Exercise set

Medical Imaging Dose, Exposure, and Safety Physics Exercises

Solved medical imaging dose exercises for CT DLP, SSDE, fluoroscopy time, MRI SAR, ultrasound indices, optical exposure and release gates.

Branch: Biomedical Engineering
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.0.0 · reviewed

These exercises focus on dose, exposure and safety physics in medical imaging. The goal is not to minimize every exposure number in isolation, but to control patient and operator risk while preserving the diagnostic task and avoiding repeat examinations.

Assume simplified screening models unless an exercise states otherwise. Real release evidence should follow the modality protocol, local regulatory requirements, scanner calibration, patient-size policy, quality-assurance procedure, clinical task and medical-physics review.

Release Evidence Notes

Dose evidence must identify the protocol, scanner, phantom or patient-size category, acquisition settings, reconstruction method, displayed dose metric and clinical task. A lower dose is not automatically better if it increases repeats or removes task-relevant contrast.

Safety evidence should connect the calculated value to a controlled limit. CT dose, fluoroscopy time, MRI SAR, ultrasound output and optical exposure all require traceability from scanner settings to acceptance criteria, uncertainty and release authority.

Engineering Boundary Notes

These calculations are screening exercises. They do not replace patient-specific clinical judgment, regulatory dose reporting, medical-physics commissioning, MRI safety review, ultrasound output auditing, laser safety analysis or institutional protocol approval.

Scenario Map

Scenario	Exercises	Primary check	Engineering decision
CT exposure	1, 2, 3, 4, 5, 16	DLP, effective dose, SSDE, repeat penalty and reference level	Decide whether CT protocol exposure is controlled.
Procedure and cumulative dose	6, 7, 14, 15	Fluoroscopy dose rate, cumulative air kerma and staff distance	Decide whether procedure controls are adequate.
Non-ionizing safety	8, 9, 10, 11, 12	MRI SAR, ultrasound MI/TI and optical radiant exposure	Decide whether output remains within controlled limits.
Release gate	13, 17, 18	Repeatability, dose-quality tradeoff and all-of release	Decide whether the protocol can be released.

Exercise 1: CT DLP and Effective Dose

A CT protocol reports:

CTDI_{vol}=8.5\ \text{mGy}

and scan length:

L=42\ \text{cm}

Use conversion factor $k=0.015\ \text{mSv}/(\text{mGy cm})$ . Compute DLP and effective dose.

Solution

DLP=CTDI_{vol}L=8.5(42)=357\ \text{mGy cm}

E=kDLP=0.015(357)=5.36\ \text{mSv}

Engineering Comment

Effective dose is a population-risk screening metric, not a patient-specific organ dose. It is useful for protocol comparison only when anatomy and scan length are controlled.

Plausibility Check

A CTDI below $10\ \text{mGy}$ over about forty centimeters gives a DLP of a few hundred $\text{mGy cm}$ .

Exercise 2: Size-Specific Dose Estimate

A pediatric CT has $CTDI_{vol}=5.8\ \text{mGy}$ and size conversion factor $f=1.45$ . Compute SSDE.

Solution

SSDE=fCTDI_{vol}=1.45(5.8)=8.41\ \text{mGy}

Engineering Comment

SSDE is higher than CTDIvol because the patient is smaller than the reference phantom. Protocol release should use the correct size bin.

Plausibility Check

Multiplying by a factor greater than one should raise the dose estimate.

Exercise 3: Dose-Noise Trade-Off

Image noise is approximately proportional to $1/\sqrt{D}$ . A protocol reduces dose from $10\ \text{mGy}$ to $6.4\ \text{mGy}$ . Estimate the noise increase factor.

Solution

\dfrac{\sigma_2}{\sigma_1}=\sqrt{\dfrac{D_1}{D_2}}

\dfrac{\sigma_2}{\sigma_1}=\sqrt{\dfrac{10}{6.4}}=\sqrt{1.5625}=1.25

Noise increases by $25\%$ .

Engineering Comment

The lower dose may still be acceptable if the diagnostic task tolerates the added noise or reconstruction compensates without artifacts.

Plausibility Check

Reducing dose by about one third should noticeably increase noise but not double it.

Exercise 4: Repeat Scan Dose Penalty

A low-dose protocol gives $3.2\ \text{mSv}$ per scan but has a $12\%$ repeat probability. A standard protocol gives $3.6\ \text{mSv}$ with $2\%$ repeat probability. Compare expected dose per completed study.

Solution

Expected dose using one possible repeat is:

E_{low}=3.2(1+0.12)=3.584\ \text{mSv}

E_{std}=3.6(1+0.02)=3.672\ \text{mSv}

The low-dose protocol remains lower by:

3.672-3.584=0.088\ \text{mSv}

Engineering Comment

The dose advantage is small after repeat risk. The release decision should check whether repeats are caused by noise, motion or workflow.

Plausibility Check

The low-dose protocol saves $0.4\ \text{mSv}$ initially but loses most of that advantage through higher repeat probability.

Exercise 5: CT Dose Reference Level Margin

A protocol has DLP $420\ \text{mGy cm}$ . The local diagnostic reference level is $500\ \text{mGy cm}$ , and release requires a $10\%$ guard below the level. Decide status.

Solution

Guarded limit:

DLP_g=0.90(500)=450\ \text{mGy cm}

Since:

420<450

the protocol passes with margin:

M=450-420=30\ \text{mGy cm}

Engineering Comment

Passing a reference level does not prove diagnostic adequacy. The protocol still needs image-quality and task-performance evidence.

Plausibility Check

$420$ is below both $500$ and the guarded $450$ threshold.

Exercise 6: Fluoroscopy Time Budget

A procedure dose rate is $18\ \text{mGy/min}$ . The planned maximum entrance dose for the screening portion is $270\ \text{mGy}$ . Compute allowed fluoroscopy time.

Solution

t=\dfrac{270}{18}=15\ \text{min}

Engineering Comment

The time budget is only valid at the stated dose rate. Magnification, angulation, frame rate and patient size can change the rate.

Plausibility Check

At nearly twenty milligray per minute, a few hundred milligray allows about fifteen minutes.

Exercise 7: Cumulative Air Kerma Margin

A procedure has current cumulative air kerma $1.8\ \text{Gy}$ . The internal alert level is $2.5\ \text{Gy}$ . If the remaining step is expected to add $0.4\ \text{Gy}$ , compute alert margin after completion.

Solution

Projected kerma:

K=1.8+0.4=2.2\ \text{Gy}

Margin:

M=2.5-2.2=0.3\ \text{Gy}

Engineering Comment

The procedure stays below the alert level, but the margin is not large. The team should monitor dose rate during the remaining step.

Plausibility Check

Adding less than half a gray to $1.8\ \text{Gy}$ gives a value just above two gray.

Exercise 8: MRI SAR Average-Power Screen

An MRI sequence deposits average RF power $P=180\ \text{W}$ in a patient mass of $72\ \text{kg}$ . Compute SAR.

Solution

SAR=\dfrac{P}{m}=\dfrac{180}{72}=2.5\ \text{W/kg}

Engineering Comment

SAR depends on sequence, coil, body region and scanner model. A pass should be checked against the applicable operating mode and patient condition.

Plausibility Check

Hundreds of watts distributed over tens of kilograms gives a few watts per kilogram.

Exercise 9: MRI SAR Guard Margin

A release limit is $3.2\ \text{W/kg}$ and uncertainty allowance is $0.4\ \text{W/kg}$ . A sequence estimates $SAR=2.9\ \text{W/kg}$ . Decide status.

Solution

Guarded SAR is:

SAR_g=2.9+0.4=3.3\ \text{W/kg}

Because:

3.3>3.2

the sequence fails the guarded release screen.

Engineering Comment

The nominal estimate is below the limit, but uncertainty consumes the margin. Sequence adjustment or stronger evidence is needed.

Plausibility Check

The nominal margin is only $0.3\ \text{W/kg}$ and the guard is $0.4\ \text{W/kg}$ , so failure is expected.

Exercise 10: Ultrasound Mechanical Index

Peak rarefactional pressure is $0.9\ \text{MPa}$ at frequency $4.0\ \text{MHz}$ . Estimate mechanical index:

MI=\dfrac{p_r}{\sqrt{f}}

Solution

MI=\dfrac{0.9}{\sqrt{4.0}}=\dfrac{0.9}{2}=0.45

Engineering Comment

MI is a screening index for cavitation-related risk. The clinical mode, tissue, contrast agents and dwell time still matter.

Plausibility Check

At $4\ \text{MHz}$ the square-root denominator is $2$ , so the result is half the pressure value.

Exercise 11: Ultrasound Thermal Index Guard

An ultrasound mode displays $TI=0.8$ . A procedure allows $TI\le1.0$ but requires a $0.15$ guard for long dwell. Check status.

Solution

Guarded value:

TI_g=0.8+0.15=0.95

Since:

0.95<1.0

the mode passes.

Engineering Comment

The display index passes, but dwell time and probe position should be controlled for long examinations.

Plausibility Check

The guard leaves only $0.05$ of margin, so the pass is narrow.

Exercise 12: Optical Radiant Exposure

An optical imaging source emits $12\ \text{mW}$ over an illuminated area of $4.0\ \text{cm}^2$ for $8\ \text{s}$ . Compute radiant exposure in $\text{mJ/cm}^2$ .

Solution

Energy:

E=Pt=12(8)=96\ \text{mJ}

Radiant exposure:

H=\dfrac{96}{4.0}=24\ \text{mJ/cm}^2

Engineering Comment

Optical safety should check wavelength, exposure duration, aperture, eye or tissue target and any pulsed peak limits.

Plausibility Check

Twelve milliwatts for eight seconds is just under one tenth of a joule, spread over four square centimeters.

Exercise 13: Detector Dose Output Repeatability

Five measured output values are $9.8$ , $10.1$ , $10.0$ , $9.9$ and $10.2\ \text{mGy}$ . The mean is $10.0\ \text{mGy}$ . Find maximum percentage deviation from mean.

Solution

Largest absolute deviation is:

\Delta=0.2\ \text{mGy}

Percentage deviation:

d=\dfrac{0.2}{10.0}(100)=2\%

Engineering Comment

Repeatability is acceptable only if the protocol limit allows this deviation and the dosimeter calibration is current.

Plausibility Check

The readings vary by only a few tenths around ten milligray, so a low single-digit percentage is expected.

Exercise 14: Cumulative Protocol Dose

A patient pathway includes CT effective dose $4.8\ \text{mSv}$ , nuclear medicine dose $3.2\ \text{mSv}$ and a fluoroscopy component $1.1\ \text{mSv}$ . Compute total pathway dose.

Solution

E_{total}=4.8+3.2+1.1=9.1\ \text{mSv}

Engineering Comment

Summing effective dose is a broad pathway screen. It should not be used as patient-specific organ-dose evidence.

Plausibility Check

The CT component is about half the total, and the sum is just over nine millisievert.

Exercise 15: Staff Distance Reduction

Scatter dose rate is $80\ \mu\text{Sv/h}$ at $1\ \text{m}$ . Estimate dose rate at $2\ \text{m}$ using inverse-square behavior.

Solution

\dot D_2=\dot D_1\left(\dfrac{r_1}{r_2}\right)^2

\dot D_2=80\left(\dfrac{1}{2}\right)^2=20\ \mu\text{Sv/h}

Engineering Comment

Distance is a strong control, but shielding, room scatter, orientation and procedure time still need review.

Plausibility Check

Doubling distance reduces inverse-square dose rate by a factor of four.

Exercise 16: Dose Reduction and CNR Guard

A proposed CT protocol reduces dose by $25\%$ . Baseline CNR is $6.0$ . Assuming CNR scales with $\sqrt{D}$ , estimate new CNR and decide if it meets a release minimum of $5.0$ .

Solution

Dose fraction:

\dfrac{D_2}{D_1}=0.75

New CNR:

CNR_2=6.0\sqrt{0.75}=6.0(0.866)=5.20

Since:

5.20>5.0

the CNR screen passes.

Engineering Comment

The margin is small. The final decision needs phantom and clinical-task evidence, not just square-root scaling.

Plausibility Check

A $25\%$ dose reduction should reduce CNR by about $13\%$ , from $6$ to just above $5$ .

Exercise 17: Repeat-Rate Release Trigger

A protocol has $240$ studies and $18$ repeats due to insufficient image quality. The release trigger is repeat rate below $5\%$ . Check status.

Solution

r=\dfrac{18}{240}=0.075=7.5\%

Since:

7.5\%>5\%

the trigger fails.

Engineering Comment

Dose per scan may look acceptable while repeat rate makes the pathway unsafe or inefficient. Root-cause review is required.

Plausibility Check

Eighteen repeats out of two hundred forty is more than one in twenty.

Exercise 18: Imaging Dose Safety Release Gate

A release gate requires CT reference-level pass, guarded SSDE pass, MRI SAR pass, ultrasound output pass and repeat-rate pass. Results are pass, pass, conditional pass, pass and pass. Decide status.

Solution

The rule is all-of:

G=R_{CT}\land R_{SSDE}\land R_{SAR}\land R_{US}\land R_{repeat}

A conditional pass is not a full pass, so release is blocked.

Engineering Comment

Medical-imaging safety gates should not average across modalities. A weak SAR condition cannot be offset by good CT or ultrasound results.

Plausibility Check

An all-of rule fails when one required element is conditional rather than accepted.

Common Release Mistakes

Comparing dose values without protocol, scanner, patient-size and reconstruction context.
Lowering dose without checking repeats and task performance.
Treating SSDE, effective dose or SAR as exact patient-specific truth.
Ignoring uncertainty guards near safety limits.
Averaging failed repeat-rate or output-control evidence into a pass.
Separating exposure control from QA and image-quality evidence.

Validation Package Checklist

Protocol identifier, scanner, software version and acquisition settings.
CTDIvol, DLP, SSDE, dose reference and repeat-rate evidence.
Fluoroscopy air kerma, dose-rate and procedure-time monitoring.
MRI SAR estimate, operating mode and uncertainty guard.
Ultrasound MI/TI setting, dwell condition and clinical mode.
Optical exposure, wavelength, aperture and duration evidence.
Medical-physics review, acceptance limits and release authority.

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