## 1 Introduction  
As energetic ions interact with matter, they dissipate energy to atoms and molecules in a unique and stochastic manner. To a good approximation, non-relativistic energetic ions interact with the medium mostly by inelastic Coulomb-force collisions with atomic electrons, leading to ionizations and electronic excitations. An important observation is that radiation effects depend upon the spatial pattern of energy deposition which is distinct for each type and energy of radiation. This is termed radiation quality [1; 2; 3; 4]. Radiation quality plays a major role in the radiobiological effects observed in living matter, and it is commonly described by relative biological effectiveness (RBE). To a first approximation, RBE is a function of the (unrestricted) Linear Energy Transfer (LET), which is defined as the mean electronic energy-loss by a primary charged particle per unit path length. LET may be conveniently calculated from [PERSON]'s stopping-power theory [5]. The official connection of LET with radiation quality (or RBE) has been made by the International Commission on Radiation Protection (ICRP) in a series of reports [3; 4; 6]. According to the ICRP, the quality factor, \(Q\), is defined as the low-dose RBE (or RBEmax)for stochastic effects [3] and may be expressed as a continuous function of LET [4]. More recently, LET has been used in several empirical RBE models for tissue reactions (or deterministic effects) in the context of hadron therapy [7; 8; 9; 10; 11]. Despite its simplicity and wide applicability, there are well-known physical limitations of the LET concept [12], which are more profound when the site of interest is reduced to cellular and subcellular structures (micrometres to nanometres). The most notable limitation of LET is that it neglects the stochastic nature of energy deposition (energy-loss straggling) from the charged particle to the medium while no consideration is given to the finite range of secondary electrons (\(\delta\)-rays). Although the restricted LET, L\({}_{\Delta}\), which excludes those \(\delta\)-rays with energy above a cut-off value \(\Delta\), can be used to better approximate the energy retained in the site, it still refers to a mean value and does not account for straggling [13]. The above limitations of LET can be overcome using microdosimetric approaches [13; 14; 15]. For example, classical (or regional) microdosimetry considers a fixed-size critical site in which energy deposition distributions are linked to radiobiological effects. More advanced microdosimetric approaches (e.g., structural microdosimetry [14]) are available with the cost of increased calculation complexity and computing time. Lineal energy has been widely exploited in radiation quality studies as a measurable physical quantity by tissue-equivalent proportional counters (TEPC), also called Rossi counters [16]. The biophysical justification of connecting lineal energy with RBE (or \(Q\)) is provided by the Theory of Dual Radiation Action (TDRA) [17]. The official connection of lineal energy with \(Q\) has been made by the International Commission on Radiation Units and Measurements (ICRU) in its Report 40 [2]. The lineal energy has also been used in RBE studies of cell toxicity in the context of hadron therapy through the Microdosimetric-Kinetic Model (MKM) [18; 19; 20]. Other microdosimetric-based models of \(Q\) used by NASA for space radiation can be found in several published works [21; 22; 23; 24; 25]. \(Q\) is used to determine the quantity effective dose which corresponds to the stochastic (mainly carcinogenic) risk of irradiated individuals. In turn, the effective dose is used to calculate the number of safe days in space, a vital quantity for manned missions such as those to the Moon and Mars, i.e., the maximum number of days that astronauts can spend in space without exceeding their radiation dose limits. In proton radiotherapy, the effective dose is used to quantify the secondary cancer risk in healthy tissues that are exposed to stray proton radiation.