Time-explicit life cycle assessment: a flexible framework for coherent consideration of temporal dynamics

Vol.:(0123456789) The International Journal of Life Cycle Assessment https://doi.org/10.1007/s11367-025-02539-3 LCI METHODOLOGY ANDDATABASES Time-explicit life cycle assessment: aflexible framework forcoherent consideration oftemporal dynamics AmelieMüller1,2 · TimoDiepers3 · ArthurJakobs4 · GiuseppeCardellini2 · NiklasvonderAssen3 · JeroenGuinée1 · BernhardSteubing1 Received: 14 February 2025 / Accepted: 24 August 2025 © The Author(s) 2025 Abstract Purpose A well-known limitation of conventional Life Cycle Assessment (LCA) is the lack of temporal considerations, particularly the temporal distribution and evolution of processes, emissions, and environmental responses. While these aspects have been explored to some extent in dynamic and prospective LCA, a comprehensive approach for considering both temporal distribution and evolution is currently missing. We introduce a novel framework for time-explicit LCA that integrates the temporal distribution and evolution of product systems in the Life Cycle Inventory (LCI) phase and supports dynamic characterization of emissions in the Life Cycle Impact Assessment (LCIA) phase. Methods The proposed approach expands the conventional LCA matrices to incorporate timing and time-based changes. We use a best-first graph traversal to derive an absolute timeline of intermediate flows by convolving relative temporal distributions at the process level. These timings are then integrated into the LCA matrices by adding time-specific row-column pairs in the technology matrix. Temporal markets are used to distribute product demands to the most-suitable processes in timespecific background databases. New rows in the biosphere matrix represent time-specific elementary flows. By preserving the timing of elementary flows during inventory calculation, time-explicit LCA enables dynamic alongside conventional LCIA. The proposed framework can be used for assessing any product system and impact category. An implementation of time-explicit LCA is provided in the open-source python package bw_timex, part of the Brightway ecosystem. Results We demonstrate the framework with a simplified case study of an electric vehicle (EV). For a Paris-Agreementcompatible scenario, which assumes strong decarbonization over time, time-explicit LCA determines the EV's total Global Warming Impact to be half that of a 2020 conventional LCA and nearly double that of a 2040 prospective LCA. These differences arise because time-explicit LCA uses time-specific inventory data for each timestep, depending on the timing of processes in the supply chain, contrasting the conventional or prospective cases, which rely on a single inventory database. To further demonstrate dynamic characterization, we show the instantaneous and cumulative radiative forcing over the EV life cycle. Conclusions Overall, time-explicit LCA can provide more representative results compared to conventional LCA, by considering when processes and emissions occur and what the state of the systems is at these timings. This is particularly valuable for long-lived products in temporally variable or fast-evolving systems. Future research should focus on filling data gaps and connecting time-explicit LCA with spatial LCA or dynamic material flow analysis. Communicated by Masaharu Motoshita. Amelie Müller and Timo Diepers contributed equally to this study. * Amelie Müller [email protected].nl 1 Institute ofEnvironmental Sciences (CML), Leiden University, P.O. Box9518, Leiden, RA2300, TheNetherlands 2 Flemish Institute forTechnology Research (VITO), EnergyVille, Thor Park 8310, Genk3600, Belgium 3 Institute ofTechnical Thermodynamics (LTT), RWTH Aachen University, Schinkelstrasse 8, Aachen52062, Germany 4 Technology Assessment Group, Laboratory forEnergy Analysis (LEA), Center forNuclear Engineering andSciences & Center forEnergy andEnvironmental Sciences, Paul Scherrer Institute PSI, Forschungsstrasse 111, Villigen5232, Switzerland The International Journal of Life Cycle Assessment Graphical Abstract Keywords Temporal distribution· Temporal evolution· Dynamic LCA· Prospective LCA· Open-source software· Time-differentiated· Time-resolved· Dynamic modelling 1 Introduction Like all models, life cycle assessment (LCA) models simplify the real world, reducing complexity to accommodate data and modelling constraints. One increasingly questioned simplification is that LCA typically treats processes, emissions, and environmental responses as static, disregarding temporal considerations (ISO 140402006). The importance of temporal considerations in LCA has been demonstrated in many studies and summarized in multiple reviews (BeloinSaint-Pierre etal. 2020; Lueddeckens etal. 2020; Sohn etal. 2020; Su etal. 2021b). To structure the multitude of temporal considerations in both the life cycle inventory (LCI) and the life cycle impact assessment (LCIA) phases in LCA, we generally distinguish two categories: temporal distribution and temporal evolution. 1.1 Temporal distribution In the real world, supply chains must obey a certain temporal sequence, as products must be produced before they can be consumed. In other words, there is a time lag between demand and supply (Beloin-Saint-Pierre etal. 2014; TirutaBarna etal. 2016). This time lag can originate not only from the processes themselves taking a certain time to complete (e.g., a distinct process profile, such as a long use phase) but also from a delay between production and consumption (e.g., transport or storage processes) (Beloin-Saint-Pierre etal. 2014; Tiruta-Barna etal. 2016). Consequently, emissions and the induced environmental impacts are spread across time. We summarize temporal considerations that describe the timing of processes, emissions, and environmental responses under the term temporal distribution. Conventional LCA typically does not model the temporal distribution of real-world systems, arguing that this simplification does not significantly influence a study’s outcomes. Instead, it implicitly assumes that the entire system occurs in the present moment, which Arvidsson etal. (2023) describe as the “ever-advancing ‘now’.” This reference to current time in conventional LCA is often implied by the presumed representativeness of contemporary conditions in the data (Guinée etal. 2002). A frequently used term for LCAs that account for aspects of temporal distribution is dynamic LCA (dLCA), although this term has been used inconsistently (Beloin-Saint-Pierre etal. 2020; Lueddeckens etal. 2020; Sohn etal. 2020; Su etal. 2021a). As a key characteristic, dLCA retains the timing of LCIs, e.g., emission x occurring at time t , with various methods proposed to calculate the temporal sequence of inventories. Beloin-Saint-Pierre etal. (2014) propose the ESPA (enhanced structural path analysis) approach, which models the temporal distribution of intermediate and elementary flows using “process-relative temporal distributions” (rTDs). Dynamic inventories are derived through convolution and power-series-expansion. Building on this, Cardellini etal. (2018) also apply convolution of rTDs but prioritize processes using a “best-first” graph traversal algorithm (i.e., traversing processes with the highest impacts first), implemented in the tool Temporalis (Cardellini and The International Journal of Life Cycle Assessment Mutel 2018). Tiruta-Barna etal. (2016) propose to use the technosphere matrix as an adjacency matrix and calculate the temporal sequence of processes using a supply-demand model. This allows, in contrast to rTDs, to directly model global process behavior, such as process durations and production profiles. This approach has been operationalized in the tool DyPLCA (Pigné etal. 2020). Next to the timing of LCIs, dLCA also investigates time-dependencies of the environmental responses at the LCIA phase, often referred to as dynamic LCIA. Various approaches to dynamic characterization have been developed for the impact categories climate change (Levasseur etal. 2010; Kendall 2012; Shimako etal. 2016; Tiruta-Barna 2021; Lan and Yao 2022; Ventura 2022), including different indicators, such as global warming potential (GWP) and global mean temperature change (GMTC). Dynamic characterization for other impact categories is less common but has been studied for air pollution (Shah and Ries 2009), toxicity (Lebailly etal. 2014; Shimako etal. 2017), noise (Cucurachi and Heijungs 2014), and water use (Núñez etal. 2015). Existing dLCAs usually focus on climate change impacts, often covering biogenic carbon in bio-based materials (Levasseur etal. 2012b, 2012a; Brandão etal. 2013; Shimako etal. 2016), the built environment (Breton etal. 2018), transport (Albers etal. 2019), and CO2-based products (von der Assen etal. 2013). While the aforementioned studies consider the timing of emissions and apply dynamic characterization, they still model a steady-state operation of processes within a static supply chain configuration and a steady-state environment, assuming that omitting temporal evolution at LCI and LCIA is a reasonable modelling simplification. 1.2 Temporal evolution In reality, processes, supply chains and the state of the environment change over time. These changes may originate from variations in process operation (e.g., temporal profile of solar power production), structural shifts in supply chains (e.g., integration of novel renewable technologies into the electricity mix), or changing background conditions in the environment (e.g., increasing abundance of CO2 in the atmosphere). We summarize the time-based changes in processes, emissions and environmental responses under the term temporal evolution. Capturing the temporal evolution towards future systems is central to the field of prospective LCA (pLCA). A pLCA “models the product system at a future point in time relative to the time at which the study is conducted” (Arvidsson etal. 2023). Various methods are used in pLCA studies to adapt LCIs based on projections for the future developments of product systems (Thonemann etal. 2020). While early pLCA studies mainly focused on the projection of the technology under review (foreground system), recent studies have shifted towards modeling economy-wide projections (foreground and background system) (Mendoza Beltran etal. 2020; Sacchi etal. 2022). Concerning existing software for prospective data generation, premise (Sacchi etal. 2022) has emerged as a widely used tool to modify ecoinvent databases based on integrated assessment model output. The focus of pLCA studies is typically on changes at the LCI stage, while temporal evolution at the LCIA stage is rarely considered. Regardless of projection methods and scope, pLCA approaches have in common that they model a system as a snapshot at distinct future points in time. These snapshots can be viewed as prospective static LCAs: the entire production system is moved forward in time, but all processes in the system are still simplified to happen simultaneously at this future timestep, under these future steady-state conditions. This means that any temporal distribution effect is left unaccounted. A conceptually similar approach to pLCA is retrospective or historical LCA, which uses past data rather than future projections to adapt LCIs (Arvidsson etal. 2023; Bruhn etal. 2024). However, like pLCA, retrospective LCA typically produces steady-state snapshots in time, without consideration of temporal distribution. Table1 summarizes how existing LCA methods treat temporal distribution and evolution at the LCI and LCIA phase. Table 1 Schematic overview of how different LCA methods typically treat temporal distribution and temporal evolution at the life cycle inventory (LCI) and life cycle impact assessment (LCIA) phase Life cycle inventory Life cycle impact assessment Temporal distribution Temporal evolution Temporal distribution Temporal evolution Conventional LCA No, 1 current timestep No No No Dynamic LCA Yes, multiple timesteps Rarely Yes No Prospective LCA No, 1 future timestep Yes No Rarely Retrospective LCA No, 1 past timestep Yes No No Time-explicit LCA (proposed in this study) Yes, multiple timesteps Yes Yes Yes The International Journal of Life Cycle Assessment 1.3 Joint consideration oftemporal distribution andevolution In current literature, temporal distribution and evolution are mostly considered separately. While dLCA studies emphasize that systems are temporally distributed, they rarely account for their temporal evolution. Conversely, pLCA studies consider the temporal evolution of technologies and associated emissions at a future point in time but do not consider that processes and emissions are also distributed over time, see Table1. Although a consistent treatment of temporal dynamics is widely recognized as important (Beloin-Saint-Pierre etal. 2020), it is often constrained by the limited availability of tools and data to address both aspects simultaneously (Vance etal. 2022). Existing work that jointly considers temporal distribution and evolution usually focuses on a subset of inputs, e.g., electricity supply, and only considers the temporal distribution and evolution of this subset, but not other inputs or any upstream supply chains, or targets a single sector (e.g., buildings), while a generalizable and transparent method is missing. In a seminal early work, Collinge etal. (2013) conduct a dLCA of an institutional building and add yearly inventories for fuel and electricity and yearly emission factors to the foreground system. Zimmermann et al. (2015) add yearly prospective electricity mixes during the use phase in a pLCA study on electric mobility in Germany but apply static LCIA methods, using the term “time-resolved LCA.” A similar approach, but including static and dynamic LCIA for climate change, has been conducted by Peng etal. (2019) for a case study on compressors in China, using system dynamics to calculate yearly prospective electricity mixes. Reinert etal. (2021) optimize the costs of an energy system transition and then determine environmental impacts using different prospective databases based on the optimized deployment time of processes. Sigüenza etal. (2021) developed a time-vintage LCA model that splits the product system into life cycle stages and calculates different foreground and background LCIs per life cycle stage for each model cohort. Bruhn etal. (2023) argue that pLCAs for long-lived products, such as the built environment, should use data from different projection years for the different life cycle stages. Beloin-SaintPierre etal. (2016) use a systematic method (ESPA, cf. Beloin-Saint-Pierre etal. (2014)) to account for the temporal distribution of foreand background processes, linking a subset of processes to their temporal evolution and applying dynamic LCIA. However, the linking to the temporal evolution of processes required extensive manual work and their excel-based workflow is not publicly available. Negishi etal. (2018) and Negishi etal. (2019) present a notable example of joint consideration of temporal distribution and evolution in LCI and LCIA for the building sector. They link a static building model to a dynamic parameter database that models time-based changes at the building (e.g., performance degradation), user (e.g., occupant behavior), and system (e.g., energy mix) levels. Foreground processes are discretized into fixed time intervals (e.g., 1 or 10 years), during which parameters are assumed constant. This inventory is then processed in the DyPLCA tool (Pigné etal. 2020), connecting the foreground processes to the data of the dynamic parameter database and adding the temporal distribution of supply chain processes. Finally, the resulting dLCI is characterized with dynamic LCIA for three climate change indicators. While these two studies are a substantial step towards temporal coherence in LCI and LCIA, the underlying algorithm in DyPLCA is not made publicly accessible, hindering the comparison to our approach. Lastly, recent work on coupling dLCA and pLCA for assessing transition paths in a tool called Prosperdyn seems promising (Lang-Quantzendorff and Beernmann 2024), but at the time of writing no published information could be found on the tool. Although these approaches highlight the importance of jointly accounting for temporal distribution and evolution of processes, emissions, and environmental responses in LCAs, they have limitations, such as a lack of transparency, focus on only specific sectors, a subset of processes or life cycle assessment steps, fixed temporal scopes and scalability constraints. As outlined above, existing tools such as Temporalis and DyPLCA support modeling the temporal distribution of processes and emissions, while tools like premise enable the projection of technological evolution at discrete points in time. Although some casespecific implementations, such as Negishi etal. (2019), combine both aspects to a degree, no existing tool offers a generalizable, transparent, and scalable solution that accounts for both temporal distribution and evolution simultaneously, which is essential for time-explicit LCA. We propose a novel framework to simultaneously account for temporal distribution and temporal evolution in LCA by both considering the timing of processes and emissions as well as the state of technologies and supply chains at the respective point in time. We coin this framework “time-explicit LCA.” An implementation is available in the open-source python package bw_timex (Diepers etal. 2025b), which is part of the Brightway LCA ecosystem (Mutel 2017). In the following section, the framework is described and demonstrated with a case study of an electric vehicle (EV). We show that time-explicit LCA can yield more representative results for environmental impacts, particularly for temporally variable, fast-evolving systems or long-lived products with impacts spread considerably over their lifetime. The International Journal of Life Cycle Assessment 2 Method We first introduce the mathematical basis of time-explicit LCA. Then, we describe the time-explicit LCA framework and demonstrate it with a simple system. An in-depth description of the software implementation is given in Diepers etal. (2025b). 2.1 Mathematical basis The conventional inventory problem in LCA is described by Eq. (1)(Heijungs and Suh 2002): where: – f ( products × 1) is the demand vector of the functional unit, – A(products ×processes) is the technology matrix, whose element ak,p represents the amount of product k required or produced by process p , – B ( elementary flows × processes) is the intervention or biosphere matrix, whose element bj,p represents elementary flow j (e.g., emission or resource use) emitted or consumed by process p , – C ( impact categories × elementary flows) is the characterization matrix, whose element ci,j represents the characterization factor of elementary flow j for impact category i , and – h(impact categories ×1) is the vector of environmental impacts. Conventional LCA simplifies the complexity of realworld systems to a steady state in production technologies ( A ), their elementary flows ( B ) and translation to impacts ( C ) (Heijungs and Suh 2002). Existing temporal variation in data is handled by integrating it over time, leaving only an implicit reference to time through the temporal representativeness of the data (Guinée etal. 2002). pLCA explicitly references time by modeling the system at a distinct future point t , described by Eq. (2). where: trepresents a future point in time, e.g., year 2045. pLCAs typically modify the A and B matrices to reflect the projected state of the technology at the future point in time (Mendoza Beltran etal. 2020; Thonemann etal. 2020; Sacchi etal. 2022). The issue is that pLCA treats the entire system as occurring at a single future point, ignoring that also a future system has temporally distributed processes and emissions. This corresponds to essentially performing a conventional, static LCA with projected data for one point in time. Retrospective LCA is conceptually the same, only for distinct points of time in the past. (1) h = CBA−1f (2) ht =C t B t A −1 t f t Collinge etal. (2013) propose a mathematical formulation to account for temporal evolution across distinct timesteps, see Eq. (3): where: – t represents a distinct point in time at which the state of the system is known, and – t0 and te represent the start and end time points of the analysis, usually the beginning and ending of the product or system life cycle (Collinge etal. 2013). Equation3 is in essence the sum of Eq.2 for all points in time with available data. This means that the LCA equation is evaluated separately for each timestep: The system is split into temporal segments, each with its own distinct set of technology, biosphere and characterization matrices. While this improves upon modeling a system only at a single current (Eq.1) or future (Eq.2) point in time, it still neglects interconnections across timesteps. For example, consider an EV life cycle: Collinge etal. (2013) split the life cycle into distinct timesteps, e.g., car factory construction at t0 , car assembly at t1 , car use phase from t2 to tn−1 , and disposal at tn . According to Eq.3, each segments’ supply chain is modeled at the same time as the segment itself. For example, materials for factory construction are produced at t0 and all car components at t1 . However, in reality, these activities occur sequentially–materials required for the factory need to be produced before the factory can be built, and so on. Such time lags exist throughout supply chains, leading to complex temporal distributions in real-world systems. Collinge etal. (2013) acknowledge this limitation, noting that “a more complete formulation would involve specifying the lag time for each supply–demand linkage, which would require calculation using a tree structure rather than a matrix structure, as the number of inputs at different time lags would multiply with each step back through the supply chain” (Collinge etal. 2013, p.4). In time-explicit LCA, the results of a tree-based time lag propagation are used to extend the original matrices. This expansion allows us to reflect the timing of processes and emissions in the supply chain (temporal distribution) and, at the same time, to consider different process inventories for different points in time (temporal evolution). The resulting mathematical formulation of time-explicit LCA is structurally identical to Eq.1 but with temporally extended matrices, as denoted by the asterisks (*), see Eq. (4): where: (3) h =∑ t e t0 CtBtA−1 tft (4) h∗=C∗B∗A∗−1f∗ The International Journal of Life Cycle Assessment – f∗ (products @timesteps ×1) is the time-explicit demand vector of the functional unit, – A∗ (products @timesteps ×processes @timesteps) is the time-explicit technology matrix, whose element a∗ k,p represents the amount of product k at a specific timestep required or produced by process p at a specific timestep, – B∗ (elementary flows @timesteps ×processes @timesteps) is the timeexplicit biosphere matrix, whose element b∗ j,p represents elementary flow j at a specific timestep emitted or consumed by process p at a specific timestep, – C∗ ( impact categories @ timesteps × elementary flows @ timesteps) is the time-explicit characterization matrix, whose element c∗ i,j represents the characterization factor for impact category i at a specific timestep for elementary flow j at a specific timestep, and – h∗ (impact categories @ timesteps × 1) is the vector of time-explicit environmental impacts. The key distinction of the time-explicit LCA formulation is its ability to embed temporal information directly into the LCA matrices by adding a new element (row-column pair) for each process at a specific time (see section2.2 for details). This expansion approach allows the elements of each matrix to correspond to different points in time. By contrast, conventional LCA (Eq.1) assumes that matrix elements represent an implicitly defined “current” time, while pLCA (Eq.2) and dLCA (Eq.3) assume a single, fixed point in time–at a single future time or for each t within the summation, respectively. 2.2 Time‑explicit LCA framework The implementation of the time-explicit LCA framework (Eq.4) is outlined in the following section. Figure1 provides an overview of the steps involved in a time-explicit LCA. First, we describe how a product system is temporalized (see section2.2.1). Then, we explain how this temporal information is incorporated into the matrix structure (see section2.2.2). The approach is demonstrated for a simple example in Fig.2. 2.2.1 Temporalization ofproduct systems A time-explicit LCA must be informed of the absolute timing of processes across the system (temporal distribution) before linking them to their temporal evolution at these points in time. By absolute timing, we refer to the specific point in calendar time (i.e., the date) at which a process occurs, rather than just its relative temporal position within the product system. This absolute timing can be determined by propagating temporal information at the process level along the supply chain (Beloin-Saint-Pierre etal. 2014). For this purpose, we use rTDs as implemented in Cardellini etal. (2018). A rTD reflects how the total amount of a flow is distributed across time. For intermediate flows, rTDs describe when product k is demanded relative to the timing of its consuming process p , and for elementary flows when elementary flow b is emitted or consumed relative to the timing of its emitting or consuming process p . As time in LCA is inherently discrete (Heijungs and Suh 2002), continuous inputs or emissions can be discretized by sampling the supply or emission functions at specific intervals. To determine the absolute timing of all processes and emissions, the rTDs are convolved along the supply chain. This procedure begins at the functional unit, which is demanded at an absolute point in time defined by the LCA practitioner. From there, the supply chain graph is traversed and the rTDs are propagated through time using convolution, following the approach of Cardellini etal. (2018). The best-first traversal algorithm of Cardellini etal. (2018) is more suitable than a breadthfirst (Beloin-Saint-Pierre etal. 2014) or depth-first variant Productsystem model Temporalizedproduct systemmodel Timeline of intermediateflows Expanded timeexplicit matrices Time-explicit inventory (Time-explicit) Environmentalimpacts Temporal distributions of intermediateand elementary flows Time-specific background databases Graph traversal Relinking Solving inventory LCIA Section 2.2.1Section2.2.2 Fig. 1 Overview of the time-explicit LCA framework The International Journal of Life Cycle Assessment Fig. 2 Time-explicit LCA procedure for an illustrative example consisting of three processes X, Y, and Z The International Journal of Life Cycle Assessment (Beloin-Saint-Pierre etal. 2014; Tiruta-Barna etal. 2016) as it prioritizes the traversal of the most important contributors, covering the relevant parts of the supply chain faster. For a detailed explanation of this traversal algorithm, readers are referred to the original work. Unlike Cardellini etal. (2018), we construct an absolute timeline of intermediate flows, rather than that of elementary flows. This timeline of intermediate flows specifies the absolute timing of the producing and consuming process for each intermediate flow in the product system. The rTDs of an exemplary system and the resulting timeline of intermediate flows are shown in Fig.2a and b. In addition to the absolute timing of processes, a timeexplicit LCA requires information on the temporal evolution of processes over time. This is achieved using timespecific background databases. Time-specific is defined here as referring to a single absolute point in time. Timespecific databases, thus, represent the state of the production system (temporal evolution) at single absolute points in time, which is stored as metadata. The linking of intermediate flows to these time-specific inventory databases is described in the next section. 2.2.2 Matrix expansion To reference to multiple time points in a single matrix, we build on the approach by Lesage etal. (2019). Each process at a specific time, derived from the timeline of intermediate flows, is treated as a separate process, referred to as a “temporalized process.” These temporalized processes are added as new columns to the A∗ -matrix. Correspondingly, new rows are added for the “temporalized products” produced by these processes. To control the desired level of detail of new entries, the temporal resolution of the new entries can be harmonized by grouping them, e.g., on a yearly resolution. If a temporalized process receives an intermediate flow from a background database, the temporalized process is linked to the producing process(es) of the intermediate flow from the most temporally appropriate background database(s). This is achieved by introducing a new set of row-column pairs, called “temporal markets.” Intraditional LCA, market processes distribute a demand for a product across spatial or technological alternatives (Wernet etal. 2016). In analogy, temporal markets distribute a demand across time, linking to processes that represent different temporal evolutions. They allocate this demand across time using temporal weighting factors based on the temporal proximity between the time of the producing process and the times of the most temporally appropriate background databases (see “temporal market shares” in timeline of intermediate flows in Fig.2b). The default option is linear interpolation, which is consistent with common LCA practice and methods used in prospective background database generation. However, the option to take only the value from the nearest time-specific background database is also available, and users can implement custom interpolation methods if non-linear dynamics are more appropriate for their case or reflect non-linearities through more finely resolved timespecific background databases. A schematic representation of a time-explicit A∗ -matrix compared to a conventional A -matrix is given in Fig.3. The time-explicit matrices for the simple example are shown in Fig.2d. Next, the biosphere matrix B∗ is also reconstructed to retain the temporal information at the level of elementary flows. The timing of an elementary flow is determined by the timing of its emitting process, convolved with the rTD of the elementary flow, if available. This accounts for any additional temporal shift of the elementary flow relative to the emitting process, e.g., long-term emissions from landfills. Thus, the timesteps in the time-explicit B∗ -matrix can differ from those in the time-explicit A∗ -matrix, as demonstrated for the example in Fig.2d. To retain the correct timing of the elementary flows, elementary flows from processes in the background databases are aggregated at the corresponding temporal market, resulting in zero entries in B∗ for background processes, see Fig.3. The resulting B∗ -matrix is typically highly sparse due to the large number of timesteps. Lastly, the time-explicit LCIs are obtained by multiplying the time-explicit biosphere matrix B∗ by the time-explicit supply vector s ∗ =A ∗ −1 ⋅ f∗ , following conventional matrixbased LCA calculation. The time-explicit LCI retains temporal information of the emissions, enabling subsequent characterization with either conventional characterization factors or dynamic characterization functions. Dynamic characterization functions for the climate change metrics radiative forcing and dynamic GWP(Levasseur etal. 2010) are available in the Brightway library dynamic_characterization (Brightway 2025c). A simple software interface enables users to easily change the time horizon of the assessment and whether the time horizon should be treated as fixed or moving (Ventura 2022). In contrast to current examples of dynamic LCIA, full time-explicit LCIA would require to also consider the temporal evolution of environmental responses, e.g., due to changes of future greenhouse gas (GHG) background concentrations for GWP. 2.3 Software implementation The time-explicit LCA framework is implemented in the open-source Python software package bw_timex (Diepers etal. 2025b). bw_timex is part of the open-source LCA ecosystem Brightway (Mutel 2017), a widely used LCA software in the scientific community due to its flexibility and computational efficiency. The best-first graph traversal algorithm used in bw_timex originates from Cardellini etal. (2018), but has been updated and moved to The International Journal of Life Cycle Assessment the Brightway library bw_graph_tools (Brightway 2025a). The dynamic impact assessment methods are sourced from the Brightway library dynamic_characterization (Brightway 2025c). Additional information and instructions are available in the comprehensive and beginnersfriendly online documentation of bw_timex (Diepers etal. 2025a). By making all source code publicly accessible and by relying exclusively on other open-source frameworks, this work contributes to a higher level of transparency, quality and productivity in the Industrial Ecology research community (Pauliuk etal. 2015). 3 Case study To demonstrate the capabilities of the developed framework, we apply time-explicit LCA in a case study of an EV. The product system is described in section3.1. The case study results are presented in section3.2. The full case study code is available as an annotated Jupyter Notebook in the bw_timex GitHub repository (Brightway 2025b). 3.1 Case study setup The goal of this case study is to assess a product system using time-explicit LCA and compare the results to those of a conventional LCA, a dLCA and a pLCA. Life cycles of EVs span several years and the climate change impact of EVs is highly sensitive to the electricity supply, which makes this a well-suited example for time-explicit LCA. We consider a cradle-to-grave model of an EV and assess the Global Warming Impact (GWI) over the EV’s lifetime using GWP100 from the Environmental Footprint 3.1 impact assessment method (European Commission 2023). To additionally showcase time-explicit LCA’s capability to reflect time-resolved environmental impacts, we calculate the resulting radiative forcing over time using dLCIA functions from the library dynamic_characterization (Brightway 2025c). The calculations for radiative forcing are based on Myhre etal. (2014), with updated numerical values for radiative efficiencies and substance lifetimes from the IPCC Assessment Report 6 (Smith etal. 2021). Further details are available in the Jupyter Notebook on GitHub (Brightway 2025b). To reduce complexity and focus on methodological implications rather than subject-specific findings, the modeled EV is greatly simplified. The product system is shown in Fig.4. The foreground system consists of three processes covering the assembly, driving and dismantling of the EV. The foreground processes link to background processes from the ecoinvent 3.10 database (Wernet etal. 2016). For the background processes, we choose global average markets for the respective processes. The EV-specific parameter assumptions are listed in Table2. Figure4 shows the product system as well as the rTDs embedded in the system. All intermediate flows that Fig. 3 Construction of the time-explicit matrices A∗ and B∗ from the conventional (static) matrices A and B The International Journal of Life Cycle Assessment three-dimensional tensors and preserving this time dimension throughout inventory calculations. Alternatively, a time dimension can be introduced after determining the supply vector conventionally. The temporal information of when a process is supplied can be retrieved from the timeline of intermediate processes, yielding a supply matrix. Elementwise multiplication of each time-slice of the supply matrix with the biosphere matrix of that point in time yields a threedimensional time-explicit inventory tensor. Lastly, the computational time of a time-explicit LCA is generally higher than that of a standard LCA, though the extent depends on the system studied. In general, the number of necessary computations in a time-explicit LCA scales with the number of temporalized intermediate flows that must be traversed. This number grows linearly with the number of processes per supply chain tier, but exponentially with the number of tiers and the number of time steps in the rTDs. Each time step creates a new “virtual temporal branch” in the supply chain that needs to be followed to reach downstream processes. Benchmarking with bw_timex v0.3.1 (Diepers etal. 2025b) shows that for a test system with 4 Supply chain tiers, 5 processes per tier (fully connected across tiers), and 2 time steps per rTD, the total calculation time is 82 ± 2 s (mean and standard deviation of 3 runs using an Apple M4 Pro processor). As a comparison, for the same system with 10 time steps per rTD, the total calculation time is 865 ± 8 s. Additional benchmarking results are available on GitHub. 4.3 Link tospatial LCA As Heijungs and Suh (2002) point out, the implementation of temporal differentiation has a strong resemblance to spatial differentiation in LCA, which is a common practice for both LCI and LCIA data (Frischknecht etal. 2019; Mutel etal. 2019; Shi and Yan 2024). Existing LCI databases commonly feature separate regional processes (e.g., electricity production in Spain and Italy) (Wernet etal. 2016), which is similar to the separate temporal processes in our approach (e.g., electricity production in 2023 and 2024). Regional markets group spatially-specific processes, much like our temporal markets bridge different time periods. Similarly, elementary flows are spatially distinguished, i.e. emissions to “urban air close to ground” and “non-urban air or from high stacks” (Wernet etal. 2016), and can be paired with spatially specific characterization factors (Mutel etal. 2019). Spatial differentiation has received considerable attention in the LCA community, with various computational solutions proposed (Maier etal. 2017; Li etal. 2021; Mutel and Hellweg 2023; Peng and Pfister 2024). For instance, Peng and Pfister (2024) introduced a database-wide regionalization of activity datasets using trade data from a multi-regional input-output model. While such “fully regionalized” databases could be directly applied in a timeand region-explicit LCA using bw_timex, the level of temporal and regional resolution needs to be carefully selected to fit the research question while balancing the additional computational complexity. As for temporalization (Collet etal. 2014), prioritization is also necessary for regionalization (Patouillard etal. 2019). Future research could therefore explore adaptive frameworks that leverage contribution and uncertainty analysis to determine where a selective application of regional and temporal detail is most impactful, while maintaining higher levels of spatial and temporal aggregation for less critical parts of the supply chain. 4.4 Link toMFA LCA is often used in combination with material flow analysis (MFA), whether static or dynamic (Pauliuk and Hertwich 2016; Barkhausen etal. 2023). While a comprehensive discussion of system dynamics approaches is beyond the scope of this paper, we briefly highlight how time-explicit LCA can interface with dynamic MFA (dMFA). Time-explicit LCA, with its temporally distributed supply chains and the consideration of the specific technology landscape at each point in time, offers significant opportunities for integration with dMFA. An integrated dMFA and time-explicit LCA allows the analysis of material flows within and into the foreground system, while also calculating the elementary flows and impacts. Unlike classical dMFA software such as ODYM (Pauliuk and Heeren 2019), which is typically stockor inflowdriven, bw_timex currently only supports outflow (or final demand) driven models. Additionally, while bw_timex automatically produces material flows and impacts, the stock levels and changes must be derived separately from the timeline of intermediate flows. Typical features of dMFA are the use of lifetime distributions and age cohorts. In bw_timex, lifetime distributions can be modelled using rTDs, while differing age cohorts are represented by distinct products and producing activities at different timesteps, e.g., years. This makes bw_timex, particularly when combined with a modular LCA approach as proposed in Steubing etal. (2016), a powerful tool for combined dMFA-LCA assessments. 5 Conclusion Time-explicit LCA represents a significant advancement in accounting for temporal dynamics in LCA by jointly considering temporal distribution and temporal evolution of processes, emissions, and environmental responses at the LCI and LCIA stage. The resulting time-explicit inventory The International Journal of Life Cycle Assessment records the emissions as they occur in time, reflecting the technology landscape at each point in time. This timeexplicit framework enables more representative modeling of the emissions across the life cycle of the product under study. It is especially valuable for assessing long-lived products and supply chains with temporal variability, particularly when applied in scenarios that envision transformative technological changes. An implementation of the time-explicit LCA framework is available as the open-source python package bw_ timex(Diepers etal. 2025b)within the Brightway LCA ecosystem (Mutel 2017), with seamless integration for prospective databases generated via premise (Sacchi etal. 2022). The tool automatically propagates rTDs through the supply chain and links intermediate flows to time-specific databases according to their time of occurrence, offering substantial time savings and scalability improvements compared to manual approaches. The implementation supports a high degree of customization of the data inputs, accommodating the different temporal requirements across impact categories and the varying availability of time-specific data. Built on the conventional LCA matrix structure, bw_timex is also compatible with other LCA tools. We apply the framework in a case study of an EV, showcasing significant differences between the timeexplicit results and the results of assessments that model all processes at a single point in time. As temporal information is preserved in the LCI, dynamic LCIA methods can be applied, which we demonstrate for climate change impacts. Further research may include coupling time-explicit LCA with dynamic MFA or spatial LCA, and filling data gaps to enable time-explicit LCAs for entire supply chains and for impact categories besides climate change. Acknowledgements We especially thank Chris Mutel for discussion during initial method development and Benjamin Fuchs and Tom van Schaijk for code reviews. Author contributions Amelie Müller and Timo Diepers contributed equally to this study. Amelie Müller: Conceptualization, Methodology, Software, Writing - original draft preparation, Writing - review and editing Timo Diepers: Conceptualization, Methodology, Software, Writing - original draft preparation, Writing - review and editing Arthur Jakobs: Conceptualization, Methodology, Software, Writing - review and editing Giuseppe Cardellini: Conceptualization, Methodology, Writing - review and editing, Supervision, Funding acquisition Niklas von der Assen: Writing - review and editing, Supervision, Funding acquisition Jeroen Guinée: Writing - review and editing, Supervision, Funding acquisition Bernhard Steubing: Conceptualization, Methodology, Writing - review and editing, Supervision, Funding acquisition Funding This study received funding from the European Union’s Horizon Europe Research and Innovation Programme ForestPaths (ID No 101056755) and from the Eidgenössische Technische Hochschule (ETH) Board in the framework of the Joint Initiative Swiss Center of Excellence on Net Zero Emissions (SCENE). Code availability The source code of the software bw_timex can be accessed via the GitHub repository at:https:// github. com/ brigh twaylca/ bw_ timex. Extensive documentation of bw_timex is available at:https:// docs. br igh tway. dev/ proje cts/ bwtimex/ en/ latest/. The EV case study notebook can be accessed at:https:// docs. brigh tway. dev/ proje cts/ bwtimex/ en/ latest/ conte nt/ examp les/ paper_ case_ study. html. The notebook for benchmarking calculation time can be accessed at: https:// github. com/ brigh twaylca/ b w_ timex/ blob/ main/ not eb ooks/ run_ time_ test_ bench marki ng. ipynb Declarations Conflicts of interest The authors have no competing interests to declare. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 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