📚 A-Level AQA Medical Physics Revision Notes | A-Level AQA 物理：医疗物理 考点精讲
Medical Physics is one of the optional modules in AQA A-Level Physics. It covers the physical principles behind a wide range of diagnostic and therapeutic techniques used in modern medicine. From the production of X‑rays and their interaction with tissue, to ultrasound imaging, fibre‑optic endoscopy, nuclear medicine and radiotherapy, this topic demands a clear understanding of core physics concepts such as wave behaviour, particle interactions, attenuation, and electromagnetism. These revision notes are structured to align with the AQA specification, providing concise explanations of each key area, with all necessary equations and definitions highlighted.
医疗物理是AQA A-Level物理的可选模块之一，涵盖了现代医学中常用诊断和治疗技术背后的物理原理。从X射线的产生及其与组织的相互作用，到超声成像、光纤内窥镜、核医学和放射治疗，该专题要求清晰理解波行为、粒子相互作用、衰减、电磁学等核心物理概念。本复习笔记按照AQA考纲结构编写，简明阐释每个关键领域，并突出所有必要的方程和定义。
1. X-ray Production and the Continuous Spectrum | X射线的产生与连续谱
X‑rays are produced in an X‑ray tube when fast-moving electrons are suddenly decelerated upon striking a metal target. A heated cathode emits electrons by thermionic emission, and a high potential difference (typically 30–150 kV) between the cathode and the anode accelerates these electrons across the vacuum. The kinetic energy gained by an electron is Eₖ = eV. When the electrons hit the target (usually tungsten), most of the energy is converted into heat, but a small fraction produces X‑rays via the bremsstrahlung (braking radiation) mechanism. The resulting spectrum is continuous up to a maximum photon energy corresponding to the tube voltage: hfₘₐₓ = eV, which gives a minimum wavelength λₘᵢₙ = hc/(eV).
X射线在X射线管中产生，当高速电子撞击金属靶时突然减速。加热的阴极通过热电子发射释放电子，阴阳极之间的高电压（通常30–150 kV）在真空中加速这些电子。电子获得的动能为 Eₖ = eV。当电子撞击靶（通常为钨）时，大部分能量转化为热，但一小部分通过轫致辐射机制产生X射线。产生的能谱是连续的，最高光子能量对应于管电压：hfₘₐₓ = eV，从而得到最小波长 λₘᵢₙ = hc/(eV)。
A typical continuous spectrum shows intensity versus wavelength, with the intensity rising to a peak before falling to zero at λₘᵢₙ. Increasing the tube current (mA) increases the number of electrons and therefore the intensity of the beam without changing the shape of the spectrum. Increasing the tube voltage (kV) shifts λₘᵢₙ to shorter wavelengths and increases both the intensity and the average photon energy, producing a ‘harder’ beam that is more penetrating.
典型的连续谱显示强度随波长变化，强度上升到一个峰值，然后在λₘᵢₙ处降为零。增大管电流（mA）会增加电子数量，从而提高X射线束的强度，而不改变谱形。增大管电压（kV）会使λₘᵢₙ向更短波长移动，同时提高强度和平均光子能量，产生穿透力更强的“硬”射线束。
2. Characteristic X-rays and Filtration | 特征X射线与滤波
When an incoming electron has enough energy to eject an inner-shell electron from a target atom, the vacancy is filled by an electron from a higher energy level. The transition releases a photon with an energy equal to the difference between the two energy levels. These photons produce sharp peaks in the X‑ray spectrum, known as characteristic X‑rays, at fixed wavelengths that depend on the target material. For tungsten, the K‑series lines are prominent in the 60–70 keV region.
当入射电子具有足够能量将靶原子内壳层电子击出时，空穴会被较高能级电子填补。该跃迁释放的光子能量等于两能级差。这些光子在X射线谱中形成尖锐的峰，称为特征X射线，其固定波长取决于靶材料。对钨而言，K系谱线在60–70 keV区域显著。
To improve image quality, a thin metal filter (often aluminium) is placed in the beam path. The filter absorbs low‑energy photons more strongly than high‑energy ones because attenuation coefficients are larger at low photon energies. This process ‘hardens’ the beam, reducing patient dose from photons that would be absorbed by the skin without contributing to the image. Filtration also removes parts of the spectrum that would reduce contrast.
为提高图像质量，在射束路径中放置薄金属滤波片（通常为铝）。滤波器对低能光子的吸收远强于高能光子，因为低光子能量下衰减系数更大。这一过程“硬化”了射线束，减少了会被皮肤吸收而无助于成像的光子所导致的患者剂量。滤波还去除了会降低对比度的谱成分。
3. Attenuation and Half-Value Thickness | 衰减与半值厚度
When X‑rays pass through matter, their intensity decreases. The attenuation follows an exponential law:
I = I₀ e−μx
, where I₀ is the incident intensity, I is the transmitted intensity, x is the thickness of the material, and μ is the linear attenuation coefficient (unit: m⁻¹ or cm⁻¹). The coefficient μ depends on the photon energy and the atomic number of the material.
当X射线穿过物质时，其强度减弱。衰减遵循指数规律：
I = I₀ e−μx
，其中I₀为入射强度，I为透射强度，x为材料厚度，μ为线性衰减系数（单位：m⁻¹或cm⁻¹）。系数μ取决于光子能量和材料的原子序数。
A convenient clinical quantity is the half‑value thickness (HVT), defined as the thickness required to reduce the intensity to half its original value. Setting I = I₀/2 gives x½ = ln 2 / μ ≈ 0.693 / μ. Materials with a large μ have a small HVT, meaning they are good absorbers. Bone (high Z) absorbs X‑rays more than soft tissue, which is the basis of contrast in conventional radiography.
临床上一个方便的量是半值厚度（HVT），定义为将强度降为一半所需的厚度。令I = I₀/2可得 x½ = ln 2 / μ ≈ 0.693 / μ。μ大的材料HVT小，即它们吸收能力强。骨（高Z）比软组织吸收更多X射线，这正是常规X射线摄影对比度的基础。
4. Image Formation, Contrast and Intensifiers | 图像形成、对比度与增强器
A radiographic image is formed because different tissues attenuate the X‑ray beam by different amounts. The transmitted intensity pattern is detected by a film-screen system or a digital flat-panel detector. Contrast arises from differences in μ between adjacent structures. Contrast can be improved by using a lower tube voltage, but this increases the dose to the patient. In diagnostic radiography a balance must be struck between acceptable contrast and patient safety.
X射线图像之所以形成，是因为不同组织对射束的衰减程度不同。透射后的强度分布被胶片-屏系统或数字平板探测器检测。对比度源于相邻结构之间μ值的差异。使用较低的管电压可以改善对比度，但这会增加患者剂量。诊断X射线摄影中必须在可接受的对比度与患者安全之间取得平衡。
Image intensifiers are used in real‑time fluoroscopy to convert the dim X‑ray image into a bright visible image. The input phosphor (caesium iodide) converts X‑rays into light photons, which then strike a photocathode and release electrons. These electrons are accelerated and focused onto a smaller output phosphor, producing a much brighter image that can be captured by a video camera. The brightness gain is typically several thousand times.
影像增强器用于实时透视，将微弱的X射线影像转换为明亮的可见影像。输入荧光体（碘化铯）将X射线转换为光子，光子再打到光电阴极上释放电子。这些电子被加速并聚焦到一个较小的输出荧光体上，产生亮度高得多的图像，可由摄像机捕捉。亮度增益通常达到数千倍。
5. Computed Tomography (CT) Scanning | 计算机断层扫描（CT）
CT scanning produces cross‑sectional images of the body by rotating an X‑ray tube and a row of detectors around the patient. As the gantry rotates, a thin fan‑beam of X‑rays passes through the patient from many different angles. The attenuation data are processed by a computer using filtered back‑projection to reconstruct a map of linear attenuation coefficients within the slice. CT images display the Hounsfield scale, defined as:
HU = 1000 × (μtissue − μwater) / μwater
CT扫描通过使X射线管和一组探测器围绕患者旋转，产生人体横断面图像。当机架旋转时，一束薄的扇形X射线从许多不同角度穿过患者。衰减数据由计算机使用滤波反投影法处理，重建出切片内线性衰减系数的分布图。CT图像显示亨氏单位，定义为：
HU = 1000 × (μ组织 − μ水) / μ水
On this scale, water is 0, air is around −1000, and bone can be +1000 or more. Modern multi‑slice CT scanners can acquire several slices per rotation, greatly reducing scan time and enabling 3D imaging. The main advantage of CT over conventional radiography is its ability to distinguish overlapping tissues, although it delivers a higher radiation dose.
在此刻度上，水为0，空气约−1000，骨骼可达+1000或更高。现代多层CT扫描仪每旋转一周可采集多个切片，大幅缩短扫描时间并实现三维成像。CT相对于常规X射线摄影的主要优势在于能够区分重叠的组织，尽管它提供的辐射剂量更高。
6. Ultrasound Physics: Piezoelectric Effect | 超声物理：压电效应
Ultrasound imaging uses high‑frequency sound waves, typically 1–15 MHz. At the heart of the transducer is a piezoelectric crystal, such as lead zirconate titanate (PZT). When an alternating potential difference is applied across the crystal, it vibrates at the same frequency and emits ultrasound. Conversely, when returning echoes cause the crystal to vibrate, it generates an alternating pd. The same transducer therefore acts as both transmitter and receiver, switching between modes thousands of times per second.
超声成像使用高频声波，通常为1–15 MHz。换能器的核心是压电晶体，如锆钛酸铅（PZT）。当在晶体两端施加交变电压时，晶体以相同频率振动并发射超声波。相反，当返回的回波使晶体振动时，它会产生交变电压。因此，同一个换能器既充当发射器又充当接收器，每秒在两种模式之间切换数千次。
The intensity of the ultrasound beam is described by I = P / A, and it decreases with depth because of absorption and scattering. The resolution depends on the wavelength λ: higher frequencies give shorter wavelengths and better spatial resolution, but they are attenuated more quickly, limiting the penetration depth. A typical compromise for abdominal imaging is 3–5 MHz.
超声波束的强度用 I = P / A 描述，它随深度增加因吸收和散射而降低。分辨率取决于波长λ：频率越高，波长越短，空间分辨率越好，但它们衰减更快，限制了穿透深度。腹部成像的典型折衷频率为3–5 MHz。
7. Acoustic Impedance and Reflection | 声阻抗与反射
When ultrasound encounters a boundary between two media, part of the energy is reflected. The fraction reflected is determined by the acoustic impedances of the two tissues. Acoustic impedance Z is defined as Z = ρ c, where ρ is the density and c is the speed of sound in the medium. For normal incidence, the intensity reflection coefficient R is given by:
R = ( (Z₂ − Z₁) / (Z₂ + Z₁) )²
当超声波遇到两个介质之间的界面时，部分能量被反射。反射的分数由两种组织的声阻抗决定。声阻抗Z定义为 Z = ρ c，其中ρ为密度，c为介质中的声速。对于垂直入射，强度反射系数R为：
R = ( (Z₂ − Z₁) / (Z₂ + Z₁) )²
If the impedances are very different, R is large and most of the ultrasound is reflected, leaving little to penetrate deeper. To minimise reflection at the skin‑transducer boundary, a coupling gel is used. The gel has an impedance close to that of skin and eliminates the air gap, which would otherwise cause almost total reflection because of the huge impedance mismatch between air and tissue.
如果阻抗差异很大，R就很大，大部分超声波被反射，只有很少能进入更深部。为减小换能器-皮肤界面的反射，使用耦合凝胶。该凝胶的阻抗与皮肤接近，并消除了空气间隙，否则由于空气与组织之间巨大的阻抗失配，几乎会造成全反射。
8. Doppler Ultrasound: Blood Flow Measurements | 多普勒超声：血流测量
The Doppler effect can be used to measure the velocity of blood flow. When ultrasound of frequency f₀ is incident on moving red blood cells, the reflected frequency fᵣ is shifted. For a transducer at an angle θ to the direction of flow, the Doppler shift frequency Δf is:
Δf = fᵣ − f₀ ≈ (2 f₀ v cos θ) / c
where v is the speed of the blood and c is the speed of sound in tissue (≈ 1540 m s⁻¹). The equation assumes v ≪ c.
多普勒效应可用于测量血流速度。当频率为f₀的超声波入射到运动的红细胞上时，反射频率fᵣ发生偏移。对于换能器与血流方向成角度θ的情形，多普勒频移Δf为：
Δf = fᵣ − f₀ ≈ (2 f₀ v cos θ) / c
其中v为血流速度，c为组织中的声速（≈ 1540 m s⁻¹）。该方程假设v ≪ c。
In practice, the shift is in the audible range and is presented as a sound or as a colour flow map overlaid on a B‑mode image. Colour Doppler assigns different colours to flow towards and away from the transducer, allowing rapid diagnosis of blockages or valve malfunctions. Continuous‑wave Doppler measures high speeds but lacks depth information, whereas pulsed‑wave Doppler provides range resolution but cannot measure very high velocities.
实际中，频移落在可听范围内，以声音或叠加在B模式图像上的彩色血流图呈现。彩色多普勒为朝向和背离换能器的流动分配不同颜色，可快速诊断堵塞或瓣膜故障。连续波多普勒可测量高速血流但缺乏深度信息，而脉冲多普勒提供距离分辨率但不能测量非常高的速度。
9. Fibre Optics and Endoscopy | 光纤与内窥镜
An endoscope allows doctors to see inside the body without major surgery. It uses a bundle of flexible optical fibres. Each fibre consists of a core with refractive index n₁ surrounded by cladding with a lower refractive index n₂. Light is transmitted by total internal reflection (TIR). TIR occurs when the angle of incidence at the core‑cladding boundary exceeds the critical angle θ꜀, where sin θ꜀ = n₂ / n₁.
内窥镜使医生无需大手术就能看到体内。它使用一束柔性光纤。每根光纤由折射率为n₁的纤芯和折射率较低的包层（n₂）构成。光通过全内反射（TIR）传输。当纤芯-包层界面的入射角超过临界角θ꜀时发生TIR，其中 sin θ꜀ = n₂ / n₁。
A coherent bundle preserves the relative positions of the fibres so that a recognisable image is transmitted. An incoherent bundle can only deliver illumination. In addition to the imaging bundle, endoscopes contain a light guide for illumination, a lens system at the tip, and often channels for instruments and water or air. The flexibility of the fibre is crucial for reaching curved regions such as the colon.
相干束保持光纤的相对位置，从而传输可识别的图像。非相干束仅能提供照明。除成像束外，内窥镜还包括用于照明的光导、末端的透镜系统以及通常用于器械和水/气的通道。光纤的柔韧性对于到达诸如结肠等弯曲区域至关重要。
10. Nuclear Medicine: Gamma Camera | 核医学：伽马相机
Nuclear medicine imaging uses radioactive tracers that emit gamma photons. The most common detector is the Anger gamma camera. It consists of a collimator (usually a lead plate with many parallel holes), a large scintillation crystal (NaI doped with Tl), an array of photomultiplier tubes (PMTs), and position‑sensitive electronics. The collimator ensures only gamma rays travelling perpendicular to the crystal are accepted, forming a two‑dimensional projection of the radioactivity distribution.
核医学成像使用发射伽马光子的放射性示踪剂。最常见的探测器是Anger伽马相机。它包含一个准直器（通常为带有许多平行孔的铅板）、一个大闪烁晶体（掺铊的碘化钠）、一组光电倍增管（PMT）和位置敏感电子线路。准直器确保只有垂直于晶体传播的伽马射线被接收，形成放射性分布的二维投影。
When a gamma photon interacts with the crystal, it produces a flash of visible light. The PMTs near the interaction site receive more light than those farther away, and analogue circuits compute the (x, y) coordinates of the event. The Z‑signal (sum of all PMT outputs) is used for energy discrimination to reject scattered photons. A typical tracer is technetium‑99m, which emits 140 keV gamma rays and has a half‑life of 6 hours.
当伽马光子与晶体相互作用时，产生可见光闪光。靠近相互作用点的PMT接收到的光比远处的多，模拟电路计算事件的（x, y）坐标。Z信号（所有PMT输出之和）用于能量甄别以排除散射光子。典型的示踪剂是锝‑99m，它发射140 keV伽马射线，半衰期为6小时。
11. Positron Emission Tomography (PET) | 正电子发射断层扫描（PET）
PET uses tracers labelled with positron‑emitting isotopes, such as fluorine‑18. A positron (β⁺) travels a short distance in tissue before annihilating with an electron. The annihilation produces two identical gamma photons, each of energy 511 keV, which travel in almost exactly opposite directions due to conservation of momentum.
PET使用以正电子发射同位素（如氟‑18）标记的示踪剂。正电子（β⁺）在组织中行进很短距离后与电子湮灭。湮灭产生两个相同的511 keV伽马光子，由于动量守恒，它们几乎沿恰好相反的方向运动。
The patient is surrounded by a ring of detectors. When two detectors register photons within a narrow time window (a few nanoseconds), a coincidence event is recorded. The line connecting the two detectors is called the line of response, and many such lines are used to reconstruct the distribution of the tracer using algorithms similar to those in CT. PET provides functional information, dramatically different from the anatomical detail of CT, and modern scanners often combine PET and CT (PET‑CT) into a single unit.
患者被一圈探测器包围。当两个探测器在很窄的时间窗（几纳秒）内都记录到光子，即记录一次符合事件。连接两个探测器的线称为响应线，利用许多这样的线通过类似CT的算法重建示踪剂分布。PET提供功能信息，与CT的解剖细节有本质不同，现代扫描仪常将PET与CT（PET‑CT）整合为一台设备。
12. Radiotherapy: External Beam and Brachytherapy | 放射治疗：外照射与近距离治疗
Radiotherapy uses ionising radiation to destroy malignant cells. The most common external beam method employs a medical linear accelerator (linac) that accelerates electrons to energies of 4–25 MeV. The electrons strike a high‑Z target to produce high‑energy bremsstrahlung X‑rays, which are shaped by a multi‑leaf collimator to match the tumour’s outline. The beam is rotated around the patient so that the tumour receives the maximum dose while sparing surrounding healthy tissue.
放射治疗利用电离辐射杀灭恶性细胞。最常见的外照射方法使用医用直线加速器（linac），将电子加速到4–25 MeV。电子撞击高Z靶产生高能轫致辐射X射线，再通过多叶准直器塑形以匹配肿瘤轮廓。射束围绕患者旋转，使肿瘤接受最大剂量同时保护周围健康组织。
Brachytherapy involves placing small sealed radioactive sources (such as iridium‑192 or iodine‑125) directly into or near the tumour. This delivers a very high local dose with a rapid fall‑off over a short distance, protecting normal tissue. The inverse‑square law is fundamental here: dose from a point source falls as 1/r². Treated tumours are often in the prostate, cervix or breast. All radiotherapy treatment plans are carefully calculated to deliver the prescribed absorbed dose (measured in gray, Gy) to the target volume.
近距离治疗涉及将小型密封放射源（如铱‑192或碘‑125）直接放置在肿瘤内部或附近。这可在短距离内提供非常高的局部剂量且剂量迅速跌落，保护正常组织。此处的平方反比定律至关重要：点源的剂量随1/r²下降。治疗的肿瘤常位于前列腺、宫颈或乳腺。所有放疗计划都经过仔细计算，以将处方吸收剂量（以戈瑞Gy为单位）投照到靶区。
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