Introduction

Buildings, in which people of developed countries spend most of the time, provide ecological niches with varying nutritional and environmental conditions, which typically result from oscillations of ambient boundary conditions, actions of inhabitants and operation of technical systems. The biological activity of living organisms may cause a number of damage mechanisms ranging from the production of pigments (aesthetic issue) to the mechanical action of the biomass that colonizes the surface of materials (physical issue). Mold growth is one of the possible types of damage caused by moisture (Berger et al. 2015).

Wood is one of the primary building materials that has been used for thousands of years. Today, wood and wood-based materials are of increasing popularity and importance in the building industry and are becoming an important element of environmentally friendly and sustainable buildings. The current situation shifts from the preservation of wood by chemicals to control of hygrothermal, chemical and biological loads in the natural limits of wood. In principle, this requires thorough knowledge of moisture dynamics, correct setting of limits for the safe use and using reliable assessment procedures. The sufficient durability of wood is required to achieve the goal of sustainable constructions.

Molds are the initial microbial colonizers of wood. They do not significantly affect the strength of wood (Viitanen 1996). Problems caused by molds are unpleasant appearance of the surface and smell. The long-term presence of molds in buildings can have an impact on the health of inhabitants (World Health Organization 2009). The presence of molds also indicates that moisture state of building components is problematic. Therefore, suitable mold growth metrics, either predicted by mold growth models or observed on site, could be used as a criterion for hygrothermal assessment of building components.

The life cycle of molds consists of several stages: (1) spore germination, (2) growth (mycelium growth, sporulation) and (3) decline; see Fig. 1. The decline of mold in the last stage represents possible successive dying out due to long-term dry conditions, or due to the lack of nutrients, or due to unfavorable temperature, when mold coverage fraction might be quantified unchanged but the decay of sporangia is observed.

Fig. 1
figure 1

Life cycle of molds in stable environmental conditions: a schematic representation of hypha inspired by Ruijten et al. (2021)

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The onset of mold growth on a substrate requires a suitable combination of relative humidity, temperature, sufficient exposure time and available nutrients, non-structural wood components such as low molecular carbohydrates (Viitanen 1996). Mold growth takes place over a wide range of temperatures, with the reduction in growth rate at temperatures below 10 °C (Johansson et al. 2013). In the case of pine sapwood stored at a standard laboratory temperature of 23 °C, the lowest relative humidity of ambient air for onset of mold growth is reported to be 0.75–0.80  (Johansson et al. 2012). The time required to develop visible molds (by naked eye) on pine sapwood under favorable stable environmental conditions is in the order of weeks (Viitanen and Ritschkoff 1991; Johansson et al. 2013).

Under varying environmental conditions, molds can exist on a surface either in growing or non-growing (vegetative) state. If environmental conditions change from dry to moist, the mold uptakes moisture and can multiply itself. If the conditions change from moist to dry, the mold loses water and shrinks, i.e., stops growing. When humidity increases, regrowth starts again for the part of molds that survived the dry period. Transitions between both states occur at rates that generally depend on time and environmental conditions. The reaction to unfavorable conditions is expected to be fast (within one hour). Rehydration after an increase in humidity might take longer, but mold is still expected to react quite quickly, in the order of hours. Activation and deactivation speeds can change from generation to generation as a consequence of mold history, as discussed in Ruijten et al. (2021), and may also differ between fungi species.

Methodology of obtaining the quantitative data of mold growth metrics by experiments is a problematic part of mold growth research. Mold growth in laboratory conditions depends to some extent on the definition of the experimental protocol, i.e., on mold species used in inoculum (different growth kinetics of mold species), initial spore concentration, initial physiological state of spores and distribution of inoculum over the substrate (Vereecken et al. 2015). Mold growth data are often the result of the subjective evaluation of laboratory workers performing microscopic and visual observations. Different results might also be obtained for the same mold growth metric at different scales (observation by naked eyes × observation by magnifying glass × observation by a microscope) and for different species of fungi. The choice of mold growth metric is probably more related to the historical context and the available equipment of the laboratory performing microbial investigations rather than to the assessment of moisture safety and mathematical formulation of mold growth models.

Various mold growth metrics are used in the scientific literature to evaluate the extent of mold growth on material over time. Metrics may take the form of dimensionless mold indexes, for example, expressed as 7-degree scale (Ojanen et al. 2011), 6-degree scale (Lie et al. 2019b) or expressed as 5-degree scale (Johansson et al. 2013). The metrics may also be constructed to have physical units, for example, growth expressed in mm/day (Sedlbauer 2001) or growth expressed in m2m−2 as mold coverage fraction (Nielsen 2004). The definition of mold growth metric influences the interpretation and usability of mold growth experiments and mold growth models. It is not yet resolved whether or not unambiguous transitions between different mold growth metrics may be established to enable quantitative comparison between different experimental or modeled mold growth data.

The mold-related definition of the limit state for moisture safety of building components will influence the choice of the mold growth metrics and also impacts the formulation of the mold prediction model. For instance, if spore germination is considered as the limit state, the mold prediction model must be accurate at the first stage of the fungal life cycle and will not be so important to capture the dynamics of mycelium growth. On the contrary, if it is needed to predict the mold development in building components over several years, a mathematical model of fungi across all life stages is needed to incorporate into the model. In this case, the mold growth model should be capable to switch between microscopic and macroscopic scales and represent processes that are very different in nature (germination, growth, dying).

The Finnish mold growth model (Hukka and Viitanen 1999; Ojanen et al. 2011) is an empirical model based on experimental results. Although the mold development is expressed by a single differential equation, insight into the performance of the Finnish mold growth model is difficult to obtain due to the number of model parameters and interconnections between parameters. To our best knowledge, the optimization procedure leading to setting the published values of model parameters is not presented in the scientific papers. The mathematical formulation of the Finnish mold growth model is recapitulated in Online resource 1. The bio-hygrothermal model (Krus et al. 2007) extends the combined isopleth model (Sedlbauer 2001) for defined substrate classes by the dynamic moisture balance of the spore. The spore is here treated analogously with building material to calculate moisture content inside the spore, i.e., the spore is defined by a moisture retention curve and a moisture-dependent vapor diffusion thickness. Both mold growth models have already been incorporated in existing hygrothermal simulation tools as add-ons (WUFI-Bio 2022; WUFI-VTT 2022).

The dynamic mold growth models are used in practice to postprocess and interpret hourly values of temperature and relative humidity calculated by dynamic heat and moisture transfer models of building components. It is supposed in such calculations that molds do not have an impact on surface hygric conditions. In practice, mold growth models were used to assess building components with doubtful moisture performance (Alev and Kalamees 2016; Klõšeiko et al. 2015; Harrestrup and Svendsen 2016), to interpret measured data in real-scale experiments (Glass et al. 2017; Kopecký et al. 2019) and to predict the presence of molds using recorded temperature and relative humidity of ambient air from the large number of British households (Menneer et al. 2022). Mold growth models were also used to predict fungal colonization of historic timber structures in the future climatic conditions (Choidis et al. 2021). The common weakness of such mold growth predictions is that they usually do not account for uncertainties involved (Gradeci et al. 2017), such as the uncertainty due to the limited mathematical representation of the complex biological system, and the uncertainty associated with environmental boundary conditions.

Veerecken and Roels (2012) reviewed existing mold growth models. They found large differences between the models even in the simple cases of steady-state environmental conditions. Discrepancies between the models and the data from laboratory experiments have also been reported (Veerecken et al. 2015). In contrast, Lie et al. (2019a) concluded that the Finnish mold growth model was fairly reliable in predicting mold growth on wooden claddings, if surface boundary conditions were used as model input. The agreement of the Finnish mold growth model with field observations was also pointed out in Glass et al. (2017).

The reliability of the Finnish mold growth model was thoroughly and explicitly evaluated in Berger et al. (2018). It was concluded that the mathematical formulation of the model is not reliable due to the oversensitivity of the mold index to subtle changes in model parameters. The oversensitivity could cause problems when one attempts to fit the model over experimental mold growth data. Berger et al. (2018) therefore proposed a mathematically more robust model. The suggested model was based on the logistic differential equation (Verhulst 1845) which is often used to model growth phenomena in mathematical biology.

The objective of this paper is to investigate the logistic differential equation for modeling the time development of mold coverage fraction on the surface of wood, including the investigation of the relationship of model parameters with environmental boundary conditions. Due to the lack of high-quality experimental mold growth data, the unknown parameters of the logistic mold growth model were determined by mutual comparisons with synthetic mold growth data calculated by the current state-of-the-art Finnish mold growth model. The Finnish mold growth model was selected because it is based on extensive mold growth experiments with wood. Steady-state boundary conditions were used to estimate relations between boundary conditions and parameters appearing in the logistic mold growth model. The logistic mold growth model was verified by a comparison with the Finnish mold growth model using cases with cyclic boundary conditions. The logistic model was also compared with experimental observations of mold growth selected from the literature. The future development of mold growth models is then discussed, and finally, conclusions are drawn.

Logistic mold growth model

Relations between mold growth metrics

Relations of the mold index according to Ojanen et al. (2011) with other mold growth metrics (mold rating, mold coverage fractions) are summarized in Table 1. During the incubation period of a laboratory mold growth experiment, mold coverage fractions are primarily noted. Microscopic mold coverage fraction is estimated by microscopic observations (typical magnification ×40 to ×100). Visual mold coverage fraction is estimated from photographs (typical magnification ×10). Mold coverage fractions are usually translated to dimensionless mold indexes to obtain one scale. The mold growth experiments, observation and evaluation procedures are not standardized, and vary between laboratories.

Table 1 Relations of mold index with other mold growth metrics
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Based on the information in Table 1, both the microscopic and visual mold coverage fractions can be converted from the mold index defined in Ojanen et al. (2011). The logistic function (1) can be used to define the transition:

$$C=\frac{1}{1+\mathrm{exp}(-a(M-b))}$$
(1)

where C [m2m−2] is mold coverage fraction and M [–] is mold index. Parameters a = 2.02 and b = 4.43 correspond to visual mold coverage fraction defined according to Ojanen et al. (2011), respectively, a = 1.29 and b = 3.67 correspond to microscopic mold coverage fraction defined according to Nielsen (2004). Parameters are chosen so that the function (1) approximately passes through the midpoints of the mold coverage intervals defined in Table 1. The transition M → C is illustrated in Fig. 2.

Fig. 2
figure 2

Suggested transitions between mold index M and mold coverage fraction C (left: microscopic coverage, right: visual coverage). Gray band shows the expected uncertainty of the transition M → C (based on Table 1)

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Equation (1) can also be used for backward conversion of mold coverage fraction C to mold index scale M (C → M). Consequently, it can also be used for the derivation of the relationship between microscopic and visual mold coverage fractions. It means that the visual mold coverage fraction can be directly calculated from the microscopic mold coverage fraction and vice versa.

It is important to notice that mold growth curves expressed by the mold coverage fraction resemble a sigmoid shape (Fig. 3, right). In contrast, mold growth expressed by a dimensionless mold index may not resemble a sigmoid shape (Fig. 3, left). A sigmoid shape of mold coverage fractions results from their lower resolution in the microscopic domain of observation.

Fig. 3
figure 3

Comparison of experimental and calculated mold growth curves. Laboratory mold growth data (Viitanen and Ritschkoff, 1991) are compared with mold index M calculated by the Finnish mold growth model (left) and microscopic mold coverage fraction C calculated by Eq. (1) from mold index (right)

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Mathematical formulation

Berger et al. (2018) suggested a new formulation of a mold growth model based on the logistic equation:

$$ \frac{{{\text{d}}M}}{{{\text{d}}t}} = k \cdot M \cdot \left( {M_{\infty } - M} \right) $$
(2)

where k [s−1] is the rate of mold growth and M [–] is dimensionless mold growth index according to Ojanen et al. (2011). When M(t) approaches maximum value M∞, the component in the bracket effectively reduces the growth rate which finally stops when M(t) = M∞. On the contrary, when M(t) is close to zero, the growth is exponential.

The rate of growth rate k and the maximum value M∞ are unknown and possibly complex functions of variables that affect mold growth, for example, environmental conditions (relative humidity, temperature), availability of nutrients, mold species, acidity, etc. Berger et al. (2018) used the logistic model to fit with mold growth data observed on the surface of bamboo boards in three levels of constant relative humidity.

As observed in Fig. 3, the microscopic mold coverage fraction might be more suitable for the state variable of the logistic equation:

$$ \frac{{{\text{d}}C}}{{{\text{d}}t}} = k \cdot C \cdot \left( {C_{\infty } - C} \right)\, \left[ {{\text{m}}^{{2}} {\text{m}}^{{ - {2}}} {\text{s}}^{{ - {1}}} } \right] $$
(3)

where C [m2m−2] is the microscopic mold coverage fraction, and C∞ is the maximum value of microscopic mold coverage fraction. By setting r = k·C∞, Eq. (3) can also be written as follows:

$$ \frac{{{\text{d}}C}}{{{\text{d}}t}} = r \cdot C \cdot \left( {1 - \frac{C}{{C_{\infty } }}} \right)\,\left[ {{\text{m}}^{{2}} {\text{m}}^{{ - {2}}} {\text{s}}^{{ - {1}}} } \right] $$
(4)

where r [s−1] is called mold growth coefficient.

The analytical solution of Eq. (4) for constant r and C∞ (corresponding to steady temperature and relative humidity) can be expressed as follows:

$$ C\left( t \right) = \frac{{C_{\infty } C_{0} }}{{\left( {C_{\infty } - C_{0} } \right){\text{exp}}\left( { - rt} \right) + C_{0} }}\,\left[ {{\text{m}}^{{2}} {\text{m}}^{{ - {2}}} } \right] $$
(5)

where C0 is initial value of mold coverage fraction (C0 = C(t = 0)). The performance of a complex dynamic biological system is simplified into a single differential equation with three input parameters (r, C∞, C0).

As seen from the analytical solution, a possible drawback is that the initial value must be set nonzero to start the growth (Berger et al. 2018). The initial condition also influences M(t) or C(t) for a long time after t = 0, i.e., the initial condition effectively acts as the third parameter. To practically circumvent this disadvantage, the initial value can be fixed to a suitable positive default value. The value of initial condition should be lower than microscopic mold coverage fraction corresponding to M = 1 (the first visible signs of mold in a microscope), i.e., the initial condition should be lower than 3 × 10−2 m2m−2.

Estimation of model parameters

At first, the extended version of the Finnish mold growth model (Ojanen et al. 2011) was used to calculate reference mold growth curves. The reference mold growth curves were calculated in an hourly time step for different levels of constant temperature and relative humidity. The calculations of reference mold growth curves were performed with parameters corresponding to a very sensitive class (pine sapwood), and for both surface qualities distinguished by the Finnish model: kiln dried, respectively, resawn. The obtained reference mold growth curves expressed by mold index were then converted by Eq. (1) into the reference microscopic mold coverage fraction.

The analytical solution (5) was used in the second step to calculate the mold growth curves under the identical set of boundary conditions. The mold growth coefficient r was treated as the optimization variable, while other two input parameters C0, C∞ were treated as constants. The C∞ values were obtained by means of Eq. (1) from the maximum values of the mold index M∞ (Online resource 1, Eq. (5)). The initial condition C0 was fixed at a default value of 3 × 10−4 m2m−2 (i.e., one hundredth of C corresponding to M = 1) and represents a mold-free initial state. The initial condition was set identical for all calculated cases. The value of C0 was chosen so as to keep the ability to predict slow growth in less favorable environmental conditions. The initial condition could possibly be related to the type of the substrate material and environmental conditions occurring after t = 0.

The optimal value of r for each calculated case was determined by searching for the best fit between the reference mold growth curve and the curve produced by the analytical solution. The coefficient of determination R2 was used to quantify how well the solution fits to the reference mold growth curves. An optimization method (The Mathworks 2020) was used.

Relation of model parameters to environmental conditions

There was no prior information on the nature of relations between mold growth coefficient and environmental conditions. It was supposed that the mold growth coefficient r can be expressed as follows:

$$ r = r_{0} f\left( x \right)g\left( \theta \right)\,\left[ {{\text{s}}^{{ - {1}}} } \right] $$
(6)

where f(x) [–] is a function of normalized independent variable x which is defined by Eq. (7), g(θ) [–] is a function of temperature θ, and r0 [s−1] is mold growth coefficient in conditions of the reference temperature of 23 °C (standard laboratory temperature) and x = 0.

Normalized independent variable x [–] is defined as follows:

$$ x = \frac{{\varphi - \varphi_{{\text{c}}} }}{{1 - \varphi_{{\text{c}}} }}\,\left[ {-} \right] $$
(7)

where φ [–] is relative humidity and φc [–] is critical relative humidity corresponding to the reference temperature of 23 °C, φc = 0.8 was assumed (Online resource 1, Eq. (7)).

Functions f(x), g(θ) and r0 were found by a regression analysis of optimal values of mold growth coefficient. Function f(x) was expressed as follows:

$$ \begin{array}{*{20}c} {f\left( x \right) = {\text{exp}}\left( {k \cdot x} \right)\,{\text{for}} \, \varphi \ge \varphi_{{\text{c}}} } \\ {f\left( x \right) = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{for}} \, \varphi < \varphi_{{\text{c}}} } \\ \end{array} [ - ] $$
(8)

where k [–] is a regression parameter.

Function g(θ) was expressed as follows:

$$ \begin{array}{*{20}l} {g\left( \theta \right) = p \cdot \theta ^{q} \,\,{\text{for}} \, 0{\mkern 1mu} \le \theta \le 40} \hfill \\ {g\left( \theta \right) = 0{\mkern 1mu}\, \,\,\,\,\,\quad {\text{for}} \, \theta { < }0} \hfill \\ \end{array} [ - ] $$
(9)

where p [°C−q], q [–] are regression parameters.

The parameters appearing in Eqs. (8) and (9) are summarized in Table 2. Their values are sensitive to the initial value C0 to some extent. The lower value of the initial condition C0 leads to the increase in r0. The higher value of the initial condition C0 leads to the decrease in r0. Parameter k is not sensitive to changes of C0. Euler’s number for k is the unexpected result of the regression analysis.

Table 2 Regression parameters for pine sapwood
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Functions f(x) and g(θ) are plotted in Fig. 4.

Fig. 4
figure 4

Plot of functions f(x) and g(θ)

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The optimized values of the mold growth coefficient are depicted in Fig. 5 as a function of variable x and temperature.

Fig. 5
figure 5

Mold growth coefficient r for pine sapwood (left: resawn surface, right: kiln dried surface)

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Reliability of the logistic model

To test the sensitivity of outputs to changes of inputs, calculations were performed with different settings of mold growth coefficient. Calculations were performed with pine sapwood (resawn) which was exposed to a constant relative humidity of 0.95 and temperature of 23 °C. The first calculation uses the proposed values of parameters from Table 2 (the reference case). Then, the calculations were repeated for modified mold growth coefficient r ± 0.3r (30% relative error of r). Finally, the calculations were performed once again so that all regression parameters are simultaneously changed by ± 1% and by ± 10%. Figure 6 shows the calculated development of mold coverage fraction over 100 days.

Fig. 6
figure 6

Sensitivity of the model to change of input parameters

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The time to the first signs of visible molds by naked eyes t(C = 0.3) was evaluated. The case with 1% error of all regression parameters led to the relative error lower than ± 7% (approximately, t(C = 0.3) = 26 ± 2 days). This shows that the presented parametrization is not oversensitive to subtle changes of parameter values. However, it is not realistic to expect model parameters to be known with lower than 1% error. The real uncertainty of the model can only be assessed by comparison with mold growth experiments.

Model verification

Comparison with the reference mold growth model

The microscopic mold coverage fractions calculated by Eq. (4) are compared with the microscopic mold coverage fractions calculated by Eq. (1) from the predictions of the Finnish mold growth model. The dynamic cases with cyclic alternations of constant relative humidity, as defined in Fig. 7 and Table 3, are used. The mold growth coefficient in the logistic mold growth model was calculated on an hourly basis from Eq. (6).

Fig. 7
figure 7

Cyclic boundary conditions (wet and dry cycles of constant relative humidity). The complete cycle has a length tp and contains a wet period with a duration denoted tw

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Table 3 Cases with cyclic boundary conditions used for verification of the logistic mold growth model
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The comparison was made for pine sapwood (W = 0) with resawn surface (SQ = 0). The coefficient of determination was used to quantify the match between both models (Fig. 8, left). The difference between the times for the first microscopic signs of molds, defined as Δt = t(C = 0.03) − t(M = 1), is depicted as well (Fig. 8, right).

Fig. 8
figure 8

Coefficient of determination R2 (left) and the difference in times for the first microscopic signs of molds Δt expressed in days (time from logistic mold growth model–time from Finnish mold growth model), depicted as a function of tw/tp and φw

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The coefficient of determination decreases as the relative time of wetness tw/tp and relative humidity φw decrease. The better fit is visible in conditions more favorable for mold growth. The poor match occurs in environmental conditions less favorable for mold growth. The steady-state cases with constant relative humidity (relative time of wetness tw/tp = 1) were calculated with very good match for the whole range of relative humidity (R2 > 0.95). A good agreement between both models (R2 > 0.8) was observed for tw/tp > 0.7 and φw > 0.90. In contrast, the mold growth is overestimated by the logistic model for cases with tw/tp < 0.7. In those cases, the logistic model systematically takes less time to develop the first microscopic signs of molds, typically from 0 to 20 days. The time of the first microscopic signs of molds is slightly overestimated in cases with very good fit.

Model validation

Comparison with laboratory experiments

Steady-state mold growth experiments

The experimental mold growth data (Viitanen and Ritschkoff 1991; Johansson et al. 2013) were used. The transition between the mold index M (7-degree scale) and the microscopic mold coverage fraction C was performed according to Eq. (1). The transition between Johansson’s mold rating MR (5-degree scale) and mold index M was performed by linear interpolation of values specified in Table 1. The calculation of mold growth was performed for pine sapwood (W = 0) with planed surface (SQ = 0).

The comparison of mold growth models and the experimental observations of mold growth on the surface of pine sapwood is shown in Figs. 9 and 10. Mold growth models correspond well with the experimental data of Viitanen and Ritschkoff (1991). In contrast, the prediction of mold growth does not match with the experimental data of Johansson et al. (2013). In this case, the mold growth observed in laboratory experiment occurred much earlier than it was predicted by the models. Relative humidity of 0.97 would have to be used in models to achieve similar mold growth rate, but an overshooting of the maximum value of mold index would occur.

Fig. 9
figure 9

Comparison of the logistic mold growth model, the Finnish mold growth model and experimental mold growth observations performed with pine sapwood under constant environmental conditions (Viitanen and Ritschkoff 1991). Legend: Mexp mold index observed in the experiment; MVTT mold index calculated by the Finnish mold growth model; C microscopic mold coverage fraction calculated by Eq. (4), the band shows the uncertainty of relative humidity ± 0.01; Cexp microscopic mold coverage fraction calculated from Mexp, the band shows the uncertainty due to transition M → C

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Fig. 10
figure 10

Comparison of the logistic mold growth model, the Finnish mold growth model and experimental mold growth observations performed with pine sapwood under constant environmental conditions (Johansson et al. 2013). Legend: Mexp is mold index interpolated from experimental mold rating MR, otherwise the same meaning as symbols in Fig. 9

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Dynamic mold growth experiments

Lie et al. (2019a) reported on observations of mold growth on wood samples subjected to repeated wetting. Only four experimental cases (1, 2, 5 and 6) were used for comparison. Lie et al. (2019a) calculated the surface relative humidity from the measured moisture content of a surface layer of the samples. An analytical expression for a single sorption curve was used, accounting for the temperature effect, but it is not clear whether it corresponded to the desorption regime that dominated in the surface layer. Therefore, the calculations were performed with the uncertainty of relative humidity ± 0.015. The transition of the experimentally observed mold rating (6-degree scale) into mold index was performed according to Table 1.

The comparison of the logistic mold growth model with the experimental observations of mold growth is shown in Fig. 11. It is seen that the logistic model predicts faster mold growth in case 1 when surface conditions are less favorable for mold growth. On the other hand, slower mold growth is predicted in cases 5 and 6 when surface conditions are more favorable for mold growth. The maximum mold coverage fraction compares reasonably well. The delayed mold growth in case 6 could be attributed to the temperature correction of the mold growth coefficient (Eq. (9)).

Fig. 11
figure 11

Comparison of the logistic mold growth model (W = 0, SQ = 0) with dynamic mold growth experiments (Lie et al. 2019a). Legend: φa relative humidity of ambient air; φs surface relative humidity; Mexp mold index calculated from mold rating observed in the experiment, the band shows the uncertainty of the transition; MVTT mold index calculated by the Finnish mold growth model; Cexp microscopic mold coverage fraction calculated from Mexp, the band shows the uncertainty of the transition; C microscopic mold coverage fraction calculated by Eq. (4), the band shows the uncertainty of surface relative humidity

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Discussion

The logistic mold growth model as defined by Eq. (4) and Eq. (6–9) is the projection of the Finnish mold growth model into the space in which an alternative mold growth metric is used (M → C). Both mold growth models presented are mathematical constructs that mimic experimental mold growth data. Their scope is not to enable better understanding of microbiological processes. The deterministic mathematical formulation of both models allows us to estimate the development of different mold growth metrics on a wooden surface under arbitrary dynamic environmental conditions. The obvious main weakness of both models is the lack of comparison with mold growth experiments.

The comparison of the mold growth models with Vittanen’s experimental data showed reasonable overestimation of mold growth in φ = 0.97, respectively, reasonable underestimation in φ = 0.87. In contrast, the significant underestimation of mold growth rate was observed in the comparison with Johansson’s experimental mold growth data for φ = 0.90. These two mold growth experiments are for some reason not comparable with each other.

The observed discrepancies indicate that mold growth models and experiments should be improved. The mold growth laboratory experiments would be improved if their methodology was designed so as to minimize sources of errors associated with experimental observations of mold growth, and facilitate the more straightforward side-by-side comparison between experiments. In this respect, it could be useful to perform mold growth experiments with an artificial growing medium extracted or prepared from pine sapwood, in the first step with single mold cultures, then with a combination of molds.

Mold growth models can be improved by either better formulation of model equations (more internal aspects of mold growth are considered), or adjustment of model parameters (improvement of functions to calculate model parameters, reduction of the uncertainty in model parameters), or using more accurate boundary conditions. In the case of existing mold growth models, the most important sources of errors are considered to be the model parameters and the mathematical formulation of the model. With respect to latter, some modifications of the logistic equation could also be of interest, for example, Richards equation (Richards 1959) allows us to influence the position of inflection point on the growth curve, where mold attains the maximum growth rate.

The advantage of the logistic equation is its simple transparent structure and low number of input parameters. The deterministic mathematical formulation is considered to be the main drawback. It is important to say that mold growth models are intended to be used in the hygrothermal assessment of building components. The deterministic prediction of the quantity of molds is not necessary for assessment of mold risk. Instead, a mold growth model should estimate the probability of the presence or absence of molds (Lepage 2021).

How to choose the right values of input parameters in mold growth models remains still an open and difficult research question. The mold growth on wood can exhibit large deviations from mean values, even for the samples cut in the identical anatomical direction, from the same tree, and inoculated by the same procedure. The differences in mold growth may also be caused by variations in concentration of available nutrients on the surface, by different methods of inoculation, and by other factors (e.g., pH of the substrate and the presence of dust on the surface). The optimal values of mold growth coefficient in the logistic model should ideally estimate the average or the maximum response of wood for the most typical mixture of mold species under given environmental conditions. The future research work should hence carefully collect available mold growth data from available experimental studies and optimize parameters in mold growth models by a systematic comparison of mold growth predictions with experiments.

For pine sapwood, mold growth coefficient was found to be a rising function of relative humidity and temperature (Eqs. (8) and (9)). In particular, the exponential dependency of mold growth coefficient on relative humidity should be checked. With respect to latter, it could be important to note that wood does not have the character of standard porous stony materials since it stores moisture as bound water in cell walls, i.e., no liquid water is deposited in wood even for ambient air with relative humidity far above relative humidity 0.98 (Fredriksson 2019).

The mold growth coefficient in the logistic model was found by a regression of results of steady-state calculations performed by the Finnish mold growth model. In cyclic cases used for model verification, the mold growth coefficient was simply calculated on an hourly basis from Eq. (6). The mold growth coefficient therefore changes from hour to hour, immediately following the changes in relative humidity. Such a proportionality between mold growth and relative time of wetness is questioned in the literature. Adan and Samson (2011) demonstrated by means of a toy mold growth model, and by means of real experiments, that mold growth may not be proportional with relative time of wetness in the case of fast humidity fluctuations (hours and days). Microbial processes in molds are dynamic and may, in reality, depend on mold growth conditions accumulated over previous time. For instance, mold growth after longer dry conditions can occur with another rate if compared to initial growth after germination. Current knowledge of mold growth processes is probably far from the ability to quantify inertial effects that dampen changes of mold growth coefficient. Future mold growth research therefore should focus on dynamic mold growth experiments leading to insight into the development of mold growth coefficient during wet-to-dry and dry-to-wet transitions.

Conclusion

Mold growth is a complex microbiological phenomenon that is difficult to model due to many influencing factors. This paper used the logistic growth equation to model mold growth on wood. The microscopic mold coverage fraction was used as a state variable that represents mold development over time.

The model parameters were optimized on the basis of comparison with the current state-of-the-art Finnish mold growth model. To enable comparisons with the Finnish mold growth model, two-way transition between mold coverage fraction (state variable used in the logistic model), and mold index (state variable used in the Finnish mold growth model), was mathematically defined by using the current definition of mold index. The mold growth coefficient appearing in the logistic equation was searched for the fixed initial condition by finding the best agreement with the Finnish mold growth model. The regression analysis of optimal values of the mold growth coefficient led to a formulation of functions for estimation of the mold growth coefficient. How well both mold growth models compare with the mold growth in real situations, still remains an open question.

It was shown by comparisons with selected experimental mold growth data, that both mold growth models predict mold growth similarly with observations only to some extent. The development of mold growth models should focus on the careful experimental verification of model parameters and the evaluation of uncertainty in mold growth predictions. The further development of mold growth models will be made possible only by the high-quality experimental mold growth data. Unfortunately, there is a lack of solid experimental data on mold growth on materials exposed to constant and varying boundary conditions. There is also the lack of accurate data of the hygrothermal conditions on the surfaces of studied hygroscopic materials during dynamic mold growth experiments. The latter is especially challenging in the case of wood and wooden materials.

Last but not least, the important practical question is the definition of the reliable methodology for using existing mold growth models by building designers as supporting tools for the assessment of the moisture safety of building components containing solid wood and wood-based materials.