8** illustrates the architecture and working of SLI. It indicates that 16 identical modules

The system applicability is studied by using an experimental setup, as shown in **Figure 9**. It indicates that the radioactive tracer, *22Na*, is relocated by a robotic arm with respect to the utilized crystal scintillators matrix "6." The gap among successive relocation steps remains equivalent to 1 mm. The system includes a sensor,

3 � 3 mm2 arrays of Hamamatsu MPPCs, "7a" and "7b." MPPC array consists of

The front-end electronics modules, "8a" and "8b," manage the pulses, which come from both MPPC matrices "7a" and "7b." They realize amplification, selection, and addition of delay. Signal selection and address encoding block "10" is used to process outputs of "8a" and "8b." The selected pulses are then added using the adder circuit "9." Finally, the selected pulses supplied by "10" and their sum, supplied by "9" are digitized concurrently by using an oscilloscope. The LeCroy WavePro 7300A oscilloscope is utilized in this study. The pulses, produced by "10" are digitized at FS = 200 MHz. The sum of pulses produced by "9" is digitized at the sampling rate of 3.2GHz. Later, when characterizing the hybrid interpolation module, these pulses, obtained at a frequency of 3.2 GHz, are used as reference ones.

The designed BLR and hybrid interpolator modules are implemented in VHDL

The post-synthesis summary of the FPGA resource utilization, by one BLR and hybrid interpolator, is presented in **Table 1**. It shows that more than 10 proposed BLR and hybrid interpolator chains can be implemented on a single *xc7a200t* chip. The available DSP-blocks, on a single *xc7a200t* chip, mainly pose this limitation on the count of proposed conditioning chains. In case of need, this limitation can be resolved by using alternate logic-cells-based architectures and implementations,

The statistical features of the PET pulses such as rise-time, fall-time, bandwidth, and magnitude are extracted. It is carried out by analyzing the PET pulses delivered

and are synthesized on an FPGA. Preliminarily the 28-nm technology-based "xc7a200t FPGA" from the "Artix-7" family is considered [32]. The circuit VHDL implementation is synthesized on the selected FPGA by using the Xilinx Synthesis Technology (XST) [33, 34]. The system Register-Transfer-Level (RTL) schematics are shown in **Figure 11a** and **Figure 11b**. **Figure 11a** shows the top-level RTL schematic and **Figure 11b** shows the components-level RTL schematic.

2 (10)

, enclosed from both sides with

*(a) The top-level RTL schematic. (b) The components-level RTL schematic.*

#### **Table 1.**

*Summary of the xc7a200t resources utilization.*

#### **Table 2.**

*Summary of amplified and conditioned pulses parameters.*

by "10." Ten thousand pulses are considered through this operation. They are collected at a sampling rate of 200 MHz by utilizing the inbuilt ADCs of the oscilloscope. A summary is presented in **Table 2**.

### **4.1 The characterization of BLR**

The utilized delay lines inject 50-ns delays in the selected pulses. These are acquired with a sampling rate of 200 MHz. Thus, 10 samples are collected on the baselines of arriving pulses. While estimating *OCalc*, *N* = 8 is selected. It refers to primarily 40-ns portion of the baseline. On the accumulator output, the division with a factor of 8 is attained by executing a 3-bit right shift.

The interpolation errors are computed. Throughout this procedure, 10,000 interactions among the radioactive source and the scintillators matrix "6" are used.

*Computationally Efficient Hybrid Interpolation and Baseline Restoration of the Brain-PET Pulses*

The module "10" produces a pair of pulses as an outcome of each interaction. These pulses and their related sum are digitized with an oscilloscope. The pulses produced by the module "10" are recorded at a sampling frequency of 200 MHz. The sum of pulses produced by the adder circuit "9" is recorded at a sampling rate of 3.2 GHz. The error per interpolated sample, *Ien*, is estimated with Eq. (11), where, *xrefn* is the reference sample value with respect to the interpolation instant *trn*. It represents the summed analog signal that is sampled with an oscilloscope, at 3.2 GHz. *yrn* is the approximated value computed with respect to the interpolation

The error of the used interpolator is measured in terms of the Mean Square Deviation Error (RMSDE). The RMSDE is determined for each intended pulse by using Eq. (12), where, *C* is the number of samples that are considered for the incoming pulse. The number of summed pulses is indexed by *Np*. Eventually, the

For designed hybrid interpolator, the average RMSDE value is 13.6 μV. The interpolated signal is used for post timestamps estimation. The interaction time among the γ-ray and the crystal scintillator is computed by comparing the arriving pulse amplitude against determined thresholds. The time is estimated on the basis of thresholds crossing instants. Due to the discrete-time digitized pulses, a threshold crossing is estimated as intersection among the threshold and the straight line, crossing via two successive samples that lay across that threshold. For the

timestamp estimators, based on digital discriminators, the precession of timestamps is directly related to the temporal resolution and the magnitude accuracy of the

In this case, by using the suggested hybrid interpolator, the temporal resolution of the selected portion is enhanced 16-fold. It outputs an interpolated signal with 0.3125 ns of temporal resolution and 13.6 μV of average RMSDE. It is capable of significantly improving the accuracy of the post timestamp estimator while digitiz-

**4.3 Comparison of the hybrid interpolator with a conventional interpolator**

The effectiveness of the suggested hybrid interpolator is also compared with a counter mono-algorithm-based approach. Interpolation is achieved traditionally by using a unique algorithm. The computational load is directly related to the order of

The leading-edge selection module, in the proposed solution, allows concentrating on the pertinent portion of the signal. In the examined case, five samples per incoming pulse are picked. Without this element, however, the entire pulse duration of around 150 ns should be considered [11, 12]. The ADC will deliver 30 samples per incoming pulse, for a sampling frequency of 200 MHz. It will increase

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 *C:* X *C*

*n*¼1 *Ie*2 *n*

*Ien* ¼ *xref n* � *yrn* (11)

vuut (12)

The intended interactions are recorded at predetermined locations.

*DOI: http://dx.doi.org/10.5772/intechopen.92193*

RMSDE mean is determined as the average value of the *RMSDENp*.

ing pulses with ADCs that are economically accessible [11].

the interpolator and to the count of input samples.

the interpolator computational load with a factor of 6.

*RMSDENp* ¼

instant *trn*.

arriving pulses [11, 23].

**185**

**Figure 12** shows a typical performance of the conceived BLR. It depicts how well the offset, injected by the front-end electronics, is compensated from the received pulses. Amplification and conditioning chains in modules "8a" and "8b" add a particular offset in the received pulses. Due to the influence of various offsets, for a fixed location among the radiotracer and the crystal scintillator matrix "6," the incoming pulses demonstrate a peak amplitude dispersion of roughly 25% among them. With the application of the designed BLR, these dispersions are limited to approximately 1%. It reflects how the use of designed BLR will enhance the correctness of post energies and DOI estimators. It will also improve the performance of the scanner in terms of precision in the localization of tumor cells [11, 12].

### **4.2 The characterization of hybrid interpolator**

#### *4.2.1 The leading-edge selector*

BLRs' outputs are summed and then transferred to the signal selection unit. The selected pulses are digitized at a rate of 200 MHz. **Table 1** reveals that the incoming signal's mean [10] % rise time is 11 ns. It clarifies that there will be at least two samples, in the digitized version of the signal, on the signal leading-edge. Hence, *Q* = 3 is selected. It ensures a proper selection of the signal leading-edge, the most critical signal portion needed by the used timestamp estimator [11].

#### *4.2.2 The first and the second stage interpolators*

Signal leading-edge selector output is transferred to the first interpolation stage realized with WLSI. In this study, this interpolation stage receives five selected signal samples, initially sampled at 200 MHz. It outputs 17 samples, with a sampling rate of 800 MHz.

The SLI is selected to process the second interpolation stage. It receives 17 samples and outputs 65 samples with a sampling rate of 3.2 GHz.

**Figure 12.** *The baseline restoration process.*

*Computationally Efficient Hybrid Interpolation and Baseline Restoration of the Brain-PET Pulses DOI: http://dx.doi.org/10.5772/intechopen.92193*

The interpolation errors are computed. Throughout this procedure, 10,000 interactions among the radioactive source and the scintillators matrix "6" are used. The intended interactions are recorded at predetermined locations.

The module "10" produces a pair of pulses as an outcome of each interaction. These pulses and their related sum are digitized with an oscilloscope. The pulses produced by the module "10" are recorded at a sampling frequency of 200 MHz. The sum of pulses produced by the adder circuit "9" is recorded at a sampling rate of 3.2 GHz. The error per interpolated sample, *Ien*, is estimated with Eq. (11), where, *xrefn* is the reference sample value with respect to the interpolation instant *trn*. It represents the summed analog signal that is sampled with an oscilloscope, at 3.2 GHz. *yrn* is the approximated value computed with respect to the interpolation instant *trn*.

$$I e\_n = \varkappa r e f\_n - \mathcal{Y} r\_n \tag{11}$$

The error of the used interpolator is measured in terms of the Mean Square Deviation Error (RMSDE). The RMSDE is determined for each intended pulse by using Eq. (12), where, *C* is the number of samples that are considered for the incoming pulse. The number of summed pulses is indexed by *Np*. Eventually, the RMSDE mean is determined as the average value of the *RMSDENp*.

$$\text{RMSDE}\_{Np} = \sqrt{\frac{1}{C} \cdot \sum\_{n=1}^{C} I e\_n^2} \tag{12}$$

For designed hybrid interpolator, the average RMSDE value is 13.6 μV. The interpolated signal is used for post timestamps estimation. The interaction time among the γ-ray and the crystal scintillator is computed by comparing the arriving pulse amplitude against determined thresholds. The time is estimated on the basis of thresholds crossing instants. Due to the discrete-time digitized pulses, a threshold crossing is estimated as intersection among the threshold and the straight line, crossing via two successive samples that lay across that threshold. For the timestamp estimators, based on digital discriminators, the precession of timestamps is directly related to the temporal resolution and the magnitude accuracy of the arriving pulses [11, 23].

In this case, by using the suggested hybrid interpolator, the temporal resolution of the selected portion is enhanced 16-fold. It outputs an interpolated signal with 0.3125 ns of temporal resolution and 13.6 μV of average RMSDE. It is capable of significantly improving the accuracy of the post timestamp estimator while digitizing pulses with ADCs that are economically accessible [11].

#### **4.3 Comparison of the hybrid interpolator with a conventional interpolator**

The effectiveness of the suggested hybrid interpolator is also compared with a counter mono-algorithm-based approach. Interpolation is achieved traditionally by using a unique algorithm. The computational load is directly related to the order of the interpolator and to the count of input samples.

The leading-edge selection module, in the proposed solution, allows concentrating on the pertinent portion of the signal. In the examined case, five samples per incoming pulse are picked. Without this element, however, the entire pulse duration of around 150 ns should be considered [11, 12]. The ADC will deliver 30 samples per incoming pulse, for a sampling frequency of 200 MHz. It will increase the interpolator computational load with a factor of 6.

**4.1 The characterization of BLR**

*Integrated Circuits/Microchips*

The utilized delay lines inject 50-ns delays in the selected pulses. These are acquired with a sampling rate of 200 MHz. Thus, 10 samples are collected on the baselines of arriving pulses. While estimating *OCalc*, *N* = 8 is selected. It refers to primarily 40-ns portion of the baseline. On the accumulator output, the division

**Figure 12** shows a typical performance of the conceived BLR. It depicts how well the offset, injected by the front-end electronics, is compensated from the received pulses. Amplification and conditioning chains in modules "8a" and "8b" add a particular offset in the received pulses. Due to the influence of various offsets, for a fixed location among the radiotracer and the crystal scintillator matrix "6," the incoming pulses demonstrate a peak amplitude dispersion of roughly 25% among them. With the application of the designed BLR, these dispersions are limited to approximately 1%. It reflects how the use of designed BLR will enhance the correctness of post energies and DOI estimators. It will also improve the performance of the scanner in terms of precision in the localization of tumor cells [11, 12].

BLRs' outputs are summed and then transferred to the signal selection unit. The selected pulses are digitized at a rate of 200 MHz. **Table 1** reveals that the incoming signal's mean [10] % rise time is 11 ns. It clarifies that there will be at least two samples, in the digitized version of the signal, on the signal leading-edge. Hence, *Q* = 3 is selected. It ensures a proper selection of the signal leading-edge, the most

Signal leading-edge selector output is transferred to the first interpolation stage realized with WLSI. In this study, this interpolation stage receives five selected signal samples, initially sampled at 200 MHz. It outputs 17 samples, with a sampling

The SLI is selected to process the second interpolation stage. It receives 17

critical signal portion needed by the used timestamp estimator [11].

samples and outputs 65 samples with a sampling rate of 3.2 GHz.

with a factor of 8 is attained by executing a 3-bit right shift.

**4.2 The characterization of hybrid interpolator**

*4.2.2 The first and the second stage interpolators*

*4.2.1 The leading-edge selector*

rate of 800 MHz.

**Figure 12.**

**184**

*The baseline restoration process.*

The computational complexity for the suggested case is determined by adding the computational costs of both stages. The computational complexity of the optimized WLSI with an *IF* = 4 is 15 multiplications and 3 additions (cf. **Figure 10**). The computational complexity of SLI is 3 additions and 3 binary-weighted divisions (cf. **Figure 8**). Relative to the addition and multiplication operations, the circuit level complexity of the binary weighted division is insignificant. However, the complexity of a WLSI with an IF = 16 is 45 multiplications and 15 additions for approximating 15 samples among two consecutive originals [12].

tomographic resolution are directly linked to the computational accuracy of timestamps [11, 23]. The timestamp can be measured in either digital or analog worlds. The analog timestamp calculators require the development of complex, integrated circuits for specific applications. Using digital timestamp calculators can result in a cost-effective solution [23]. It allows a solution to be realized using regular ADCs and Field Programmable Gate Arrays (FPGAs). Comparing the amplitude of digitized versions of PET pulses to established thresholds, they measure timestamps [23, 24, 35]. The time is stamped by using the instants of the threshold crossing. Hence, the exactitude of timestamp calculations is directly related to the incoming pulses' temporal resolution and precision of magnitude. The incoming pulses are digitized at a sampling rate of 200 MHz in the examined case. It provides 5 ns of temporal resolution. Later, with the suggested hybrid interpolator, the temporal resolution of the selected portion of pulses is 16-fold enhanced. It results in an interpolated signal with a temporal resolution of 0.3125 ns and with RMSDE of 13.6 μV. It aptitudes a significant improvement in the precision of the post timestamp

*DOI: http://dx.doi.org/10.5772/intechopen.92193*

*Computationally Efficient Hybrid Interpolation and Baseline Restoration of the Brain-PET Pulses*

calculator while acquiring the pulses with economically available ADCs.

effectively in improving their tomographic resolution.

help in preparing figures for this book chapter.

The author declares no conflict of interest.

to be attained [12].

**Acknowledgements**

**Conflict of interest**

**187**

Component-level architectures of the suggested BLR and hybrid interpolator are described. The proposed chain is implemented in VHDL, and synthesis is realized on the *xc7a200t* FPGA. It is shown that more than 10 proposed BLR and hybrid interpolator chains can be implemented on a single *xc7a200t* chip, which costs around 260US\$. This reveals that the suggested concept can be developed, unlike traditional predecessors, by using cost-effective ADCs and FPGAs [12]. It prevents the production of complex high-performance specific integrated circuits and thus results in effective realization. In addition to cost-effectiveness, it also facilitates the device reconfiguration compared to hardwired circuits and allows similar precision

These results demonstrate the potential applicability of the proposed BLR and hybrid interpolator in current brain-PET scanners. They can be easily incorporated into contemporary PET scanners based on a simple architecture and can contribute

The author is thankful to Dr. D. Liksonov, Dr. A. Bakkali, and Dr. N. Tamda for the fruitful discussions, Mme. Rabia Qaisar, Ms. M. Tayeb, and Ms. E. Noor for their

By the grace of leading-edge selector, in the proposed case, the WLSI needs to interpolate 5 selected samples and it delivers 17 interpolated samples at its output. The overall computational cost of WLSI becomes 75 multiplications and 15 additions. The complexity of used SLI for processing 17 samples is 51 additions. It results in an overall cost of 75 multiplications and 66 additions. Contrary, in conventional case, the system has to process 30 samples for the whole pulse length, between 10% rise-time to 10% fall-time, of around 150 ns should be considered [1, 2]. It results in a computational cost of 1350 multiplications and 450 additions.

These results demonstrate that the suggested solution reaches an 18-fold gain in the count of multiplications and a 6.8-fold gain in the count of additions over the mono WLSI based solution.

The proposed hybrid interpolator's interpolation precision is also contrasted with the mono WLSI-based solution. Characterization is achieved by using the same pair of pulses registered and used during the measurement of the hybrid interpolator error. The interpolation error of the up-sampled signal outputs by the mono WLSI interpolator is estimated by using Eq. 12. It results in 11.4-μV RMSDE. For the case of designed hybrid interpolator, 13.6-μV RMSDE is attained. It reveals that the designed solution achieves substantial computational advantage over the mono WLSI-based tactic while achieving an analogous precision.

#### **5. Discussion and conclusion**

A brain-PET scanner's resolution and sensitivity rely on how accurate it is to measure the depths and times of interactions among scintillators and the γ-rays [1–3]. The exact measurement of the DOI requires an accurate estimate of the arriving PET pulses' energies [12]. A precise measurement of interaction times needs an exact calculation of timestamps [11]. This helps in accurate reconstruction of the LORs. Accurate DOIs and LORs contribute to the reconstruction of threedimensional high-resolution tomography, which allows a precise location of tumor cells in the patient's brain.

The readout electronics of PET pulses produce unpredictable offsets. It causes an incorrect estimation of these pulses' energies, which decreases the accuracy of measurement of the DOI and timestamp. As a result, it diminishes the scanner's ability to assess the location of tumor cells accurately. An efficient BLR is proposed for minimizing the impact of offsets.  By the grace of the suggested BLR, the peak amplitude dispersions for a determined position of the radioactive source with respect to the matrix of scintillators "6" is diminished from ≤25% to ≤1%. This assures that the use of designed BLR would enhance the precision of post energies and DOIs' estimators. This should improve the scanner's accuracy in the localization of tumor cells [1–3, 11].

After energies, the second significant parameter to be measured in brain-PET scanners is the timestamp of the annihilated γ-rays [11]. It is conducted to calculate LORs, which allow the tumor cells to be located. A PET scanner's sensitivity and

#### *Computationally Efficient Hybrid Interpolation and Baseline Restoration of the Brain-PET Pulses DOI: http://dx.doi.org/10.5772/intechopen.92193*

tomographic resolution are directly linked to the computational accuracy of timestamps [11, 23]. The timestamp can be measured in either digital or analog worlds. The analog timestamp calculators require the development of complex, integrated circuits for specific applications. Using digital timestamp calculators can result in a cost-effective solution [23]. It allows a solution to be realized using regular ADCs and Field Programmable Gate Arrays (FPGAs). Comparing the amplitude of digitized versions of PET pulses to established thresholds, they measure timestamps [23, 24, 35]. The time is stamped by using the instants of the threshold crossing. Hence, the exactitude of timestamp calculations is directly related to the incoming pulses' temporal resolution and precision of magnitude. The incoming pulses are digitized at a sampling rate of 200 MHz in the examined case. It provides 5 ns of temporal resolution. Later, with the suggested hybrid interpolator, the temporal resolution of the selected portion of pulses is 16-fold enhanced. It results in an interpolated signal with a temporal resolution of 0.3125 ns and with RMSDE of 13.6 μV. It aptitudes a significant improvement in the precision of the post timestamp calculator while acquiring the pulses with economically available ADCs.

Component-level architectures of the suggested BLR and hybrid interpolator are described. The proposed chain is implemented in VHDL, and synthesis is realized on the *xc7a200t* FPGA. It is shown that more than 10 proposed BLR and hybrid interpolator chains can be implemented on a single *xc7a200t* chip, which costs around 260US\$. This reveals that the suggested concept can be developed, unlike traditional predecessors, by using cost-effective ADCs and FPGAs [12]. It prevents the production of complex high-performance specific integrated circuits and thus results in effective realization. In addition to cost-effectiveness, it also facilitates the device reconfiguration compared to hardwired circuits and allows similar precision to be attained [12].

These results demonstrate the potential applicability of the proposed BLR and hybrid interpolator in current brain-PET scanners. They can be easily incorporated into contemporary PET scanners based on a simple architecture and can contribute effectively in improving their tomographic resolution.

#### **Acknowledgements**

The computational complexity for the suggested case is determined by adding the computational costs of both stages. The computational complexity of the optimized WLSI with an *IF* = 4 is 15 multiplications and 3 additions (cf. **Figure 10**). The computational complexity of SLI is 3 additions and 3 binary-weighted divisions (cf. **Figure 8**). Relative to the addition and multiplication operations, the circuit level complexity of the binary weighted division is insignificant. However, the complexity of a WLSI with an IF = 16 is 45 multiplications and 15 additions for

By the grace of leading-edge selector, in the proposed case, the WLSI needs to interpolate 5 selected samples and it delivers 17 interpolated samples at its output. The overall computational cost of WLSI becomes 75 multiplications and 15 additions. The complexity of used SLI for processing 17 samples is 51 additions. It results in an overall cost of 75 multiplications and 66 additions. Contrary, in conventional case, the system has to process 30 samples for the whole pulse length, between 10% rise-time to 10% fall-time, of around 150 ns should be considered [1, 2]. It results in

These results demonstrate that the suggested solution reaches an 18-fold gain in the count of multiplications and a 6.8-fold gain in the count of additions over the

The proposed hybrid interpolator's interpolation precision is also contrasted with the mono WLSI-based solution. Characterization is achieved by using the same pair of pulses registered and used during the measurement of the hybrid interpolator error. The interpolation error of the up-sampled signal outputs by the mono WLSI interpolator is estimated by using Eq. 12. It results in 11.4-μV RMSDE. For the case of designed hybrid interpolator, 13.6-μV RMSDE is attained. It reveals that the designed solution achieves substantial computational advantage over the mono

A brain-PET scanner's resolution and sensitivity rely on how accurate it is to measure the depths and times of interactions among scintillators and the γ-rays [1–3]. The exact measurement of the DOI requires an accurate estimate of the arriving PET pulses' energies [12]. A precise measurement of interaction times needs an exact calculation of timestamps [11]. This helps in accurate reconstruction of the LORs. Accurate DOIs and LORs contribute to the reconstruction of threedimensional high-resolution tomography, which allows a precise location of tumor

The readout electronics of PET pulses produce unpredictable offsets. It causes an

After energies, the second significant parameter to be measured in brain-PET scanners is the timestamp of the annihilated γ-rays [11]. It is conducted to calculate LORs, which allow the tumor cells to be located. A PET scanner's sensitivity and

incorrect estimation of these pulses' energies, which decreases the accuracy of measurement of the DOI and timestamp. As a result, it diminishes the scanner's ability to assess the location of tumor cells accurately. An efficient BLR is proposed for minimizing the impact of offsets.  By the grace of the suggested BLR, the peak amplitude dispersions for a determined position of the radioactive source with respect to the matrix of scintillators "6" is diminished from ≤25% to ≤1%. This assures that the use of designed BLR would enhance the precision of post energies and DOIs' estimators. This should improve the scanner's accuracy in the

approximating 15 samples among two consecutive originals [12].

a computational cost of 1350 multiplications and 450 additions.

WLSI-based tactic while achieving an analogous precision.

mono WLSI based solution.

*Integrated Circuits/Microchips*

**5. Discussion and conclusion**

cells in the patient's brain.

localization of tumor cells [1–3, 11].

**186**

The author is thankful to Dr. D. Liksonov, Dr. A. Bakkali, and Dr. N. Tamda for the fruitful discussions, Mme. Rabia Qaisar, Ms. M. Tayeb, and Ms. E. Noor for their help in preparing figures for this book chapter.