From Photon to Number: Measurement reliability, medical physics and ISO quality standards

On the occasion of the International Day of Medical Physics, celebrated on 7 November, Tomasz Młynik, PhD in Physics and Lead Software Developer at EmbeddedSystems.do, explores what makes a physical measurement truly trustworthy. Drawing on examples from medical imaging, radiation therapy and environmental monitoring, he examines how physics, electronics and software contribute to measurement reliability. With EmbeddedSystems.do recently achieving ISO 9001 and ISO 13485 certification, he also reflects on the role of quality management, verification and validation in developing systems whose results can be trusted.

Every measurement begins with a physical interaction and ends with an estimate of a quantity we cannot usually observe directly. Between these two points lies a sequence of transformations involving detectors, electronics, mathematical models and software. Each transformation introduces assumptions and potential sources of error. In medical physics, the reliability of this process is particularly important. A measurement may contribute to a diagnosis, determine a radiation dose or support a clinical decision. Consequently, the accuracy of an individual detector is not sufficient to establish the reliability of the final result. On 7 November, the birthday of Marie Skłodowska-Curie, the scientific community celebrates the International Day of Medical Physics [1]. As a physicist working in electronics and embedded systems, I find this an opportunity to consider a question that extends well beyond medical applications: what makes a physical measurement trustworthy?

From physical interaction to measurement

A detector does not directly provide the value of a physical quantity. Instead, it produces a signal that must be interpreted through a measurement model. Consider a photon-counting detector. If the detected count N follows a Poisson distribution with expected count λ, its standard deviation is

σ_N = √(λ)

The corresponding relative statistical uncertainty is

σ_N / λ = 1 / √(λ)

Here N is the observed count, $λ$ is its expected value, and $σN$ describes the spread of counts over repeated measurements. In practice, $λ$ is often estimated by N, giving an estimated counting uncertainty of $√(N)$ and a relative uncertainty of $1/√(N)$ for sufficiently large counts. Very small counts require more careful interval estimation; observing zero events does not imply zero uncertainty.

Increasing the number of detected photons from 10² to 10⁴ reduces the relative counting uncertainty from approximately 10% to 1%. This result is elementary, but its interpretation requires care. It assumes an ideal counting process and describes only the statistical fluctuations of the detected events. A real measurement also depends on detector efficiency, calibration, background contributions, geometry, electronic effects and other factors. In general, if the estimated quantity is expressed as y = f(x₁,…,x_n) [2], its combined standard uncertainty, for uncorrelated input quantities, can be approximated to first order by

u_c²(y) = ∑_{i = 1}ⁿ((∂f) / (∂x_i))²u²(x_i)

where:

• y is the estimated quantity and f is the measurement model that computes it from the inputs,
• x_i is the i-th input quantity and n is the number of inputs,
• u(x_i) is the standard uncertainty of x_i,
• ∂f/∂x_i is the sensitivity coefficient, showing how strongly y changes when x_i changes,
• u_c(y) is the combined standard uncertainty of y.

This expression illustrates an important property of physical measurements: reducing statistical noise does not necessarily improve the overall accuracy if systematic effects remain insufficiently controlled. The interpretation of a measured value therefore requires more than the value itself. It requires knowledge of the measurement procedure, calibration conditions and assumptions used to obtain the result.

Photon detection in medical imaging

Positron emission tomography (PET) provides an instructive example of how fundamental physics determines the architecture of a measurement system. Following positron emission and annihilation with an electron, two photons with energies close to

E_γ = m_ec² ≈ 511 keV

are emitted in approximately opposite directions. Here E_γ is the photon energy, m_e is the electron rest mass, c is the speed of light, and keV (kiloelectronvolt) is a unit of energy. Detecting both photons within a sufficiently narrow coincidence window allows the system to identify a line of response along which the annihilation occurred [4]. However, identifying the coincidence is only part of the problem. In time-of-flight PET (TOF-PET), the difference between photon detection times contains additional spatial information. If the temporal uncertainty is Δt, the corresponding uncertainty in the position along the line of response is

Δx = (cΔt) / 2,

where:

• Δx is the uncertainty of the event position along the line of response,
• Δt is the uncertainty of the measured time difference,
• c is the speed of light.

For example, a timing uncertainty of 200 ps corresponds to approximately 3 cm of spatial uncertainty along the line of response, using the same uncertainty convention for time and position. This value describes localisation along a single line of response, rather than the final reconstructed image resolution. Consequently, the quality of the reconstructed image depends not only on detecting individual photons, but also on the timing, energy and spatial resolution of the complete detection system. A related development can be found in photon-counting computed tomography (CT). Unlike conventional energy-integrating detectors, photon-counting detectors produce individual electrical pulses associated with photon interactions. Their amplitudes contain information about the deposited energy, enabling energy discrimination. The first clinical photon-counting CT system received FDA clearance in September 2021 [5]. Both examples demonstrate that quantum interactions provide the physical basis for medical imaging, while electronics and signal processing determine how much of the available information can be recovered.

Electronics, software and the limits of measurement

Even when the underlying physical interaction is well understood, detector electronics introduce additional limitations. One example is detector dead time. Following the registration of an event, a counting system may remain temporarily unable to detect subsequent events. For a non-paralysable detector with dead time τ, the relationship between the observed counting rate m and the true event rate n (both in events per second) is

m = n / (1 + nτ).

Solving for n, we obtain

n = m / (1 − mτ).

This correction is valid within the assumptions of the non-paralysable model, with mτ < 1 [6]. The distinction is important. Applying a mathematically correct expression to a detector governed by a different response model can produce an incorrect result. Furthermore, the dead-time parameter must be determined experimentally, and its stability must be considered during operation. As mτ approaches one, the correction becomes highly sensitive to uncertainties in both the observed rate and the dead time. In modern measurement systems, these corrections are often implemented in software. Therefore, an incorrect parameter, an undocumented firmware modification or a processing error can affect the final result even when the detector hardware functions as intended. The importance of reliable control becomes particularly clear in radiation therapy. In the FAST-01 clinical trial, proton FLASH radiotherapy was investigated using a prescribed absorbed dose of 8 Gy and a nominal dose rate of 60 Gy s ^{− 1} [7]. The gray (Gy) is the SI unit of absorbed dose: one gray corresponds to one joule of energy absorbed per kilogram of matter. Assuming a constant dose rate of 60 Gy s ^{− 1}, delivering 8 Gy would take approximately

t = D / Ḋ ≈ 0.13 s,

where:

• t is the irradiation time,
• D is the prescribed absorbed dose (8 Gy; 1 Gy = 1 J kg ^{− 1}),
• Ḋ = (dD) / (dt) is the assumed constant dose rate (60 Gy s ^{− 1}).

Such short irradiation times illustrate why treatment delivery cannot depend on manual intervention during exposure. Instead, the system requires appropriately designed control mechanisms, independent safety interlocks, verification procedures and reliable software. The physical process may last only a fraction of a second, but its safety depends on decisions made throughout the entire engineering and clinical workflow.

The same measurement problem in a different environment

The principles discussed above are not restricted to medical applications. At EmbeddedSystems.do, I am involved in the Bentos project, an open-source project focused on monitoring water quality in the Gulf of Gdańsk [8]. The system combines environmental sensors, embedded electronics and data-processing methods to obtain information about the marine environment. Although Bentos is an environmental monitoring project, several challenges are similar to those encountered in medical measurement systems. Marine sensors are exposed to biological fouling, changing environmental conditions and long-term calibration drift. A sensor may continue transmitting data while gradually losing its measurement accuracy. Consequently, successful communication with a sensor is not equivalent to obtaining a reliable physical measurement. This becomes important when the resulting data are used to estimate quantities that cannot be measured directly. Consider the attenuation of light in water. For a specified wavelength or spectral band, the depth dependence of downwelling irradiance can be expressed in terms of a depth-dependent diffuse attenuation coefficient K_d(z) as

E_d(z) = E_d(0)exp[ − ∫ K_d (z′) dz′],

where:

• E_d(z) is the downwelling irradiance (light power per unit area arriving from above) at depth z,
• E_d(0) is the irradiance just below the surface,
• z is the depth,
• K_d(z) is the local diffuse attenuation coefficient (per metre), describing the fractional decrease in downwelling irradiance per unit depth.

If K_d is constant over the interval, this reduces to E_d(z) = E_d(0)e ^{− K_dz}. Measurements at two different depths, z₁ and z₂, with z₂ > z₁, allow us to estimate the mean attenuation coefficient over the corresponding interval [10]

K̄_d = (ln[E_d(z₁)/E_d(z₂)]) / (z₂ − z₁).

Here E_d(z₁) and E_d(z₂) are the irradiances measured at depths z₁ and z₂, ln is the natural logarithm, and K̄_d is the mean attenuation coefficient between those two depths. This immediately establishes a measurement requirement: a single irradiance measurement at one depth is insufficient to determine K_d without additional information. The system must provide depth-resolved measurements, either through multiple sensors or an appropriate profiling procedure. The measurements must be positive and comparable with respect to illumination conditions, spectral response and radiometric calibration. Interestingly, an analogous exponential relationship appears in the attenuation of a narrow, mono-energetic X-ray beam in matter [11]

I(x) = I₀e ^{− μx},

where:

• I(x) is the beam intensity after passing a thickness x of material,
• I₀ is the incident X-ray beam intensity,
• x is the thickness of material traversed,
• μ is the linear attenuation coefficient (inverse length), accounting for the removal of photons from the original beam through absorption and scattering.

The two expressions have a similar mathematical form, but describe different quantities. Underwater downwelling irradiance includes both direct and scattered light arriving from the upper hemisphere. In the narrow-beam X-ray model, attenuation describes the loss of photons from the original beam through absorption or scattering out of the beam. This comparison illustrates a broader principle: a measurement system must be designed around the physical quantity it is intended to estimate, rather than around the mere availability of a sensor.

From measurement uncertainty to quality management

The previous examples concern different applications, but they share a common structure. A physical quantity is estimated from observations using a model. The validity of the estimate depends on the assumptions of that model, the properties of the instrument and the procedures used during measurement. For medical devices, these requirements are embedded within a broader regulatory and quality-management framework. ISO 13485:2016 specifies requirements for quality management systems used by organisations involved in the medical-device life cycle [9]. Its purpose is not to certify that an individual measurement is correct. Rather, it establishes a framework for controlling processes and demonstrating that applicable requirements have been addressed. This includes design and development controls, verification and validation, management of changes, and appropriate documentation. From an engineering perspective, the distinction between verification and validation is especially useful.

Verification asks whether the system meets its specified requirements. Validation asks whether the resulting system satisfies its intended use.

Returning to the Bentos example, if determining K_d is a system requirement, verification could demonstrate that the instrument acquires appropriately calibrated irradiance measurements at different depths. Validation would address the broader question of whether the complete measurement procedure provides sufficiently reliable attenuation estimates under the environmental conditions for which the system was designed. The Bentos project is not subject to ISO 13485 simply because this analogy is useful. Nevertheless, the distinction between requirements, verification and validation remains fundamental to sound engineering practice. An equally important aspect is data provenance: documenting how a measurement record was acquired and processed. Each result should be linked to the instrument used, its calibration history, acquisition settings, software version and subsequent processing steps. This documentation does not eliminate errors, but helps us identify their sources and reproduce the analysis. In radiation oncology, this process-oriented perspective is strongly reflected in the AAPM Task Group 100 report, which shows that significant risks may arise from failures of the overall clinical workflow rather than from equipment malfunction alone [3].

Why this matters for our work

EmbeddedSystems.do has recently achieved certification to ISO 9001:2015 and ISO 13485:2016. For our work in electronics and embedded systems, this represents an important step towards more systematic management of development processes, technical requirements, risk and documentation. It does not mean that every device or measurement automatically becomes reliable. Instead, it provides a structured approach through which reliability can be investigated, demonstrated and continuously improved.

As a physicist, I find this particularly meaningful. In theoretical physics, the validity of a result depends on the assumptions and mathematical arguments from which it follows. In experimental physics, it additionally depends on the measurement procedure and its uncertainty. In engineering, these considerations must extend to the complete system, including hardware, firmware, software and the processes through which the system is developed and maintained. The applications may differ considerably, from detecting annihilation photons in a PET scanner to measuring optical properties of seawater. Nevertheless, the underlying requirement remains the same.

A numerical result should not be separated from the physical model, uncertainty and measurement history that give it meaning.

On the International Day of Medical Physics, this is the principle I would like to emphasise: whether we develop instruments for medical applications or environmental monitoring, our responsibility extends beyond making a device work. We must also understand, and be able to demonstrate, when its results can be trusted.

That is the foundation on which meaningful measurement systems should be built.

References

1. International Organization for Medical Physics, International Day of Medical Physics 2026.
2. Joint Committee for Guides in Metrology (JCGM), Evaluation of measurement data — Guide to the expression of uncertainty in measurement. JCGM 100:2008 (2008; corrected version 2010). JCGM 100:2008.
3. Huq, M. S. et al., The report of Task Group 100 of the AAPM: Application of risk analysis methods to radiation therapy quality management. Medical Physics 43, 4209–4262 (2016). DOI: 10.1118/1.4947547.
4. Cherry, S. R., Sorenson, J. A. & Phelps, M. E., Physics in Nuclear Medicine, 4th ed. (Elsevier, 2012); Surti, S., Update on Time-of-Flight PET Imaging. Journal of Nuclear Medicine 56, 98–105 (2015).
5. U.S. Food and Drug Administration, 510(k) clearance K211591 — NAEOTOM Alpha (2021).
6. Knoll, G. F., Radiation Detection and Measurement, 4th ed. (Wiley, 2010).
7. Mascia, A. E. et al., Proton FLASH Radiotherapy for the Treatment of Symptomatic Bone Metastases: The FAST-01 Nonrandomized Trial. JAMA Oncology 9, 62–69 (2023). DOI: 10.1001/jamaoncol.2022.5843.
8. BENTOS, Open-source environmental monitoring project.
9. ISO 13485:2016, Medical devices — Quality management systems — Requirements for regulatory purposes.
10. Mueller, J. L., Fargion, G. S. & McClain, C. R. (eds.), Ocean Optics Protocols for Satellite Ocean Color Sensor Validation, Revision 4, Volume III: Radiometric Measurements and Data Analysis Protocols. NASA (2003). NASA/TM-2003-21621/Rev-Vol III.
11. National Institute of Standards and Technology (NIST), X-Ray Mass Attenuation Coefficients, Section 2: The Mass Attenuation Coefficient.