© 2019 The Authors. Biotechnology and Bioengineering published by Wiley Periodicals, Inc.
Biotechnology and Bioengineering . 2019;116:3360 – 3371. 3360
|
wileyonlinelibrary.com/journal/bit
Received: 5 July 2019
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Revised: 30 August 2019
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Accepted: 2 September 2019
DOI: 10.1002/bit.27166
ARTICLE
From three ‐ dimensional morphology to effective diffusivity in
filamentous fungal pellets
Stefan Schmideder
1
|
Lars Barthel
2
|
Henri Müller
1
|
Vera Meyer
2
|
Heiko Briesen
1
1
Chair of Process Systems Engineering,
Technical University of Munich, Freising,
Germany
2
Department of Applied and Molecular
Microbiology, Institute of Biotechnology,
Technische Universität Berlin, Berlin,
Germany
Correspondecne
Heiko Briesen, Technical University of
Munich, Chair of Process Systems
Engineering, 85354 Freising, Germany.
Email: [email protected]
Funding information
Deutsche Forschungsgemeinschaft,
Grant/Award Number: 198187031,
315305620,315384307
Abstract
Filamentous fungi are exploit ed as cell fact ories in biotechn ology for the produc tion of
proteins, orga nic acids, and natural pr oducts. Hereby , fungal macromo rphologies adop ted
during submerged cultivations in b ioreactors strongly impact the produ ctivity. In
particular, fun gal pellets are k nown to limit th e diffusivity of oxygen, substrates, and
products. To inve stigate the spa tial distribut ion of substances in side fungal pellets, the
diffusive mass transport must be locally reso l v e d .I nt h i ss t u d y ,w ep r e s e n tan e wa p p r o a c h
to obtain the effective diffusivity in a fungal pellet based on its three ‐ dimensional
morphology. Freeze ‐ dried Aspergillus niger pellets were studied by X ‐ ray microcomputed
tomography, and t he results were r econstructed t o obtain three ‐ dimensional images. After
processing the se images, representa tive cubes o f the pellets were sub jected to diffu sion
computations. T he effective diffusion factor and the to rtuosity of each cube were
calculated using t he software GeoDict. Aft e rwards, th e effectiv e diffusion factor was
correlated with the amount of hyphal material inside the cubes (hyphal fraction). The
obtained correlat ion between the effective d iffusion factor an d hyphal fraction shows a
large deviation from the c orrelations repor te d in the literature so far, giving new and more
accurate insights. This knowledge can be u sed for morphological o ptimization of
filamentous pellets to increase the y ield of biotech nological processes.
KEYWORDS
Aspergillus niger , effective diffusion, filamentous fungal pellets, tortuosity, X ‐ ray microcomputed
tomography
1
|
INTRODUCTION
Filamentous fungi are widely used cell factories for the pro duction of a
variety of compounds such as enzymes, organic acids, or antibiotics
(Meyer, 2008). As just one example, plant ‐ biomass ‐ degrading enzymes
produced by filamentous fungi, have a g lobal market value o f
€

4.7 billion
(Meyer et al. , 2016). During su bmer ged cultivation, filamentous fungi
adopt different macromorphological entities such as non ‐ aggr egated
hyphae (disperse mycel ia), loosely aggregated hyphal clump s, and densely
aggregated spherical str uctures (pellets; Pirt, 1966). This morphology is
influenced by the fungal species and cultivation paramet ers (Papagianni,
2004). Depending on the predominant morphology in a submerged
fungal culture, the substances produced by the organ ism can differ
significantly. As fungal growth an d protein secretion are coupled
processes, it is for example known that the highest protein secret ion
normally occurs during rapid hyphal growth, which takes place in
disperse mycelia and the outer layers of fungal pellets where nutrien t
supply is not limited ( Cairns, Zheng, Zheng, Sun, & Meyer, 2019). On the
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other hand, the production of secondary metabolites peaks when the
producing organism shows a n extreme ly low or zero growth (Brakhage,
2013). These conditions can be observed for example in the dense center
of fungal pellets, where the restrict ed diffusion of oxygen and nu trients
leads to limitations, thus inhibiti ng growth (Veiter, Rajamanickam, &
Herwig, 2018). This dem onstrates th at a d etailed understanding of t he
limitations and diffusion proce sses in fungal pellets is of crucial
importance for many biotech nological applications.
The effective diffusion coefficient is required to calculate the
diffusive mass transport. For component
i
in a porous medium, this
parameter can be expressed as (Becker, Wieser, Fell, & Steiner, 2011)
=⋅
D

Dk
,

ii ,e f f ,b u l k e f f
(1)
where D
i,bulk
is the diffusion co efficient of component i in th e bul k
medium without geometrical hind rance and k
eff
describes the reduction
o ft h ef r e eb u l k d i f f u s i o n D
i, bulk
to the effective diffusion
D
i, eff
and is solely dep endent on the pore geometry and in dependent
of the diffu sing substance i (Becker et al., 2011). Hereafter, k
eff
is called
the effective diffusion fa ctor. In the case of diffusion in fila mentous
pellets, D
i, bulk
corresponds to t he molecular diffu sion coefficient of
components such as oxygen, glucose, or products in t he fermentat ion
medium and is strongly dependent on the medium, diffus ing substan ce,
and process conditions such as temperature. Temperature ‐ dependent
bulk diffusion coefficients fo r gl ucose, oxygen, and several other
compounds i n aqueous solu ti ons can be estimated, fo r example, from
Yaws (2014). The ge ometrically caused reduction of the diffusi on, k
eff
,
can be expressed as (Epstein, 1989):
= ϵ
k
,

eff 2
τ (2)
where the porosity,
ϵ

, is defined as the ratio of volume of voids to the
total volume. The tortuosity,
τ

, is a geometrical parameter and can be
taken as the ratio of the average pore length to the length of the
porous medium along the major flow or diffusion axis. Thus, in
general it is >
1

(Epstein, 1989).
So far, no experimental or modeling approaches to exactly pre dict the
diffusive mass transport i nside whole f ungal pellets are available. Several
modeling approaches of filament ous microorganisms consider the
diffusive mass transport of oxygen and/ or substrates like
glucose (Buschulte, 1992; Celle r, P icioreanu, van Loosdrecht, & van
Wezel, 2012; Cui, Van der Lan s, & Luyben, 1998; Lejeune &
Baron, 1997; Meyerhoff, Tiller, & Be llgardt, 1995; Table 1). They include
effective diffusion coefficients that are dependent on the molecular
TABLE 1 Applied correlations between effective diffusion coefficients ( D
i,eff
), effective diffusion factor ( k
eff
), bulk diffusion coefficients
( D
i,bulk
), porosity (
ϵ

), hyphal fraction ( =− ϵ
c

1
h ), solid fraction (
ϕ
), and tortuosity (
τ

) in the literature to model the diffusive transport
mechanisms
Eq. for k
eff
in
D

Dk
ii ,e f f ,b ul k e f f
=
Origin
structure Diffusion direction Field Source
− c 1
h
–– Filamentous microorganisms Celler et al. (2012)
–– Filamentous microorganisms Cui et al. (1998) adopted from Van ’ t
Riet and Tramper (1991)
–– Filamentous microorganisms Buschulte (1992) adopted from Aris
(1975)
(− ) c 1 h
τ with
=
2

τ
–– Filamentous microorganisms Lejeune and Baron (1997)
−
+
c
c
22
2
h
h
–– Filamentous microorganisms Meyerhoff et al. (1995) adopted from
Neale and Nader (1973)
(− ) c
e

xp 2.8
h
–– Filamentous microorganisms Buschulte (1992) adopted from Vorlop
(1984)
⎛
⎝
⎜
⎜

−
⎞
⎠
⎟
⎟

+− −
−
1 2
1 0.01336
0.30583 4
1 1.40296 8
8
ϕ
ϕϕ
ϕ
ϕ
– Perpendicular to
fibers
Square array of parallel fibers Perrins et al. (1979)
ϵ
2
τ with
=( ( − ϵ ) ) exp 1.16 1
2
τ
Simulated Perpendicular to
fibers
Nonoverlapping parallel fibers Tomadakis and Sotirchos (1993),
Tomadakis and Robertson (2005)
ϵ
2
τ with
()
= −
ϵ−
2 10 . 3 3
0.33
0.70
7

τ Simulated Perpendicular to
fibers
Overlapping parallel fibers Tomadakis and Sotirchos (1991)
ϵ
2
τ with
()
= −
ϵ−
2 10 . 0 4
0.04
0.10
7

τ μ CT Parallel to fibers Parallel carbon fibers Vignoles et al. (2007)
ϵ
2
τ with
()
=
−
ϵ−
2 10 . 0 4
0.04
0.46
5

τ μ CT Perpendicular to
fibers
Parallel carbon fibers Vignoles et al. (2007)
ϵ
2
τ with
()
=
−
ϵ−
2 1 0.037
0.037
0.66
1

τ
Simulated All directions Overlapping nonparallel fibers Tomadakis and Sotirchos (1991)
ϵ 1.05
3

Simulated All directions Overlapping nonparallel fibers He et al. (2017)
SCHMIDEDER ET AL .
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3361

diffusion coefficient, tort uosity, and porosity/hyphal fraction. Thereby, the
hyphal fraction, =− ϵ
c

1
h , is defined as the ratio of the volume of
hyphae to the total volume. It is important to note that t he modeling
approaches published so far (Busc hulte, 1 992; Celler et al., 2012; Cui
et al., 1998; Lejeune & Baron, 199 7; Meyerhoff et al., 1995) either neglect
the tortuosity of the hyphal structure or assume a constant tortuosity for
different porosities. These si mplifications may be caused by the absence
of information about the tortuosity. In the materials science of fibers,
correlations b etween the porosity an d the effective diffusion factor for
bulk diffusion that include knowled ge about the tortuosity have been
described. Perr ins, McKenzie, and Mc Phedran (1979) derived an analytic
expression for ideal square arrays of parallel fibers, Tomadakis and
Sotirchos (1991) a correlation for rando m distributed overlapping parallel
fibers, Tomadakis and Sotirchos (1 993) and Tomadaki s and Robert son
(2005) a relation for random distribu ted nonoverlapping parallel fibers,
Tomadakis and Sotirchos (1991) and He, Guo, Li, Pan, and Wang (2017) a
relation for 3D random distributed overlapping fibers, and Vignoles,
Coindreau, Ahmadi, and Bernard (2007) a correlation for parallel fibrous
carbon – carbon composite preforms (Tabl e 1). The only publis hed
experimental approach to predict the diffusive mass transport is based
on the measurement of oxygen concen trations using microelectrodes
inside fungal pellets (Hille, Neu, Hempel, & Horn, 2009; Wittier,
Baumgartl, Lübbers, & Schügerl, 1986). The researchers correlated the
mass transport of oxygen into t he pellets with microscopic information
from cryo ‐ slices. However, the microscopy images of the cryo ‐ slices
missed the three ‐ dimensional information of the hyphal network and
hyphae superimposed in the two ‐ dimensional project ions. Therefore,
the appropriate correlation betw een pellet morphology and diffusion was
not possible in this experimental ap proach. We could recently overcome
the limitations of two ‐ dimensional image generation by performing X ‐ ray
microcomputed tomography ( μ CT) measurements on whole fungal pellets
(Schmideder et al., 2019). The investigated Aspergillus niger pellet showed
av e r yc o m p l e xt h r e e ‐ dimensional structure. With a diameter of
6

33
m

μ
,
the A. niger pellet had a hyphal length of 1.5 m and 15,425 tips in total.
In materials science, μ CT measurements and subseque nt
mass ‐ transfer co mputations ma tured into a widely used me thod
to determine the effect ive diffusivit y of porous/fibrous mate rials.
Exemplarily, Panerai et al. (2017), Becker et al. (2011), Coin-
dreau, Mula t, Germain, L achaud, and Vignoles (201 1), and
Foerst et al. (2019) investigated the effective diffusivity of
fibrous insulators, fuel cell media, carbon ‐ carbon composites, and
maltodextrin solutions, respectively. Thereby, μ CT appeared as a
non ‐ destructive technique to measure three ‐ dime nsional mi cro ‐
structures.
In this study, we computed the effective diffusivity of A. niger
pellets based on the micro ‐ structural characterization gained from
μ CT measurements and subsequent image analysis. The newly
developed technique for fungal pellets combines the experimental
acquisition of three ‐ dimensional images with the locally resolved
calculation of the effective diffusion coefficient and tortuosity within
this structure for the first time. This results in an unprecedented
potential for the determination of diffusion processes inside fungal
pellets.
2
|
MATERIALS AND METHODS
2.1
|
Preparation of pellets
The A. niger hyperbranching strain MF22.4, which has been shown to
be a better protein ‐ secretion strain than the wild ‐ type strain (due to
deletion of the racA gene; Fiedler, Barthel, Kubisch, Nai, & Meyer,
2018), was used in this study. Pellets were obtained by submerged
cultivation of MF22.4 for 48 hr and freeze ‐ drying following the
previously described protocol (Schmideder et al., 2019).
2.2
|
X ‐ ray microcomputed tomography
To obtain three ‐ dimensional images of the freeze ‐ dried filamentous
pellets, μ CT measurements were performed based on the method
reported by Schmideder et al. (2019). Two ‐ dimensional projections
from different angles were reconstructed to generate three ‐ dimen-
sional images with a custom ‐ designed software (Matrix Technologies,
Feldkirchen, Germany) that uses CERA (Siemens, Munich, Germany).
The image resolution was 1 μ m (i.e., the edge length of the voxels was
1 μ m), and to generate the beam, 60 kV and 25 μ A were applied.
Depending on the size of the pellets, 1 ‐ 5 pellets can be measured
with one μ CT ‐ measurement (3 hr including the time for image
reconstruction). An instant adhesive (UHU, Bühl, Germany) was used
to fix the freeze ‐ dried fungal pellets on top of a sample holder. In
contrast to the previous study (Schmideder et al., 2019), in this case,
the instant adhesive dried 5 min before placing the pellets on top of
the holder. This procedure resulted in a smooth surface of the instant
adhesive, while it remained sticky enough to fix the pellets. The
smooth surface facilitated the segmentation of the instant adhesive
in the subsequent image processing.
2.3
|
Image processing
Image processing aimed at differentiating between hyphal material
and background. The background included the instant adhesive used
for sample fixation, the air between the hyphae, and small impurities.
In general, image processing is leaned to the one reported in the
section “ Preprocessing ” by Schmideder et al. (2019)). The image
processing result for one pellet is exemplarily illustrated in Figure 1.
As the instant adhesive showed similar gray values as the pellets,
we did not implement an automated segmentation. Instead, the
instant adhesive at the bottom of the pellets was cropped manually
using the commercial software VGSTUDIO MAX (version 3.2,
Volume Graphics GmbH, Heidelberg, Germany) in a first processing
step. The further image processing steps were carried out auto-
matically using MATLAB (version 2018a, MathWorks, Natick, MA).
To differentiate between hyphae and air voxels, a threshold –
calculated by Otsu ’ s method (Otsu, 1979) – was applied on the gray
value images. Finally, small connected objects with a maximum size of
1,000 μ m
3
were deleted to eliminate objects that were not part of
the pellet. The processed three ‐ dimensional pellets were used for
further diffusion computations.
3362
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SCHMIDEDER ET AL .

2.4
|
Representative cubes for diffusion
computations of A. niger pellets
To comp ut e the spati ally reso lv ed effe ct ive diff us iv ity in fung al pell et s,
we extr acted re pres enta tive cub ic sub ‐ volu me s of the pr oc esse d
thre e ‐ di mens ional im ages . The diff usi ve ma ss tran sp ort wa s comp uted ,
as de scr ib ed in Se ction 2. 6 using thes e cube s. The ce nt ers o f the cu bes
us ed for di ffu si on co mput at ions we re sele cted a long the ma in a xis
orig inat ing fr om the ca lc ula ted ma ss ce nte r of the A. ni ger pe llet s. Th e
dis ta nce be tween th e cent er s of th e cu bes al ong th e main axis wa s se t
to 25 μ m. In Fi gure 2, th e dist an ce betw een the cu be s was in crea se d
for th e sake of cl arit y. To iden tify th e in flue nce of th e cu be ‐ size
on th e di ffu sio n comp utat ions , the ed ge le ngt h of th e cu bes wa s varied
to 30 , 50 , 70, an d 90 μ m.
2.5
|
Beam ‐ Pellet
A common assumpti on for the effective d iffusion coeff icient
of filamento us microorganisms is the direct proport ionality
to the poros ity
ϵ

(Buschulte, 1992; Celler et al., 2012; Cui
et al., 1998; Sil va, Gutierrez, Dend ooven, Hugo, & Ocho a ‐ Tapia,
2001):
=⋅ ϵ
D

D .
ii ,e f f ,b u l k
(3)
Thereby, the effective diffusion factor k
eff
is assumed to be equal to
the porosity, and the tortuosity is neglected. To imitate a filamentous
spherical object, where the tortuosity can be nearly neglected, we
simulated a so ‐ called “ Beam ‐ Pellet, ” which was used to validate the
FIGURE 1 Processed three ‐ dimensional μ CT image of an Aspergillus niger pellet. The images were rendered using VGSTUDIO MAX.
(a) Projection of the whole pellet. (b) Projection of a central slice with a depth of 25 μ m [Color figure can be viewed at wileyonlinelibrary.com]
FIGURE 2 Processed three ‐ dimensional μ CT image of the Aspergillus niger pellet of Figure 1 and cubes for the diffusion computations:
(a) Transparent: projection of a whole pellet; red: exemplary cubes that were used for the diffusion computations. (b – e) Morphology of a
single cube from different viewing directions; the gray boundaries in (c – e) illustrate the boundaries parallel to the diffusion computation.
(b) Cube without illustration of boundaries for the diffusion computations. (c) Cube for diffusion in the x ‐ direction. (d) Cube for diffusion in the
y ‐ direction. (e) Cube for diffusion in the z ‐ direction [Color figure can be viewed at wileyonlinelibrary.com]
SCHMIDEDER ET AL .
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3363

method of the diffusion computation and critically scrutinize the
tortuosity neglect in the literature.
The “ Beam ‐ Pellet ” (Figure 3) was built up from equally sized
filaments (in diameter and length) with their origin in the pellet
center and having a radial orientation. To guarantee a uniform
distribution of the filaments in space, their orientation was calculated
using the HEALPIx (hierarchical equal area iso ‐ latitude pixelization)
discretization (Gorski et al., 2005). In this way, 50,700 representa-
tive, equally distributed points were calculated on the pellet surface;
all of them were connected to the pellet center. Then, the connected
lines were dilated with MATLAB to obtain a “ Beam ‐ Pellet ” with a
defined diameter of the filaments. The diameter was chosen to be
3 μ m, similar to the average hyphal diameter of A. niger (Colin,
Baigorí, & Pera, 2013; Nielsen, 1993; Schmideder et al., 2019). To
investigate the influence of the image resolution on the subsequent
diffusion computations, similar “ Beam ‐ Pellets ” with different resolu-
tions were simulated. Starting from a ” Beam ‐ Pellet ” with a radius of
700 voxels and a dilation of one voxel, the resolution of the filaments
was increased. Thereby, the number of filaments was kept constant,
whereas the radius of the “ Beam ‐ Pellet ” was set to 1,167, 1,633,
2,567, and 3,500 voxels and the dilation was set to 2, 3, 5, and
7 voxels, respectively. Thus, the “ Beam ‐ Pellets ” only differed in the
scaling and the resolution of the filaments. The effective diffusivity of
the “ Beam ‐ Pellet ” was analyzed, as described in Section 2.6. This
analysis required cubes that represent the whole pellet. Similar to the
case of the A. niger pellets, cubes located along the main axes were
chosen to investigate the effective diffusivity (Figure 3). Identical to
the final analysis of the A. niger pellets, the cube ‐ edge length was set
to 50 μ m, and the cubes were selected along the main axis. Thus, the
cube edge length was 50 voxels for the “ Beam ‐ Pellet ” with a radius of
700 voxels and a dilation of one voxel. To guarantee that the same
structure (50 μ m×5 0 μ m×5 0 μ m) was analyzed for higher
resolutions, we increased the cube edge length to 83, 117, 183,
and 250 voxels, respectively. In Figure 4 , the same representative
cube is shown for different resolutions.
2.6
|
Computation of the effective diffusivity
To compute the effective diffusion factor an d tortuosity of
filamentous structures (Equa tion (2)), the module DiffuDi ct of the
commercial software GeoDict (Becker et al., 2011; Velichko,
Wiegmann, & Mücklich, 2009; Wiegmann & Zemitis, 2006;
Math2Market Gmbh, Kaiserslautern, Germany) was used. As
DiffuDict allows fo r the voxel ‐ based solut ion of transport equa-
tions, the processed three ‐ dimensio nal image data o f the μ CT
measurements as well as the si mulated “ Beam ‐ Pellet, ” could be
used for the dif fusion compu tations. Diff uDict requires cubic
domains for th e diffusion co mputations, and t herefore, cubic
sub ‐ volumes of the filamentous pe lle ts we re extra cted f rom the
three ‐ dime nsional images fo r further analysis. Th e selections of
representa tive sub ‐ v o l u m e sa r ed e s c r i b e di nS e c t i o n s2 . 4a n d
2.5 for A. n ige r pell ets an d the “ Beam ‐ Pellet, ” r espe ctive ly.
Concept ual ly, as sho wn in Fi gur es 2c – e and 3c – e, the di ffusi on
compu tation s in Di ffuD ict w ere exec uted in one of the three main
axes f or each co mput ation . As th e repre sent ati ve cub es were chos en
along th e mai n axes , we w ere ab le to ap ply three com putati ons on
each cu be an d coul d thus analy ze th e effec tive diff usiv ity of each
cube in th e radi al and in tw o tange nti al dire cti ons. The computa -
tiona l effor t to obta in th e diffus ivi ty of on e cube in the th ree
direc tions was abou t 1 mi n for cu bes with 50 × 50 × 5 0 vox els with
an Inte l Xeon E5 ‐ 16 60 CPU (3.7 GH z). The detail s of th e computa -
tions ar e desc ribe d in the fo ll owing.
FIGURE 3 Simulated “ Beam ‐ Pellet ” and cubes for diffusion computations: (a) Transparent: projection of the whole pellet; red: exemplary
cubes that were used for the diffusion computations. (b – e) Morphology of a single cube from different viewing directions; the gray boundaries
in (c – e) illustrate the boundaries parallel to the diffusion computation. (b) Cube without illustration of boundaries for the diffusion
computations. (c) Cube for diffusion in the x ‐ direction. (d) Cube for diffusion in the y ‐ direction. (e) Cube for diffusion in the z ‐ direction [Color
figure can be viewed at wileyonlinelibrary.com]
3364
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SCHMIDEDER ET AL .

We computed the diffusion in the spa ce/liquid between the
hyphae (porous medium). The p redominant diffusion regime in
liquids is bulk diffusion (Becker et al., 2011; Panerai et al., 2 017),
that is, mass transport is mainly driven by collisions between fluid
molecules. In our ap proach, we negle cted surface effect s on the
solid – liquid interface that could influence diffusion. One possible
effect could be surf ace diffusion, that is molecules can diffuse on
the surface of pores. This phenomenon is known to be an
important transpo rt mechanism in reversed ‐ phase liquid chroma-
tography. However, predictions are difficult and depend on the
temperature, s urface concentr ation, and surface chemistry
(Medve ď & Č ern ỳ , 2011; Miyabe & Guiochon, 2010). Electrical
double ‐ layers o n solid – liquid interfaces could change the diffusiv-
ity through porous media, especially the diffusivity of ions (Gabitto
& Tsouris, 2017). In the present study we assumed pure bulk
diffusion. In the case of pure bulk diffusion, the diffusion in the
pores can be modeled by the Laplace equation, with Neumann
boundary conditions on the pores ‐ to ‐ solids boundaries and a
concentration drop of the diffusing component in the diffusion
direction (Becker et al., 2011). Thus, we applied bulk diffusion in
DiffuDict and applied a concentration drop of the diffusing
component between the inlet and outlet. To avoid a bias in the
diffusion computations, we c hose Dirichlet boundary conditions on
the in ‐ and outlet and symmetric boundary conditions on the other
four faces. Computing the total diffusion flux through the porous
structures and applying Fick ’ s law, DiffuDict calculates the
effective diffusion factor (Becker et al., 2011; Wiegmann &
Zemitis, 2006). Ac cor ding to Equation (2), the tortuosity can be
calculated from the porosity and the e ffective diffusion factor.
Therefore, both the effective diffusion factor and the tortuosity
were calculated fo r eac h representative sub ‐ volume of the
filamentous pellets.
3
|
RESULTS AND DISCUSSION
3.1
|
Effective diffusivity of the “ Beam ‐ Pellet ”
The “ Beam ‐ Pellet ” introduced in Sectio n 2.5 (Figu re 3) imita tes a
hypothetical filamentous pell et that grew only in the radial
direction and is similar but not identical to parall el fibers. We
compared the diffusion behavior of the “ Beam ‐ Pellet ” with
literature correlations for the di ffu sion th rough p aral lel fib ers
to validate th e diffusio n computations. Ad ditionall y, we
critically scrutinized the li terature assumption that
=ϵ k
eff
(Buschulte, 1992; Celler et a l., 2012; Silva et al., 2001; Van ’ t
Riet & Tramper, 1991), where the tortuosity is neglected and the
effective diffusi on factor is assumed to be only depen dent on the
porosity.
FIGURE 4 Same representative cube of
“ Beam ‐ Pellets ” with different resolutions:
(a) Three voxels per filament (dilation 1).
(b) Five voxels per filament (dilation 2).
(c) Seven voxels per filament (dilation 3).
(d) 15 voxels per filament (dilation 7)
[Color figure can be viewed at
wileyonlinelibrary.com]
SCHMIDEDER ET AL .
|
3365

The results of the diffusion computations on the “ Beam ‐ Pellet ”
are shown in Figure 5 for the hyphal fraction range 0.05 – 0.4. A
hyphal fraction of 0.4 was the maximum value for the A. niger pellets
investigated in this study. Figure 5a shows the relation between the
hyphal fraction of cubic sub ‐ volumes and their corresponding
effective diffusion factor. In the radial diffusion direction, the
effective diffusion factor decreased almost linearly with increasing
hyphal fractions. The radial effective diffusion factors showed a
similar behavior as that of the often applied assumption for
filamentous microorganisms:
=ϵ k
eff (Buschulte, 1992; Celler et al.,
2012; Silva et al., 2001; Van ’ t Riet & Tramper, 1991). This formula is
the prediction of the “ law of mixtures ” for the flow along parallel
fibers (Tomadakis & Robertson, 2005). In the tangential directions,
the effective diffusion factors were much smaller than their radial
counterparts – and the slope was by no means in the order of minus
one. Figure 5b shows the tortuosities for the radial and tangential
diffusions, and it can be seen that they increase with increasing
hyphal fractions. As expected, the tortuosities in the tangential
diffusion direction were much higher than their radial counterparts.
To investigate the influence of the image resolution on the
diffusion computations, we simulated “ Beam ‐ Pellets ” with different
resolutions (Section 2.5). With increasing resolutions, the effective
diffusion factor increased and the tortuosity decreased, which
was probably caused by the increased circularity of the filaments
(Figure 4). As shown in Figure 3 and 4, representative cubes of the
“ Beam ‐ Pellets ” closely resemble parallel nonoverlapping fibers.
Literature correlations for bulk diffusion through parallel fibers
(Perrins et al., 1979; Tomadakis & Sotirchos, 1991, 1993; Tomadakis
& Robertson, 2005) suggest, that the applied diffusion computations
underestimate the diffusivity in the case of low resolutions. With
increased resolutions, the computed diffusion results approach the
literature correlations. However, it has to be mentioned that the
literature correlations are for a square array of parallel fibers
(Perrins et al., 1979), parallel random distributed overlapping fibers
(Tomadakis & Sotirchos, 1991), and random distributed nonoverlap-
ping fibers (Tomadakis & Robertson, 2005; Tomadakis & Sotirchos,
1993) and thus similar but not identical to our “ Beam ‐ Pellets. ”
In contrast to the tangent ial diffusion pat hs, the radial paths of the
“ Beam ‐ Pellet ” were not windi ng (Figure 3). In the ory, increased pa th
lengths result in an increased tortuosity and a decre ased effective
diffusion factor ( Epstein, 1989). This b ehav ior could be observed for the
“ Beam ‐ Pellet ” in the radial and tangential diffu sion directi ons as well.
The small difference between the computed radial effective diffusion
factors/tort uosi ties and the “ law of mixtures ” for the flow al ong parallel
fibers (Tomadakis & Rob ertson, 2005)
=ϵ k
eff wa s e vo ked b y th e ra dia l
direction of the filaments of the “ Beam ‐ Pellet. ” Thus, the filaments were
not completely parallel to each other. Further investigations with
simulated perfectly parallel fi laments resulted in
=ϵ k
eff and a consta nt
tortuosity of o ne (data now shown).
To sum up, the diffusi on behavior of the “ Beam ‐ Pelle t ” was
consistent with the diffusion theo ry described by Epstein (1989) and
could be used to verify the applied diff usion computations. The
comparison to literature correlation sa b o u tt h ed i f f u s i v i t yo fp a r a l l e l
fibers suggests t hat our diffusion computations of fibers with a low
image resolution tend to underesti mate the diffusivity, whereas high
resolutions approach the l iterature correlation s. The radial diffusi on
behavior of the “ Beam ‐ Pellet ” illustrated the o ften applied lit erature
assumpt ion for fi lamento us micro organism s:
=ϵ k
eff (Buschul te, 1992;
Celler et al., 2012; Silva et al., 2001; Van ’ t Riet & Tramper, 1991).
However, the morphology of the idealized “ Beam ‐ Pellet ” (Figure 3) was
very different from the μ CT data of A. nige r p e l l e t s( F i g u r e2 ) .T h u s ,t h e
actual correlation between the eff ective diffusiv ity and the hyphal
fract ion o f A. niger pellets was investigated in further detail.
3.2
|
Effective diffusivity of A. niger pellets
Contrary to the “ Beam ‐ Pellet ” (Figure 3), in the case of the A. niger
pellets, the voids, that is, the spaces between the hyphae (Figure 2),
(a) (b)
FIGURE 5 Diffusion computations of “ Beam ‐ Pellets ” in the radial and tangential directions and compa rison to existing literature correlations for
parallel fibers (Table 1). The “ Beam ‐ Pelle ts ” differ in the fiber ‐ diameter in voxels, whereas the diameter in μ m stayed constant (Figure 4). Each data point
results from a single diffusion c omputation of a cubic sub ‐ volume with a cube ‐ edge length of 50 μ m. Cros ses and circles correspond t o computed
diffusion properties in the radial and tangential directions, respectively. The black lines represent literature correlations. The hyphal fraction corresponds
to the ratio of the volume of hyphae to the total volume in the cubic sub ‐ volumes [ Color figure can be vi ewed at wileyonlinelibrary.com]
3366
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SCHMIDEDER ET AL .

are strongly winded. According to Epstein (1989) that should result
in an increased tortuosity, and therefore, in a decreased effective
diffusivity. The tortuosity and effective diffusion factors of five A.
niger pellets are investigated in this section. The diameters of the
pellets were between 410 and 570 μ m.
To investigate the influence of the size of the cubic sub ‐ volumes
on the diffusion computations, the edge length of the cubes was
varied. Figure 6a shows the relation between the effective diffusion
factor and the hyphal fraction for different cube sizes. The data
include the diffusion computations of the five investigated A. niger
pellets in the radial direction for cube ‐ edge lengths of 30, 50, 70, and
90 μ m. Generally, the effective diffusion factors for different cube ‐
edge lengths showed similar behavior. When no hyphal material is
present, that is, when the hyphal fraction is zero, the effective
diffusion factor is one, and thus, the diffusion is not geometrically
hindered. With increasing hyphal fraction, the effective diffusion
factor decreases. The applied cube ‐ edge lengths did not have a high
impact on the diffusion results. Thus, a cube ‐ edge length of 50 μ m
was applied for further computations in this study. Figure 6b) shows
the effective diffusion factors of the five A. niger pellets studied
herein for a cube ‐ edge length of 50 μ m. It can be seen that the values
for the different pellets varied only slightly. As five pellets were
investigated, this very subtle scattering of the data points implies
that the method is quite reproducible when applied to different
pellets obtained from the same cultivation sample.
Figure 7 shows the results of the diffusion computations for the
five studied A. niger pellets in the radial and tangential directions for
a cube ‐ edge length of 50 μ m. Figure 7a shows that the computed
effective diffusion factors of the A. niger pellets were much smaller
than the values expected from the literature assumption
=ϵ k
eff , and
therefore, also much smaller than the values obtained for the
investigated “ Beam ‐ Pellet. ” A slight anisotropy was observed when
comparing the radial and tangential diffusion directions because the
effective diffusion factors in the radial direction were slightly higher
(a) (b)
FIGURE 6 Correlations between the radial effective di ffusion factor and hyphal fraction (ratio of the volume of hyphae to the total volume) for five
Aspergillus niger pellets. Each data point results from a single d iffusion computation of a cubic sub ‐ vol ume: (a) Diffusion computations with different cube
sizes. (b) The cube ‐ edge lengt h was 50 μ m; each color rep resents the inves tigated cube s of one pellet [Color figure can be viewed at
wileyonlinelibrary.com]
(a) (b)
FIGURE 7 Diffusion computations of five Aspergillus niger pellets in the radial and tangential directions. Each data point results from a single
diffusion computation of a cubic sub ‐ volume with a cube ‐ edge length of 50 μ m; k
eff
is the effective diffusion factor and
ϵ

is the porosity: (a) The
blue and green data points correspond to effective diffusion factors in the radial and tangential directions, respectively; the black line
represents the literature assumption
=ϵ k
eff . (b) The blue and green data points represent the tortuosities in the radial and tangential
directions, respectively [Color figure can be viewed at wileyonlinelibrary.com]
SCHMIDEDER ET AL .
|
3367

than their counterparts in the tangential direction. In the absence of
hyphal material, that is, when the hyphal fraction is zero, the
tortuosity is one (Figure 7b). With increasing hyphal fraction, the
tortuosity increases as well. Again, a slight difference was observed
between the radial and tangential diffusion directions, with the radial
diffusion computations resulting in lower tortuosities than their
tangential counterparts. According to Equation (2), the lower
tortuosities explain the higher effective diffusion factors in the radial
direction. In the model assumption
=− + =ϵ kc 1
eff h
, the tortuosity
is assumed to be constantly one. This simplification explains
the differences between our computed effective diffusion factors
for the A. niger pellets and the values expected from the literature
(Figure 7a) as well as those reported for the “ Beam ‐ Pellet ” in the
previous section.
The local hyphal fraction range of the five investigated A. niger
pellets was 0 – 0.4. To the best of our knowledge, there are no
reports in the literature, in which the hyphal fraction of filamentous
pellets is higher than the maximum hyphal fraction measured in this
study, for example, Cui, Van der Lans, and Luyben (1997, 1998)
reported average hyphal fractions between 0.07 and 0.30 for whole
Aspergillus awamori pellets. Thus, the hyphal fraction ranges of the
investigated A. niger pellets could already be representative for
realistic pellets.
3.3
|
Correlation between effective diffusivity and
hyphal fraction
The diffusion computations of the five investigated A. niger pellets
(Section 3.2) are compared to literature assumptions (Table 1) for the
correlation between the effective diffusion factor ( k
eff
) and the
hyphal fraction ( c
h
)/solid fraction/porosity (
ϵ

=− c 1
h

) of filamentous
microorganisms (Figure 8a) and fibers (Figure 8b). As the diffusion of
spherical pellets should be driven mainly by radial diffusion, we have
investigated it herein. Additionally, a new modeling approach was
introduced to correlate the effective diffusion factor and the hyphal
fraction.
The computed diffusion factors differed strongly from the
assumption
=ϵ k
eff , which has been used for simulation/modeling
studies of filamentous microorganisms by Celler et al. (2012),
Buschulte (1992), and Silva et al. (2001). In fact, the computed
effective diffusion factors were far below the expected values. In
those previous models, the effective diffusion factor was assumed to
be only dependent on the porosity of the material, while neglecting
the tortuosity. Thus, the difference between our computed effective
diffusion factors and previous assumptions (Buschulte, 1992; Celler
et al., 2012; Silva et al., 2001) was not surprising. The second linear
literature assumption for filamentous microorganisms was =ϵ / k 2
eff
(Lejeune & Baron, 1997). In their work, Lejeune and Baron (1997)
considered the tortuosity to be consistently 2, without explaining
that assumption. As shown, their model differed strongly from the
computed data. Their model would result in a geometrically hindered
diffusion for a hyphal fraction of zero. Thus, especially for low hyphal
fractions, that assumption seems to be untenable. In their growth
modeling approach for filamentous microorganisms, Meyerhoff et al.
(1995) applied a nonlinear correlation between the effective
diffusion factor and the hyphal fraction: = ( −) / ( +) kc c 22
eff h h . This
approach seemed to approximate our computed data better than the
two previous model assumptions from the literature. Additionally,
the effective diffusion factor is 1 and 0 for hyphal fractions of 0 and
1, respectively. In theory, these conditions should be fulfilled.
However, the model seemed to overestimate the effective diffusion
factors. Besides the assumption
=ϵ k
eff
, Buschulte (1992) deduced a
second model for filamentous microorganisms: = −
k e c
eff 2.8
h

.I n
comparison with the other literature assumptions for filamentous
microorganisms, this approach fitted our computed data best.
However, in the hyphal fraction range of 0 – 0.4, the model seemed
to underestimate the computed data. Additionally, for a hyphal
fraction of 1, the model would result in an effective diffusion factor of
(a) (b)
FIGURE 8 Correlations between hyphal fraction ( c
h
)/solid fraction/porosity (
ϵ

=− c 1
h

) and effective diffusion factor ( k
eff
). The blue data
points correspond to the computed effective diffusion factors of five Aspergillus niger pellets in the radial direction, with the cube ‐ edge length for
the diffusion calculations being 50 μ m. The solid bold blue line shows the new correlation between the hyphal fraction and the effective
diffusion factor; the black lines represent existing correlations in the literature (Table 1) for (a) filamentous microorganisms and (b) 3D random
distributed overlapping fibers [Color figure can be viewed at wileyonlinelibrary.com]
3368
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SCHMIDEDER ET AL .

0.06. In theory, for a hyphal fraction of 1, the effective diffusion
factor has to be 0 (Epstein, 1989).
The correlations for 3D random distributed overlapping fibers
(Figure 8b) fitted our data better than the correlations existing for
filamentous microorganisms. Our data fall in between the correlation
of Tomadakis and Sotirchos (1991) and He et al. (2017). In both
studies, the computation of the effective diffusivity was well
validated for other structures like parallel fibers. However, they
differ significantly. Tomadakis and Sotirchos (1991) applied a
modification of Archie ’ s law (Archie, 1942) to correlate the tortuosity
factor = ( ( − ϵ) / ( ϵ − ϵ) ) 1
2 pp
τ α , with the percolation porosity
ϵ

= 0.03
7

p and α = .661. It has to be mentioned that the modification
of Archie ’ s law also fitted well for Vignoles et al. (2007) for bulk
diffusion of μ CT ‐ generated images of parallel fibrous carbon ‐ carbon
composite preforms. The percolation porosity was
ϵ

= 0.04
p and
=∕ = 0.107 0.46
5 α

α for diffusion parallel/perpendicular to the
fibers, respectively. According to Nam and Kaviany (2003), the
effective diffusivity of isotropic structures is often estimated using a
power function of porosity. Thus, He et al. (2017) fitted their
diffusion results of 3D random distributed overlapping fibers and
found =ϵ k n
eff α , with = 1.0
5 α

and =
n 3

.
To overcome the limitations and/or inaccuracies in the correla-
tions between the effective diffusion factor and hyphal fraction for
filamentous microorganisms and to obtain a relation with only one
fitting parameter, we propose a new correlation:
=( − ) kc 1
,

a
eff h (4)
where a is the only fitting parameter. This rather simple expression
guarantees theory ‐ consistent effective diffusion factors of 1 and 0 at
hyphal fractions of 0 and 1, respectively, and provides an excellent fit
to our data. Minimizing the error squares,
=± a 2.02 0.0
2

(with 95%
confidence bounds):
=( − ) ±
kc 1.
eff h 2.02 0.02 (5)
In conclusion , the literature assumptions for mo deling diffusion in
filamentous pellet s failed to fit our computed results, whereas
correlations for fibers in material science fitted our data better. Thus,
we set up a nonlinear equation (5) approach with only one fitting
parameter, which is a special case for the power function of porosity, for
=
1 α

as well as for the modified Archie ’ sl a w ,w h e n
ϵ

=
0

p .T h i s
approach modeled the correl ation between the effective dif fusion factor
and the hyphal fraction for five inv estigated A. niger pellets qui te well.
3.4
|
Proposed workflow in bioprocess
development
Depending on the product of i nter est, a saturation or a limitation
of substrates inside fungal pellets is pursued (Veiter et al., 2018).
To predict the spatial distributio n of substrates inside p ellets, the
effective dif fusivity through t he fibrous network has to be known
(Buschulte, 1992; Celler et al., 2012). Thus, our newly proposed
method to determine the effective diffusivity of filamentous fungal
pellets with μ CT measurements and subsequent diffusion compu-
tations through the three ‐ dimensional morphology has consider-
able benefits for bioprocess development. We propose the
following idealized workflow to achieve an optimal/suitable pellet
morphology:
(1) Generate pellets through experiments or simulations
a) Cultivate different strains at different process conditions;
apply μ CT measurements and subsequent image analysis of
pellets (Schmideder et al., 2019)
b) Simulate various three ‐ dimensional pellet ‐ networks with
algorithms similar to Celler et al. (2012); model calibration
could be carried out based on μ CT measurements with
subsequent image analysis (Schmideder et al., 2019)
(2) Compute the correlation between the hyphal fraction and the
effective diffusivity for each existing and simulated pellet of
Step 1, as described in the present study
(3) Compute the proportion of substrate ‐ limited and substrate ‐
saturated regions of each pellet based on the consumption ‐ and
diffusion ‐ terms of models such as Buschulte (1992)); apply
correlation of the hyphal fraction and effective diffusivity in
diffusion terms (this study; Step 2)
( 4 ) Assemble data base of experimentally or simulatively generated
pellets including substrate ‐ s u p p l ya n dm o r p h o l o g i c a lf e a t u r e s
( 5 ) Select optimal/suitable pellet fo r the desired process from data base
(6) Realize optimal/suitable pellet in bioprocess, for example,
through the upscale of previous experiments (Step 1a), genetic
modifications, or process control
Obviously, some of these steps have to be investigated and
elaborated in much more detail to reach the proposed optimum
macromorphology through this workflow. However, we consider the
investigation of the effective diffusivity as an important step towards
morphological engineering. In our study, we investigated five pellets
of one process and the correlation between the effective diffusivity
and hyphal fraction scattered only slightly. Thus, we propose, that
our method is at least reproducible for a certain strain at certain
process conditions. If the observed correlation between the hyphal
fraction and the effective diffusivity is representative for the applied
A. niger strain in general, other fungal strains, and/or theoretically all
filamentous microorganisms should be the focus of future studies.
Thereby, as described, μ CT measurements are suitable to detect the
three ‐ dimensional morphology used for diffusion computations.
However, other three ‐ dimensional methods such as confocal laser
scanning microscopy of pellets slices or even simulated pellets are
also conceivable to explore other strains and processes.
4
|
CONCLUSIONS
The findings described in this manuscript unveil the actual relation
between the hyphal fraction ( c
h
, i.e., the ratio between the volume of
SCHMIDEDER ET AL .
|
3369

hyphae and the total volume) and the effective diffusion factor ( k
eff
)
and tortuosity inside filamentous fungal pellets. They also uncover a
discrepancy with the assumptions made in the literature for
filamentous microorganisms so far. We propose a new correlation,
which is inspired by correlations for fibers, rather simple, consistent
with theoretical expectations, and shows an excellent fit to the
investigated A. niger pellets: keff=(1 − c
h
)
2
. Our μ CT images resulted
in hyphae with a diameter of approximately five voxels. As indicated
in Section 3.1 and Figure 5, that might lead to a slight under-
estimation of the diffusivity. For future studies of fibers/filamentous
microorganisms, we recommend μ CTs, that could resolve fibers/
hyphae to a diameter of approximate 11 voxels. Alternatively, a
correction factor could be calculated based on diffusion computa-
tions of simulated filamentous pellets. The method described in this
study is not limited to A. niger but can also be applied to a variety of
fungal species as well as to other organisms forming filamentous
structures. It is conceivable to use the described workflow not only
based on μ CT data, but also on other three ‐ dimensional data such as
confocal laser scanning microscopy (CLSM) images of smaller
structures and pellet slices, or even simulated pellets. The computed
diffusion parameters, and thus the diffusion rates of substrates and
products inside fungal pellets, could in combination with a consump-
tion and production model, be applied to predict the actual metabolic
flux inside filamentous pellets. With this information, it would be
possible to propose an ideal fungal macromorphology for the
production of a certain substance, which could then be achieved by
genetic engineering and control of the process parameters during
fermentation.
ACKNOWLEDGMENTS
The authors want to thank Markus Bet z and Christian Preischl for
preliminary studies on diffusion computations of filamentous
pellets and Clarissa Schulze for assistance with μ CT measure-
ments. We also wish to thank Christoph Kirse and Michael Kuhn
for helpful discussions about diffusion mechanisms. This study
made use of equipment that was funded by the Deutsche
Forschungsgemeinschaft (DF G, German Research Founda-
tion) – 198187031. The authors thank the Deutsche Forschungsge-
meinschaft for financial support for this study within the SPP 1934
DiSPBiotech – 3153843 07 and 315305620 .
CONFLICT OF I NTERESTS
The authors declare that there is no conflict of interests.
ORCID
Stefan Schmideder http://orcid.org/0000-0003-4328-9724
Lars Barthel http://orcid.org/0000-0001-8951-5614
Henri Müller http://orcid.org/0000-0002-4831-0003
Vera Meyer http://orcid.org/0000-0002-2298-2258
Heiko Briesen http://orcid.org/0000-0001-7725-5907
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Why organizations use Identific for document trust, entry 48

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