Circularity and LCA - material pathways: cascade potential and cascade environmental impact of an in-use building product

Improving circularity in the building sector entails ensuring greater material efficiency to avoid virgin material extraction. To assist stakeholders in decisions regarding salvaging an in-use building product, requires to predict and assess the potential further productive uses of that product and its materials. The range of possible cascade material paths originating from the in-use building product X and their assessments comprise the cascade potential of product X. Method: To determine the cascade potential and impact, we work further on existing efforts done in the field of circularity and life cycle assessment (LCA). This entails discussing scenario models to predict cascade material pathways over time, and multifunctionality solutions to assess those pathways. Due to the fact that the environment is a complex system and long term forecasting is required, the cascade potential can never be exactly determined. Therefore, we first set up conceptual formulas and then discuss steps to make these formulas feasible. Furthermore, the effort to generate the cascade paths originating from a product, can also be used to form circular systems that adhere to carbon mitigation pathways.


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In appendix 3, the following text appears: "The following formula utilizes substitution to model usage duration benefits in the case that temporal differentiation is simplified, and the stakeholder requires a single LCA score. : [55]. , ℎ ℎ ℎ ℎ [30]. : Reference service life of that product, according to EN15804, or average life of a virtual representative product." The previous text and formulas should be replaced with: When only accounting once for the substitution that a salvaged component can do, market value can be used as a factor to indicate quality loss when comparing the virgin component to the salvaged component [53]. However, when applying repetitive substitution, i.e. considering all substitution effects caused by reusing a single salvaged component multiple times, an alternative quality factor, partially based on usage duration, may be found. This latter case is here discussed. For each product Q, that holds a component with mass Mi originating from product X, the substitution effect caused by using this salvaged component with mass Mi in product Q can be accounted for by utilizing formula F4a. Formula 4a is an alternative to Formula 3.

: = − ( ) •
EIRstrategy Mi for product Q : environmental impact arising from the R-strategy processes to get to the point of substitution resulting in a mass Mi, which is able to substitute a component in product Q. These processes may include disassembly, sorting and transformative processes. : actual usage time of mass Mi in product Q, which may differ from the market average.
: Reference service life of the substituted product or component, according to EN15804, or average life of a virtual representative product. EIcomponent substituted by Mi : The environmental impact by substituting marginal usage [55] accountable to the component substituted by Mi. Or, it is the environmental impact of the production phase of the component that has been substituted by Mi in product Q. We remark that, in this EI term, other aspects of quality difference between Mi and the component it substitutes, complimentary to usage duration difference, may still be considered. We note that due to changes in technologies over time, the EI caused by substitution can change over time [55]. [12] F. Suter [55] S. Schaubroeck, T. Schaubroeck, P. Baustert, T. Gibon, and E. Benetto, "When to replace a product to decrease environmental impact?-a consequential LCA framework and case study on car replacement," Int J Life Cycle Assess, May 2020, doi: 10.1007/s11367-020-01758-0.

Circularity
The initial outset of circular economy (CE) is to develop an economic environmental system in which waste is turned into resources to achieve a constant or increasing stock of natural resources [1]. This can be done through a technological and natural ecosystem feedback mechanism [1]. Seeing as this effort could regenerate nature, circularity is considered to be one of the pathways to sustainability [2]. To improve circularity, a paradigm shift is needed in society. Hereto, circular strategies, such as rethink, re-use, repurpose and recycle, need to be adequately implemented and combined [3]. These strategies generate a system where resources are used over multiple product cycles, i.e. cascaded [4], and in which additional interdependencies are formed. The resulting system of flows is complex, which entails that a part of the system can never fully represent the whole [5]. To manage circular material flows, a combination of tools is thus needed that study different scales, time horizons and outsets. Scales can, e.g., be geographical. The required interplay between models on micro and macro scale has been studied in many fields, e.g., in logistics and socio-economics. Per scale, a different type of monitoring and data is available [6]. In the building sector, different scales are the scale of the building stock, building and building product. The outset of this study is to predict and assess the impact of possible further uses of in-use building products and their materials. Hereto, different scales are to be understood as discussed in the next section. We refer to this potential of different possible further product uses as cascade potential. In existing literature the cascade potential is referred to as the reuse potential [7], [8].
Cascades have been assessed by utilizing dynamic Material Flow Analysis and Life Cycle Assessment [9]. In LCA, cascades have been predicted and assessed of wood [10], [11], [12], [13], [14], of flooring [9], of water, of plastics, etc. We work further on these efforts to develop a framework and formulas to determine the cascade potential. Data on the possible further uses of products , and the environmental impact, can assist stakeholders in disassembly and design decisions. We note that, to predict further uses of a product, a database is needed, which is addressed in a complimentary article [15].

Cascade potential in the building sector
In the building sector, cascading products is one of the strategies required to adhere to decarbonization pathways in the building sector [16]. Other complimentary strategies are: utilizing certain bio-based materials, extending usage of buildings, and changing building utilization patterns [16]. Studying cascades is needed for modeling on different scales in the building sector, such as on the scale of the building stock and on building product level. On the scale of the building stock, studying past or possible cascades of building products may reveal material flows that cross sectors. For example, metal products retrieved from cars may be used in the construction sector [17] or concrete debris from the housing sector may be used in the transport sector [18]. Extending the usage duration of concrete houses, thus affects resource usage in the transport sector. The carbon budgets of each sector can thus not be set separately, and iteration is needed to set the budgets for connected sectors. Once a decarbonization pathway is set for the building stock, an optimal cascade pathway can then be distilled for each product present in these buildings. This pathway indicates how long products should be used and whether they should be recycled or repurposed several times in other buildings and where. For example, excess steel scrap from demolished buildings can be recycled and used in the Global South to lower GHG emissions [16]. A possible additional intermediate step to achieve decarbonization of the building stock, is to allocate the carbon budget of the building stock to each building [19], [20]. Once a top-down derived pathway is set for a building product, info on this desired sustainable pathway can guide stakeholders, such as contractors, in their disassembly decision. However, in the case that info on such desired pathway is missing, a range of possible further usages of the product and their impact should be compared to aid the decision taker. Furthermore, if a desired product pathway is solely calculated based on global environmental targets, this pathway might not be financially desirable for the contractor. If the stock model does not consider sufficient data, the derived desired product pathway might not even be possible given available technologies. Comparing possible and desirable cascade choices on micro and macro scale, will lead to updating or rethinking models and scenarios. Comparisons of cascade choices to desired cascade choices can be interpreted as benchmarking cascade choices.
Studying the cascades of an in-use building product is crucial on different scales. In order to address the cascade potential, this paper consists of several sections. First, an overview of circular product strategies is provided. These strategies generate product pathways. Second, to predict and assess product pathways, we discuss LCA of building products and scenarios. Third, cascade potential and impact are discussed in their specificity. Fourth, applications of the cascade potential in the building sector are presented. Finally, the feasibility of the cascade potential formulas are discussed.

Circular strategies on building products
In the building sector, circular strategies can be applied to a whole building, building spaces or to components [21] [22], and these strategies can be anticipated during the design of a building to ensure value retention. For example, design for disassembly can be applied to ensure that products can be reused. Several circular strategies can be applied to products, which generates product cascades. A wellknown hierarchy of circular strategies is the 10 R's: refuse, rethink, reduce, re-use, repair, refurbish, remanufacture, repurpose, recycle and recover [3]. This list is explained in figure 1. Note that, the hierarchy is merely a rule of thumb [3]. Each circular strategy can entail a range of actions. For   In the case of wood, the bigger the shape of the resulting pieces after applying actions, such as cutting, the higher its quality because the wood can still be cut in smaller pieces afterwards and cutting wood is an irreversible process [23]. Note that, this quality maximization is only one type of value maximization applied in the CE [4]. Value is also dependent on sufficient demand, and thus business strategies are important in circularity. The 10 R's can further be categorized in open and/or closed loop strategies. Open loop strategies, such as repurpose, will use a product's materials in products with a different function or assembly. In figure 1, a representation of the single life cycle of a building product is shown with circular strategies, and simplified in figure 2. During the life cycle phase of production, circular material flows can serve as input. After usage of the product, circular material flows can serve as output. The latter requires disassembly of the product. Depending on a building product's properties and its assembly characteristics, certain disassembly types will be possible [22], [24]. For example, in the case of a soft material (e.g., insulation) glued to a hard material (e.g., concrete) the former will most certainly break during disassembly. Different types of disassembly (e.g. destructive or non-destructive) can render a product and the surrounding products either undamaged, damaged but repairable, damaged beyond repair or damaged beyond repurpose [25] [26]. After disassembly, sorting takes place, during which, quality of materials and product states may be mixed thus determining potential future uses [27]. To summarize, depending on product properties, assembly characteristics, disassembly type and sorting, certain circular strategies will be possible after product usage. When applying disassembly and circular strategies multiple times on an in-use building product and its materials, different material pathways can be generated. For example, a part of wooden boarding can be cut and used in furniture, and the other part can be chipped and then turned into particle board which in turn can be cut. These paths go until the material is transformed into a different substance, e.g., energy. Each possible set of material pathways originating from an in-use building product forms a cascade tree graph. See the green striped lines in figure 4, which depicts a cascade graph that occurs over time.
A group of products that cascade materials between them, can be viewed as a circular system of stocks and flows (figure 3). Ample data is needed to model and predict technically possible cascade tree graphs and circular systems. Information is needed on a products technical lifetime, on how each strategy may alter material properties (e.g., recycling of wood, plastic or steel [28]) , and possible uses for this material given its changed properties [15]. Several aspects discussed in this section, (circular strategies, product quality, disassembly) have been studied in Life Cycle Assessment (LCA). Therefore, in order to assess the cascade tree graphs originating from an in-use building product, we further discuss LCA.

State of the art of product environmental LCA
Environmental LCA (E-LCA) is a method for quantifying environmental impacts and trade-offs among goods and services considering their life cycles [29], thus informing stakeholders on the impact of product choices. An LCA can assess the impact of single or groups of products [30], and is thus relevant to our question to predict and assess the impact of cascades of multiple products. In general, to perform an LCA study, the ISO 14040-4 standards set out several steps. See figure 5 for a modified version of these steps. The goal and scope of the specific LCA study is firstly expressed, which contains a timing and amount of the goods and services. For example, what is the environmental impact of a possible cascade tree graph originating from a wooden board used in 2022 ? Secondly, a functional unit, which quantifies the product related change or performance, can then be related to the goal and scope [30], [31]. For example, considering a cascade of wood, this can be x m² wooden boarding in a wall, then y m² particle boards for a closet, and z m² of paper. We note that these quantities can vary over time in circular systems [11]. Starting from the functional unit, processes are then selected following mass or energy flows, e.g., to produce a particle board, wood needs to be chipped. These processes can be categorized according to the life cycle phases they occur in: usage, production or end-of-life. To predict which processes will occur at which location, a (scenario) model is required. For example, a model may predict in which country the production of particle boards will occur. Furthermore, boundary rules for the product system are needed, because certain processes might serve multiple products, e.g., oil can be used for multiple products. To address this multifunctionality issue, several approaches exist [32]. See appendix 1. The resulting collection of industrial processes is referred to as the product system [33] [34]. The inclusion of processes occurs until it reaches elementary flows that impact three areas of protection: Human Health, Natural Resources and Ecosystems. The quantification of this impact is referred to as Life Cycle Impact Assessment (LCIA). It is then possible to compare the impacts of products to each other, or to a treshold value. Examples of threshold value are setting a carbon budget for each product [19] or a Safe Operating Space for each product. The latter sets targets within which a product should stay to remain within planetary boundaries [35]. Given our aim to determine the cascade potential of a product, the following aspects of LCA require further attention: the prediction of material pathways that form cascade tree graphs, and how to assess the environmental impact of these cascade tree graphs.
Prior to this, we discuss European LCA standards for building products. In the building sector, the European standard EN15804+A2(2019) [36] defines specific rules for the LCA of construction products following a modular approach. This standard indicates the environmental impact to be declared for each life cycle stage. Module D of this approach holds a formula to account for further usage of a product and its materials after its end-of-life. Aside from this standard, the Product Environmental Footprint (PEF) [37] method is a general assessment method, for all products, developed by the European Commission and includes detailed methodological LCA rules. In the PEF, to account for end-of-life multifunctionality, the Circular Footprint Formula (CFF) [38] [37] is proposed. To assess the environmental impact of buildings as a whole, the EN15978 (2011) standard [39] can be followed.

Timing of processes
To predict cascade graphs of building products, requires data on how material pathways and processes of cascades occur over time. See figure 4, for a cascade To differentiate processes and their impact over time in an LCA is referred to as dynamic LCA [40], [41]. This data on the timing of cascades is required to compare cascades, and to assist in invest and return decisions. For example, consider the decision to invest extra material to ensure a reversible connection in a structural element, such as a concrete beam, to enable its further use. This decision will only lead to a reduction of material use when this structural element is disassembled decades later. Additionally, understanding the timing of cascades is important to understand how long it will take before all products in a cascade have been affected by a change [9], [11]. Changes in supply to one product of a cascade system of timber products will require some time before it affects the entire system [8]. See appendix 2, for more on time differentiation. We note that in a stock and flow model of a cascade timber products, the forest growth rate should be in line with the demand for timber products in the future. Hereto a prospective model is required.

Scenario models and integrated assessment
To predict processes over time, requires an adequate prospective model [42], [43]. Utilizing a model to predict future processes and impacts in LCA is referred to as prospective LCA. The chosen prospective model should be in line with the goal, scale and time horizon of the LCA study [44]. For example, changing the further usage of a relatively large group of building products, can cause a large scale effect in an economy thus requiring a model that can assess this change. Seeing as the industry and environment form a complex system [5], predicting material pathways over decades is beyond the capability of pure deterministic models to predict accurately [45]. Scenarios are thus needed to work together with the models [43]. A scenario in LCA presents the development or pathway from the current situation to a future situation by relying on a specific set of assumptions [45]. A scenario provides a narrative in a prospective model. These scenario models will determine the occurrence of future processes and impacts, and thus determine the product system and impact assessment of cascades in LCA. See figure 5. Furthermore, the scenario used for assessment of cascades and for setting carbon budgets of cascades, if applied, should be the same. This is necessary to ensure that comparison is possible between budgets and impacts. We here shortly discuss existing prominent scenarios and models. On prospective models: In existing LCA studies many economic models have been used to predict the occurrence of processes. Examples of economic forecast models used in LCA are stock-flow consistent models, agent based models, supply and demand analysis and computable general or partial equilibrium models [44]. Each model predicts, to some extent, human decisions. Due to our rising impact on planetary systems, recent work has expanded the scope of prospective models to consider socio-economic climate models. These models are referred to as Integrated Assessment Models. [46] IAM models differ in the extent to which they combine economic complexity and climate/environmental complexity. "IAMs model the long-term effects(time horizon usually until 2100) of human activities and the natural environment of prospective scenarios following various future socio-economic and climate change narratives." [47] On scenarios: A prominent example of scenarios are the Shared Socioeconomic Pathways, which set how society will act in the future (e.g. will certain policies be introduced that ensure longer product use) [46]. These can then be combined with Representative Concentration Pathways (RCP) that set pathways to stay under a certain degree of temperature change [46]. To conclude, within each scenario model, a certain set of cascade tree graphs are possible. For example, in a scenario where politics set strict rules on which fractions to reuse, certain cascade paths of materials will be possible. We note that, ecoinvent, i.e. a prominent LCA database, applies for most data a Business As Usual Scenario, which entails that ecoinvent projects average past data or sets marginal market behavior to predict process occurrence in the present [48]. The applicability of that data, without modification or setting additional assumptions, is bound to BAU scenarios. In general, we remark that scenario models should be quantitative and qualitative evaluated [49].

Framework and cascade potential
Based on insights of previous sections, we have set up a framework to predict and assess the cascade tree graphs and thus the cascade potential of an in-use building product. See Figure 5. This framework consists of a goal and scope (top), data input (left column), a scenario model (centre column) and evaluation (right column). An iterative workflow between data input, modeling and evaluation is required to find those cascade tree graphs possible per scenario. This iteration consists of generating cascade graphs according to scenario model and data input, and filtering which are possible within the bounds of the (scenario) targets. To extensively discuss iterative workflows is out of the scope of this paper, but a first possible step is discussed in section 5.2. The cascade potential of a product P per scenario S is then the set of possible cascade graphs allowed within the bounds of this scenario and adhering to targets. See description 1. For example, in case the scenario is an RCP, a constraint is then a maximum Greenhouse Gas concentrations. The evaluation of each cascade graph can thus be according to environmental budgets or they can also be evaluated according to its characteristics, such as resource efficiency. See section 5.3. The cascade potential can thus also be seen as a set of scores, with each score connected to a possible cascade tree graph. See description 2. A set of scores and graphs can be represented by an average or by a maximum and minimum value. We note that, each cascade graph, can be seen as a set of cascade paths,. See description 3. In the building sector, to find optimal cascade graphs, first a decarbonization pathway can be set on a building stock scale to derive optimal cascade graphs on building product level, as discussed in section 1.2.

Technical cascade potential
A first possible step to determine the cascade potential, is to generate all technically possible cascade trees of an in-use building product, i.e. technical cascade potential of a product. To form the technical cascade potential entails generating all the material paths of an in-use building product that are possible, not considering supply or demand constraints, but only considering technical limitations ,such as technical usage duration, and certain technologies. Those technologies can be technologies available within a certain geographic scope [7]. In the resulting network of technically possible paths, several graphs can then be modelled and checked if they are possible within the scenario. We note, that the network of possible paths might be too big to be considered, and can be narrowed by first determining parts of paths and their probability (in a short time horizon) with (scenario) models.

Product system characteristics
Prior to discussing the environmental impact assessment of graphs, we discuss their characteristics. These characteristics can also serve as scores to find optimal graphs. We only discuss a limited set of characteristics. A first characteristic is the resource loss over time per cascade path or tree graph. For example, incinerating wood after a single usage, entails a fast loss of the resource. An existing circularity indicator, i.e. the re-use potential [7], can be viewed as a simplification of this characteristic. The re-use potential determines, based on available technologies in a specific area, the mass percentage of a product that can be re-used or recycled [7]. This indicator, however, does not take the aspect of rate of material Examples hereof are (i) the amount of paths in a cascade graph that hold open loops or closed loops, (ii) the (amount of ) industrial sectors or geographic regions which the cascade graph crosses. These characteristic are important to determine system dependencies of the cascade. For example, a cascade graph that only occurs locally, will be strongly dependent on local supply and demand. Another possible characteristic is entropy change over time of cascade graphs [23]. This has been calculated utilizing Statistical Entropy Analysis (SEA) for cascading of cars [50] and wood products [51]. Comparing the entropy change of each cascade graph, can reveal which cascade graph provides the slowest gain of entropy and could keep materials in the loop longer. We note that SEA does not consider aspects, such as the environmental impact of the energy mix utilized, as is done in LCA [52]. A similar observation can be made for resource efficiency and system dependency characteristics. This entails that these characteristics can not replace life cycle impact assessment as an evaluation tool for cascade graphs. However, for example, in the case that data is missing for impact assessment, these characteristics might provide a faster way to find desirable cascade graphs. To better understand the role these characteristics can play, the exact interaction between LCA and these characteristics needs to be researched. Hereto, LCA and SEA results can be compared [14], or SEA could be integrated directly into LCA. Regarding integration: in LCA quality change due to, e.g., recycling, is modelled in the end-of-life multifunctionality formulations [53]. However, by comparing entropy scores of cascade graphs, relative entropy scores could be formed, which could be used as an additional quality factor in the end-of-life multifunctionality formulations. Including this additional quality factor in these formulation entails that more environmental burdens would be allocated to products with an end-of-life choice that no longer allows for high quality retention cascade paths. However, merely integrating entropy of further product paths, still dismisses the impact assessment of these further paths.

System expansion merely as starting point
In this section, we discuss the environmental impact assessment of cascade tree graphs of an in-use building product P following a scenario model S. Each time a material of product P is cascaded through circular strategies, such as repurposing or recycling, that material is utilized in several product life cycles, and thus serves multiple products. Several options exist to deal with multifunctionality [13], [32], [34], [53]. Seeing as we aim to consider the complete potential burdens and benefits caused by materials of product P, system expansion is one of the possible multifunctionality solution for those occurrences. In the next section, we will discuss modifications to this. We note that, linearky degressive [32], [54] is not chosen as an approach, because it does not consider all burdens associated with the cascade of materials. Nevertheless, it may be relevant in certain situations. See appendix 2. Choosing system expansion as a starting point entails the following: we consider system expansion each time a material of product P is cascaded, thus forming a single product system containing all the products and their functions that utilize a material of product P. This occurs up and until the materials originating from product P are turned to energy or until they are landfilled. For all materials, aside from those originating from product P, the same multifunctionality solution should be chosen. We note that this encompassing product system of the multiple cascade products, can also be interpreted as a set of smaller product systems. Each smaller product system than adheres to a single product that utilizes a material of product P. These product systems should then apply the same multifunctionality solution consistently. For example, for all products in the cascade tree the PEF CCF method could be chosen [13]. Seeing as further operations on these approaches are possible, e.g. sequence of substitutions [14] , we will set a starting formula and then discuss these changes.

Modifications to the starting point
In this section, we set up a starting formula and then discuss modifications. Consider a cascade path starting from product P according to scenario model S that contains two products: product X and product Y. Product X utilizes material part M1 of product P , this material M1 is then reused in product Y in a smaller fraction M1' due to recycling. Each product also consists of other materials. Product X, with a mass of Mx, consists of material M1 and material M2. Product Y, with a mass of My, consists of material M1' and material My. The environmental impact (EI) of this cascade path is then expressed in formula 1. For the sake of simplification we consider three life cycle phases: production (PROD), usage (USE) and end-of-life (EOL). The EI of each cascade product can e.g. be calculated according to the PEF 50:50 [13].
This ideal formula requires that processes and impacts are considered over time and that a scenario model is utilized to predict the occurrence of these processes. This formula takes all the impact of the materials present in product X and product Y into account, even if they do not originate from product P. This approach can help to pinpoint those products in the cascade sequence that have a high negative EI, even due to materials not originating from product P. This can help pinpoint lock-ins, i.e. situations in which the need to cascade materials forces users to utilize products with a high negative EI. However, this also means assigning a significant impact to the cascade disproportionate to the amount of material of product P that it is in the cascade. To alleviate this, we can propose to utilize mass allocation. See formula 2. This is a common multifunctionality solution in LCA [33], and is here applied to the cascade.
Aside from mass allocation, other alternatives to account only for the impact of materials, originating from product P, exist. The EUI of the cascade path can be calculated as the difference between (a) the EI calculated with the presence of the cascade material originating from in-use product P, as mentioned in formula 1, and (b) the EI calculated where the cascaded input material from in-use product P is replaced with virgin or alternative material input. A further simplification is to only account for material substitution effects that occur when the cascaded material replaces virgin or alternative material input, and to ignore the impacts occurring during the use phase altogether. This is expressed in formula 3. This formula of repeated substitution has been utilized to determine a wood cascade [14]. It has been argued that only the function of the original product P should then be accounted for [14]. We argue against this. It is important to consider the functions of all products, containing materials from product P, when comparing cascades. Otherwise you can compare a cascade of product P that has been cascaded two times (products X and Y), to a cascade that has occurred three times (products A,B and C), without realizing this difference. Comparisons without mentioning sufficient cascade data should be avoided.

Challenges in LCA when simplifying time
The formulas of the previous sections, all consider processes over time. Hereby, they thus also take into account usage duration and thus the circular strategy of extending usage. However, in certain cases time differentiation is not possible. This can be due to lack of data, or if the stakeholder desires a single score to represent the EI, instead of an impact over time. Furthermore, simplification of time differentiation is often utilized to compare products (or cascades) with different usage duration [55]. In such events, additional measures should be taken to perform this simplification, but we stress that these can never fully remediate the loss of data when dismissing time differentiation. An example of such a measure to  [12]. Other possible measures to simplify time differentiation is to utilize allocation or substitution. For example, these can be utilized to account for usage duration [55]. The conditions for allocation to be acceptable are discussed in appendix 2. Substitution to account for each usage instance, can be modelled according to the formulas presented in appendix 3.

Challenges in LCA in comparisons and substitution
In general to compare products is a challenge in LCA, because functions of products are never the same [55] and the usage duration of different products can differ. For the latter we have discussed in the previous session that allocation and discounting or substitution can hereto be used, but these remain simplifications that are only valuable under certain conditions. We stress that when modeling substitution the choice of the substitution product significantly affects the EI [14]. To compare cascades, several options are available. An option is to ensure that when comparing a similar group of functions are compared over a similar time horizon. For example, if we compare the cascade potential of a wooden beam, to that of a steel beam, then we need to ensure that the functions provided over time are the same in both systems, which might entail that for one of the two products, additional products outside of the cascade sequence will need to be considered. Another option is, that if a comparison is made between cascades with different time and products, data on these differences should be mentioned, and simplification of time differentiation, as mentioned in previous section can be applied.

Simplified cascade paths.
To apply the cascade potential in practice, simplifications might be considered. For example, while the cascade potential can consist of a set of possible cascade tree graphs, we can propose to limit this set to four possible technical cascade paths: the reuse path, the repurpose path, the recycle path and the incineration path. The reuse path consists of choosing the option to reuse first, and if no reuse is possible, opt for repurpose while considering longest technical usage duration. The repurpose path consists of choosing the option to repurpose for each cascade, while considering longest technical usage duration. The recycle path consists of choosing the option to recycle for each cascade, while considering longest technical usage duration. Incineration consists of incinerating the product. These four paths might provide a fast way to characterize a cascade potential.

Disassembly potential.
In this section, we discuss the application of the cascade potential when performing disassembly. Each different disassembly and sorting option (see figure1), results in a different product state and thus a different cascade potential and impact can be returned. In other words, for each investment of time, labor and energy related to a certain disassembly and sorting, a possible return of EI savings due to cascading can be provided (see description 4). This data helps stakeholders in their decision whether to disassemble or not. We note that building products are often part of an assembly, so any attempt to salvage it for re-use or recycle, may damage the product itself and other elements connected to it [22]. the impact according to the usage duration of each product per building compared to the product's total usage duration. However, the cascade potential also accounts for reuse of the in-use building product. Seeing as we should avoid double counting impact, how to handle this overlap still needs to be addressed.

Discussions and further outlooks
A further step to determine cascade potentials and cascade impacts consists of setting-up a cascade database containing data for each product on which other products can utilize its re-used, recycled or repurposed parts, and the recycling efficiency and quality loss caused by each circular strategy and technology. The formula developed in this study should be applied to in-use building products to test their validity. Furthermore, we stress the importance of looking beyond single product LCA studies, seeing as a part of the system can never represent the whole. The efforts to find the right combinations of tools to reach sustainability are as important as harmonizing the efforts of single product LCA.

Appendix 1
The chosen multifunctionality approach should be in accordance with the goal and scope of the study [33]. Introducing circular strategies, such as repurpose or recycle, entails that materials are utilized in several product life cycles and their functions. This thus generates a multifunctionality issue. The question is then how to assign the burdens and benefits of these circular processes to each or a selection of the products involved [53] [32].Various end-of-life allocations are being used in LCA studies to assign the burdens and benefits of recycling or reuse over the various related products. Some only take the material pathways into account between the product under study and its directly succeeding and preceding product, as is done in the CFF of the PEF. Other methods take the pathways of materials over multiple cycles (and products) into account, such as system expansion [33] and the linearly degressive method [32]. The former has for example been applied to assess a cascading systems of timber products [11]. Depending on the multifunctionality solution, quality loss of materials due to circular strategies several solutions might be modelled differently [53].

Appendix 2
Processes in a product system all occur at different times, however only under certain conditions it is possible to consider these processes as all flowing simultaneously. This case only holds true when a significant amount of products are considered, and at the moment of study there is an optimal spread of lifetimes. This simplification is, e.g., present in the ecoinvent database [48]. This simplification is also utilized in practice in some studies when applying allocation of processes. For example, in studies that utilize the linearly degressive [54], the impact of an end-of-life process of steel, that is cascaded four times, is allocated to the end-of-life of the product under study. However, there is a huge time difference of four steel cascades between these processes. This dismissal of time is only acceptable under the conditions mentioned in the beginning of this paragraph.

Appendix 3
The following formula utilizes substitution to model usage duration benefits in the case that temporal differentiation is simplified, and the stakeholder requires a single LCA score.