Biomedical Imaging and Diagnostic Systems Formula Sheet

This formula sheet collects first-pass calculations used to review biomedical imaging and diagnostic systems. Use it to make image quality, dose, detector signal, sampling, latency, uncertainty and post-deployment quality assurance traceable to an intended diagnostic or measurement task.

These equations are screening tools. They do not replace modality-specific standards, radiation protection review, acoustic output review, optical safety analysis, clinical validation, software verification, usability evidence, regulatory requirements, or qualified professional judgement.

Before calculating, state the task: detection, measurement, guidance, monitoring, segmentation, screening, diagnosis, device inspection, or quality assurance. A metric is useful only when it is tied to a task, acquisition protocol, reconstruction or processing version, patient or sample population, and acceptance criterion.

How to Use This Formula Sheet

Use this sheet as a task-based imaging review tool. Start with the clinical, laboratory or engineering decision: detect a low-contrast target, measure a dimension, guide a procedure, monitor a physiological signal, qualify a detector, release a QA phantom result or compare protocol changes. The same CNR, dose or latency value can be acceptable for one task and unsafe or useless for another.

Then define the modality boundary: source, detector, geometry, acquisition settings, reconstruction algorithm, display or classifier input, patient or phantom population, and acceptance criterion. Apply image-quality, dose, detector, sampling and uncertainty formulas only after that boundary is explicit.

Use the dose and safety metrics to control exposure, not to prove diagnostic adequacy. Use CNR, SNR, resolution, latency and drift metrics to support a task claim, not as universal quality scores. Use the validation package before changing protocols, releasing a diagnostic system, accepting a QA drift trend or comparing software versions.

Symbols and Basis

Symbol	Meaning	Common unit
FOV	field of view	\text{mm}
N	matrix samples or pixels per dimension	count
p	pixel size	\text{mm/pixel}
d	target size	\text{mm}
CNR	contrast-to-noise ratio	dimensionless
\mu_t,\mu_b	target and background mean intensity	image units
\sigma_b	background noise standard deviation	image units
CTDI_{vol}	volume CT dose index	\text{mGy}
DLP	dose-length product	\text{mGy cm}
SSDE	size-specific dose estimate	\text{mGy}
L	scan length or path length, depending on context	\text{cm} or \text{mm}
f	frequency	\text{Hz}
c	acoustic speed	\text{m/s}
\lambda	wavelength	\text{m} or \text{mm}
B	bandwidth	\text{Hz}
P	optical or signal power	\text{W}
A	illuminated or detector area	\text{cm}^2 or \text{m}^2
R_\lambda	photodiode responsivity	\text{A/W}
I	detector photocurrent	\text{A}
V_{FS}	ADC full-scale voltage range	\text{V}
\Delta	quantization step	\text{V}
U	expanded uncertainty	unit of measurand

Basis and Validity Limits

The formulas in this sheet are first-pass engineering screens. Pixel size, Nyquist frequency, CNR and SNR assume that the chosen region, reconstruction and processing pipeline are representative of the task. They do not fully capture point-spread function, slice thickness, motion, scatter, speckle, beam hardening, iterative reconstruction texture, display mapping, segmentation bias or reader variability.

Dose metrics such as CTDI, DLP, SSDE and effective-dose screens are useful for protocol review and quality assurance, but they do not by themselves prove patient-specific risk or diagnostic adequacy. Ultrasound formulas depend on acoustic path, focusing, attenuation, operator technique and acoustic output limits. Optical and detector formulas depend on wavelength, responsivity, geometry, saturation, ambient light, analog front end and calibration.

Validation is task-dependent. A phantom result, detector bench check or software metric is not automatically transferable to clinical workflow, patient population, device-inspection workflow or a changed reconstruction version. When the result supports safety, diagnosis or release, record the population, protocol, version, acceptance rule and evidence that would invalidate the calculation.

Pixel Size and Spatial Sampling

Pixel size for a square matrix is:

Approximate number of pixels spanning a target is:

Nyquist spatial frequency for pixel pitch p is:

where f_N is in cycles per millimetre when p is in millimetres.

Mini-Check

For:

and:

pixel size is:

A target with:

spans:

The Nyquist spatial frequency is:

Engineering Comment

Pixel size is not the same as true resolution. Point-spread function, focal spot, detector aperture, reconstruction kernel, motion, slice thickness, optical blur, speckle and segmentation method can all reduce effective resolution.

Contrast-to-Noise Ratio

A common contrast-to-noise ratio is:

where \mu_t is target mean intensity, \mu_b is background mean intensity, and \sigma_b is background noise.

Mini-Check

Use:

Then:

Engineering Comment

CNR is meaningful only for the phantom, region of interest, acquisition protocol, reconstruction setting, display processing and task. A CNR of 4.0 may be adequate for one high-contrast task and inadequate for low-contrast detection.

Signal-to-Noise Ratio

Power signal-to-noise ratio is:

Decibel form for power ratio:

For equal-impedance amplitude ratios:

Validation Use

State where SNR is measured: detector output, analog front end, reconstructed image, displayed image, segmented feature, or diagnostic classifier input. Processing can increase apparent SNR while changing texture, edges, bias or detectability.

CT Dose Metrics

Dose-length product is:

Size-specific dose estimate is:

A rough effective-dose screen is:

where k is a region-specific conversion factor. This is a population-level screen, not patient-specific dosimetry.

Mini-Check

For:

and:

the DLP is:

If:

then:

For:

the effective-dose screen is:

Engineering Comment

These metrics support protocol review and quality assurance. They do not prove diagnostic adequacy or individual patient risk. The diagnostic task, patient size, automatic exposure control, reconstruction version, scan length and image-quality acceptance must also be checked.

Dose-Noise Trade-Off

For a simplified quantum-noise dominated x-ray model:

If relative exposure is:

then:

Mini-Check

If exposure is reduced to:

then:

Noise is expected to increase by about 25\%.

Engineering Comment

Lower dose is not automatically better. A dose reduction is acceptable only if the task still meets contrast, resolution, artifact and diagnostic performance requirements.

Ultrasound Wavelength and Pulse-Echo Attenuation

Ultrasound wavelength is:

For a simplified pulse with n_c cycles, spatial pulse length is:

and approximate axial resolution is:

Soft-tissue attenuation is often screened as:

For pulse-echo travel:

where \alpha is in \text{dB/(cm MHz)}, f is in \text{MHz}, and z is depth in \text{cm}.

Mini-Check

For:

and:

wavelength is:

For a two-cycle pulse:

and:

If:

and:

then:

Engineering Comment

Higher frequency improves wavelength-limited resolution but reduces penetration. Actual ultrasound performance also depends on aperture, focusing, beamforming, speckle, tissue path, acoustic window, operator technique and acoustic output limits.

Optical Power Density and Photodiode Current

Optical power density is:

Photodiode current is:

Shot-noise current over bandwidth B is:

where q=1.602\times10^{-19}\ \text{C}.

Mini-Check

For:

and:

power density is:

With:

photocurrent is:

For:

shot-noise current is:

Engineering Comment

Shot noise may be much smaller than ambient light variation, motion artefact, saturation, skin optical variability, amplifier noise, source drift or calibration error. Always identify the dominant noise source.

Frame Rate and Latency

Frame interval is:

Total displayed latency can be screened as:

Mini-Check

For:

the frame interval is:

If:

and network latency is negligible, then:

Engineering Comment

Latency requirements depend on task. A static diagnostic review can tolerate more delay than catheter guidance, ultrasound needle tracking, robotic assistance or alarmed physiological monitoring.

Detector Quantization

ADC step size is:

Ideal quantization noise RMS is:

Mini-Check

For:

and:

the step size is:

Quantization noise RMS is:

Engineering Comment

More bits do not fix detector saturation, poor gain staging, motion blur, aliasing, reconstruction error or uncalibrated display mapping.

Measurement Uncertainty and Guard Bands

Combined standard uncertainty for independent contributors is:

Expanded uncertainty is:

For an upper limit:

is a conservative pass condition.

Mini-Check

If a measured feature diameter is:

with:

and the acceptance limit is:

then:

Since:

the result passes the guarded limit.

Engineering Comment

Uncertainty should include acquisition, calibration, segmentation, operator, reconstruction, display, reference phantom, and repeatability effects when they affect the decision.

QA Drift

Percent drift from a baseline is:

Mini-Check

If a monthly phantom CNR falls from:

to:

then:

If the QA trigger is a 10\% drop, this result requires investigation.

Engineering Comment

QA drift can be caused by detector gain, reconstruction software, calibration, source output, probe damage, phantom setup, operator practice, display changes or environmental conditions. The response should identify the source before changing clinical protocols.

Common Formula Mistakes

Misuse	Why it is risky
treating pixel size as true resolution	blur, reconstruction and motion can dominate
optimizing CNR without a task	image quality metrics are task-dependent
reducing CT dose without checking diagnostic adequacy	noise and artifact may hide findings
using ultrasound frequency alone to claim resolution	aperture, focusing, speckle and attenuation matter
using detector SNR as clinical validation	processing and workflow can alter decision quality
ignoring latency in guidance tasks	delay can create spatial or procedural error
comparing QA metrics across software versions without control	algorithm changes can mimic performance drift
quoting uncertainty without source terms	the number may exclude the dominant error

Additional common mistakes include treating dose reduction as automatically beneficial, accepting a high SNR while ignoring motion or artifacts, and validating a detector output while ignoring the reconstructed image or diagnostic classifier input. Metrics should follow the task, not the other way around.

Validation Evidence Package

A task-based imaging calculation package should state:

1. intended diagnostic or measurement task;
2. modality, source, detector, geometry and acquisition settings;
3. reconstruction or processing version;
4. image-quality metric and region-of-interest definition;
5. safety metric such as dose, acoustic output or optical exposure;
6. calibration and phantom basis;
7. uncertainty contributors;
8. user workflow and latency constraints;
9. QA trigger values and response path;
10. evidence that would invalidate the calculation.

Also include software version, processing settings, display or classifier boundary, user workflow assumptions, sample or patient population, operator or reader role, reference phantom, calibration status, repeatability evidence, uncertainty or guard band, safety review basis and post-release monitoring plan.

The formulas are useful when they support a controlled claim: the image or diagnostic output is adequate for the task, inside safety limits, stable over time, and traceable to validation evidence.