## 2 Materials and Methods
### UUUM and the Base Unit Representation  
The Unified Code for Units of Measure is a coding system developed by the Regenstrief Institute that aims to facilitate the unambiguous electronic communication of all physical units and their associated quantities currently used in international science [4]. Moreover, it plays an important role within the LOINC standard as well being the recommended way of reporting quantitative results [12]. All valid UUUM units are exclusively part of the seven-bit US-ASCII character set. The coding system consists of a set of simple terminal symbols, also called unit atoms, which can be combined by following the rules set out by the UCDM expression syntax ([[https://ucum.org/ucum.html#section-Syntax-Rules](https://ucum.org/ucum.html#section-Syntax-Rules)]([https://ucum.org/ucum.html#section-Syntax-Rules](https://ucum.org/ucum.html#section-Syntax-Rules)) (accessed on 29 April 2022)), resulting in an infinite amount of possible unit terms with precise semantics. Unit atoms can be connected via the division (\"/\") or multiplication operator (\".\"). Metric units may be prefaced with a metric prefix (e.g., \"mg\" for milligrams) and integer suffixes signify the dimensionality of a unit (e.g.,\"cm2\" and \"s-1\"). Additionally, there are two important types of brackets used. Round brackets retain their established usage of ordering operations, whereas curly brackets at the end of a valid unit are used to add annotations which may also be used in place of a unit symbol or integer. The content  
Figure 1: This figure shows all six LOINC axes that end up forming a specific LOINC code for “Erythrocytes [#/Volume] in Urine by Test Strip”. Concatenating the parts using colons yields the fully specified name of a LOINC code. of these annotations carries no semantic value within UCUM and is disregarded when processing units. The possibility to append such notes exists in order to accommodate traditions and habits in chemistry and biomedical sciences (e.g., mg{creat} conveying \"milligrams of creatinine\") [15]. An important aspect of UCUM is the commensurability of units. Each quantity \(\mathbf{Q}\) is the product of a measurement \(\mathbf{\mu}\) and its unit \(\mathbf{u}\). Moreover, the same kind of quantity can be expressed in a different unit \(\mathbf{u}^{\prime}\) with an accompanying measurement \(\mathbf{\mu}^{\prime}\). For example, one could use both meters or yards to determine a previously unknown length (e.g., \(3.6\cdot\text{m}=3.937\cdot\text{yd}\)). In this way they would be described as being commensurable. \[\mathbf{Q}=\mathbf{\mu}\mathbf{\cdot}\mathbf{u}=\mathbf{\mu}^{\prime}\mathbf{\cdot}\mathbf{u}^{\prime} \tag{1}\]  
One could also use the meter stick to measure the yard stick's length or vice versa. Therefore, the following equation holds:  
\[\mathbf{u}^{\prime}=\mathbf{y}\mathbf{\cdot}\mathbf{u} \tag{2}\]  
where \(\mathbf{y}\) is the magnitude of the unit \(\mathbf{u}^{\prime}\) measured as a quantity by unit \(\mathbf{u}\) (e.g., \(\text{yd}=3600/3937\cdot\text{m}\)). It follows that any fixed quantity can be used as a unit to measure all other quantities that are of the same kind. The formula for unit conversion is then derived from Equations (1) and (2):  
\[\mathbf{\mu}^{\prime}=\mathbf{\mu}\mathbf{\cdot}\frac{\mathbf{u}}{\mathbf{u}^{\prime}}=\mathbf{\mu} \mathbf{\cdot}\frac{\mathbf{1}}{\mathbf{y}} \tag{3}\]  
where \(\frac{\mathbf{1}}{\mathbf{y}}\) can be described as a conversion factor between different but related units. UCUM proposes seven metric base units from which to derive all other measurements. A valid unit may also have no dimensions at all. In this case, it describes a numeric value. A special numeric value is being described by the default unit \"1\" (or \"the unity\").