Bridging Microdosimetry and Macrodosimetry: A Comparative Review of Computational Approaches

Tatsuhiko Sato

doi:10.14407/jrpr.2025.00164

J. Radiat. Prot. Res > Volume 51(1); 2026 > Article

Sato: Bridging Microdosimetry and Macrodosimetry: A Comparative Review of Computational Approaches

Review Article

J. Radiat. Prot. Res 2026; 51(1): 8-18.

Published online: December 30, 2025

DOI: https://doi.org/10.14407/jrpr.2025.00164

Bridging Microdosimetry and Macrodosimetry: A Comparative Review of Computational Approaches

Tatsuhiko Sato

Nuclear Science and Engineering Center, Japan Atomic Energy Agency, Ibaraki, Japan

Corresponding author: Tatsuhiko Sato Nuclear Science and Engineering Center, Japan Atomic Energy Agency, Shirakata 2-4, Tokai, Ibaraki 319-1195, Japan E-mail: sato.tatsuhiko@jaea.go.jp https://orcid.org/0000-0001-9902-7083

Received May 9, 2025 Revised July 7, 2025 Accepted August 18, 2025

This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Microdosimetry does not simply refer to dosimetry with small targets, but rather to dosimetry that accounts for the stochastic nature of energy deposition. The International Commission on Radiation Units and Measurements has proposed using the probability densities of microdosimetric quantities such as lineal energy (y) and specific energy (z) to characterize the radiation fields instead of relying solely on linear energy transfer (LET) and absorbed dose. However, the practical application of microdosimetry remains limited due to significant computational costs associated with evaluating these stochastic quantities in macroscopic contexts such as the human body. To bridge the gap between microdosimetry and macrodosimetry, three approaches have been proposed and implemented: direct integration of track-structure and macroscopic radiation transport codes; integration of a database containing dose-mean values of microdosimetric quantities into macroscopic radiation transport codes; and integration of the analytical microdosimetric function for instantaneously calculating the probability densities of microdosimetric quantities within macroscopic radiation transport codes. This paper comprehensively reviews the conceptual and practical differences between LET and microdosimetric quantities, including their respective computational methodologies. The advantages and limitations of the three approaches are then discussed, followed by a summary of previous research efforts and suggestions for future research directions.

Keywords: Microdosimetry, Lineal Energy, Specific Energy, Linear Energy Transfer, Analytical Microdosimetric Function

Introduction

Estimating biological effects from exposure to high charge and energy (HZE) particles is critically important for treatment planning in charged-particle therapy and for the risk estimate of astronauts. For these purposes, it is essential to evaluate not only the physical dose but also the relative biological effectiveness (RBE), which is generally expressed as a function of unrestricted linear energy transfer (LET) in water. However, it is well recognized that the RBE of HZE particles cannot be uniquely determined by LET alone [1], as the LET concept does not account for the spatial heterogeneity of ionization density due to high-energy secondary electron (δ ray) production, nor for the stochastic nature of energy deposition at microscopic scales.

To overcome these limitations of LET, the International Commission on Radiation Units and Measurements (ICRU) has proposed using microdosimetric quantities such as lineal energy (y) and specific energy (z) to characterize the radiation fields [2, 3]. Unlike LET, they are stochastic quantities expressed through their probability density (PD), reflecting spatial heterogeneity as well as the discrete and random nature of individual ionization and electronic excitation events. As a result, the ion-species dependence observed in LET-based analyses disappears when y is adopted as the index for expressing the RBE of HZE particles [4, 5]. Furthermore, the microdosimetric analysis provided a better interpretation of RBE compared to the LET-based analysis, even in the case of proton therapy [6]. However, significant computational costs associated with evaluating these stochastic quantities in macroscopic contexts such as the human body have created a gap between microdosimetry and macrodosimetry, limiting the practical application of microdosimetry in medical physics and radiological protection.

In this paper, the conceptual and practical differences between LET and microdosimetric quantities are comprehensively reviewed, including their respective computational methodologies. Subsequently, potential approaches to bridging the gap between microdosimetry and macrodosimetry are discussed, with a summary of previous research efforts and proposals for future research directions. It should be noted that this paper focuses exclusively on computational microdosimetry; experimental aspects, including nanodosimetry, have been discussed in other review articles [7–9].

Difference among LET, Lineal Energy, and Specific Energy

Table 1 summarizes the characteristics of LET, y, and z. In this section, the features of each quantity are described, with an emphasis on the differences among them.

1. Lineal Energy Transfer

The original definition of LET is the restricted LET, L_Δ, which is defined as the mean energy lost by a charged particle due to electronic interactions in traversing a unit distance dl, excluding the sum of the kinetic energies exceeding a cutoff value Δ for secondary electrons (δ rays) released by the charged particles. The unrestricted LET (typically referred to simply as LET) is a special case of L_Δ in the limit as Δ→∞. Both LET and L_Δ are the deterministic quantities, and thus the stochastic nature of the energy deposition is ignored in their concepts. On the other hand, the lateral spread of energy deposition due to δ-ray productions can be approximated in the case of L_Δ, although only empirical models can be established by using L_Δ as the index for expressing RBE.

The numerical values of LET can be calculated using stopping power calculation codes, such as SRIM [10] and ATIMA [11], by providing the energy E, charge Z, and atomic number A of the charged particle. These codes, or databases derived from them, have been integrated with macroscopic radiation transport simulation codes based on the Monte Carlo method such as Geant4 [12] and PHITS [13], enabling the straightforward estimation of the LET distributions in macroscopic contexts, even in mixed radiation fields. It should be noted that few codes are capable of calculating L_Δ for arbitrary Δ values, and therefore L_Δ is rarely used in practical computational dosimetry studies.

2. Lineal Energy

Lineal energy y is defined as the quotient of the energy imparted to a target volume by a single particle track, ε_s, by the mean chord length of the target volume, l¯. For a spherical target, l¯=4r/3, where r is the radius of the sphere. Although lineal energy shares the same units as LET, its concept is fundamentally different from LET. Lineal energy is a stochastic quantity and enables accurate consideration of the lateral spread of energy deposition due to δ-ray production. Therefore, y is generally described by its PD, either frequency distribution f(y) or dose distribution d(y).

To calculate f(y) or d(y), microscopic radiation transport simulations that explicitly model each ionization and electronic excitation—so-called track-structure simulations—are generally required. Various track-structure simulation codes have been developed worldwide, particularly over the past decade [14–22], as summarized in ICRU Report 98 [3]. However, the significant computational costs associated with track-structure simulations limit their applicability to macroscopic contexts, resulting in less frequent use of y compared to LET in medical physics and radiological protection.

3. Specific Energy

The specific energy z is defined as the quotient of the energy imparted to a target volume, ε, by its mass, m. Although specific energy shares the same units as absorbed dose, its concept is more closely related to that of y, with the key difference being its ability to account for multiple-hit events within the target. Accordingly, the PD of z for a given number of events, n, is denoted as f_n(z) or d_n(z). For example, f₁(z) represents the single-event frequency distribution of z, which is equivalent in shape to f(y). Therefore, f₁(z) and d₁(z) can be calculated by the track-structure simulation in the same as f(y) and d(y).

Additional computation procedures are required to account for the multiple-hit events. For n≥2, f_n(z) can be calculated by the convolution of f₁(z) and f_n-1(z) as written below Equation (1):

(1)

fn(z)=∫0∞f1z´fn-1z-z´dz´

where z´ is the variable of integration. Then, the PD of z for a given absorbed dose D, f(z, D), can be obtained from Equation (2):

(2)

fz,D=Σn P(λD;n)fnz

where P(λ(D);n) represents the Poisson distribution with an expected value λ(D), which can be determined from Equation (3):

(3)

λD=D/z¯1F

where z−1F is the frequency-mean value of f₁(z). It is desirable to evaluate f(z, D) in macroscopic contexts in order to apply the concept of specific energy to medical physics and radiological protection. However, numerically solving these equations is computationally expensive, particularly at higher doses where λ(D) becomes large. As a result, the practical use of z is even more limited than that of y, except for its single-event distribution f₁(z). Note that the computational cost of the convolution can be significantly reduced by applying the fast Fourier transform when the PD is expressed on a linear scale with respect to z [23].

Examples of Lineal-Energy and Specific-Energy Distributions

To clarify the differences among LET, y, and z, selected examples of y and z distributions for a 3 MeV proton and a 300 MeV/nucleon (hereafter MeV/n) ¹²C ion, both of which have a similar LET of around 12 keV/μm, are presented in this section. Fig. 1 shows the spatial distributions of the deposition energies around the trajectories of the 3 MeV proton and the 300 MeV/n ¹²C ion, calculated using a track-structure model named Ion Track-Structure model for Arbitrary Radiation and Targets (ITSART) [24] in PHITS. The schematic images of the spherical targets used for scoring lineal energy are overlaid on the distributions. It is evident from the figure that the energy-deposition events are more widely spread in the lateral direction around the trajectory of the ¹²C ion due to the production of δ rays, while they are concentrated around the trajectory of the proton. Consequently, many targets are hit solely by δ rays in the case of the ¹²C ion, and their lineal energies tend to be lower compared with those directly hit by the primary particle. This phenomenon contributes to the ion-species dependence observed in RBE, as heavier particles generally have higher kinetic energies and produce more δ rays compared with lighter particles with the same LET.

Fig. 2 shows the lineal-energy distributions, yf(y), obtained from the track-structure simulations by setting the target diameter to 100 μm. The corresponding data calculated using the analytical microdosimetric function (AMF) are also plotted in the figure, which will be described later in the subsection entitled ‘Integration of AMF with macroscopic radiation transport codes.’ Two peaks are observed in the ¹²C ion data, where the higher and lower y peaks are attributed to the contributions from the primary ion and δ rays, respectively. In contrast, only the primary peak is observed in the proton data because of the negligible contribution from δ rays. In addition, the lineal energy of the primary peak for the ¹²C ion is slightly lower than that for proton, as is its LET, owing to the lateral spread of energy deposition. Based on these two observations, RBE of 300 MeV/n ¹²C ion is expected to be lower than 3 MeV proton when y is used as the index for representing the radiation quality. This tendency is consistent with the general trend observed in the large cell survival database Particle Irradiation Data Ensemble (PIDE), which explicitly suggests that lighter particles provide higher RBEs for a fixed LET [25].

Fig. 3 shows the specific-energy distributions for various absorbed dose D, denoted as zf(z, D), obtained from yf(y) plotted in Fig. 2, in combination with Equations (1) to (3). In this calculation, yf(y) calculated by AMF was first converted to zf₁(z) using their proportional relationship as described previously in the subsection entitled ‘Specific energy (z).’ Then, f_n(z) was evaluated for n up to a few hundred by numerically solving Equation (1), and zf(z, D) for various D was determined using Equation (2) by varying λ(D) obtained from Equation (3). It is evident from the figure that the relative shapes of zf(z, D) are almost independent of D for absorbed doses below z−1F, which are approximately 15.9 Gy and 3.4 Gy for 3 MeV proton and 300 MeV/n ¹²C ion, respectively. This behavior arises because most of the targets are either not hit or hit only once when λ(D) <1. Unlike f(z) or f₁(z), f(z, D) can be defined at z=0, although it cannot be depicted in the figure due to the logarithmic scale of the x-axis. For example, f(0,1) was evaluated to be 0.92 and 0.69 for 3 MeV protons and 300 MeV/n ¹²C ions, respectively, indicating that approximately 92% and 69% of targets are not hit during 1 Gy irradiation by these particles. In contrast, as the absorbed dose increases beyond 10 Gy, zf(z, D) gradually approaches a Gaussian distribution, regardless of the original shape of zf₁(z). Thus, under such high-dose irradiation conditions, the RBE is expected to become less dependent on the microdosimetric profile and approach unity. This may be one contributing factor to another general trend observed in the PIDE database—namely, the decrease in RBE with increasing dose—although the primary cause of this trend is the linear-quadratic relationship between cell survival and dose. This insight can be deduced only when f(z, D) is used as the index for representing radiation quality.

Computational Approaches to Bridging the Gap between Microdosimetry and Macrodosimetry

Fig. 4 illustrates the computational approaches to bridging the gap between microdosimetry and macrodosimetry. The details of each approach are described below.

1. Direct Integration of Track-Structure and Macroscopic Radiation Transport Codes

From a technical perspective, integrating track-structure and macroscopic radiation transport codes is relatively straightforward, as their underlying algorithms are quite similar: both employ random sampling techniques and interaction cross sections to simulate the behavior of radiation in matter. In fact, such integrated codes are already available, including combinations such as Geant4-DNA [14] with Geant4 [12], TOPAS-nBio [21] with TOPAS [26], and ITSART [24], ETS [27], ETSART [28], and KURBUC [29] with PHITS [13]. However, as previously mentioned, it is practically infeasible to perform the track-structure simulations in macroscopic contexts due to their significant computational time. For example, ITSART implemented in PHITS requires approximately 9,000 seconds to simulate the track structure generated by a single 300 MeV/n ¹²C ion within a 1 mm³ water target, when executed on a standard Windows PC equipped with an Intel Core i7-9700K central processing unit (CPU) @ 3.60GHz with parallelization disabled. This computational time is equivalent to that required for a corresponding macroscopic radiation transport simulation with approximately 3×10⁷ particle histories. It should be noted that the computational time largely depends on computer specifications.

To address this substantial computational cost of track-structure simulations, the use of graphics processing units (GPUs) has recently been proposed as an alternative to CPUs. GPUs offer significant advantages in parallelization compared to CPUs, although they are less efficient at handling complex branching and data structures. As a result, GPUs are particularly suitable for radiation transport simulations in specific applications, such as treatment planning for certain types of radiotherapy [30–36]. In this context, track-structure simulation is considered one of the most suitable applications for GPU utilization, as its algorithms are generally much simpler than those used in general-purpose macroscopic radiation transport simulations, which must account for various radiation types, materials, and complex geometries. For example, MPEXS-DNA [37], a GPU-based track-structure and radiolysis code, has demonstrated speedup factors of up to 2,900 compared to Geant4-DNA executed on a single CPU. Nevertheless, even such a high speedup is still insufficient for enabling practical track-structure simulations in macroscopic contexts, and further dramatic advancements in computing technology will be necessary.

2. Integration of the Mean Values of Microdosimetric Quantities with Macroscopic Radiation Transport Codes

Not only the PD but also the dose-mean values of the microdosimetric quantities—commonly denoted as y_D and z−1D–can be used as indices for representing radiation quality [2, 3]. Although these are deterministic quantities, they offer advantages over LET and L_Δ, as they can accurately consider the lateral spread of energy deposition due to δ-ray productions through explicit specification of the target size. Additionally, the occurrence of overkill effects can be approximately accounted by introducing their saturation-corrected values, y* and z¯1D∗, defined as Equation (4):

(4)

y∗=y02∫0∞{1−exp⁡[−(y/y0)2]}f(y)dy/∫0∞yf(y)dyz¯1D∗=z02∫0∞{1−exp⁡[−(z/z0)2]}f1(z)dz/∫0∞zf1(z)dz

where y₀ and z₀ are the saturation parameters [2]. A typical value of y₀ is approximately 100 keV/μm, which corresponds to the threshold at which the overkill effect on cell survival tends to become apparent.

Since y_D and z−1D as well as their saturation-corrected values, are deterministic quantities—similar to LET—they can be directly integrated into macroscopic radiation transport codes through the construction of appropriate databases. Ideally, such databases should be generated by calculating yf(y) or zf₁(z) using track-structure simulation codes. However, the substantial computational cost of track-structure simulations makes it impractical to create databases covering various radiation types over wide energy ranges. As a result, analytical approaches [38–40] or amorphous track-structure models based on the radial dose distribution around ion trajectories [41–43] are commonly employed for database construction, as the stochastic nature of energy depositions has minimal impact on the estimation of the dose-mean values.

A successful example of this integration is the treatment planning system for carbon-ion therapy developed at the National Institute of Radiological Science. Inaniwa et al. [44] first generated a database of z¯1D∗ using an amorphous track-structure model and integrated it with the Geant4-based macroscopic radiation transport code, PTSsim [45]. Using the integrated code, they subsequently calculated the absorbed dose and z¯1D∗ in a water phantom irradiated by all ion species and kinetic energies relevant to carbon-ion therapy [46]. The resulting data were then implemented into the treatment planning system, which employs a modified microdosimetric kinetic (MK) model [47] that uses z¯1D∗ as the index for RBE estimation [4, 44]. Another widely used RBE estimation model for carbon-ion therapy treatment planning, the local effect model developed at GSI Helmholtz Center for Heavy Ion Research [48], also relies on the microscopic dose distributions calculated by an amorphous track-structure model [43]. Importantly, the time-consuming Monte Carlo method is not used in the actual treatment planning process, even in the macroscopic radiation transport simulation, yet the essential features of microdosimetry are retained in the resulting dose distributions through this approach.

3. Integration of Analytical Microdosimetric Functions with Macroscopic Radiation Transport Codes

Although the mean-value approach is successful in bridging the gap between microdosimetry and macrodosimetry for certain applications, such as the treatment planning of carbon-ion therapy, it is still desirable to fully account for the stochastic nature of energy depositions in macroscopic radiation transport simulations. To achieve this, a fast and simple method for calculating the PD of microdosimetric quantities is required. The AMF, which was originally proposed by Olko and Booz [49] and later refined by Sato et al. [50, 51] and Parisi et al. [52], is well suited for this purpose, as it is simply composed of a few theoretical distribution functions with several free parameters that depends on the ion species, energies, and target sizes. By appropriately tuning the parameters, the AMF can reproduce the PD around the trajectories of various particles obtained from the track-structure simulations very well, as illustrated in Fig. 2. Moreover, it can also predict the PD for particles whose track-structure simulations have not been performed, by interpolating the parameters from known cases. The AMF has been incorporated into PHITS [53] as the [t-sed] tally, and in TOPAS through its specific extensions [54, 55].

The AMF in PHITS has been applied to various fields, including medical physics and radiological protection. For medical applications, it has been combined with various versions of MK models and used for the biological dose estimation for X-ray [56, 57], proton [58, 59], ion [53, 60], boron-neutron capture [61–65], and targeted alpha therapy [66, 67]. As previously mentioned, the MK model based on z¯1D∗ [4, 44] performs sufficiently well; however, an extended version that fully incorporates f(z, D) calculated by the AMF with the convolution—referred to as the double-stochastic MK model—can more accurately capture complex features of cell survival, such as the linear dose-response observed in high-dose photon irradiations and in extremely high-LET irradiations [23, 68]. A methodology for abridging the PD calculated by the AMF for use in the treatment planning of carbon-ion therapy has also been proposed [69], as the direct implementation of the AMF remains challenging due to memory constraints. Furthermore, the AMF enables the direct estimation of RBE-weighted dose in the human body when combined with the Q(y) relationship established from cell survival data [70–72].

For radiological protection applications, fluence-to-organ dose conversion coefficients based on Q(y) defined in ICRU Report 40 [73] were calculated using PHITS coupled with the ICRP adult reference voxel phantoms [74], utilizing the AMF [75, 76]. Unlike other Q(y) relationships, the ICRU Q(y) was established primarily based on the human chromosome abbreviation data and is therefore applicable to radiological protection against stochastic effects. Baiocco et al. [77] calculated the neutron RBE for inducing complex DNA damage in the ICRU soft-tissue spherical phantom using the AMF in PHITS coupled with PARTRAC [17] and compared the results with the radiation weighting factor w_R defined in ICRP Publication 103 [78]. These studies provided information on the effective quality factors in macroscopic contexts and can contribute to the scientific discussion on the adequacy of the current w_R value, especially given that ICRP has not yet documented the underlying rationale or the specific biological endpoints used in its determination. Additionally, recent analyses on the RBE for skin reactions [79] and diseases of circulatory system [80], conducted using the MK model coupled with the AMF, suggested that the RBE for these biological endpoints tend to be lower than the corresponding w_R value. This finding supports the ICRP’s decision not to apply w_R for radiological protection against tissue reactions [81] and may be useful for future proposals regarding new weighting factors for tissue reactions.

Furthermore, the AMF has been applied to detector science, even though it was established based on the track-structure simulation in water. For example, Parisi et al. [82, 83] demonstrated that the ion-species dependences observed in the detection efficiency of thermoluminescence and optically stimulated luminescent detectors can be explained by the differences in the PDs of z at the nano-meter scale. In addition, Hirata et al. [84] suggested that the best-fit target size of z for reproducing the measured detection efficiency of BaFBr:Eu is comparable to the mean distance between trap centers. These studies provide insights into the mechanism of the quenching effect observed in detectors irradiated with HZE particles.

Summary and Perspectives

There are three computational approaches to bridging the gap between microdosimetry and macrodosimetry: (1) direct integration of track-structure codes; (2) integration of a database containing dose-mean values of microdosimetric quantities; and (3) integration of the AMF for instantaneously calculating the PD of microdosimetric quantities. The first approach is the most straightforward but the least practical due to computational time constraints, even with GPU-based parallelization. The second approach has already been employed in the treatment planning of carbon-ion therapy, although it sacrifices information on the stochastic nature of energy deposition—one of the fundamental distinctions between microdosimetry and macrodosimetry. The third approach is the most sophisticated and has been successfully applied to various research fields.

One drawback of the third approach is the difficulty of developing a reliable AMF. This process requires performing track-structure simulations for various ion species over a wide energy range, along with the accurate estimation of their stopping powers. However, the ionization and electronic excitation cross-section models currently implemented in track-structure simulation codes often fail to reproduce the stopping powers calculated by specialized codes such as SRIM and ATIMA, particularly for heavier ions [55]. In the case of ITSART in PHITS, which was used in the latest AMF development, scaling factors were introduced to adjust the cross sections to match the stopping powers calculated by ATIMA. Therefore, improvements to the cross-section models as well as their experimental verification are desirable for future updates of AMF. Furthermore, the evaluation of the free parameters in AMF is laborious, as the PDs of microdosimetric quantities under numerous conditions must be fitted through least-squares fitting. The use of recent deep-learning techniques may help address this issue in the future.

To date, only one AMF (in two versions [50, 51]) has been available in macroscopic simulation codes; it has been used in PHITS for many years and was recently implemented in TOPAS. As more AMFs are developed and more macroscopic radiation transport codes gain the capability to calculate the PDs of microdosimetric quantities in macroscopic contexts, the application scope of microdosimetry is expected to expand not only to medical physics and radiological protection but also detector science.

Article Information

Funding

This work was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI 23K21426.

Conflict of Interest

No potential conflict of interest relevant to this article was reported.

Ethical Statement

This article does not contain any studies with human participants or animals performed by any of the authors.

Data Availability

The raw data are protected and are not available due to data privacy laws.

Author Contribution

Conceptualization, Supervision, Funding acquisition, Project administration, Writing - original draft, Writing - review & editing, Approval of final manuscript: Sato T.

Acknowledgements

The author is thankful to Dr. Y. Matsuya of Hokkaido University and Dr. A. Parisi of the Mayo Clinic for their advice on writing this review article.

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Fig. 1.

Spatial distributions of the deposition energies around the trajectories of (A) 3 MeV proton and (B) 300 MeV/n ¹²C ion, together with schematic images of the spherical targets used for scoring lineal energy. Note that the actual simulations and scorings were performed in three dimensions.

Fig. 2.

Lineal-energy distributions, yf(y), obtained from the track-structure (TS) simulations drawn in Fig. 1, by setting the target diameter to 100 μm. The corresponding data obtained from the analytical microdosimetric function (AMF) are also depicted.

Fig. 3.

Specific-energy distributions for various absorbed dose D, denoted as zf(z, D) for (A) 3 MeV proton and (B) 300 MeV/n ¹²C ion, obtained from yf(y) plotted in Fig. 2, in combination with Equations (1) to (3).

Fig. 4.

Illustration of the computational approaches to bridging the gap between microdosimetry and macrodosimetry. TS, track-structure; AMF, analytical microdosimetric function.

Table 1.

Characteristics of LET, Lineal Energy y, and Specific Energy z

	LET and L_Δ	Lineal energy (y)	Specific energy (z)
Definition	dEΔdl, where Δ→∞ for LET	εsl¯	εm
Unit	J/m or keV/μm	J/m or keV/μm	J/kg or Gy
Dependent	E, Z, A, and Δ for L_Δ	E, Z, A, r	E, Z, A, r, D
Stochastic or deterministic	Deterministic	Stochastic	Stochastic
Lateral spread of energy deposition	Ignored in LET Empirically considered in L_Δ	Considered	Considered
Typical calculation method	Stopping power calculation code	Track-structure simulation	Track-structure simulation with convolution

Δ, cutoff value for secondary electrons (δ rays) released by the charged particles; ε_s, energy imparted to a target volume by a single particle track; ε, energy imparted to a target volume; l¯ m, r, mean chord length, mass, and radius of a target volume, where l denotes chord length; E, Z, A, energy, charge, and atomic number of the charged particle; D, absorbed dose; LET, linear energy transfer.