医学成像物理学入门：理论与临床应用详解

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"《医学成像物理学、工程与临床应用简介》是一本精心编写的高级/第一年研究生水平教材，旨在为学生提供全面的医学成像基础知识。该书内容涵盖了X射线、计算机断层扫描（CT）、正电子发射断层扫描（PET）、核医学、超声成像以及磁共振成像（MRI）等核心领域。书中注重将复杂的物理概念以易于理解的方式讲解，每章后都附有详尽的习题集，包括解答例题，这种设计让学生既能深入理解理论知识，又能在实践中锻炼解决问题的能力。作者关注最新技术的发展，如联合CT/PET成像、多层CT、四维超声和并行成像MRI技术，确保学生紧跟行业前沿。对于临床实例的引入，使学生能够将物理学和工程技术原理应用于实际医疗情境中，提升其理论与实践结合的能力。此外，每个章节末尾还提供了专门参考文献、近期综述文章和相关科学期刊链接，使得本书成为一学期课程的理想教材，适合对医学成像感兴趣的高级本科生和初学者进行系统学习。《医学成像物理学、工程与临床应用简介》不仅注重基础理论教学，还强调了实际操作技能的培养，是医学影像专业学生和研究人员不可或缺的学习工具，有助于他们在这个快速发展的领域内不断深化理解和应用。"

medical images, namely the Fourier transform and ﬁltered backprojection, are

introduced.

1.2 Speciﬁcity, sensitivity and the receiver operating characteristic

(ROC) curve

The accuracy of clinical diagnoses depends critically upon image quality, the

higher the quality the more accurate the diagnosis. Improvements in imaging

techniques and instrumentation have revolutionized early diagnosis and treatment

of a number of different pathological conditions. Each new imaging technique or

change in instrumentation must be carefully assessed in terms of its effect on

diagnostic accuracy. For example, although the change from planar X-ray ﬁlm

Figure 1.1

(a) MR image showing two small white-matter lesions indicated by the arrows. Corresponding

images acquired with (b) four times poorer spatial resolution, (c) four times lower SNR, and (d) a

reduced CNR between the lesions and the surrounding healthy tissue. The arrows point to lesions that

can be detected.

General image characteristics

to digital radiography clearly has many practical advantages in terms of data

storage and mobility, it would not have been implemented clinically had the

diagnostic quality of the scans decreased. Quantitative assessment of diagnostic

quality is usually reported in terms of speciﬁcity and sensitivity, as described in the

example below.

Consider an imaging study to determine whether a group of middle-aged

patients has an early indication of multiple sclerosis. It is known that this

disease is characterized by the presence of white matter lesions in the brain.

However, it is also known that healthy people develop similar types of lesion as

they age, but that the number of lesions is not as high as for multiple sclerosis

cases. When analyzing the images from a particular patient there are four

possible outcomes for the radiologist: a true positive (where the ﬁrst term ‘true’

refers to a correct diagnosis and the second term ‘positive’ to the patient having

multiple sclerosis), a true negative, a false positive or a false negative. The four

possibilities can be recorded in either tabular or graphical format, as shown in

Figure 1.2. The receiver operating characteristic (ROC) curve plots the number of

true positives on the vertical axis vs. the number of false positives on the horizontal

axis, as shown on the right of Figure 1.2. What criterion does the radiologist use to

make his/her diagnosis? In this simple example assume that the radiologist simply

counts the number of lesions detectable in the image. The relative number of true

positives, true negatives, false positives and false negatives depends upon the

particular number of lesions that the radiologist decides upon as being the thresh-

old for diagnosing a patient with multiple sclerosis. If this threshold number is very

high, for example 1000, then there will be no false positives, but no true positives

either. As the threshold number is reduced then the number of true positives will

increase at a greater rate than the false positives, providing that the images are

giving an accurate count of the number of lesions actually present. As the criterion

for the number of lesions is reduced further, then the numbers of false positives

and true positives increase at a more equal rate. Finally, if the criterion is

dropped to a very small number, then the number of false positives increases

much faster than the true positives. The net effect is to produce a curve shown

in Figure 1.2.

Three measures are commonly reported in ROC analysis:

(i) accuracy is the number of correct diagnoses divided by the total number of

diagnoses;

(ii) sensitivity is the number of true positives divided by the sum of the true

positives and false negatives; and

(iii) speciﬁcity is the number of true negatives divided by the sum of the number of

true negatives and false positives.

1.2 Speciﬁcity, sensitivity and the ROC curve

The aim of clinical diagnosis is to maximize each of the three numbers, with an

ideal value of 100% for all three. This is equivalent to a point on the ROC curve

given by a true positive fraction of 1, and a false positive fraction of 0. The

corresponding ROC curve is shown as the dotted line in Figure 1.2. The closer

the actual ROC curve gets to this ideal curve the better, and the integral under

the ROC curve gives a quantitative measure of the quality of the diagnostic

procedure.

A number of factors can inﬂuence the shape of the ROC curve, but from the

point-of-view of the medical imaging technology, the relevant question is: ‘what

fraction of true lesions is detected using the particular imaging technique?’. The

more lesions that are missing on the image, then intuitively the poorer the resulting

diagnosis. ‘Missing lesions’ may occur, referring to Figure 1.1, due to poor SNR,

CNR or spatial resolution. In turn, these will lead to a decreased percentage

accuracy, sensitivity and speciﬁcity of the diagnostic procedure.

Example 1.1 Suppose that a criterion used for diagnosis is, in fact, completely unrelated to the

actual medical condition, e.g. as a trivial example, trying to diagnose cardiac

disease by counting the number of lesions in the brain. Draw the ROC curve for

this particular situation.

Solution Since the criterion used for diagnosis is independent of the actual condition,

effectively we have an exactly equal chance of a true positive or a false positive,

irrespective of the number of lesions found in the brain. Therefore, the ROC curve

is a straight line at an angle of 45° to the main axes, as shown below.

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Figure 1.2

The receiver operating characteristic curve. (left) The 2 3 2 table showing the four possible

outcomes of clinical diagnosis. (right) A real ROC curve (solid line), with the ideal curve also shown

(dotted line).

General image characteristics

1.3 Spatial resolution

The spatial resolution of an imaging system is related to the smallest feature that

can be visualized or, more speciﬁcally, the smallest distance between two features

such that the features can be individually resolved rather than appearing as one

larger shape. The two most common measures in the spatial domain are the line

spread function (LSF) and point spread function (PSF). These measures can be

represented by an equivalent modulation transfer function (MTF) in the spatial

frequency domain. The concept of spatial frequency is very useful in characteriz-

ing spatial resolution, and is explained in the following section.

1.3.1 Spatial frequencies

One familiar example of spatial frequencies is a standard optician’s test. In one test,

patients are asked to look at a series of black lines on a white background, and then

to tell the optician if they can resolve the lines when a series of lenses with different

strengths are used. As shown in Figure 1.3, the lines are of different thickness and

separation. The spatial frequency of a particular grid of lines is measured as

the number of lines/mm, for example 5 mm

for lines spaced 200 lmapart.The

closer together are the lines, the higher is the spatial frequency, and the better the

spatial resolution of the image system needs to be to resolve each individual line.

For each medical imaging modality, a number of factors affect the resolving

power. One can simplify the analysis by considering just two general components:

ﬁrst the instrumentation used to form the image, and second the quantity of data

that is acquired i.e. the image data matrix size. To take the everyday example of a

digital camera, the lens and electronics associated with the charge-coupled device

(CCD) detector form the instrumentation, and the number of pixels (megapixels) of

the camera dictates the amount of data that is acquired. The relative contribution of

each component to the overall spatial resolution is important in an optimal

Figure 1.3

Grid patterns with increasing spatial frequencies going from (a) to (c).

1.3 Spatial resolution

engineering design. There is no advantage in terms of image quality in increasing

the CCD matrix size from 10 megapixels to 20 megapixels if the characteristics of

the lens are poor, e.g. it is not well focused, produces blur, or has chromatic

aberration. Similarly, if the lens is extremely well-made, then it would be sub-

optimal to have a CCD with only 1 megapixel capability, and image quality would

be improved by being able to acquire a much greater number of pixels.

1.3.2 The line spread function

The simplest method to measure the spatial resolution of an imaging system is to

perform the equivalent of the optician’s test. A single thin line or set of lines is

imaged, with the relevant structure made of the appropriate material for each

different imaging modality. Examples include a strip of lead for an X-ray scan,

a thin tube of radioactivity for nuclear medicine imaging, or a thin wire embedded

in gel for an ultrasound scan. Since the imaging system is not perfect, it introduces

some degree of blurring into the image, and so the line in the image does not appear

as sharp as its actual physical shape. The degree of blurring can be represented

mathematically by a line-spread function (LSF), which is illustrated in Figure 1.4.

The LSF of an imaging system is estimated by measuring a one-dimensional

projection, as shown in Figure 1.4, with y deﬁned as the horizontal direction. The

width of the LSF is usually deﬁned by a parameter known as the full-width-at-half-

maximum (FWHM). As the name suggests, this parameter is the width of the

particular function at a point which is one-half the maximum value of the vertical

axis. From a practical point-of-view, if two small structures in the body are

LSF(y)

Figure 1.4

The concept of the line-spread function. A thin object is imaged using three different imaging systems.

The system on the left has the sharpest LSF, as deﬁned by the one-dimensional projection measured

along the dotted line and shown above each image. The system in the middle produces a more

blurred image, and has a broader LSF, with the system on the right producing the most blurred image

with the broadest LSF.

General image characteristics

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