Modeling of Fibrin Gels Based on Confocal Microscopy and Light-Scattering Data

Abstract

Fibrin gels are biological networks that play a fundamental role in blood coagulation and other patho/physiological processes, such as thrombosis and cancer. Electron and confocal microscopies show a collection of fibers that are relatively monodisperse in diameter, not uniformly distributed, and connected at nodal points with a branching order of ∼3–4. Although in the confocal images the hydrated fibers appear to be quite straight (mass fractal dimension D_m = 1), for the overall system 1<D_m<2. Based on the confocal images, we developed a method to generate three-dimensional (3D) in silico gels made of cylindrical sticks of diameter d, density ρ, and average length 〈L〉, joined at randomly distributed nodal points. The resulting 3D network strikingly resembles real fibrin gels and can be sketched as an assembly of densely packed fractal blobs, i.e., regions of size ξ, where the fiber concentration is higher than average. The blobs are placed at a distance ξ₀ between their centers of mass so that they are overlapped by a factor η = ξ/ξ₀ and have D_m ∼1.2–1.6. The in silico gels’ structure is quantitatively analyzed by its 3D spatial correlation function g_3D(r) and corresponding power spectrum I(q) = FFT_3D[g_3D(r)], from which ρ, d, D_m, η, and ξ₀ can be extracted. In particular, ξ₀ provides an excellent estimate of the gel mesh size. The in silico gels’ I(q) compares quite well with real gels’ elastic light-scattering measurements. We then derived an analytical form factor for accurately fitting the scattering data, which allowed us to directly recover the gels’ structural parameters.

Introduction

Fibrin gels have been the subject of countless studies due to their fundamental importance in the process of blood coagulation. They constitute the network inside which platelets and other blood components are trapped, forming the hemostatic plug that stops bleeding (1). Fibrin gels also play important roles in other pathological and physiological situations, such as thrombosis and cancer (the latter because some tumoral cells are known to migrate through fibrin networks (2)). They exhibit extraordinary physical and mechanical characteristics (e.g., elasticity, extendibility, and resistance), and, given their natural origin, are fully biocompatible materials. As a consequence, they are ideal substrates for many biotechnological applications, ranging from surgery (e.g., adhesives and sealants called fibrin glues) to tissue engineering (specific cell growth for tissue regeneration) and drug delivery. A recent review on fibrin gels and their biotechnological applications can be found in Janmey et al. (3).

Fibrin gels are grown from polymerization of the macromolecule fibrinogen, after activation by the enzyme thrombin. According to the currently accepted mechanism, the activated monomers aggregate forming half-staggered, double-stranded fibrils that initially grow in length and then, when they are long enough to interact with each other, start to branch and aggregate laterally; eventually, a three-dimensional (3D) network is obtained (see Weisel (1) and references therein). The final structure of the gel depends on the physical-chemical conditions of the solution in which the gel is grown, such as the concentration of fibrinogen and thrombin, pH, ionic strength, and presence of Ca²⁺ ions. Fibrin gels can be schematically classified as either thin gels made of narrow pores and thin fibers, or coarse gels made of large pores and thick fibers. In the first case, fibrin gels behave as semiflexible polymers networks in which the thin fibers (bundles of a few fibrils linked together) have a persistence length comparable to the gel mesh size, whereas in the second case, the thick fibers (bundles of hundreds of fibrils) are much stiffer, with persistence lengths that are much longer than the gel mesh size. In both cases, however, the fibers are very soft (if compared with equal-diameter fibers of other biopolymer networks, such as F-actin or collagen), can be deformed or stretched to a quite large extent (4), and exhibit an impressive strain-stiffening behavior (5), a feature that is essential for the gel’s ability to function as an efficient hemostatic plug and a wound-healing matrix.

The mechanical and elastic properties of fibrin gels, and of biopolymer fibrous networks in general, have been intensively studied via rheological (6,7) and imaging techniques such as electron microscopy and, more recently, confocal and atomic force microscopy (6,8,9). These studies, motivated by the still-unclear physical mechanism underlying the highly nonlinear elasticity that is the basis of the strain-stiffening behavior that characterizes almost all known biopolymer fibrous networks, have spurred the development of many network models aimed at reproducing the observed mechanical properties of the various systems. Most of these models have been implemented in two dimensions (10–15), but with the advent of confocal imaging, 3D models are becoming more and more popular (16–21). The common feature of all of these models is that they are validated by comparing mainly their mechanical properties with those of the real network they aim to reproduce, and very little attention is paid to the agreement between the static and structural/morphological properties of the real gel versus those of the in silico network. In this work, we followed the complementary approach of developing a method for generating 3D in silico networks that is focused on matching the static structural properties with those of real fibrin gels as revealed by confocal microscopy and elastic light scattering (ELS) measurements (22–24).

When observed on a confocal microscope under near-native conditions, a fibrin gel appears as a collection of mostly straight fibers that are relatively monodisperse in diameter, not uniformly distributed in space, and connected together at nodal points with a branching order that is not easily quantifiable due to resolution issues (25,26). However, electron microscopy determinations have constrained the branching order to ∼3–4 (27). At the same time, ELS data show that the gel structure exhibits a fractal morphology, with a mass fractal dimension 1<D_m<2, and is characterized by a long-range spatial order whose length scale is comparable to the gel mesh size (22–24). Taking into account these structural features, we implemented a simple iterative algorithm that allows one to generate in silico gels as an assembly of cylindrical sticks of equal diameter d, joined together at randomly distributed nodal points. The resulting 3D network bears a striking resemblance to that of real fibrin gels and, when quantitatively analyzed by means of its 3D spatial correlation function g_3D(r) and corresponding power spectrum I(q) = FFT_3D[g_3D(r)], it exhibits a fractal morphology with D_m close to that of actual fibrin gels and a spatial structure ordered on length scales comparable to the average length of the gel fibers. Based on these results, we were able to describe the gel as an assembly of densely packed fractal blobs (see Ferri et al. (22,23) and references therein), i.e., regions of mass fractal dimension D_m and size ξ where the fiber concentration is higher than average. The blobs are placed at a distance ξ₀ between their centers of mass so that they are overlapped by factor η = ξ/ξ₀. This blob model allowed us to refine a previously developed analytical expression (23) of a form factor that is capable of accurately fitting the ELS data of real fibrin gels. Furthermore, it provided, for the first time to our knowledge, the key interpretation that ξ₀ is the physical quantity that must be used to correctly estimate the gel mesh (or pore) size, which can be geometrically defined as the average diameter of the largest spheres that can be accommodated in the pore zones of the gels and are tangent to the surrounding fibers (28).

Materials and Methods

In silico generation of fibrin gels

We generated 3D in silico gels aimed at mimicking the structures of real fibrin gels, according to the following procedure:

• 1.A set of N points of coordinates (x_i,y_i,z_i), i = 1,2..N were spread randomly in a volume V. These points are the nodal points from which the fibers will depart and join together at other nodal points.
• 2.A nodal point is selected at random and tentatively connected to its first three nearest-neighbors points with straight cylindrical fibers of (constant) diameter d and (different) length L. Although the first two shortest fibers are always annexed as part of the network, the third one is accepted only if the sum of the largest two angles formed between the three fibers is >240°. If this condition is not fulfilled, the third fiber is rejected, a next nearest-neighbors point is found, and the above criterion is applied iteratively until a third fiber is annexed to the network. The angle-based acceptance criterion ensures that unphysical nodal points at which three fibers form acuminate angles are rejected and, at the same time, that Y-shaped nodal points, which are very common in real fibrin gels, can be formed.
• 3.The procedure outlined in step 2 is repeated for all of the nodal points generated at step 1, regardless of the fact that the nodal point being considered was already connected to other point(s) at some previous step(s). In that case, previous connections are ignored and the three fibers found at the current step might be either duplicates of already selected fibers, or truly new fibers that are connected to the point. The resulting effect is to have a nodal point whose branching order might be >3.

Note that in the procedure outlined above, all of the nodal points are treated in the same way, regardless of any previously established connection. Thus, the resulting network is univocally determined only by the nodal point positions, independently of the order in which the points are connected.

A typical gel obtained in this way is depicted in Fig. 1, which shows a 3D rendering of the entire network. The gel structure was characterized in terms of the statistical properties of its single fibers and the morphological properties of the entire network. We studied the former by computing the fiber length distribution P_L(L), the branching order distribution P_k(k), and the angle distribution P_α(α), between fibers. We recovered the latter by computing, over the entire gel volume, the 3D density-density correlation function g_3D(r), which allowed us to investigate the self-similarity of the overall gel structure and characterize it by means of its mass fractal dimension D_m.

Figure 1.

3D rendering of an in silico fibrin gel generated with the algorithm described in the text. (a) Cube side = 25 μm, fiber diameter d = 0.2 nm. (b) Zoom of a portion of panel a, cube side = 10 μm.

Confocal microscopy of fibrin gels

Fully formed fibrin gels polymerized under physiological conditions from solutions of fibrinogen (FG) and thrombin (Thr), at FG concentration c_F = 0.5 mg/ml and Thr/FG ratio 1/100, were imaged with an Olympus Fluoview 500 confocal microscope (Olympus Biosystems, Hamburg, FRG), equipped with a 60×, NA 1.40 oil immersion objective. For a detailed description of the confocal microscopy data and sample preparation, see Molteni et al. (28).

Results and Discussion

Structure of in silico gels

The procedure outlined in the section “In silico generation of fibrin gels” leads to the formation of a 3D network of cylindrical fibers of different lengths connected together at the nodal points generated at point 1. Depending on the nodal points’ concentration, the resulting network may have an appearance resembling that of a real gel. This is illustrated in Fig. 2, where a 3D rendering of a fibrin gel images stack obtained from our confocal microscope (Fig. 2 a) is compared with the same rendering of an in silico gel (Fig. 2 b). The in silico gel was generated with a fiber diameter d = 200 nm, and the rendering of Fig. 2 b was obtained after convolution of the 3D stack with the PSF of the confocal microscope (28) used to obtain the data of Fig. 2 a. The similarities between the two images are quite remarkable and show that in each case the structure is characterized by a collection of straight fibers joined together at some nodal points whose branching order is mainly ∼3–4. Moreover, in spite of the fact that the nodal points are randomly spread in space, the fibers forming the two gels are not distributed uniformly in space, and correlated spatial fluctuations are formed (see gray highlighted rectangles in Fig. 2). This feature is sketched in Fig. 3, where the mechanism underlying the in silico gel formation is illustrated. Wherever the local concentration of the nodal points is higher than average, the fibers are shorter and the branching is higher than 3. Conversely, in the regions of lower nodal-point concentration, the fibers are longer and the branching order is equal to 3. The resulting effect is the formation of spatially correlated blobs of denser fiber concentration, characterized by an average size ξ and an effective average branching order k >3. For the gel of Fig. 2 b, we had 〈k〉 = 4.0 ± 0.9.

Figure 2.

(a) 3D rendering of a real fibrin gel obtained from confocal microscopy images. The gel was prepared at a fibrinogen concentration c_F = 0.5 mg/ml, as in Ferri et al. (23). (b) The same rendering for an in silico gel obtained as described in the text. In both cases, the gel structure can be sketched as an assembly of mass fractal blob regions of size ξ (see text).

Figure 3.

Sketch of the mechanism underlying the formation of an in silico gel. Nodal points are randomly spread in space and sequentially connected to their nearest neighbors with straight fibers (red solid lines). Each point is connected independently of the other ones, and previous fiber connections are ignored (see text). This mechanism leads to the formation of a network in which fibers are connected at points with different branching order, equal to 3 (blue or black), 4 (yellow or gray) or 5 (green or white). Nodal points and fibers are also correlated in space inside regions of size ξ, where there is a higher density of shorter fibers.

As a further comment about Fig. 2, a and b, it must be said that their similarities hold down to length scales in the submicrometer range, which correspond to the microscope resolution (∼200 nm (28)). However, when observed with an electron microscope (27), fibrin gels grown under physical-chemical conditions similar to the one in Fig. 2 a show fibers with a relatively small diameter polydispersity (∼25–30%) and their nodal points appear to be mainly trifunctional, with average branching orders of 〈k〉 = 3.4 ± 0.8 or 〈k〉 = 3.1 ± 0.3 (see Table 1 of Ref. 27). Thus, our model fails to reproduce these two features of a real fibrin gel. However, as discussed in Appendix A of the Supporting Material and shown below in Fig. 7, such discrepancies do not invalidate the ability of our in silico gels to reconstruct the μm-length-scale structural features of real fibrin gels.

Figure 7.

(a) Power spectra I(q) for all the gels of Fig. 5 (set A) obtained by using Eq. 3. (b) ELS data R(q) from fibrin gels grown under the same quasi-physiological conditions (adapted from Ferri et al. (23)). (c) Master curve I(q)/c obtained by rescaling all the spectra of panel a by their concentration, c. The similarity between the I(q) and R(q) curves in the fractal range is quite remarkable.

We carried out a quantitative analysis of the overall gel structure by computing its 3D spatial density-density correlation function g_3D(r) = 〈δn(x) δn(x + r)〉_x, where δn(x) is the gel density fluctuation at the 3D point x defined as δn(x) = n(x) – 〈n〉. Here n(x) is the gel density, which assumes the binary values n(x) = 1 in correspondence to the gel fibers and n(x) = 0 elsewhere, and 〈n〉 is the average gel density computed over the entire gel volume. We carried out the computation by exploiting the radial symmetry of g_3D(r) [g_3D(r) = g_3D(r), where r = |r|], which allows us to obtain g_3D(r) as the z average of the 2D spatial correlation functions computed on horizontal sections of the gel structure at the various heights z. To do that, we sliced the 3D gel volume into a stack of N_z images, each made of N_x × N_y pixels. The voxel size was 25 × 25 × 25 nm and the side of the cubic gel volume was 25.6 μm or 51.2 μm, corresponding to N_x = N_y = N_z = 1024 or 2048. A 2D image representing a horizontal section of the in silico gel of Fig. 1 a at a given height z is sketched in Fig. 4 a. Since the fibers are randomly oriented cylindrical segments of diameter d, their horizontal sections are randomly oriented ellipses characterized by the same short axis (equal to d) and by different long axes whose length and orientation depend, respectively, on the incident and azimuthal angles between the fiber axis and the normal to the horizontal plane. If we indicate with ρ the 2D vector lying on the plane at height z, we can write the 2D-spatial correlation function as g_2D(r, z) = 〈δn(ρ, z) δn(ρ + r, z)〉_ρ, which is clearly not symmetric, as sketched in Fig. 4 b. However, by averaging over different z-planes, we obtain a correlation function that depends only on r = |r|, and we can write

$${\text{g}}_{3\text{D}}\left(\text{r}\right)={\langle {\text{g}}_{2\text{D}}\left(\text{r},\text{z}\right)\rangle }_{\text{z}}$$ (1)

Figure 4.

Sketch of the method for computing the 3D correlation function of the gel. (a) Horizontal section of the gel at a given height, z. (b) Corresponding 2D correlation function. (c) Mean 2D correlation function averaged over many sections at different z-values.

Notice that the procedure indicated by Eq. 1 is quite simple and can be easily implemented on confocal microscopy data, provided that the convolution effects associated with the microscope PSF are negligible, as in the case of a 4 Pi-STED microscope (29,30).

We synthesized in silico fibrin gels of different volume fraction ϕ by changing the concentration n_p of the nodal points at which the gel fibers are connected and the fiber diameter d. A quantitative relation between ϕ, n_p, and d can be found in Eq. S1 of the Supporting Material, Appendix A, where all these gels are shown to be characterized by the same distributions of fiber length, angle between fibers, and branching order. We synthesized two data sets: set A, in which we fixed the fiber diameter d = 200 nm and changed the nodal point concentration n_p so as to span a volume fraction range between 2.8×10⁻⁴ < ϕ < 3.2×10⁻² (see Figs. S2 and S3 of Appendix A), and set B, where both d and n_p were varied so as to reproduce a behavior of d as a function of ϕ (d ∼ ϕ ^0.15) similar to what was observed in ELS data (23). In the case of set B, we varied the diameter between 150 ≤ d ≤ 250 nm and the volume fraction between 3.9×10⁻⁴ < ϕ < 1.1×10⁻².

Fig. 5 shows the 3D correlation functions g_3D(r) of the gels corresponding to set A. In the main panel (Fig. 5 a), all of the curves exhibit a power-law decay characterized, for the lower concentration gels, by the same exponent α∼1.8, which corresponds to a mass fractal dimension D_m = 3 − α∼1.2. All of the curves are also perfectly superimposed for r ≤ 0.1 μm and exhibit the same full width at half-maximum δ_FWHM ∼ 0.22 μm (indicated by the horizontal arrow in inset b). At values of r ≥ 0.5 μm, the curves deviate from their power-law decay and turn toward negative values, where they exhibit their minima. The positions at which the g_3D(r) functions cross zero, r_zxs, and exhibit their minima, r_min (see inset c), depend on ϕ, with higher-concentration gels exhibiting more pronounced minima with smaller r_zxs and r_min. Because the determination of r_min is somewhat troublesome (due to the shallow and broad minimum), we focused only on r_zxs. The behavior of r_zxs (open black circles) as a function of ϕ is reported in Fig. 6 a and compared with the average fiber lengths 〈L〉 (solid green circles) found from the distributions of Fig. S2. As one can see, the two curves are quite similar and the ratio between the two is fairly constant, i.e., r_zxs / 〈L〉 ∼1 (see inset, filled circles). We repeated the same analysis for the gels of set B and found quite similar results: the behavior of g_3D(r) versus ϕ is similar to that reported in Fig. 5 a (data not shown) and δ_FWHM scales linearly with d, with a ratio δ_FWHM/d ∼ 1.07 equal to the one found for set A. Similarly, r_zxs (open black triangles) and 〈L〉 (solid green triangles) are quite comparable and their ratio is fairly constant (∼1; see inset, filled triangles) over the entire ϕ range. We can therefore state that the following relations hold independently for ϕ and d:

$$\frac{{\mathit{r}}_{\text{zxs}}}{\langle \mathit{L}\rangle }=0.97\pm 0.04$$ (2a)

$$\frac{{\delta }_{\text{FWHM}}}{\mathit{d}}=1.07\pm 0.02$$ (2b)

Equations 2a and 2b provide an easy and statistically robust method for recovering accurate estimates of the average fiber length 〈L〉 and fiber diameter d from the 3D density-density correlation function of the gel. The uncertainties reported in Eqs. 2a and 2b refer to a statistical analysis carried out over all the gels of set A and B. It should be pointed out, however, that whereas Eq. 2a is independent of fiber diameter polydispersity, Eq. 2b is not, because it has been recovered numerically for in silico gels characterized by different but monodisperse fiber diameters. For example, in the case of a typical (number) relative diameter polydispersity of ∼25% (27), Eq. 2b becomes ${\delta }_{\text{FWHM}}\text{/}{\langle \mathit{d}\rangle }_{\text{nb}}\sim 1.1$ or ${\delta }_{\text{FWHM}}\text{/}{\langle \mathit{d}\rangle }_{\text{wt}}\sim 1.0$, where 〈d〉_nb and 〈d〉_wt are the number and weight averaged diameters, respectively. Although it would be relatively easy to implement in our in silico models a polydisperse distribution of fiber diameters similar to that observed in real gels, given the minor effect it has on the recovered parameters, we elected to avoid further complicating the modeling at this stage.

Figure 5.

(a) Correlation function g_3D(r) for a series of gels (set A) generated at different volume fractions in the range of 2.8×10⁻⁴ < ϕ < 3.2×10⁻² with the same diameter d = 200 nm, reported on a log-log plot. The slope α is related to the mass fractal dimension by D_m = 3 − α and, for the lower concentration gels, α ∼1.8, corresponding to D_m ∼1.2. (b) At small r-values, all curves are superimposed and characterized by the same FWHM, δ_FWHM, which allows us to estimate the fiber diameter as δ_FWHM ∼1.07 d. (c) At larger r-values, all curves turn negative at different zero-crossing positions, r_zxs, and exhibit shallow minima at positions r_min. The values of r_zxs are related to the average fiber length 〈L〉.

Figure 6.

Comparison of the average fiber length 〈L〉 and the zero crossing (r_zxs) of the gel correlation functions for the two sets of gels (sets A and B) as a function of gel volume fraction ϕ. (Inset) the ratio r_zxs / 〈L〉 for all the data point of gels of sets A and B is fairly constant over the entire ϕ range, with average value equal to 0.97 ± 0.04.

As a final test, we also compared the power spectra I(q) associated with the in silico gels of sets A and B with typical ELS data obtained from real fibrin gels formed under the same conditions used for the gel in Fig. 2 a. As is known, I(q) is given by the 3D Fourier transform (FT) of g_3D(r) and, thanks to its isotropy, can be written in terms of the 1D sine-transform:

$$\mathit{I}\left(\text{q}\right)\sim \int {\text{g}}_{3\text{D}}\left(\text{r}\right){\text{e}}^{\text{i}\mathbf{q}\cdot \mathbf{r}}\text{d}\mathbf{r}=4\pi \underset{0}{\overset{\infty }{\int }}{\text{r}}^2\frac{\text{sin}\left(\text{qr}\right)}{\text{qr}}{\text{g}}_{3\text{D}}\left(\text{r}\right)\text{dr}$$ (3)

which one can easily compute numerically by using a 1D fast FT (FFT) algorithm. Notice that aside from a proportional factor, I(q) is the same quantity that is recovered in an ELS experiment, that is, the scattered intensity distribution R(q). Fig. 7 a reports the power spectra I(q) for the in silico gels of Fig. 5 (set A), while Fig. 7 b shows the ELS data obtained from fibrin gels at different concentrations grown under quasi-physiological conditions (see Fig. 4 of Ferri et al. (23), from which the data were taken). The similarities between the in silico and real data are quite remarkable, particularly in the fractal regime where both I(q) and R(q) decay as power laws, with exponents that appear to be fairly similar when gels at the same concentrations are compared. Indeed, the concentrations of real gels range from c₁ = 0.064 mg/ml to c₅ = 0.61 mg/ml, which correspond to volume fractions ϕ₁ ∼3.2 × 10⁻⁴ to ϕ₅ ∼3.0 × 10⁻³ (fiber density ∼0.2 g/cm³; see next section). Thus, the real gels’ concentrations are comparable to the lower concentrations of the in silico gels, and in both cases the decay exponent is D_m∼1.2. We observe also that the extension of the fractal q-range decreases with increasing concentration, going from ∼2 decades (∼0.1–10 μm⁻¹) at ϕ ∼10⁻³, to <1 decade for the highest (in silico) gel concentration at ϕ ∼3 × 10⁻². Correspondingly, the fractal dimension increases from D_m∼1.2 to D_m∼1.6.

A comparison of Fig. 7, a and b, also shows that in silico and real gels exhibit a similar intensity dependence on concentration. In particular, if we divide all the I(q) data of Fig. 7 a by ϕ, we obtain a single master curve showing that the I(q) values of all the gels are perfectly superimposed at large wavevectors (q≥ 30 μm⁻¹) where they exhibit equal oscillations due to the same fiber diameter d = 200 nm (Fig. 7 c). When the same type of rescaling is applied to real gels, no single master curve is obtained (data not shown), consistent with the fact that the gels of Fig. 7 b appear to be characterized by fibers of different diameters. This feature can be easily deduced from the data because gels at lower concentrations cross over to a much higher faster decay at larger q-values, indicating that their fibers have smaller diameters. A similar behavior is observed in the in silico gels of set B (data not shown). Unfortunately, the data of real gels do not cover wavevectors q ≥ 30 μm⁻¹, so it is not possible to observe oscillations as in the in silico gels.

One further (but not essential) difference between the real and in silico data occurs at low q-vectors, where the in silico gels do not present a clear peak as the real gels do. The latter feature probably reflects the fact that the spatial correlations are stronger in real gels than in in silico gels, and thus produce a sharper peak in the scattering intensity distribution and, in turn, a deeper minimum in the correlation function.

In conclusion, we can say that our simple representation of a fibrin gel, described as a network of straight fibers of different lengths connected together at some randomly distributed nodal points, is fairly good and is able to describe the main structural features of a real fibrin gel over length scales down to the submicrometer range. We will exploit this result in the next section to implement a form factor function that is capable of fitting real light-scattering data.

Form factor of a fibrin gel

Starting from the results of the previous section, we can describe the complex structure of a fibrin gel by following an approach similar to one originally suggested in Ferri et al. (22,23). As depicted in Fig. 8, the gel can be described as a collection of spatially correlated fractal blobs of size ξ, centered on the regions where the fiber concentration is higher. Because the blobs appear to be densely packed, their average distance ξ₀ can be related to their size by means of the overlapping factor η = ξ/ξ₀. Each blob is made of an assembly of n segments or building blocks that can be sketched as cylindrical objects with diameter d and length ℓ. These segments are stacked together to form straight fibers of length L, but they can also branch off at some bonding site, producing a fully interconnected ramified structure. According to this picture, the segment length ℓ represents the minimum fiber length between two nodal points, whereas a fiber is made of many segments joined together and its length L can be much longer than ℓ. As described in Appendix A, the length of these fibers is fairly polydisperse, ranging between a minimum value of the order of d and a maximum value of the order of ξ. Thus, in our blob model, we fix the segment length to the value ℓ = d.

Figure 8.

Sketch of the gel structure based on the blob model described in the text (ξ is the blob size and ξ₀ is the average distance between blobs). Blobs are densely packed and can overlap by a factor η = ξ/ξ₀. Each blob is a fractal collection of straight fibers of the same diameter d and density ρ, linked together at few nodal points. Each fiber can be thought as a stack of cylindrical segments of length ℓ = d, so that the minimum fiber length is equal to d. Finally, each segment is obtained by packing together many protofibrils of diameter d₀<<d and ρ₀>ρ.

The intensity scattered by such a system can be written as the product of a structure factor S(q) that describes the spatial correlations between the blobs’ centers of mass, and a form factor P(q) that describes the blobs’ internal structure. Assuming that all the blobs have the same size ξ and that all the fibers have the same diameter d, the scattered intensity R(q), which when expressed in absolute units [cm⁻¹] is called the Raleigh ratio (31), reads:

$$R\left(q\right)=\text{K}{\mathit{c}}_{\text{F}}\text{M}\left[\stackrel{\text{S}\left(\mathit{q}\right)}{\overbrace{1-\beta \text{exp}(-\frac{q^2{\xi }^2}{4\pi {\eta }^2})}}\right]\times \left[\underset{\text{A}\left(\mathit{q}\right)}{\underbrace{\underset{{\text{A}}_1}{\underbrace{\frac{1}{{\left[1+{\left(\text{q}\xi /{\delta }_{{\mathit{D}}_{\text{m}}}\right)}^2\right]}^{{\mathit{D}}_{\text{m}}/2}}}}-\underset{{\text{A}}_2}{\underbrace{\frac{{\left(\mathit{d}/\xi \right)}^{{\mathit{D}}_{\text{m}}}}{{\left[1+{\left(\mathit{q}\mathit{d}/{\delta }_{{\mathit{D}}_{\text{m}}}\right)}^2\right]}^{{\mathit{D}}_{\text{m}}/2}}}}+\underset{{\text{A}}_3}{\underbrace{{(\mathit{d}/\xi )}^{{\mathit{D}}_{\text{m}}}}}}}\right]{\text{P}}_{\text{seg}}\left(\mathit{q},\mathit{d}\right)$$ (4)

where K [cm²/g²] is the usual optical constant ($\text{K}=4{\pi }^2{\mathit{n}}^2{\left(\text{d}\mathit{n}/\text{d}{\mathit{c}}_{\text{F}}\right)}^2/{\text{N}}_{\text{A}}{\lambda }^4$, where n is the solvent refraction index, N_A is Avogadro’s number, and λ is the vacuum wavelength of light) and c_F [g/cm³] is the blobs’ concentration equal to the fibrinogen concentration. In Eq. 4, M [g] is the blobs’ molecular mass:

$$\text{M}\sim {\text{N}}_{\text{A}}\frac{\rho \pi }{4}{\xi }^{{\mathit{D}}_{\text{m}}}{\mathit{d}}^{3-D_{\text{m}}}$$ (5)

where ρ [g/cm³] is the segment (or fiber) density. The term ${\delta }_{\mathit{D}\text{m}}$ appearing in Eq. 4 is a numerical parameter that has been introduced heuristically to improve the fitting accuracy in the fractal regime for any value of D_m between 1 < D_m < 2 (see Supporting Material, Appendix B-a). The last term appearing in Eq. 4, P_seg(q,d), is the segment form factor equal to the form factor of randomly oriented cylinders (32) of diameter d and length ℓ = d (see Supporting Material, Appendix B-c). A detailed description of the main features of Eq. 4 and its relationship to the blob model sketched in Fig. 8 is provided in the Supporting Material, Appendices B and C.

Fitting results: comparison of real and in silico gels

We tested the reliability of the in silico networks and corresponding blob model that led us to the fitting function development (Eq. 4) by comparing the fitting results of the in silico gels of sets A and B with the ones obtained from the experimental data of Ferri et al. (23). The latter refer to two different data sets: the first one, in which classical (CLS) and low-angle ELS (LAELS) were combined together to cover almost 3 decades in q, corresponds to the data reported in Fig. 7 b; and the second, for which only LAELS data were taken, includes data reported in Fig. 4 of Ferri et al. (23) but not shown here. The procedure for the data fitting (both real and in silico) is described in Ferri et al. (23) and recalled in Appendix D of the Supporting Material.

Fig. 9 shows the results of this analysis for both real and in silico gels as a function of the gel volume fraction ϕ. For the real gels, the concentration was obtained as ϕ = c_F/ρ, where c_F is the fibrinogen concentration and ρ is the fiber density obtained by fitting the CLS+LAELS data of Fig. 7 b. From these fittings (shown as solid lines in Fig. 7 b) we obtained an average value ρ = 0.2 ± 0.1 g/cm³, which is smaller by a factor of ∼2 with respect to the value obtained with the old fitting function used by Ferri et al. (22,23). We then used the value ρ = 0.2 g/cm³ as a fixed parameter for ρ when fitting the LAELS data of Ferri et al. (23) and calculating their volume fraction concentrations. Fig. 9 a shows that the mass fractal dimensions D_m of all the in silico and real gels are quite similar, going from D_m ∼1.2 for the lower concentrations to D_m ∼1.6 for the highest concentration of the in silico gels. Fig. 9 b shows an excellent agreement between the fiber diameters of real gels and those of in silico gels (set B). Notice that, with respect to the previous data reported in Fig. 6 b of Ferri et al. (23), the diameters are larger by roughly a factor of √2 because now the fiber density is smaller by a factor of 2. As for set A, all of the recovered diameters are fairly consistent, with the expected value of d = 200 nm. Fig. 9 c shows that also the overlapping parameter η assumes values that are comparable between real and in silico gels, although the overlapping of the in silico gels appears to be smaller and scales somewhat differently with ϕ. Finally, Fig. 9, d and e, show that the values of ξ and ξ₀ are quite comparable between in silico and real data, but the two behaviors as a function of ϕ are fairly different. In particular, whereas ξ varies with ϕ in a somewhat irregular way, the behavior of ξ₀ versus ϕ is well described by a power-law decay with an exponent close to γ ∼0.55. This value is close to the one that would be obtained if, in the close packing condition expressed by Eq. S16 of Appendix C, one could neglect η and consider only the c_F dependence of ξ₀ $({\xi }_0\sim {\mathit{c}}_{\text{F}}^{-1/(3-{\text{D}}_m)})$. In that case, the equivalent fractal dimension would be D_m ∼1.18, a value that is consistent with the apparently asymptotic (ϕ→0) behavior exhibited by D_m in Fig. 9 a.

Figure 9.

Behavior as a function of the sample volume fraction, ϕ, of the fitted parameters that characterize in silico gel set A (open circles) and set B (open triangles), and the real gels via the CLS+LAELS (blue stars) and LAELS (solid red squares) data. (a) Mass fractal dimension D_m. (b) Fiber diameter d. (c) Overlapping parameter η. (d) Blob size ξ. (e) Distances between blobs ξ₀. In panel e the behaviors of the 3D-average pore sizes of in silico gel set A (dots) and set B (small triangles) are also reported. The accurate matching between 〈ξ₀〉 and 〈D〉_3D shows that 〈ξ₀〉 provides an excellent estimate of the gel mesh size.

However, the most important result expressed by Fig. 9 e is the matching between ξ₀ of the in silico gels, both sets A and B, and the curves corresponding to the in silico gel mesh sizes, set A (dots) and set B (triangles). The latter ones were computed with a 3D-bubble method (28) that, when the coordinates of all the fibers composing the gel are known, allows one to find the average diameter 〈D〉_3D of the largest spheres that can be accommodated in the pore zones of the gel. Thus, Fig. 9 e shows that ξ₀ provides an excellent estimate of the gel mesh size with ξ₀/〈D〉_3D = 1.00 ± 0.03. For further discussion of this subject, we refer the reader to the study by Molteni et al. (28), in which the 3D-bubble method is described in detail.

As a final test, we compared the zero crossing parameters r_zxs of the real and in silico gels. We recovered these parameters by analyzing the correlation functions g_3D(r) that we obtained by inverting Eq. 3 with the fitted form factor I(q). The comparison is shown in Fig. 10, and one can observe that real gels are characterized by r_zxs-values that are only slightly smaller than those of in silico gels at the same ϕ. This result is rather intriguing because, by using Eq. 2a, we have obtained a tool for estimating the average fiber length 〈L〉 of the real gels from ELS data only, without the need for any real-space techniques such as confocal microscopy.

Figure 10.

Comparison as a function of the sample volume fraction, ϕ, between the recovered zero crossing, r_zxs, of the correlation functions, g_3D(r), for in silico gel set A (circles) and set B (triangles), the real gels via the CLS+LAELS data (blue stars), and the LAELS data (solid red squares).

Conclusions

In this work, we have developed a simple method for generating 3D in silico networks that capture the main static structural properties of fibrin gels on submicrometer length scales, as they emerge from confocal microscopy and ELS measurements. The network formed in this way is made of straight fibers of different lengths, monodisperse in the diameter d, linked together at some at randomly distributed nodal points whose average branching order is ∼4 ± 1. The algorithm used for the network construction is based on a simple mechanism (depicted in Fig. 3) that exploits the random statistical fluctuations of the nodal points’ density. Wherever the nodal points’ density is higher than average, the fibers are shorter with a higher branching order, whereas in regions of lower nodal point density, the fibers are longer and the branching order is equal to 3. The final result is the formation of a network that can be modeled as a close-packed assembly of mass fractal blobs of size ξ placed at distance ξ₀, overlapped by factor η = ξ/ξ₀, characterized by a mass fractal dimension D_m.

We quantitatively assessed the validation of the blob model by comparing ELS data obtained from real fibrin gels with the power spectra I(q) = FFT_3D[g_3D(r)] of the in silico gels obtained by (fast) Fourier transforming their 3D density-density correlation functions g_3D(r). The fairly good match between the two allowed us to refine a previously developed analytical expression (22,23) of a form factor that is capable of fitting the ELS data with high accuracy. Thus, we could retrieve all the parameters ρ, d, D_m, η, ξ₀, and, indirectly (via g_3D(r)), the average fiber length 〈L〉. The latter is accurately determined by the position r_zxs at which g_3D(r) crosses zero (see Eq. 2a).

A quite important application of the model is the possibility of relating the parameters ξ or ξ₀ derived from the fit with the average gel pore size, which is one of the most important physical parameters affecting the biological functions of filamentous networks. Indeed, because the coordinates of the fibers composing an in silico gel are known, it is possible to find (28) the geometrical average pore size of the network by determining the average diameter 〈D〉_3D of the largest spheres that can be accommodated in the pore zones of the gel and are tangent to the surrounding fibers (28). Thus, as shown in Fig. 9 e, we have provided the key interpretation that ξ₀ (i.e., the average distance between blobs) is the physical quantity that gives the correct estimate of the gel mesh size.

In conclusion, we believe that our simple algorithm, which allows us to represent a fibrin gel as a network of straight fibers of different lengths connected together at some randomly distributed nodal points, is rather faithful and quite accurately reproduces many of the geometrical properties that characterize a gel structure. We also believe that such a method for generating in silico gels with controlled known geometrical properties can be very useful for developing networks with tunable elastic and mechanical properties, or for assessing and benchmarking other pore-sizing techniques. An example of such an application is reported in Molteni et al. (28), where we propose a 2D analysis for estimating the pore size of 3D gels. Further work is also in progress to extend the modeling procedure to include semiflexible fibers and fiber diameter polydispersity, and to constrain the branching order to better match experimentally observed values.