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A mathematical model of mould growth
on wooden material
A. Hukka, H. A. Viitanen
Summary A mathematical model for the simulation of mould fungi growth on
wooden material is presented, based on previous regression models for mould
growth on sapwood of pine and spruce. Quanti®cation of mould growth in the
model is based on the mould index used in the experiments for visual inspection.
The model consists of differential equations describing the growth rate of the
mould index in different ¯uctuating conditions including the effect of exposure
time, temperature, relative humidity and dry periods. Temperature and humidity
conditions favourable for mould growth are presented as a mathematical model.
The mould index has an upper limit which depends on temperature and relative
humidity. This limiting value can also be interpreted as the critical relative
humidity needed for mould growth depending also on the mould growth itself.
The model enables to calculate the development of mould growth on the surface
of small wooden samples exposed to arbitrary ¯uctuating temperature and
humidity conditions including dry periods. The numerical values of the
parameters included in the model are ®tted for pine and spruce sapwood, but
the functional form of the model can be reasoned to be valid also for other
wood-based materials.
List of symbols
M mould index [)]
T temperature [°C]
SQ surface quality (0 = resawn, 1 = original kiln-dried)
t time [d]
k
1
,k
2
correction coef®cients [)]
W wood species (0 = pine, 1 = spruce)
RH relative humidity [%]
Wood Science and Technology 33 (1999) 475±485 ÓSpringer-Verlag 1999
Received 18 May 1997
A. Hukka, H. A. Viitanen (&)
VTT Building Technology, Wood Technology,
P.O. Box 1806, FIN-02044 VTT, Finland
Present addresses:
A. Hukka
Valmet-Utec Oy,
P.O. Box 43, FIN-20251, Turku, Finland
H. A. Viitanen
FFRI Joensuu Research Station,
P.O. Box 68, FIN-80101 Joensuu, Finland
475
Introduction
Mould fungi is a heterogeneous and not a particularly well de®ned group of fungi.
Most of the mould fungi, as also the blue-stain and soft rot fungi, belong to Asco-
mycotina-(Ascomycetes) fungi. Mould and bluestain fungi are often both called
discolouring fungi. Discolouring fungi are often the initial microbial colonisers of
wood: logs can be infected in forests and in storage, sawn goods can be contami-
nated before and during drying, at storage or at the building site if the conditions for
fungal growth are favourable. In buildings, mould fungi cause problems in different
structures and materials: roofs, basements, ¯oors and walls. Except of wooden
substrates, surfaces of many other materials support growth of microbes and mould
problems are more common than decay damages. Typical mould fungi found in
moisture damaged wood are Alternaria alternata,Aspergillus species, Aur-
eobasidium pullullans,Cladosporium cladosporioides,Chaetomium globosum,
Paecilomyces variotii,Penicillium species and Trichoderma viride.
The growth of mould fungi on wooden material has been the subject of ex-
perimental research for a long time, but the knowledge thus gathered has been
mostly qualitative in nature (Henningsson 1980; Block 1953; Park 1982). The aim
has been to describe the material response and ®nd the critical conditions for
mould growth on surfaces of different materials with different treatments. Most of
this previous extensive research has been carried out in constant temperature and
humidity conditions but even such models are usually not applicable in arbi-
trarily varying conditions. Concerning ¯uctuating conditions, no mathematical
models seem to be available at all in the literature.
Especially dry periods have posed a problem in the published regression
models, as experimental knowledge concerning the effect of non-favourable
conditions for mould growth has been very limited. The few results published on
this subject are not applicable to humidity histories other than those used in the
experiments. From a practical point of view, it has not been possible to analyse
real building structures in actual varying climatic conditions.
Recently, Viitanen (1997a) has published comprehensive regression models for
mould growth in constant humidity and temperature conditions for pure pine
and spruce sapwood. Those regression models along with the experiments in
¯uctuating conditions are the basis for the present mathematical model. It is a
material model describing the response of pure wooden material to arbitrary
temperature and humidity conditions and will form an essential part of a more
complete structural model simulating the moisture behaviour of building struc-
tures in actual measured weather conditions.
Experimental material
The experimental material consisted of small samples (7 ´15 ´50 mm) of pure
kiln dried pine and spruce sapwood. Two different surface qualities differing in
nutrient content were studied: original kiln-dried and resawn. The experimental
results have been presented in detail by Viitanen and Ritschkoff (1991), Viitanen
and Bjurman (1995) and Viitanen (1997a).
Certain preconditions must be assumed concerning these experiments:
The small samples used do describe the mould growth on wooden material
reaching the equilibrium moisture content without delay. The ®nite size of the
samples may thus be neglected as well as the delay in the wood cell wall
attaining the equilibrium moisture content prescribed by the surrounding
relative humidity.
476
The growth of mould fungi takes place only on the material surface and it may
thus be modelled using only surface temperature and moisture content as input
data.
The existence of mould fungi on the material surface does not in¯uence the
moisture behaviour of the material, e.g. sorption properties.
The mould was measured applying an existing standard index based on the visual
appearance of the surface under study. Some re®nement has been made concerning
the scale and as a result this mould index assumes following integer values:
0 no growth
1 some growth detected only with microscopy
2 moderate growth detected with microscopy (coverage more than 10%)
3 some growth detected visually
4 visually detected coverage more than 10%
5 visually detected coverage more than 50%
6 visually detected coverage 100%
The same scale is applied in the present mathematical model, but the index is not
limited to integer values.
Model development
Conditions favourable to initiation of mould growth
Moisture content of wood depends on ambient humidity, temperature, exposure
time, dimensions and moisture absorption capacity of wood; water can also exist
in wood as free water in cavities or bound water within cell walls (Siau 1981;
Cloutier and Fortin 1991; Hartley et al. 1992). Being a hygroscopic material, the
equilibrium moisture content of wood is easily affected by the ambient humidity.
At low moisture content, the most direct measure of water availability is water
potential (w) de®ned as the free energy of water in a system relative to that of a
reference pool of pure water (Schniewind 1980). Water activity (a
w
) is, like water
potential, related to actual availability of water and it is determined by both matric
and osmotic components. Relative humidity (RH) is a percentage relation of
actual vapour pressure (p) and saturated vapour pressure (p
0
). The a
w
can also be
de®ned as the relative humidity at equilibrium (ERH) divided by 100, i.e. rela-
tive vapour pressure (p/p
0
) of the atmosphere in equilibrium with the substratum.
The water activity in the ambient air or in the cavities of the material is critical
for active stages of mould growth. Growth of mould fungi and time period needed
for the initiation of mould growth is mainly regulated by water activity, tem-
perature, exposure time and surface quality of the substrate. The experiments
suggest that the possible temperature and relative humidity conditions favouring
initiation of mould growth on wooden material can be described as a mathe-
matical diagram in Fig. 1. The favourable temperature range is 0±50 °C, and the
critical relative humidity required for initiation of mould growth is a function of
temperature. Based on experiments covering the temperature range 5±40 °C this
boundary curve can be described using a polynomial function
RHcrit ÿ0:00267T30:160T2ÿ3:13T 100:0 when T 20
80% when T >20
1
477
The behaviour of RH
crit
in the vicinity of the upper end of the temperature range
is only an approximation, but it is of only very little importance in practical
applications.
The largest possible mould growth
As known from experience, mould growth once initiated does not necessarily lead
to visually detectable mould (Viitanen and Bjurman 1995). Also, the ®nal coverage
of mould fungi on a surface is dependent on the temperature and humidity con-
ditions suggesting that a certain limiting value exists above which the mould index
does not rise irrespective of time available in basically favourable conditions. To
construct this limit, it is natural to assume that in conditions critical for the ini-
tiation of mould growth, Eq. (1), this upper limit for growth is 1, i.e. just some
growth can be detected microscopically no matter how much time passes. In the
other extreme it may be concluded that at 100% relative humidity the mould will
eventually cover the whole surface regardless of temperature (in the range 0±50 °C)
and M thus reaches a value of 6. In between these two ®xed points the experiments
suggest that the largest possible value of the mould index assumes a parabolic form:
Mmax 17RHcrit ÿRH
RHcrit ÿ100 ÿ2RHcrit ÿRH
RHcrit ÿ100
2
2
The contents of Eq. (2) may also be interpreted by stating that the critical RH
needed for mould growth does not only depend on temperature but also on the
Fig. 1. Conditions favourable for
initiation of mould growth on wooden
material as a mathematical model
Fig. 2. Temperature-
dependent critical relative
humidity needed for mould
growth at different values of
mould index
478
stage of mould development, i.e. the mould index itself. This result is arrived at by
solving Eq. (2) for RH, which now represents the temperature-dependent RH
needed for mould index reaching a value of M
max
. This result is depicted in Fig. 2.
Growth rate in favourable conditions
The present model is based on mathematical relations for the growth rate of
mould index in different conditions. The model is purely mathematical in nature
and as mould growth is only investigated by visual inspection, so it does not have
any connection to biology in the form of modelling the number of live cells. Also,
the mould index resulting from computation with the model does not re¯ect the
visual appearance of the surface under study, because traces of mould growth
remain on wood surfaces for a long time. The correct way to interpret the results
is that the mould index represents the possible activity of the mould fungi on the
wood surface.
As a basis for the growth model, Viitanen (1997a) presents a regression
equation for the response time (weeks) needed for the initiation of mould growth
on wooden material in constant temperature and humidity conditions:
tmexpÿ0:68 ln T ÿ13:9 ln RH 0:14W ÿ0:33SQ 66:02 3
If the mould index M is presumed to increase linearly in time and time is mea-
sured in days, Eq. (3) may be interpreted as a differential relation
dM
dt 1
7 expÿ0:68 ln T ÿ13:9 ln RH 0:14W ÿ0:33SQ 66:02;M<1
4
This conversion extends the applicability of Eq. (3) into variable conditions such
that the relative humidity is constantly above the critical value de®ned by Eq. (1)
and the temperature is in the range 0±50 °C. Linear growth in the range M < 1 is
only a mathematical description and in principle any other growth model could
be utilised. When interpreting the results of the model all values of M below 1
indicate no growth.
As the growth proceeds above the initial stage (M = 1), Eq. (4) is no longer
valid. For a larger growth Viitanen (1997a) presents another regression model
describing the response time needed for the ®rst visual appearance of mould
growth (M = 3):
tvexpÿ0:74 ln T ÿ12:72 ln RH 0:06W 61:50:5
If growth of the mould index is presumed to proceed from M = 1 to M = 3 on
a constant rate in constant conditions, Eqs. (3) and (5) can be combined to
give the growth rate on that range. The result is a correction coef®cient if
Eq. (4) is used as a basis:
k11 when M<1
2
tv=tmÿ1when M >1
6
Although based on constant conditions, the experiments suggest that Eq. (6) is
valid also for mould growth in ¯uctuating conditions as long as the conditions are
continually favourable to growth. Based on data from growth after the visual
479
appearance of mould fungi it may be concluded that the same correction for
growth rate applies for the entire range M > 1.
Taking into account the upper limit for mould growth de®ned by Eq. (2) may
also be accomplished by using a correction coef®cient. Assuming the delay to
affect the growth rate by 10% at 1 unit below the maximum value of the index
gives this coef®cient to the following form
k21ÿexp2:3MÿMmax 7
The complete model in conditions favourable for mould growth consists of
Eqs. (4), (6) and (7):
dM
dt 1
7 expÿ0:68 ln T ÿ13:9 ln RH 0:14W ÿ0:33SQ 66:02k1k28
As an initial condition M must be a known value, usually equal to zero imme-
diately after arti®cial drying of wood in a kiln with wood temperature exceeding
50 °C.
Model during non-favourable conditions
In ¯uctuating humidity conditions Viitanen (1997a) states that the cumulative
time in high-humidity conditions can be used to a limited extent to quantify the
response time needed for the initiation of mould growth. This simpli®cation,
however, eventually always leads to a large mould activity as humidity cycles are
repeated. Instead of remaining on a constant level the activity of mould can be
thus regarded as decreasing during dry periods. Of course, the visual appearance
of the surface does not necessarily change during the dry period, but a ®nite delay
in growth after the dry period can be clearly observed. This delay does exist as
soon as after 6 h in dry conditions, but extending the dry period to 24 h does not
seem to signi®cantly affect the delay, if growth will initiate at all. After that the
delay is again prolonged. A mathematical description of the delay can be written
by using the time passed from the beginning of the dry period (t )t
1
):
dM
dt
ÿ0:032 when t ÿt16 h
0 when 6 h tÿt124 h
ÿ0:016 when t ÿt1>24 h
(9
The experimental data behind this equation cover dry periods between 6 h and 14
days, but the functional form of expression (9) is based only on a small number of
experiments and thus must not be seen as the best possible one. Knowledge of the
in¯uence of longer dry periods on mould growth is very limited as is the data
concerning the effect of temperatures below 0 °C. In lack of better information
Eq. (9) may be applied also for such situations, although the validity of such an
application must be questioned.
Comparing the model results against experimental data
The largest possible value of the mould index, Eq. (2), is based on 12-week ex-
periments in constant conditions. Figure 3 presents the values calculated using
Eq. (2) versus the experimental observations. Only points showing a clear limit of
mould index have been taken into account. It can be seen that the largest error in
480
the maximum value of mould index is 1.2 and that only one point includes an
error larger than 0.5 in value.
To compare the model results to the original experimental data, Figs. 4 and 5
present the experimental and simulated response times needed for the initiation
and the ®rst visual appearance of mould growth on the surface of resawn pine
sapwood. Similar results for spruce are presented in Figs. 6 and 7. The experi-
mental results represent the average of 6 parallel samples. It can be seen that there
are no major systematic errors in the model and that in all cases most of the
errors in response time are smaller than 25% in numerical value. However, also
some points with very large errors can be detected, indicating that a model with
only a very few numerical parameters may not be suf®cient for describing the
phenomenon of mould growth in the whole temperature and RH range, especially
in higher temperatures.
Figure 8 presents the results obtained when using the model to predict the
response time needed for the initiation of mould growth on pine sapwood in
Fig. 3. Comparison of the largest
possible value of mould index
produced by Eq. (2) against the
experimenta
Fig. 4. Comparison of
simulated and experimental
response times needed for
initiation of mould growth on
pine sapwood in constant
conditions
481
¯uctuating humidity conditions. The relative humidity has been kept at two
constant values, 75% and 95%, the period in each condition varying between
6 and 196 h. The temperature has been constant at 20 °C. It can be seen that
most of the simulated points (80%) are such that the error in response time is less
than 25%. The average error in simulated response time is only 1%, indicating
that there is essentially no systematic error in the model.
Discussion and conclusion
The mathematical model presented is throughout formulated in a differential
form. As such it allows to calculate the development of mould growth on a
wooden surface exposed to arbitrary temperature and humidity histories in-
cluding also dry periods. The numerical values of the parameters in the model
apply only for pure pine and spruce sapwood and aim at describing the average
response of the material based on a small number of parallel samples. The
Fig. 5. Comparison of
simulated and experimental
response times needed for
visual appearance of mould
growth on pine sapwood in
constant conditions
Fig. 6. Comparison of
simulated and experimental
response times needed for
initiation of mould growth on
spruce sapwood in constant
conditions
482
response does, however, exhibit a very large variation between samples origi-
nating from different stems of the same wood species. This variation is of the
same order in magnitude as is the variation in results between samples repre-
senting different species of wood. Based on this it may be reasoned that the same
functional form of the model could be utilised also for prediction of mould
growth on other wood-based materials, only the numerical values of the coef®-
cients must be re-evaluated.
The experimental data behind the model covers temperatures between 5 and
40 °C and relative humidities between 75 and 100%. The exposure time in con-
stant conditions has been at least 12 weeks and the time in ¯uctuating conditions
has varied between 6 and 24 weeks. This temporal scale is clearly shorter than will
be the application area of the model. This requires an extrapolation of experi-
mental results and causes an uncertainty in the results produced by the model in
such situations. Also the nature and range of the experiments conducted in
¯uctuating conditions will certainly need some revision as the numerical result
proposed by Eq. (9) is not totally satisfactory.
Fig. 7. Comparison of
simulated and experimental
response times needed for
visual appearance of mould
growth on spruce sapwood in
constant conditions
Fig. 8. Comparison of
simulated and experimental
response times needed for
initiation of mould growth on
pine sapwood in ¯uctuating
humidity conditions. Values of
surface quality SQ correspond
to original kiln-dried (1) and
resawn (0) surfaces
483
The maximum value of the mould index used in the model, Eq. (2), has been
deduced from experiments conducted in constant humidity conditions. It is
known that this parameter is lower in ¯uctuating humidity conditions, but the
numerical value of the maximum growth in such situations on a general level is
not comprehensively known. In this sense the present model obviously needs
further development. The same applies for ¯uctuating temperature conditions
with constant RH.
The present model describes mould growth on wooden material surfaces. It is a
pure material model trying to quantify the experimental results of mould expo-
sures conducted on small samples. The next natural step in the evolution of the
model is to insert the mathematical equations into a simulation program for
calculation of mould growth on an actual wooden structure. This would create a
new tool for prediction of the service life of a certain structural design in critical
temperature and humidity conditions. Another possible direction of further de-
velopment is to apply the model for quanti®cation of mould growth in materials
other than pure wood, which combined to a moisture simulation program would
give a powerful tool for analysing the moisture behaviour of a very wide range of
structures.
It has been found in laboratory studies and in practice that mould fungi are
more rapid invaders than decay fungi in wooden structures subjected to excess
moisture stresses (Viitanen 1997a, b). Also, the minimum humidity and tem-
perature requirements are in general lower for mould fungi than for brown rot
fungi. Growth of mould fungi and time period needed for the initiation of mould
growth are mainly regulated by water activity, temperature, exposure time and
surface quality of the substrate. Especially at low water activity and low tem-
perature, the latent period preceding the initial stages of mould growth and the
appearance of mould fungi on wood have been found to be several weeks al-
though the equilibrium moisture content of the small samples used was reached
in two weeks (Viitanen 1997a). However, Viitanen and Paajanen (1986) found no
mould growth at RH 75% and according to recent studies, the lowest humidity
condition allowing mould growth is RH 80±85% (i.e. water activity 0.80±0.85)
providing that the temperature is above 5 °C (Bjurman 1989; Park 1982; Wang
1992; Adan 1994; Hocking et al. 1994; Viitanen 1997a). Modeling of the critical
humidity and temperature conditions, and especially of the critical time required
for spore germination and the growth of mould fungi mycelium on pine and
spruce sapwood was presented by Viitanen (1997a). Temperature conditions
have a minor effect on the water activity and the main effect is concerned with
the growth and metabolic activity of the fungi. Often the selection of mould
species at high temperatures (above +30 to +35 °C) consists mainly in thermo-
tolerant or thermophilic mould fungi (Henningsson 1980). At ¯uctuating hu-
midity conditions, the growth of mould fungi is clearly retarded and latent period
is even longer than at constant favourable conditions (Adan 1994; Viitanen
1997a).
In general, the higher the temperature and the more favourable the nutrition at
a given relative humidity, the less time is required for spore germination. After
kiln drying, the concentration of nitrogen and low-molecular hydrocarbon
compounds on the surface layers of sawn timber are often higher than inside the
wood (Boutleje 1990; Theander et al. 1993). Terziev et al. (1994) and Viitanen and
Bjurman (1995) showed, that the growth of mould is clearly more rapid and
vigorous on the original kiln dried wood surface than on a resawn surface.
484
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