**Radiation Protection in Medical Imaging**

Horaţiu Colosi1, Dan Colosi2, Vlad Mureşan3 and Marius Roman3

*1Iuliu Haţieganu University of Medicine and Pharmacy, Cluj-Napoca 2State University of New York, School of Dental Medicine, Stony Brook, New York 3Technical University, Cluj-Napoca 1,3Romania 2USA* 

## **1. Introduction**

24 Will-be-set-by-IN-TECH

66 Medical Imaging

Zhan, Y. & Shen, D. (2006). Deformable segmentation of 3d ultrasound prostate image

Zhang, J. & Tan, T. (2002). Brief review of invariant texture analysis methods, *Pattern*

Zhu, S. & Yuille, A. (1996). Region competition: unifying snake/balloon, region growing,

Zucker, S. W. (1976). Toward a model of texture, *Computer Graphics and Image Processing*

*Pattern Analysis and Machine Intelligence* 18(9): 884–900.

25(3): 256–272.

5: 190–202.

*Recognition* 35: 735–747.

using statistical texture matching method, *IEEE Transactions on Medical Imaging*

and Bayes/MDL/energy for multi-band image segmentation, *IEEE Transactions on*

Life on Earth has developed continuously in the presence of natural radiation. Radiation is a form of energy transfer through space and matter. The ability of radiation to transfer energy at the molecular level to living organisms is at the root of current efforts to understand and limit the effects of radiation on humans.

The radiation spectrum includes forms of radiation that carry various energy levels. Radiation forms, such as x-radiation, which carry sufficient energy to overcome binding energies of electrons and to eject them from their atomic orbits, can cause the formation of ions in the matter through which they travel, and are termed ionizing radiation. Ionizing radiation may also dissociate molecular bonds, leading to the formation of free radicals.

#### **1.1 Biologic effects of ionizing radiation**

A great proportion of the effects of ionizing radiation on living matter are a result of the interaction of these forms of radiation with water, a ubiquitous molecule that is present in most tissues and organs. This interaction leads to the generation of highly reactive radical species, mainly from the hydroxyl and hydroperoxyl categories, which, in turn, can interact with cellular DNA, leading to the formation of strand breaks and base damage (Hall & Giaccia 2005). To a lesser extent, radiation can also directly ionize or damage DNA and other biologic molecules: RNA, proteins, lipids. Many occurrences of damage limited to one DNA strand are rapidly detected by the surveillance mechanisms of the cell and the DNA integrity is restored using the intact complementary DNA strand as template. However, double-strand breaks, in which an intact DNA template is not available, are more difficult to repair accurately (Mladenov & Iliakis 2011). As a result, radiation-induced DNA double strand breaks may often lead to misrepaired DNA, resulting in point mutations, deletions or chromosomal translocations. Radiation-induced DNA damage that escapes detection by cell cycle checkpoints during cellular mitosis can result in an uncontrolled, accelerated rate of mitotic division of the affected cell (Hall & Giaccia 2005), a cellular hallmark of cancer. The association between exposure to radiation and cancer induction is well documented through epidemiological studies (Ron 2003).

Radiation Protection in Medical Imaging 69

and tends to accumulate in basements, from where it may leak into adjacent living spaces. Its radioactive decay products are inhaled, taking residence in the respiratory epithelium and contributing to the occurrence of lung cancer. Radiation exposure from radon varies

Human activity has led to the development of new sources of radiation over the past decades. Medical radiation is the greatest peaceful contributor to radiation exposure in the general

Medical use of radiation has increased continuously throughout the world, irrespective of the economic level of various regions. While therapeutic use of radiation in the treatment of cancer contributes a significant amount of radiation to a small number of individuals, the use of diagnostic radiation imparts a smaller amount of radiation to a large, and constantly

The development of computed tomography (CT) over the past few decades has revolutionized diagnostic imaging. Its use has increased exponentially, and it is estimated that in the United States more than 62 million CT scans have been performed in 2006, compared to less than 20 million in 1995, and just 3 million in 1980 (Brenner & Hall, 2007). Due to the relatively higher radiation dose from a typical CT scan compared to that from conventional imaging modalities (table 2), and due to its widespread and increasing use, medical CT is currently estimated to contribute a significant amount of radiation exposure

Head plain radiograph 27 – 100 **4 – 15**  Dental radiographs (set of 19-20) \* 33 – 150 **5 – 23**  Chest plain radiographs (PA + lateral) 100 – 290 **15 – 44**  Mammography 230 – 400 **35 – 123**  Abdomen plain radiograph 370 – 810 **56 – 123**  Head computed tomography 900 – 2800 **137 – 426**  Abdomen and pelvis computed tomography 3100 – 15300 **1.3 – 6.4 years**  Table 2. Medical diagnostic imaging procedures and their contribution range to patients' radiation exposure (data: United Nations Scientific Committee on the Effects of Atomic

Due to the large number of CT scans being administered in the present, and the trend of this number to increase in the future, radiation exposure from CT may gain significance as a

As a result, current radiation safety efforts carry a significant focus on monitoring and

limiting medical radiation exposure from CT to the general population.

**Effective Dose (µSv)**

**Background Equivalent (days)**

population, with an average annual effective dose of approximately 0.6 mSv (table 1).

**2. Radiation exposure and current practices for radiation protection** 

widely with the geologic composition throughout the world.

**2.1 Radiation exposure in diagnostic imaging** 

Radiation Report 2008, \* White and Pharoah, 2009)

public health concern (Brenner and Hall, 2007).

growing, number of individuals.

to the general population.

## **1.2 Radiation doses**

Various quantities are used to describe the radiation dose to organisms and the resulting risk. The absorbed dose refers to the radiation energy absorbed by tissues per unit of mass. It is measured in grays (Gy), with 1 gray equivalent to 1 joule of radiation energy absorbed per kilogram.

The effective dose accounts for the type of radiation and the biologic tissue that is exposed, by applying a radiation weighting factor and a tissue-weighting factor to the absorbed dose. The radiation-weighting factor of x-radiation is one. Tissue weighting factors have been assigned to various tissues and organs based on their estimated susceptibility to radiation damage. The effective dose, measured in Sieverts (Sv), is used to estimate risks from radiation exposure. For exposures to limited parts of the body, as is often the case in medical imaging, the calculated effective dose is equivalent to the whole body exposure that carries the same risk.


Table 1. Sources of natural radiation and their average contribution to human exposure (adapted from United Nations Scientific Committee on the Effects of Atomic Radiation Report 2008)

Natural radiation contributes approximately 2.4 milliSieverts (mSv) annually to the average human radiation exposure of 3.0 mSv (United Nations Scientific Committee on the Effects of Atomic Radiation [UNSCEAR] Report 2008). Main natural sources of ionizing radiation are cosmic radiation, mainly from the sun and other stars, and terrestrial radiation, originating from naturally occurring radioactive elements (table 1).

Because the Earth's atmosphere acts as a radiation absorber and attenuator, exposure to cosmic radiation is greatly impacted by altitude. The exposure value to cosmic radiation at sea level approximately doubles with each 2000 m of elevation. While the yearly exposure to cosmic radiation at sea level is on average 0.24 mSv, at an altitude of 1600 m it is approximately 0.5 mSv. Similarly, airplane travel contributes to radiation exposure in the general population, with doses estimated to approximately 0.025 mSv for every 5 hours flown at 12,000 m (White & Pharoah, 2009).

Terrestrial radiation includes that originating from radionuclides in the soil, such as potassium 40, uranium 238 and thorium 232, some of which are taken up by ingestion. Together, these sources result in an annual dose of approximately 0.8 mSv. However, the main single source of natural radiation exposure is radon, a gas and uranium decay product, which contributes roughly 50% of the total natural radiation exposure to the general population (table 1). Radon may enter buildings through imperfectly sealed foundations

Various quantities are used to describe the radiation dose to organisms and the resulting risk. The absorbed dose refers to the radiation energy absorbed by tissues per unit of mass. It is measured in grays (Gy), with 1 gray equivalent to 1 joule of radiation energy

The effective dose accounts for the type of radiation and the biologic tissue that is exposed, by applying a radiation weighting factor and a tissue-weighting factor to the absorbed dose. The radiation-weighting factor of x-radiation is one. Tissue weighting factors have been assigned to various tissues and organs based on their estimated susceptibility to radiation damage. The effective dose, measured in Sieverts (Sv), is used to estimate risks from radiation exposure. For exposures to limited parts of the body, as is often the case in medical imaging, the calculated

> **Effective radiation dose (µSv)**

**Background Equivalent (days)** 

effective dose is equivalent to the whole body exposure that carries the same risk.

Cosmic radiation 390 59 Radon and other radioactive gases 1260 190 Other terrestrial radiation 770 116 **Total natural 2.420** 1 year Medical exposure **600** 90 **Total annual average dose 3020** 455

Table 1. Sources of natural radiation and their average contribution to human exposure (adapted from United Nations Scientific Committee on the Effects of Atomic Radiation

from naturally occurring radioactive elements (table 1).

flown at 12,000 m (White & Pharoah, 2009).

Natural radiation contributes approximately 2.4 milliSieverts (mSv) annually to the average human radiation exposure of 3.0 mSv (United Nations Scientific Committee on the Effects of Atomic Radiation [UNSCEAR] Report 2008). Main natural sources of ionizing radiation are cosmic radiation, mainly from the sun and other stars, and terrestrial radiation, originating

Because the Earth's atmosphere acts as a radiation absorber and attenuator, exposure to cosmic radiation is greatly impacted by altitude. The exposure value to cosmic radiation at sea level approximately doubles with each 2000 m of elevation. While the yearly exposure to cosmic radiation at sea level is on average 0.24 mSv, at an altitude of 1600 m it is approximately 0.5 mSv. Similarly, airplane travel contributes to radiation exposure in the general population, with doses estimated to approximately 0.025 mSv for every 5 hours

Terrestrial radiation includes that originating from radionuclides in the soil, such as potassium 40, uranium 238 and thorium 232, some of which are taken up by ingestion. Together, these sources result in an annual dose of approximately 0.8 mSv. However, the main single source of natural radiation exposure is radon, a gas and uranium decay product, which contributes roughly 50% of the total natural radiation exposure to the general population (table 1). Radon may enter buildings through imperfectly sealed foundations

**1.2 Radiation doses** 

absorbed per kilogram.

Report 2008)

and tends to accumulate in basements, from where it may leak into adjacent living spaces. Its radioactive decay products are inhaled, taking residence in the respiratory epithelium and contributing to the occurrence of lung cancer. Radiation exposure from radon varies widely with the geologic composition throughout the world.

Human activity has led to the development of new sources of radiation over the past decades. Medical radiation is the greatest peaceful contributor to radiation exposure in the general population, with an average annual effective dose of approximately 0.6 mSv (table 1).

## **2. Radiation exposure and current practices for radiation protection**

## **2.1 Radiation exposure in diagnostic imaging**

Medical use of radiation has increased continuously throughout the world, irrespective of the economic level of various regions. While therapeutic use of radiation in the treatment of cancer contributes a significant amount of radiation to a small number of individuals, the use of diagnostic radiation imparts a smaller amount of radiation to a large, and constantly growing, number of individuals.

The development of computed tomography (CT) over the past few decades has revolutionized diagnostic imaging. Its use has increased exponentially, and it is estimated that in the United States more than 62 million CT scans have been performed in 2006, compared to less than 20 million in 1995, and just 3 million in 1980 (Brenner & Hall, 2007). Due to the relatively higher radiation dose from a typical CT scan compared to that from conventional imaging modalities (table 2), and due to its widespread and increasing use, medical CT is currently estimated to contribute a significant amount of radiation exposure to the general population.


Table 2. Medical diagnostic imaging procedures and their contribution range to patients' radiation exposure (data: United Nations Scientific Committee on the Effects of Atomic Radiation Report 2008, \* White and Pharoah, 2009)

Due to the large number of CT scans being administered in the present, and the trend of this number to increase in the future, radiation exposure from CT may gain significance as a public health concern (Brenner and Hall, 2007).

As a result, current radiation safety efforts carry a significant focus on monitoring and limiting medical radiation exposure from CT to the general population.

Radiation Protection in Medical Imaging 71

3. Shielding of the examination room, to limit or eliminate radiation exposure to

4. Shielding the patient's anatomic areas that are not in the region of interest of the examination, to reduce radiation exposure to sensitive organs, such as the thyroid,

5. Filtration of the x-ray beam, which removes low energy photons that would otherwise be absorbed by the patient, and hence would not contribute to the radiographic image; 6. Optimal film-screen combinations that maximize photon-to-image conversion, while

9. Optimal combination of exposure factors (kVp, mA, seconds) to achieve low radiation

10. Quality assurance of the imaging and film processing equipment, to ensure consistent optimal quality of images with minimum exposure to patients and personnel. 11. Personal dose monitors and area dose monitors that assist in documenting individual exposure and area radiation levels. Personal dose monitors used in clinical practice include thermoluminescent dosimeters (TLD) and, more recently, optically stimulated dosimeters (OSLD). TLDs use lithium fluoride or calcium sulphate crystals laced with magnesium, titanium, copper or phosphorus impurities, which store the energy from radiation to which they are subjected. When the crystals are heated, the stored energy is released as luminescence proportional to the radiation exposure. While TLDs are small and they allow measurements of radiation doses ranging from microGy to kGy, accurate and reproducible readings are heavily technique-dependent (Kron 1994). OSLDs use aluminium oxide laced with carbon impurities to store radiation energy, which is released as luminescence proportional to the radiation exposure in response to stimulation light. OSLDs have a linear dose-response relationship in the 1-300 cGy range but can be calibrated for a significantly wider range of radiation exposures that

Due to the exponential growth in its utilization and its higher radiation effective dose compared to plain film radiographs, computed tomography (CT) is one of the most significant contributors to medical diagnostic radiation exposure today. While many of the general principles outlined above apply to CT as well, several measures specific to CT can

1. Bismuth shields are used in the exposure path in front of sensitive organs, such as the breast. They partially absorb the direct radiation beam to underlying anatomic structures, without significantly impacting the image quality of adjacent structures. Bismuth shields can reduce radiation dose to the adult female breast by up to 40% (Yilmaz et al., 2007). However, when bismuth shields are used together with automatic exposure modulation, the latter may compensate for the radiation attenuation caused by the shield, offsetting the radiation savings benefits of the shield. To circumvent this situation, bismuth shields may be applied after exposure of the scout view, and before

2. Optimization of CT imaging protocols involves tailoring imaging protocols to the patient and the diagnostic task. The radiation dose can be modified by adjusting one or

achieving the image quality required for the diagnostic task;

8. Collimation of the exposed field strictly to the region of interest;

are relevant in medical imaging (Jursinic 2007, Yukihara, 2008).

acquisition of the volumetric data (Coakley et al., 2011).

be implemented to reduce patients' radiation risk:

several scan parameters:

doses, while maintaining diagnostic image quality;

7. Appropriate x-ray source –to-skin distance, whenever this factor is adjustable;

neighbouring areas;

gonads, eye lens;

## **2.2 Radiation protection measures in diagnostic imaging**

The main concern from medical imaging involving ionizing radiation is the risk of cancer induction. Radiation-induced cancer may occur at any time, including after decades, following the radiation exposure event, and, if it occurs, it is indistinguishable from spontaneously induced cancer.

The progressive understanding of biologic effects of radiation led to the development of guidelines for radiation protection by the international community. Evidence-based guidelines are outlined in the 2007 Recommendations of the International Commission for Radiological Protection (International Commission for Radiological Protection, 2007) and in the recommendations of the National Council on Radiation Protection and Measurements (NCRP) of the United States (Kase 2004).

The overarching goal of radiation protection in medical diagnostic imaging is to obtain a maximum of information relevant to the diagnostic task with the least amount of exposure to ionizing radiation (White & Pharoah, 2009). In practice, economic and individual factors, such as availability or access to imaging modalities of choice must also be taken into account.

The application in diagnostic imaging of this principle, also referred to "As Low As Reasonably Achievable" (ALARA), involves the implementation of general protective measures, including the following:


The main concern from medical imaging involving ionizing radiation is the risk of cancer induction. Radiation-induced cancer may occur at any time, including after decades, following the radiation exposure event, and, if it occurs, it is indistinguishable from

The progressive understanding of biologic effects of radiation led to the development of guidelines for radiation protection by the international community. Evidence-based guidelines are outlined in the 2007 Recommendations of the International Commission for Radiological Protection (International Commission for Radiological Protection, 2007) and in the recommendations of the National Council on Radiation Protection and Measurements

The overarching goal of radiation protection in medical diagnostic imaging is to obtain a maximum of information relevant to the diagnostic task with the least amount of exposure to ionizing radiation (White & Pharoah, 2009). In practice, economic and individual factors, such as availability or access to imaging modalities of choice must also

The application in diagnostic imaging of this principle, also referred to "As Low As Reasonably Achievable" (ALARA), involves the implementation of general protective

1. Imaging decision. Selection of the imaging examination to answer a specific diagnostic question should take into consideration all modalities that are likely to provide the sought information. When the diagnostic question can be answered through a study that involves no ionizing radiation - such as magnetic resonance or ultrasound imaging - the added radiation dose from the diagnostic procedure is reduced to zero. The selection of the imaging modality involving exposure to ionizing radiation should follow a justification process based on the patients' individual clinical circumstances and on scientific evidence. This process should suggest that the potential diagnostic benefit of the examination outweighs the estimated risk carried by the specific radiation exposure. Decision support software, which can assist the clinician in determining the appropriateness of a given imaging study to the clinical situation, has recently become available (Rosenthal et al., 2006). Preliminary studies have suggested that its utilization can contribute to the decreased prescription of examinations that are deemed of "low-

2. Patient information: Providing easy-to-understand information to patients is an effective measure for bringing radiation exposure into context and for dispelling exaggerated fear of radiation in patients. Radiation doses from medical examinations can be compared with the amount of background radiation to which we are all exposed by living on Earth (see table 2). Risk comparison with common daily activities, such as driving or air travel, may also be effective (Coakley et al., 2011). For instance, travelling 64,000 km by car is associated with a risk of death from a motor vehicle accident of approximately 1:2000 in the United States (National Highway Traffic Safety Administration Fatality Analysis Reporting System, 2011). This risk is roughly equal to the risk of developing a fatal cancer from an effective radiation dose of 10 mSv, a dose

**2.2 Radiation protection measures in diagnostic imaging** 

utility" to the diagnostic task (Coakley et al., 2011).

five times higher than that from a typical CT scan of the head.

spontaneously induced cancer.

be taken into account.

measures, including the following:

(NCRP) of the United States (Kase 2004).


Due to the exponential growth in its utilization and its higher radiation effective dose compared to plain film radiographs, computed tomography (CT) is one of the most significant contributors to medical diagnostic radiation exposure today. While many of the general principles outlined above apply to CT as well, several measures specific to CT can be implemented to reduce patients' radiation risk:


Radiation Protection in Medical Imaging 73

Automated control of radiation field intensity generated during medical imaging and radiation therapy, constitutes an efficient method to reduce unnecessary radiation exposure

This heading presents an original method for automated control of radiation field intensity. Its implementation in medical imaging devices employing ionizing radiation can offer

The goal of this study has been to develop an original and flexible model for radiation field control, in order to allow an exact administration of the minimum necessary radiation dose at any desired tissue depth, even through non-homogeneous tissue layers that intensify or attenuate the electromagnetic field generated by an imaging device. Therefore, we modelled and simulated the spread of radiation through non-homogeneous media, as is the case when

A general model of radiation field spread using Cartesian coordinates (0p; 0q; 0r) can be represented as seen in figure 1. Patterns of spread have been considered, in relation to three

 **a.** y on 0p axis **b.** y on 0q axis; **c.** y on 0r axis

Fig. 1. General model of radiation field intensity, y(t, p, q, r) in Cartesian coordinates

Radiation field intensity y(t, p, q, r) shown in figures 1b, 1c, and 1d, can be expressed as:

The output variable y=y[t, s, u(t)] is the intensity of the radiation field, u(t) is the input (command) signal applied for the radiation source, Ky is a weighting coefficient and (2) stands for the usual form of compression of the Cartesian space variable on the axes of

2 22 2 22

y <sup>00</sup> y 0T 0S (t,p,q,r)=y [t,s,u(t)]=K ×F (t)×F (s)×u(t) (1)

f ff <sup>f</sup> s = ± p +q +r ; s = ± p +q +r (2)

**3. Modelling and simulation for radiation control of imaging devices** 

for both patients and device operators (Sajin et al., 2003, McCollough et al., 2006).

important exposure savings to both patients and medical personnel.

using medical imaging devices to investigate human tissues.

coordinates (0p), (0q) and (0r), as illustrated in figure 2.

spatial axes and in relation to time (t).

**3.1 Preliminaries** 


Lower values can be selected for both parameters when scanning children or small adults, or in cases in which noisier images are unlikely to have a significant impact on the diagnostic task outcome (e.g. a study to evaluate renal calculi).


## **3. Modelling and simulation for radiation control of imaging devices**

#### **3.1 Preliminaries**

72 Medical Imaging

radiation dose, albeit at the expense of increased image noise.

the diagnostic task outcome (e.g. a study to evaluate renal calculi).

tube current.

scanned (McCollough et al., 2005).

significantly reduced patient radiation exposure.

tracking CT radiation exposure to patients.

i. The maximum tube electric potential (kVp): while the typical exposure setting used in routine CT scanning is between 100 and 140 kVp, a lower kVP can reduce

ii. Tube current (mA): a CT scan conducted with lower mA results in noisier images, but radiation exposure is diminished by an amount proportional to the decreased

Lower values can be selected for both parameters when scanning children or small adults, or in cases in which noisier images are unlikely to have a significant impact on

3. X-ray tube current modulation and automatic exposure control refer to systems in which the user (the radiologist, through the radiological technologist) indicates the desired level of image quality, based on which the CT software calculates the radiation output required to match that image quality in all regions of the scanned volume (McCollough et al., 2005). During acquisition of each sectional image, as the gantry rotates around the patient, angular tube current modulation works to modulate x-ray tube current to equalize the average photon flux reaching the detectors (x-, y-axis tube current modulation). As the patient progresses through the gantry, longitudinal (z-axis) tube current modulation acts to maintain constant image noise levels along the longitudinal axis of the patient as body regions with different attenuation properties are

4. Enhanced CT reconstruction algorithms: most current CT scanners utilize reconstruction algorithms based on the filtered back-projection (FBP) method. When used in combination with low tube currents, this method tends to generate noisy images. The implicit radiation exposure threshold, below which the diagnostic image quality decreases to unacceptable levels, limits the potential for radiation savings with FBP reconstruction algorithms. Alternative solutions include iterative reconstruction algorithms, based on complex mathematical models that aim to correct the noise in images acquired with low tube currents. These algorithms are computation-intensive and lead to longer reconstruction times (Coakley et al., 2011). A recent, enhanced, iterative reconstruction method, known as adaptive statistical iterative reconstruction (ASIR), shows great promise for improved reconstruction speeds. Furthermore, by attempting to identify and correct for photon flux fluctuations that are unlikely to be caused by anatomic features, ASIR can reduce image noise, as well as certain types of image artefacts (Marin et al., 2010, Miéville et al., 2011, Renker et al., 2011). As a result, images reconstructed using ASIR from raw data acquired with low tube current may show similar or improved image quality compared to FBP reconstructions, while low current settings allow radiation savings in the range of 32-65% for a variety of clinical utilization scenarios (Cornfeld et al., 2011, Flicek et al., 2010, Hara et al., 2009). Together, these characteristics of ASIR protocols may lead to CT studies of overall similar or better diagnostic quality compared to FBP reconstruction methods, with a potential for

5. Calculation and reporting of radiation dose: Most CT scanners have the integrated capability to calculate radiation exposure as a dose-length product (measured in mGy/cm), which can be recorded in the patient's clinical record. This facilitates Automated control of radiation field intensity generated during medical imaging and radiation therapy, constitutes an efficient method to reduce unnecessary radiation exposure for both patients and device operators (Sajin et al., 2003, McCollough et al., 2006).

This heading presents an original method for automated control of radiation field intensity. Its implementation in medical imaging devices employing ionizing radiation can offer important exposure savings to both patients and medical personnel.

The goal of this study has been to develop an original and flexible model for radiation field control, in order to allow an exact administration of the minimum necessary radiation dose at any desired tissue depth, even through non-homogeneous tissue layers that intensify or attenuate the electromagnetic field generated by an imaging device. Therefore, we modelled and simulated the spread of radiation through non-homogeneous media, as is the case when using medical imaging devices to investigate human tissues.

A general model of radiation field spread using Cartesian coordinates (0p; 0q; 0r) can be represented as seen in figure 1. Patterns of spread have been considered, in relation to three spatial axes and in relation to time (t).

Fig. 1. General model of radiation field intensity, y(t, p, q, r) in Cartesian coordinates

Radiation field intensity y(t, p, q, r) shown in figures 1b, 1c, and 1d, can be expressed as:

$$\mathbf{y}\text{ (t.p.q.r)} = \mathbf{y}\_{\text{ou}}[\mathbf{t}, \mathbf{s}, \mathbf{u}(\mathbf{t})] = \mathbf{K}\_{\text{y}} \times \mathbf{F}\_{\text{or}}(\mathbf{t}) \times \mathbf{F}\_{\text{os}}(\mathbf{s}) \times \mathbf{u}(\mathbf{t})\tag{1}$$

The output variable y=y[t, s, u(t)] is the intensity of the radiation field, u(t) is the input (command) signal applied for the radiation source, Ky is a weighting coefficient and (2) stands for the usual form of compression of the Cartesian space variable on the axes of coordinates (0p), (0q) and (0r), as illustrated in figure 2.

$$\mathbf{s} = \pm \sqrt{\mathbf{p}^2 + \mathbf{q}^2 + \mathbf{r}^2}; \ s\_\mathbf{s} = \pm \sqrt{\mathbf{p}\_\mathbf{f}^2 + \mathbf{q}\_\mathbf{f}^2 + \mathbf{r}\_\mathbf{f}^2} \tag{2}$$

Radiation Protection in Medical Imaging 75

in which the final abscises f ff (p ), (q ), (r ) correspond to negligible values (for example ≤ 0.05). For F (p), F (q) , 0P 0Q F (r) 0R , the abscises d dd (p ), (q ), (r ) may show disturbances due to

> 1 2 2 1 1 2 S S F (s)= <sup>ε</sup> + ×<sup>ε</sup>

> > f

S S

f ff

f ff PQR

0S S S

0S 0 2

f <sup>μ</sup> (1+<sup>λ</sup> ) C = s

0vd vd

The results of F (s) 0S in (4), (10), (13) and (14) also represent spatial curves deformed by multiple degrees of freedom, for example S S (μ ), (λ ) and f (s ) , while the "length constants" (P…), (Q…) and (R…) may exhibit "inertia" or "attenuation" of radiation field propagation

The disturbance index (d) refers to the homogeneity disturbance due to tissue heterogeneity located on the abscissa <sup>d</sup> (v ), while vd (A ) in (16) represents the amplitude of heterogeneity

A more or less steep, respectively a more or less symmetrical slope can result from the convenient choice of parameter vd (K ) . The deformation effect of function F (v) 0vd on the

1-<sup>λ</sup> ; S S 1

Function components F (p), F (q) or 0P 0Q F (r) 0R can be approximated by:

right abscissa d (v ) over the component F (v) 0v can be seen in figure 3.b.

in which variable (v) becomes (p), (q) or (r), according to case.

f f <sup>1</sup> F (s)= × <sup>ε</sup> -<sup>λ</sup> <sup>×</sup><sup>ε</sup>

μ ×(1+λ )

abs(s) abs(s) - - 1 2 S S

; 2 22 S

<sup>μ</sup> (1+<sup>λ</sup> ) <sup>μ</sup> (1+<sup>λ</sup> ) - ×abs(s) - ×abs(s) s s

S S S S

C × p +q +r C p +q +r


1 11

<sup>222</sup> <sup>222</sup> 1 3 F (s)=C × <sup>ε</sup> +C ×<sup>ε</sup> (14)

; C =- 2 S λ ; 3 1

S <sup>1</sup> C = ×C

<sup>2</sup> vd d F (v)=A ×<sup>ε</sup> (16)

<sup>λ</sup> (15)

S -S S -S (10)

; S =2S1 λ ×S (11)

P +Q +R <sup>λ</sup> <sup>=</sup> P +Q +R (12)

(13)

discontinuities in tissue structure, estimated by:

Therefore, F (s) 0S can also be written:

0

through different tissue layers.

illustrated in figure 3.a.

<sup>1</sup> C =

S

where

or

in which

0S

S

1

1-λ

<sup>s</sup> S =

p +q +r <sup>μ</sup> <sup>=</sup> pq r + + μμμ

Fig. 2. Cartesian coordinates system and Cartesian space variable (s)

$$\mathbf{F\_{os}(s) = \pm \sqrt{\mathbf{F\_{op}}^2(\mathbf{p}) + \mathbf{F\_{op}}^2(\mathbf{q}) + \mathbf{F\_{on}}^2(\mathbf{r})}}\tag{3}$$

where:

$$\mathbf{F}\_{\rm av}(\mathbf{p}) = \frac{\mathbf{P}\_1}{\mathbf{P}\_1 \cdot \mathbf{P}\_2} \times \varepsilon^{\frac{\mathrm{abs}\,\mathrm{(p)}}{\mathrm{p}\cdot\mathrm{P}}} + \frac{\mathbf{P}\_2}{\mathbf{P}\_2 \cdot \mathbf{P}\_1} \times \varepsilon^{\frac{\mathrm{abs}\,\mathrm{(p)}}{\mathrm{p}\cdot\mathrm{2}}} \tag{4}$$

$$\mathbf{F}\_{\rm oq}(\mathbf{q}) = \frac{\mathbf{Q}\_1}{\mathbf{Q}\_1 \cdot \mathbf{Q}\_2} \times \boldsymbol{\varepsilon}^{\frac{\text{abs}\,\text{(q)}}{\text{Q}\Omega}} + \frac{\mathbf{Q}\_2}{\mathbf{Q}\_2 \cdot \mathbf{Q}\_1} \times \boldsymbol{\varepsilon}^{\frac{\text{abs}\,\text{(q)}}{\text{Q}\Omega}} \tag{5}$$

$$\mathbf{F}\_{\rm on}(\mathbf{r}) = \frac{\mathbf{R}\_1}{\mathbf{R}\_1 \cdot \mathbf{R}\_2} \times \boldsymbol{\varepsilon}^{\frac{\mathrm{abs}\,(r)}{\mathrm{R}\cdot 1}} + \frac{\mathbf{R}\_2}{\mathbf{R}\_2 \cdot \mathbf{R}\_1} \times \boldsymbol{\varepsilon}^{\frac{\mathrm{abs}\,(r)}{\mathrm{R}\cdot 2}} \tag{6}$$

The axes (0p) and (0q) define the horizontal plane, in which the electromagnetic field generator is located. Field intensity expressed in (1), and axis (0r) correspond to the depth of field propagation. The constants in (4), (5) and (6) can be approximated by expert procedures, namely:

$$\mathbf{P}\_{\mathbf{i}} = \frac{\mathbf{P}\_{\mathbf{i}}}{\mu\_{\mathrm{p}}(\mathbf{1} + \lambda\_{\mathrm{p}})} \; ; \; \mathbf{P}\_{\mathbf{i}} = \lambda\_{\mathrm{p}} \times \mathbf{P}\_{\mathbf{i}} \; ; \; \text{where: } \; \mu\_{\mathrm{p}} = (\mathbf{4} + \mathbf{6}) \text{ and } \; \lambda\_{\mathrm{p}} \ge \mathbf{1} \tag{7}$$

$$\mathbf{Q}\_{i} = \frac{\mathbf{q}\_{i}}{\mu\_{\mathrm{q}}(1 + \lambda\_{\mathrm{q}})} \; ; \; \mathbf{Q}\_{i} = \lambda\_{\mathrm{q}} \times \mathbf{Q}\_{i} \; \; \; \text{ where: } \mu\_{\mathrm{q}} = (4 \star 6) \text{ and } \lambda\_{\mathrm{q}} \ge 1 \tag{8}$$

$$\mathbf{R}\_{\mathbf{i}} = \frac{\mathbf{r}\_{\mathbf{i}}}{\mu\_{\mathbf{k}}(\mathbf{1} + \lambda\_{\mathbf{k}})} \; ; \; \mathbf{R}\_{\mathbf{z}} = \lambda\_{\mathbf{z}} \times \mathbf{R}\_{\mathbf{i}} \; ; \; \text{where: } \mu\_{\mathbf{z}} = (\mathbf{4} + \mathbf{6}) \text{ and } \lambda\_{\mathbf{z}} \ge \mathbf{1} \tag{9}$$

in which the final abscises f ff (p ), (q ), (r ) correspond to negligible values (for example ≤ 0.05). For F (p), F (q) , 0P 0Q F (r) 0R , the abscises d dd (p ), (q ), (r ) may show disturbances due to discontinuities in tissue structure, estimated by:

$$\mathbf{F}\_{\rm es}(\mathbf{s}) = \frac{\mathbf{S}\_1}{\mathbf{S}\_1 \cdot \mathbf{S}\_2} \times \varepsilon^{\frac{\mu \mathbf{b}(\mathbf{s})}{\mathbf{S}\_1}} + \frac{\mathbf{S}\_2}{\mathbf{S}\_2 \cdot \mathbf{S}\_1} \times \varepsilon^{\frac{\mu \mathbf{b}(\mathbf{s})}{\mathbf{S}\_2}} \tag{10}$$

where

74 Medical Imaging

2 22 F (s)=± F (p)+F (q)+F (r) 0S 0P 0Q 0R (3)

P -P P -P (4)

Q -Q Q -Q (5)

R -R R -R (6)

<sup>μ</sup> (1+<sup>λ</sup> ) ; P =2 P1 <sup>λ</sup> ×P , where: μP=(4÷6) and λP>1 (7)

; Q =2Q 1 λ ×Q , where: μ<sup>Q</sup> =(4÷6) and Q λ >1 (8)

; R =2R 1 λ ×R , where: μ<sup>R</sup> =(4÷6) and λ<sup>R</sup> >1 (9)

abs (p) abs (p) - - 1 2 P P

abs (q) abs (q) - - 1 2 Q Q

abs (r) abs (r) - - 1 2 R R

1 2 2 1 1 2 P P F (p)= <sup>ε</sup> + ×<sup>ε</sup>

1 2 2 1 Q Q 1 2 F (q)= <sup>ε</sup> + ×<sup>ε</sup>

1 2 2 1 R R 1 2 F (r)= <sup>ε</sup> <sup>+</sup> <sup>ε</sup>

The axes (0p) and (0q) define the horizontal plane, in which the electromagnetic field generator is located. Field intensity expressed in (1), and axis (0r) correspond to the depth of field propagation. The constants in (4), (5) and (6) can be approximated by expert

Fig. 2. Cartesian coordinates system and Cartesian space variable (s)

0P

0Q

0R

f

P P

f

Q Q

f

R R

μ (1+λ )

μ (1+λ )

1

1

1

<sup>r</sup> R =

<sup>q</sup> Q =

<sup>p</sup> P =

where:

procedures, namely:

$$\mathbf{S}\_{\mathbf{i}} = \frac{\mathbf{s}\_{i}}{\mu\_{\mathbf{s}} \times (1 + \lambda\_{\mathbf{s}})}; \quad \mathbf{S}\_{\mathbf{i}} = \lambda\_{\mathbf{s}} \times \mathbf{S}\_{\mathbf{i}} \tag{11}$$

$$\mathbf{\mu}\_{\rm s} = \frac{\mathbf{p}\_{\rm i} + \mathbf{q}\_{\rm i} + \mathbf{r}\_{\rm i}}{\frac{\mathbf{p}\_{\rm i}}{\mathbf{p}\_{\rm p}} + \frac{\mathbf{q}\_{\rm i}}{\mathbf{p}\_{\rm Q}} + \frac{\mathbf{r}\_{\rm i}}{\mathbf{p}\_{\rm R}}}; \quad \lambda\_{\rm s} = \frac{\mathbf{P}\_{\rm z} + \mathbf{Q}\_{\rm z} + \mathbf{R}\_{\rm z}}{\mathbf{P}\_{\rm i} + \mathbf{Q}\_{\rm i} + \mathbf{R}\_{\rm z}}\tag{12}$$

Therefore, F (s) 0S can also be written:

$$F\_{\rm es}(\mathbf{s}) = \frac{1}{1 \cdot \lambda\_{\rm s}} \times \left[ \varepsilon^{\frac{\rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm$$

or

C × p +q +r C p +q +r 0S 0 2 <sup>222</sup> <sup>222</sup> 1 3 F (s)=C × <sup>ε</sup> +C ×<sup>ε</sup> (14)

in which

$$\mathbf{C}\_{o} = \frac{1}{1 \cdot \lambda\_{s}} \; ; \; \mathbf{C}\_{i} = \frac{\mu\_{s} \left(1 + \lambda\_{s}\right)}{\mathbf{s}\_{i}} \; ; \; \mathbf{C}\_{z} = \lambda\_{s} \; ; \; \mathbf{C}\_{s} = \frac{1}{\lambda\_{s}} \times \mathbf{C}\_{i} \tag{15}$$

The results of F (s) 0S in (4), (10), (13) and (14) also represent spatial curves deformed by multiple degrees of freedom, for example S S (μ ), (λ ) and f (s ) , while the "length constants" (P…), (Q…) and (R…) may exhibit "inertia" or "attenuation" of radiation field propagation through different tissue layers.

Function components F (p), F (q) or 0P 0Q F (r) 0R can be approximated by:

$$\mathbf{F}\_{\rm vol}(\mathbf{v}) = \mathbf{A}\_{\rm vl} \times \boldsymbol{\varepsilon}^{\rm \&vd/vd \cdot v^2} \tag{16}$$

in which variable (v) becomes (p), (q) or (r), according to case.

The disturbance index (d) refers to the homogeneity disturbance due to tissue heterogeneity located on the abscissa <sup>d</sup> (v ), while vd (A ) in (16) represents the amplitude of heterogeneity illustrated in figure 3.a.

A more or less steep, respectively a more or less symmetrical slope can result from the convenient choice of parameter vd (K ) . The deformation effect of function F (v) 0vd on the right abscissa d (v ) over the component F (v) 0v can be seen in figure 3.b.

Radiation Protection in Medical Imaging 77

For a slow to fast ascending evolution we considered μT T 4÷6, λ >1 , resulting in the

If, with respect to time, the spread of radiation field intensity y(t,p,q,r) is not identical on all three axes (0p), (0q) and (0r) depicted in figures 1a, 1b and 1c, then condition (17) is no

i

12 2

F (t) F (t) F (t) 0Tp 0Tq 0Tr (20)

; T =2p Tp 1p λ ×T (21)

; T =2q Tq 1q λ T (22)

; T =2r Tr 1r λ T (23)

<sup>λ</sup> <sup>=</sup> T +T +T (24)

; T =2 λτ×T1 (25)

T -T T -T (26)

(19)

21 1 T ×T T t = ×ln T -T T

f 1 2 T1 T T t T= ; T=λ ×T ; μ ×(1+λ )

In this case, formally identical to relation (19), each axis will exhibit:

1p

1q

T =

1r

where (t…f), (μT…) and (λT…) are specifically established for each axis. In order to transform into Cartesian coordinates it can be shown that:

1

0T

<sup>t</sup> T =

τ

propagations are different along the three axes (p, q, r). Following this method, F0T(t) in equation (1) becomes:

defined by (1) and depicted in figure (1).

finally resulting in:

<sup>t</sup> T =

pf qf rf

1p 2p 1q 2q 1r 2r t +t +t

f

τ τ

μ ×(1+λ )

T =

pf

Tq Tp t

qf

Tq Tq t

rf

Tr Tr

<sup>μ</sup> <sup>=</sup> T +T +T +T +T +T ; 2p 2q 2r

which are the equivalent time constants of the Cartesian space, assuming that inertial

t t - - 1 2 T T

1 2 2 1 1 2 T T F (t)= ×<sup>ε</sup> + ×<sup>ε</sup>

which together with F0S (s) from (10), allows the computation of the radiation field intensity

τ

1p 1q 1r T +T +T

μ ×(1+λ )

μ ×(1+λ )

μ ×(1+λ )

where (ti) is the moment corresponding to the inflexion F (t ) <sup>0</sup><sup>τ</sup> <sup>i</sup> .

following time constants:

longer accomplished, resulting in:

 **a.** Amplitude **b.** Trajectory in Cartesian space

Fig. 3. Tissue heterogeneity may lead to homogeneity disturbances of field intensity

If, with respect to time, the spread of radiation field intensity y(t,p,q,r) is identical on all three axes (0p), (0q) and (0r) depicted in figures 1a, 1b and 1c, then:

$$\mathbf{F}\_{\alpha\mathbf{r}\_P}(\mathbf{t}) = \mathbf{F}\_{\alpha\mathbf{r}\_\mathbf{q}}(\mathbf{t}) = \mathbf{F}\_{\alpha\mathbf{r}\_\mathbf{r}}(\mathbf{t}) = \mathbf{F}\_{\alpha\mathbf{r}}(\mathbf{t}) \tag{17}$$

which may also be approximated as a customary form of step response, related to a control signal u(t) applied to the radiation source.

Thus:

$$\mathbf{F}\_{\rm or}(\mathbf{t}) = \mathbf{1} \cdot \frac{\mathbf{T}\_1}{\mathbf{T}\_1 \cdot \mathbf{T}\_2} \times \boldsymbol{\varepsilon}^{\frac{\mathbf{t}}{\mathbf{T}\_1}} \cdot \frac{\mathbf{T}\_2}{\mathbf{T}\_2 \cdot \mathbf{T}\_1} \times \boldsymbol{\varepsilon}^{\frac{\mathbf{t}}{\mathbf{T}\_2}} \tag{18}$$

The final values f ff f (p ,q ,r ,v ) in figures 1a, 1b, 1c, 2 and 3 show the depth of radiation field penetration, which will depend on this control signal u(t) and on the more or less homogeneous structure of the tissue. The length f (t ) of this transient phenomenon (figure 4) reflects a radiation propagation inertia, identical on axes (p, q, r, s) for which f t>t tends to a unitary asymptote.

Fig. 4. Transient phenomenon

 **a.** Amplitude **b.** Trajectory in Cartesian space Fig. 3. Tissue heterogeneity may lead to homogeneity disturbances of field intensity

three axes (0p), (0q) and (0r) depicted in figures 1a, 1b and 1c, then:

0T

signal u(t) applied to the radiation source.

Thus:

unitary asymptote.

Fig. 4. Transient phenomenon

If, with respect to time, the spread of radiation field intensity y(t,p,q,r) is identical on all

which may also be approximated as a customary form of step response, related to a control

t t - - 1 2 T T

1 2 2 1 1 2 T T F (t)=1- <sup>ε</sup> - <sup>ε</sup>

The final values f ff f (p ,q ,r ,v ) in figures 1a, 1b, 1c, 2 and 3 show the depth of radiation field penetration, which will depend on this control signal u(t) and on the more or less homogeneous structure of the tissue. The length f (t ) of this transient phenomenon (figure 4) reflects a radiation propagation inertia, identical on axes (p, q, r, s) for which f t>t tends to a

F (t)=F (t)=F (t)=F (t) 0Tp 0Tq 0Tr 0T (17)

T -T T -T (18)

For a slow to fast ascending evolution we considered μT T 4÷6, λ >1 , resulting in the following time constants:

$$\mathbf{T\_i = \frac{\mathbf{t\_i}}{\mu\_r \times (1 + \lambda\_r)}}; \quad \mathbf{T\_z = \lambda\_r \times \mathbf{T\_i}}; \quad \mathbf{t\_i = \frac{\mathbf{T\_i \times T\_z}{\mathbf{T\_z - T\_i}} \times \ln\left(\frac{\mathbf{T\_z}}{\mathbf{T\_i}}\right)}{\mathbf{T\_z - T\_i}}\tag{19}$$

where (ti) is the moment corresponding to the inflexion F (t ) <sup>0</sup><sup>τ</sup> <sup>i</sup> .

If, with respect to time, the spread of radiation field intensity y(t,p,q,r) is not identical on all three axes (0p), (0q) and (0r) depicted in figures 1a, 1b and 1c, then condition (17) is no longer accomplished, resulting in:

$$\mathbf{F}\_{\rm or\_{P}}(\mathbf{t}) \neq \mathbf{F}\_{\rm or\_{q}}(\mathbf{t}) \neq \mathbf{F}\_{\rm or\_{T}}(\mathbf{t}) \tag{20}$$

In this case, formally identical to relation (19), each axis will exhibit:

$$\mathbf{T}\_{\mathbf{t}\_{\mathrm{P}}} = \frac{\mathbf{t}\_{\mathrm{pr}}}{\boldsymbol{\mu}\_{\mathrm{r}\_{\mathrm{I}}} \times (\mathbf{1} + \boldsymbol{\lambda}\_{\mathrm{r}\_{\mathrm{P}}})}; \quad \mathbf{T}\_{\mathbf{2p}} = \boldsymbol{\lambda}\_{\mathrm{r}\_{\mathrm{P}}} \times \mathbf{T}\_{\mathbf{t}\_{\mathrm{P}}} \tag{21}$$

$$\mathbf{T}\_{\mathrm{t}q} = \frac{\mathbf{t}\_{\mathrm{q}^{\mathrm{d}}}}{\mu\_{\mathrm{r}q} \times (\mathbf{1} + \lambda\_{\mathrm{r}q})}; \quad \mathbf{T}\_{\mathrm{2q}} = \lambda\_{\mathrm{r}q} \mathbf{T}\_{\mathrm{1q}} \tag{22}$$

$$\mathbf{T\_{\tiny tr}} = \frac{\mathbf{t\_{\tiny tr}}}{\mu\_{\rm tr} \times (1 + \lambda\_{\rm tr})} ; \ \mathbf{T\_{\tiny tr}} = \lambda\_{\rm tr} \mathbf{T\_{\tiny tr}} \tag{23}$$

where (t…f), (μT…) and (λT…) are specifically established for each axis.

In order to transform into Cartesian coordinates it can be shown that:

$$\mathbf{M}\_{\mathbf{v}} = \frac{\mathbf{t}\_{\mathrm{p}t} + \mathbf{t}\_{\mathrm{q}t} + \mathbf{t}\_{\mathrm{r}t}}{\mathbf{T}\_{\mathrm{i\_{p}}} + \mathbf{T}\_{\mathrm{2p}} + \mathbf{T}\_{\mathrm{i\_{q}}} + \mathbf{T}\_{\mathrm{2q}} + \mathbf{T}\_{\mathrm{i\_{r}}} + \mathbf{T}\_{\mathrm{2r}}}; \quad \lambda\_{\mathrm{v}} = \frac{\mathbf{T}\_{\mathrm{2p}} + \mathbf{T}\_{\mathrm{2q}} + \mathbf{T}\_{\mathrm{2r}}}{\mathbf{T}\_{\mathrm{i\_{p}}} + \mathbf{T}\_{\mathrm{i\_{q}}} + \mathbf{T}\_{\mathrm{i\_{r}}}} \tag{24}$$

finally resulting in:

$$\mathbf{T\_i = \frac{\mathbf{t\_i}}{\mu\_r \times (1 + \lambda\_r)}} ; \quad \mathbf{T\_i = \lambda\_r \times T\_i} \tag{25}$$

which are the equivalent time constants of the Cartesian space, assuming that inertial propagations are different along the three axes (p, q, r).

Following this method, F0T(t) in equation (1) becomes:

$$\mathbf{F}\_{\rm or}(\mathbf{t}) = \frac{\mathbf{T}\_1}{\mathbf{T}\_1 \cdot \mathbf{T}\_2} \times \boldsymbol{\varepsilon}^{\frac{\mathbf{t}}{\mathbf{T}\mathbf{T}}} + \frac{\mathbf{T}\_2}{\mathbf{T}\_2 \cdot \mathbf{T}\_1} \times \boldsymbol{\varepsilon}^{\frac{\mathbf{t}}{\mathbf{T}\mathbf{T}}} \tag{26}$$

which together with F0S (s) from (10), allows the computation of the radiation field intensity defined by (1) and depicted in figure (1).

Radiation Protection in Medical Imaging 79

Thus, the complex phenomenon of radiation spread can be estimated and adjusted by

in which partial derivatives ( F0... ), ( F11 ) and ( F...0 ) are obtained from (32). Only signal ( u0 ) is included in the block diagram. The homogeneity disturbance ( u0 ) is an independent

From the equation system (27)…(31), the matrix containing partial derivatives of the state vector (Mdpx) represented in (36) has been deduced for the entire adjustment system

y00 y 00 S0 [t,s,u(t)]=K ×F (t,s)×u (t) (32)

φ01.00 10 00 20 10 11 01 =a ×F +2×a ×F +a ×F (34)

φ2.00 20 00 =a ×F (35)

φ0.00 00 00 10 10 01 01 20 20 11 11 02 02 =a ×F +a ×F +a ×F +a ×F +a ×F +a ×F (33)

f f ff s = p +q +r ; 2 22

d d dd s = p +q +r (31)

222

Fig. 6. FOT and FOS including homogeneity disturbance at sd

where F (t,s)=F (t)×F (s) <sup>00</sup> 0T 0S and ( K ) is a weighting coefficient. y

This resulted for the right side of equation (29) in:

In the left side of this equation we introduced the approximated solution

partial differential equation (29).

external signal.

depicted in figure 5.

f pf qf rf t = t +t +t ; 2 22

#### **3.2 Analogical modelling**

Figure 5 presents the proposed control block diagram for adjusting radiation fields. It's components have been modeled by algebraic and differential equation systems (27) and (28), as well as by partial differential equation (29).

Fig. 5. Control block diagram of the proposed adjustment system

$$\mathbf{a\_{o} = w\_{o} - m\_{o}} ; \quad \mathbf{K\_{m} \times y\_{o0} = m\_{o} + T\_{m} \times m\_{i}} \tag{27}$$

2 2 K ×a +K ×a +K ×a =c +T ×c a +b IR 0 PR 1 DR 2 1 R 2 ; K ×c =u +T ×u E0 0 E 1 (28)

$$\mathbf{a}\_{\mathrm{u}} \times \mathbf{y}\_{\mathrm{u}} + \mathbf{a}\_{\mathrm{u}} \times \mathbf{y}\_{\mathrm{u}} + \mathbf{a}\_{\mathrm{u}} \times \mathbf{y}\_{\mathrm{u}} + \mathbf{a}\_{\mathrm{u}} \times \mathbf{y}\_{\mathrm{u}} + \mathbf{a}\_{\mathrm{u}} \times \mathbf{y}\_{\mathrm{u}} + \mathbf{a}\_{\mathrm{u}} \times \mathbf{y}\_{\mathrm{u}} = \mathbf{K}\_{\mathrm{y}} \times (\boldsymbol{\upmu}\_{\mathrm{u}\boldsymbol{u}} \times \mathbf{u}\_{\mathrm{s}0} + \boldsymbol{\upmu}\_{\mathrm{1}\boldsymbol{u}0} \times \mathbf{u}\_{\mathrm{s}1} + \boldsymbol{\upmu}\_{\mathrm{2}\boldsymbol{u}0} \times \mathbf{u}\_{\mathrm{s}2}) \tag{29}$$

The indexes in relations (27), (28) and the first parameter in relation (29), correspond to the order of derivative with respect to time (t). The second parameter in (29) corresponds to the order of derivative with respect to variable (s), defined in relation (2).

The radiation field intensity ( y00 ) defined by (1) is converted by the measurement transducer M into electrical signal ( m0 ) found in (27). The control error ( <sup>0</sup> a ) results from (27), in which ( w0 ) is the reference signal.

In the above presented adjustment scheme, the R controller with proportional integrative derivative (PID) behaviour defined in (28) emits a control signal ( <sup>0</sup> c ) which adjusts the electromagnetic field generator E, as defined in relation (28).

The radiation emitting unit, E, generates the execution signal ( u0 ) representing the incident radiation field applied to various environments (e.g. air, tissues), together with a resultant disturbance signal (ữ0), resulting in:

$$\mathbf{u}\_{s0} \equiv \mathbf{u}\_0 \pm \tilde{\mathbf{u}}\_0 \tag{30}$$

In the above equation, ( uS0 ) is the resulted electromagnetic field and ( u0 ) corresponds to the homogeneity disturbance Fovd(v) from (16), where v=sd , which may be defined in Cartesian space as seen in figure 6 and expressed in (31).

Figure 5 presents the proposed control block diagram for adjusting radiation fields. It's components have been modeled by algebraic and differential equation systems (27) and (28),

0 0o a =w -m ; K ×m 00 0 m 1 y =m +T ×m (27)

2 2 K ×a +K ×a +K ×a =c +T ×c a +b IR 0 PR 1 DR 2 1 R 2 ; K ×c =u +T ×u E0 0 E 1 (28)

a ×00 00 10 10 01 01 20 20 11 11 02 02 y 0.00 S0 1.00 S1 2.00 S2 y +a ×y +a ×y +a ×y +a ×y +a ×y =K ×( ×u + ×u + ×u ) φφφ (29)

The indexes in relations (27), (28) and the first parameter in relation (29), correspond to the order of derivative with respect to time (t). The second parameter in (29) corresponds to the

The radiation field intensity ( y00 ) defined by (1) is converted by the measurement transducer M into electrical signal ( m0 ) found in (27). The control error ( <sup>0</sup> a ) results from

In the above presented adjustment scheme, the R controller with proportional integrative derivative (PID) behaviour defined in (28) emits a control signal ( <sup>0</sup> c ) which adjusts the

The radiation emitting unit, E, generates the execution signal ( u0 ) representing the incident radiation field applied to various environments (e.g. air, tissues), together with a resultant

u =u ±u S0 0 0 (30)

In the above equation, ( uS0 ) is the resulted electromagnetic field and ( u0 ) corresponds to the homogeneity disturbance Fovd(v) from (16), where v=sd , which may be defined in

**3.2 Analogical modelling** 

as well as by partial differential equation (29).

Fig. 5. Control block diagram of the proposed adjustment system

order of derivative with respect to variable (s), defined in relation (2).

electromagnetic field generator E, as defined in relation (28).

Cartesian space as seen in figure 6 and expressed in (31).

(27), in which ( w0 ) is the reference signal.

disturbance signal (ữ0), resulting in:

$$\mathbf{t}\_i = \sqrt{\mathbf{t}\_{pl}^2 + \mathbf{t}\_{q\ell}^2 + \mathbf{t}\_{rl}^2} \quad ; \quad \mathbf{s}\_i = \sqrt{\mathbf{p}\_i^2 + \mathbf{q}\_l^2 + \mathbf{r}\_i^2} \quad ; \quad \mathbf{s}\_d = \sqrt{\mathbf{p}\_d^2 + \mathbf{q}\_d^2 + \mathbf{r}\_d^2} \tag{31}$$

Fig. 6. FOT and FOS including homogeneity disturbance at sd

Thus, the complex phenomenon of radiation spread can be estimated and adjusted by partial differential equation (29).

In the left side of this equation we introduced the approximated solution

$$\mathbf{y}\_{\rm{o}0}[\mathbf{t}, \mathbf{s}, \mathbf{u}(\mathbf{t})] = \mathbf{K}\_{\rm{y}} \times \mathbf{F}\_{\rm{o}0}(\mathbf{t}, \mathbf{s}) \times \mathbf{u}\_{\rm{o}0}(\mathbf{t})\tag{32}$$

where F (t,s)=F (t)×F (s) <sup>00</sup> 0T 0S and ( K ) is a weighting coefficient. y

This resulted for the right side of equation (29) in:

$$\mathbf{q}\_{0.00} = \mathbf{a}\_{\alpha} \times \mathbf{F}\_{\alpha} + \mathbf{a}\_{\alpha} \times \mathbf{F}\_{\alpha} + \mathbf{a}\_{\alpha} \times \mathbf{F}\_{\alpha} + \mathbf{a}\_{\alpha} \times \mathbf{F}\_{\alpha} + \mathbf{a}\_{\alpha} \times \mathbf{F}\_{\alpha} + \mathbf{a}\_{\alpha} \times \mathbf{F}\_{\alpha} \tag{33}$$

$$\mathbf{q}\_{\rm out,00} = \mathbf{a}\_{10} \times \mathbf{F}\_{\rm no} + \mathbf{2} \times \mathbf{a}\_{20} \times \mathbf{F}\_{\rm no} + \mathbf{a}\_{11} \times \mathbf{F}\_{\rm no} \tag{34}$$

$$\mathbf{a}\_{2,0} = \mathbf{a}\_{20} \times \mathbf{F}\_{0} \tag{35}$$

in which partial derivatives ( F0... ), ( F11 ) and ( F...0 ) are obtained from (32). Only signal ( u0 ) is included in the block diagram. The homogeneity disturbance ( u0 ) is an independent external signal.

From the equation system (27)…(31), the matrix containing partial derivatives of the state vector (Mdpx) represented in (36) has been deduced for the entire adjustment system depicted in figure 5.

Radiation Protection in Medical Imaging 81

For each integration step ( Δt ) we operated with a number of 20 Taylor series, of which 6 for

It can be observed that 8 elements (signals) included in Taylor series are from the state vector component x while the other 12 are from the matrix component xS , all of them

The truncation error at each integration step ( Δt ) is proportional to <sup>7</sup> (Δt /7!) , for

In order to illustrate the radiation control that would result from an electronic implementation of the above adjustment system, we present the simulated results regarding

2. **heterogeneous** tissue with an intensity-**increasing** homogeneity disturbance located

3. **heterogeneous** tissue with both an intensity-**decreasing** and an intensity-**increasing** homogeneity disturbance, located between body surface and the targeted tissue depth For every case, the intensity and spread of electromagnetic radiation has been simulated in open loop (without radiation field control) and in closed loop (with radiation field

In all simulated cases the adjustment goal was to ensure a predefined, constant and uniform intensity level of 5 units (e.g. 5 mGy) at a tissue depth of 50 units (e.g. 50 mm), regardless of potential intensity disturbers (radiation absorbing or intensifying environments) localized

Since our goal has been to develop a general model, we did not impose certain scales or units of measurement. The scales and units of measurement constitute flexible choices that may be conveniently defined for every imaging technique for which our method would be implemented. Given that initialization parameters need to be conveniently chosen and experimentally calibrated for every targeted imaging technique (CT, cone-beam CT, conventional radiography, etc.), we will not undergo a presentation regarding the initial parameters used for these simulated examples. Details regarding the choice of these initial

Simulation results regarding field intensity have been plotted against time and tissue depth, as seen in figures 7-12. Colour coding has been employed in order to highlight different

For every simulated case (figures 7 and 8; figures 9 and 10; figures 11 and 12), significant changes regarding electromagnetic field intensity at different tissue depth can be observed

field intensity and its automatic adjustment for three simulated tissue types:

between body surface and the tissue depth targeted for imaging

between body surface and the targeted tissue depth of 50 units.

parameters may be found in Roman et al., 2009, 2010 and Colosi et al., 2010.

between the unadjusted systems and the ones adjusted by the PID controller.

Numerical simulation was performed using Matlab 7.5.0.

intensity levels of the electromagnetic radiation field.

( m ,m ,c ,c ,u ,u 0k 1k 0k 1k 0k 1k ) and 14 for ( y00k 01k 06k ,y ,...,y ).

( m, c, u, y00 ), as well as to 6 (Δt /6!) , for ( y10 ).

**3.4 Examples of simulated radiation control** 

included in Mdpx .

1. homogenous tissue

control).

Because the signals m=m(t), c=c(t) and u=u(t) are functions only of time, all their partial derivatives with respect to (s) are equal to zero. Details concerning analogical modelling of systems through Mdpx may be found in T. Colosi et al., 2009.

#### **3.3 Numerical simulation**

Numerical simulation has been performed using Mdpx and Taylor series, which required knowledge of initial conditions CI(t0,s), for x =x(t ,s)=x CI 0 k-1 . Hence, by partial derivative with respect to (s), x =x (t ,s)=x S,CI s 0 S,k-1 , in which sequence (k-1) corresponds to the moment k-1 t =(k-1)×Δt , and sequence (k) corresponds to k t =k×Δt , where ( Δt ) is a sufficiently small integration step.

By means of xCI and xS,CI we computed the composing elements of xT and xTS , using specific operations based on symbolic derivatives by indices: x =x t ,s =x T,CI T 0 T,k-1 and x =x t ,s =x . TS,CI TS 0 TS,k-1

After setting these initial conditions, we approximated through iterative calculus:

$$\mathbf{x}\_{\mathbf{k}} = \mathbf{x}\_{\mathbf{k}\cdot 1} + \sum\_{\mathbf{T}=1}^{6} \frac{\Delta \mathbf{t}^{\top}}{\mathbf{T}!} \times \mathbf{x}\_{\mathbf{T}\_{\mathrm{I},\mathrm{k}-1}}; \qquad \mathbf{x}\_{\mathbf{s},\mathbf{k}} = \mathbf{x}\_{\mathbf{s},\mathbf{k}-1} + \sum\_{\mathbf{T}=1}^{6} \frac{\Delta \mathbf{t}^{\top}}{\mathbf{T}!} \times \mathbf{x}\_{\mathbf{I}\mathbf{s},\mathbf{k}-1} \tag{37}$$

m0 0 m1 0 c0 0 c1 0 u0 0 u1 0 y00 y01 y02 ... y08 y10 y11 y12 ... y18

u2 0 y20 y21 y22 ... y28 m3 0 c3 0 u3 0 y30 y31 y32 ... y38 ... m6 0 c6 0 u6 0 y60 y61 y62 ... y68

Because the signals m=m(t), c=c(t) and u=u(t) are functions only of time, all their partial derivatives with respect to (s) are equal to zero. Details concerning analogical modelling of

Numerical simulation has been performed using Mdpx and Taylor series, which required knowledge of initial conditions CI(t0,s), for x =x(t ,s)=x CI 0 k-1 . Hence, by partial derivative with respect to (s), x =x (t ,s)=x S,CI s 0 S,k-1 , in which sequence (k-1) corresponds to the moment k-1 t =(k-1)×Δt , and sequence (k) corresponds to k t =k×Δt , where ( Δt ) is a sufficiently small

By means of xCI and xS,CI we computed the composing elements of xT and xTS , using specific operations based on symbolic derivatives by indices: x =x t ,s =x T,CI T 0 T,k-1 and

T! ; <sup>6</sup> <sup>T</sup>

S,k S,k-1 TS,k-1 T=1 Δt x =x + ×x

T! (37)

After setting these initial conditions, we approximated through iterative calculus:

6 T k k-1 T,k-1 T=1 Δt x =x + ×x

systems through Mdpx may be found in T. Colosi et al., 2009.

**3.3 Numerical simulation** 

integration step.

x =x t ,s =x . TS,CI TS 0 TS,k-1

(36)

x xS <sup>=</sup> m2 0 xT xTS c2 0

**M**dpx =

For each integration step ( Δt ) we operated with a number of 20 Taylor series, of which 6 for ( m ,m ,c ,c ,u ,u 0k 1k 0k 1k 0k 1k ) and 14 for ( y00k 01k 06k ,y ,...,y ).

It can be observed that 8 elements (signals) included in Taylor series are from the state vector component x while the other 12 are from the matrix component xS , all of them included in Mdpx .

The truncation error at each integration step ( Δt ) is proportional to <sup>7</sup> (Δt /7!) , for ( m, c, u, y00 ), as well as to 6 (Δt /6!) , for ( y10 ).

#### **3.4 Examples of simulated radiation control**

In order to illustrate the radiation control that would result from an electronic implementation of the above adjustment system, we present the simulated results regarding field intensity and its automatic adjustment for three simulated tissue types:


For every case, the intensity and spread of electromagnetic radiation has been simulated in open loop (without radiation field control) and in closed loop (with radiation field control).

In all simulated cases the adjustment goal was to ensure a predefined, constant and uniform intensity level of 5 units (e.g. 5 mGy) at a tissue depth of 50 units (e.g. 50 mm), regardless of potential intensity disturbers (radiation absorbing or intensifying environments) localized between body surface and the targeted tissue depth of 50 units.

Numerical simulation was performed using Matlab 7.5.0.

Since our goal has been to develop a general model, we did not impose certain scales or units of measurement. The scales and units of measurement constitute flexible choices that may be conveniently defined for every imaging technique for which our method would be implemented. Given that initialization parameters need to be conveniently chosen and experimentally calibrated for every targeted imaging technique (CT, cone-beam CT, conventional radiography, etc.), we will not undergo a presentation regarding the initial parameters used for these simulated examples. Details regarding the choice of these initial parameters may be found in Roman et al., 2009, 2010 and Colosi et al., 2010.

Simulation results regarding field intensity have been plotted against time and tissue depth, as seen in figures 7-12. Colour coding has been employed in order to highlight different intensity levels of the electromagnetic radiation field.

For every simulated case (figures 7 and 8; figures 9 and 10; figures 11 and 12), significant changes regarding electromagnetic field intensity at different tissue depth can be observed between the unadjusted systems and the ones adjusted by the PID controller.

Radiation Protection in Medical Imaging 83

Fig. 10. Adjustment of the electromagnetic radiation field towards a constant intensity of 5 units at a tissue depth of 50 units, simulated in closed loop, for a heterogeneous tissue with

Fig. 11. Intensity of the electromagnetic radiation field, simulated in open loop (without radiation field adjustment), for a heterogeneous tissue with an intensity-decreasing homogeneity disturbance at a depth of 15 units and an intensity-increasing homogeneity

Figures 7, 9 and 11 illustrate the fact that, due to radiation absorption from a dense environment, an unadjusted systems can not achieve the targeted dose of 5 intensity units at the desired depth of 50 units, unless a significantly higher field intensity is administered to

On the other hand, when the systems are adjusted by the PID controller, the electromagnetic field intensity at a tissue depth of 50 units is stabilized at the targeted value of 5 units, in all

Thus, the controlled systems in figures 8, 10 and 12 ensure a minimization of the total administered radiation dose necessary to achieve 5 intensity units at 50 units of tissue depth. As may be inferred from the examples above, this general and highly flexible model offers a foundation for very precise electromagnetic field adjustments. Experimental validation and calibration for concrete imaging techniques, as well as an electronic implementation of the

model, remain to be performed in order to implement this model for practical use.

an intensity-increasing homogeneity disturbance at a depth of 15 units.

disturbance at a depth of 35 units.

begin with.

simulated cases.

Fig. 7. Intensity of the electromagnetic radiation field, simulated in open loop (without radiation field adjustment), for a homogenous tissue.

Fig. 8. Adjustment of the electromagnetic radiation field towards a constant intensity of 5 units at a tissue depth of 50 units, simulated in closed loop, for a homogenous tissue.

Fig. 9. Intensity of the electromagnetic radiation field, simulated in open loop (without radiation field adjustment), for a heterogeneous tissue with an intensity-increasing homogeneity disturbance at a depth of 15 units.

Fig. 7. Intensity of the electromagnetic radiation field, simulated in open loop (without

Fig. 8. Adjustment of the electromagnetic radiation field towards a constant intensity of 5 units at a tissue depth of 50 units, simulated in closed loop, for a homogenous tissue.

Fig. 9. Intensity of the electromagnetic radiation field, simulated in open loop (without radiation field adjustment), for a heterogeneous tissue with an intensity-increasing

homogeneity disturbance at a depth of 15 units.

radiation field adjustment), for a homogenous tissue.

Fig. 10. Adjustment of the electromagnetic radiation field towards a constant intensity of 5 units at a tissue depth of 50 units, simulated in closed loop, for a heterogeneous tissue with an intensity-increasing homogeneity disturbance at a depth of 15 units.

Fig. 11. Intensity of the electromagnetic radiation field, simulated in open loop (without radiation field adjustment), for a heterogeneous tissue with an intensity-decreasing homogeneity disturbance at a depth of 15 units and an intensity-increasing homogeneity disturbance at a depth of 35 units.

Figures 7, 9 and 11 illustrate the fact that, due to radiation absorption from a dense environment, an unadjusted systems can not achieve the targeted dose of 5 intensity units at the desired depth of 50 units, unless a significantly higher field intensity is administered to begin with.

On the other hand, when the systems are adjusted by the PID controller, the electromagnetic field intensity at a tissue depth of 50 units is stabilized at the targeted value of 5 units, in all simulated cases.

Thus, the controlled systems in figures 8, 10 and 12 ensure a minimization of the total administered radiation dose necessary to achieve 5 intensity units at 50 units of tissue depth.

As may be inferred from the examples above, this general and highly flexible model offers a foundation for very precise electromagnetic field adjustments. Experimental validation and calibration for concrete imaging techniques, as well as an electronic implementation of the model, remain to be performed in order to implement this model for practical use.

Radiation Protection in Medical Imaging 85

Hara, A.K. et al. (2009). Iterative reconstruction technique for reducing body radiation dose

International Commission on Radiological Protection, 2007. The 2007 Recommendations of

Jursinic, P.A., 2007. Characterization of optically stimulated luminescent dosimeters, OSLDs, for clinical dosimetric measurements. Medical Physics, 34(12), pp.4594-4604. Kase, K.R., 2004. Radiation protection principles of NCRP. Health Physics, 87(3), pp.251-257. Kron, T., 1994. Thermoluminescence dosimetry and its applications in medicine--Part 1:

Marin, D. et al. (2010). Low-Tube-Voltage, High-Tube-Current Multidetector Abdominal CT:

McCollough, C.H., Bruesewitz, M.R. & Kofler, J.M., Jr, (2006). CT dose reduction and dose

Miéville, F.A. et al. (2011). Paediatric cardiac CT examinations: impact of the iterative

Mladenov, E. & Iliakis, G. (2011). Induction and repair of DNA double strand breaks: The

National Highway Traffic Safety Administration Fatality Analysis Reporting System,

Renker, M. et al. (2011). Evaluation of Heavily Calcified Vessels with Coronary CT

http://www.ncbi.nlm.nih.gov/pubmed/21693660 [Accessed July 1, 2011], ISSN

Roman, N.M.; Colosi, H. & Pusca, M. (2009). Digital Simulation for Computer Control of

Roman, N.M.; Colosi, H. & Pusca, M. (2010). Numerical Simulation Using Partial

fars.nhtsa.dot.gov/Main/index.aspx [Accessed July 1, 2011].

pp.764-771, ISSN 1546-3141

pp.175-199.

254, No. 1, pp.145 -153

ISSN 1527-1323

1527-1315

Annals of the ICRP, 37(2-4), pp.1-332.

[Accessed July 1, 2011]. ISSN 1432-1998

Vol. 711 No. 1-2, pp.61-72, ISSN 0027-5107

Reconstruction. *Radiology*. Available at:

Romania, September 23-26, 2009

at CT: feasibility study. AJR. American Journal of Roentgenology, Vol. 193, No. 3,

the International Commission on Radiological Protection. ICRP publication 103.

Physics, materials and equipment. Australasian Physical & Engineering Sciences in Medicine / Supported by the Australasian College of Physical Scientists in Medicine and the Australasian Association of Physical Sciences in Medicine, 17(4),

Improved Image Quality and Decreased Radiation Dose with Adaptive Statistical Iterative Reconstruction Algorithm—Initial Clinical Experience1. *Radiology*, Vol.

management tools: overview of available options. Radiographics: *A Review Publication of the Radiological Society of North America*, Inc, Vol. 26, No. 2, pp.503-512,

reconstruction method ASIR on image quality - preliminary findings. *Pediatric Radiology*. Available at: http://www.ncbi.nlm.nih.gov/pubmed/21717165

increasing spectrum of non-homologous end joining pathways. *Mutation Research*,

Fatality Analysis and Reporting System Encyclopedia. *National Highway Traffic Safety Administration Fatality Analysis Reporting System.* Available at: http://www-

Angiography: Comparison of Iterative and Filtered Back Projection Image

Radiological Devices: A Preliminary Model Using Partial Differential Equations, *Proceedings of MediTech 2009 International Conference on Advancements of Medicine and Health Care through Technology,* pp. 37-42, ISBN 978-3-642-04291-1, Cluj-Napoca,

Differential Equations for Modeling and Control of Medical Radiation Fields.

Fig. 12. Adjustment of the electromagnetic radiation field towards a constant intensity of 5 units at a tissue depth of 50 units, simulated in closed loop, for a heterogeneous tissue with an intensity-decreasing homogeneity disturbance at a depth of 15 units and an intensityincreasing homogeneity disturbance at a depth of 35 units.

#### **4. Conclusion**

The proposed radiation adjustment model offers a precise tool to assist in meeting the overarching goal of radiation protection: to obtain a maximum of relevant imaging information, with the least amount of exposure to ionizing radiation.

#### **5. References**


Fig. 12. Adjustment of the electromagnetic radiation field towards a constant intensity of 5 units at a tissue depth of 50 units, simulated in closed loop, for a heterogeneous tissue with an intensity-decreasing homogeneity disturbance at a depth of 15 units and an intensity-

The proposed radiation adjustment model offers a precise tool to assist in meeting the overarching goal of radiation protection: to obtain a maximum of relevant imaging

Brenner, D.J. & Hall, E.J. (2007). Computed Tomography — An Increasing Source of

Coakley, F.V. et al. (2011). CT radiation dose: what can you do right now in your practice?

Colosi, H. & Roman, N.M. (2010). Digital Simulation and Control of Electromagnetic Field

Colosi, T.; Ungureşan, M.L.; Dulf, E.H. & Cordoş, R.C. (2009). *Introduction to Anlogical* 

Cornfeld, D. et al. (2011). Impact of Adaptive Statistical Iterative Reconstruction (ASIR) on

Hall, E.J. & Giaccia, A.J. (2005). *Radiobiology for the Radiologist.* (6th ed.), ISBN 978- 0781741514 Lippincott Williams & Wilkins Publishing, Philadelphia, PA.

Quantitative Analysis. *Am. J. Roentgenol*., Vol. 196, No. 3, pp.W336-340. Flicek, K.T. et al. (2010). Reducing the radiation dose for CT colonography using adaptive

Radiation Exposure. *New England Journal of Medicine*, Vol. 357, No. 22, pp.2277-2284,

AJR. American *Journal of Roentgenology*, Vol. 196, No. 3, pp.619-625, ISSN 1546-3141.

Effects in Heterogeneous Organic Tissues, *Proceedings of 2010 IEEE International Conference on Automation, Quality and Testing, Robotics,* pp. 321-326, ISBN 978-1-

*Modeling and Numerical Simulation, with (Mpdx) and Taylor Series for Distributed Parameters Processes*, Ed. Galaxia Gutenberg, ISBN 978-973-141-192-7, Târgu-Lăpuş,

Radiation Dose and Image Quality in Aortic Dissection Studies: A Qualitative and

statistical iterative reconstruction: A pilot study. *American Journal of Roentgenology*,

increasing homogeneity disturbance at a depth of 35 units.

information, with the least amount of exposure to ionizing radiation.

4244-6722-8, Cluj-Napoca, Romania, May 28-30, 2010

Vol. 195, No. 1, pp.126-131, ISSN 1546-3141.

**4. Conclusion** 

**5. References** 

ISSN 0028-4793.

Romania


**Part 2** 

**Image Processing Techniques** 

*Annals of the Academy of Romanian Scientists, Series on Science and Technology of Information*, Vol.3, No.1, (2010), pp. 59-74, ISSN 2066-2742

