Spatio-temporal Models of Lymphangiogenesis in Wound Healing

Abstract

Several studies suggest that one possible cause of impaired wound healing is failed or insufficient lymphangiogenesis, that is the formation of new lymphatic capillaries. Although many mathematical models have been developed to describe the formation of blood capillaries (angiogenesis), very few have been proposed for the regeneration of the lymphatic network. Lymphangiogenesis is a markedly different process from angiogenesis, occurring at different times and in response to different chemical stimuli. Two main hypotheses have been proposed: (1) lymphatic capillaries sprout from existing interrupted ones at the edge of the wound in analogy to the blood angiogenesis case and (2) lymphatic endothelial cells first pool in the wound region following the lymph flow and then, once sufficiently populated, start to form a network. Here, we present two PDE models describing lymphangiogenesis according to these two different hypotheses. Further, we include the effect of advection due to interstitial flow and lymph flow coming from open capillaries. The variables represent different cell densities and growth factor concentrations, and where possible the parameters are estimated from biological data. The models are then solved numerically and the results are compared with the available biological literature.

Appendix: Parameter Estimation

1.1 Sizes, Weights, Equilibria and Velocities

1.1.1 Domain Size

We consider a full-thickness wound of length $\ell =5 \text{ mm }$, inspired by Zheng et al. (2007). For the surrounding skin, we consider a (small) variable width $\varepsilon$. Thus, we have a domain of length $5 \text{ mm } + 2\varepsilon$. In all the simulations reported in the present paper, $\varepsilon =1$; the nature of the observations does not change if a different value of $\varepsilon$ is chosen (simulations not shown).

1.1.2 TGF-$\beta$ Molecular Weight and Equilibrium $T^\mathrm{eq}$

We take TGF-$\beta$ molecular weight to be approximately 25 kDa (Boulton et al. 1997; Wakefield et al. 1988, active/mature isoform). The equilibrium value of active TGF-$\beta$ is about 30 pg/mm$^3$ (Yang et al. 1999, Figure 2).

1.1.3 Macrophage Volume and Equilibrium $M^\mathrm{eq}$

A human alveolar macrophage has a volume $V_{M\Phi }$ of approximately $5000 \mu \text{ m }^3 = 5\times 10^{-6} \text{ mm }^3$ (Krombach et al. 1997). The macrophage steady state can be estimated from Weber-Matthiesen and Sterry (1990, Figure 1), which plots typical macrophage density in the skin. This shows that there is an average of about 15 macrophages per 0.1mm$^2$ field. Assuming a visual depth of 80 $\mu$m, the macrophage density becomes 15 cells/(0.1mm$^2\times 0.08$mm) $=$ 1875 cells/mm$^3$.

1.1.4 VEGF Molecular Weight and Equilibrium $V^\mathrm{eq}$

VEGF molecular weight is taken to be 38 kDa (Kaur and Yung 2012; Yang et al. 2009, VEGF-165). The VEGF equilibrium concentration is estimated to be 0.5 pg/mm$^3$ from Hormbrey et al. (2003, Figure 1) and Papaioannou et al. (2009, Figure 2).

1.1.5 Normal Capillary Density $C^\mathrm{eq}$

In Rutkowski et al. (2006), we find that “it was not until day 60, when functional and continuous lymphatic capillaries appeared normal” and “at day 60 the regenerated region had a complete lymphatic vasculature, the morphology of which appeared similar to that of native vessels”. Hence, we assume that a capillary network that can be considered “final” appears at day 60, and we take $C^\mathrm{eq}$ to be the number of LECs present at this time. In Rutkowski et al. (2006, Figure 2E), we see that at that time there are about 80 cells. This value corresponds to a 12 $\mu$m thin section. In addition, from Rutkowski et al. (2006, Figure 2D) we can calculate the observed wound area, which is about $5.6\times 10^5 \, \mu \text{ m }^2$. In this way, we get a volume of 0.0067 mm$^3$ with 80 cells, which corresponds to $C^\mathrm{eq}=1.2\times 10^4$ cells/mm$^3$.

1.1.6 Maximum Capillary Density $C_{\max }$

First of all, we want to convert 1 capillary section into a cell number. For this purpose, we assume EC cross-sectional dimensions to be those reported in Haas and Duling (1997), namely $10\,\mu \text{ m }\times 100\,\mu \text{ m }$. We then assume that LECs lie “longitudinally” along the capillaries, and therefore, only the short dimension contributes to cover or “wrap” the circumference of the capillary. Considering a capillary diameter of 55 $\mu$m as in Fischer et al. (1996), we have that each lymphatic capillary section is made of approximately 20 LECs (taking into account some overlapping). Then, from van der Berg et al. (2003) we know that EC thickness is approximately 0.5 $\mu$m. Thus, a capillary section is a circle of about $55+2\times 0.5 \, \mu$m diameter, corresponding, as described above, to 20 cells.

If we imagine stacking 1 mm$^3$ with capillaries of this size, we see that we can pile on $1 \text{ mm }/ 56\,\mu \text{ m } \approx 18$ layers of capillaries. Then, considering an EC length of 100 $\mu$m as in Haas and Duling (1997), we have that 1 mm$^3$ fits at most a number of capillaries equivalent to the following amount of ECs:

$$\begin{aligned} 20 \text{ cells } \times 18 \times 18 \times \frac{1 \text{ mm }}{100 \, \mu \text{ m }} \approx 6.4 \times 10^4 \text{ cells } = C_{\max } \; . \end{aligned}$$

1.1.7 Lymph Velocity

Fischer et al. (1996) suggests that the high lymph flow value (0.51mm/s) is due to high pressure following die injection. This suggests that a lower value (9.7 microns/s) might be considered as typical, in agreement with Fischer et al. (1997). In both papers, the normal lymph velocity seems to be around 10 microns/s.

We thus assume lymph velocity to be $v_\mathrm{lymph}$ $=$ 10 micron/s $=$ 864 mm/day (from Fischer et al. 1996, 1997).

1.1.8 Interstitial Flow Velocity

First of all, we note that in Rutkowski and Swartz (2007) interstitial flow in the skin is calculated to be around 10 microns/s. [Note that Helm et al. (2005) is relevant for this aspect of our modelling, although it is less important for the estimation of parameters; in this reference, the synergy between interstitial flow and VEGF gradient is discussed.] Therefore, we will consider the interstitial flow to be also $v_{IF}$ $=$ 10 microns/s $=$ 864 mm/day (from Rutkowski and Swartz 2007).

1.2 Re-calculation of $s_M$ and $k_1$

$s_M$ here is calculated in the same way as in Bianchi et al. (2015), but using our amended model equations presented here. For $k_1$, we point out that in Bianchi et al. (2015) this parameter was appearing in the logistic part of the M-equation: ${{d M}/{dt} = r_2 M - {r_2}/{k_1}\cdot M^2}$. In the PDE systems, we do not include such terms because only a minor fraction of macrophages undergo mitosis (Greenwood 1973). However, death due to overcrowding is present in both models; comparing these terms, we see that our “new” $k_1$ corresponds to the “old” $k_1 / r_2$.

1.3 Diffusion Coefficients

1.3.1 VEGF Diffusion Coefficient $D_V$

In Miura and Tanaka (2009), the authors observe that “in general, the diffusion coefficient of protein molecules in liquid is of the order of $10^6\,\mu \text{ m }^2/\text{ h }=24\,\text{ mm }^2/\text{ day }$. This intuitively means that a molecule moves 10 $\mu$m/s. To generate a gradient over the order of 100 $\mu$m, the timescale of protein decay should be around 10 s. In this specific case, the protein decay time is about 1–10 h. Therefore, the observed diffusion coefficient is too large and we need some mechanism to slow down the diffusion” (where “this specific case” means that of VEGF).

In Miura and Tanaka (2009) the VEGF diffusion coefficient is estimated in three different ways: by a theoretical model ($0.24 \text{ mm }^2/\text{ day }$), and by two different empirical techniques ($24 \text{ mm }^2/\text{ day }$). The authors then suggest a diffusion coefficient of the order of $10^6\,\mu \text{ m }^2/\text{ h }=24 \text{ mm }^2/\text{ day }$. However, they also used the same technique to determine the diffusion coefficient at the cell surface; this time the diffusion coefficient is estimated to be approximately $10^4\,\mu \text{ m }^2/\text{ h }=0.24\,\text{ mm }^2/\text{ day }$. Keeping in mind all these considerations, for the model we take the intermediate value $D_V = 2.4\,\text{ mm }^2/\text{ day }$.

1.3.2 TGF-$\beta$ Diffusion Coefficient $D_T$

In Lee et al. (2014) the authors estimate a TGF-$\beta$ diffusion coefficient of 0.36 mm$^2$/h $=$ 8.64 mm$^2$/day from Brown (1999), Goodhill (1997). In Murphy et al. (2012), the authors estimate a TGF-$\beta$ diffusion coefficient of 2.54 mm$^2$/day using the Stokes–Einstein Formula.

We checked their consistency with the estimate for $D_V$ above. The Stokes–Einstein equation of these calculated values assumes spherical particles of radius r to have diffusion coefficient $D\sim {1}/{r}$; since the molecular weight w of a particle is proportional to its volume, we have that $D\sim {1}/{\root 3 \of {w}}$ and thus $D_T \approx 2.76$.

1.3.3 Macrophage Random Motility $\mu _M$

In Farrell et al. (1990), we find “Population random motility was characterised by the random motility coefficient, $\mu$, which was mathematically equivalent to a diffusion coefficient. $\mu$ varied little over a range of C5a [a protein] concentrations with a minimum of $0.86 \times 10^{-8} \text{ cm }^2/\text{ sec }$ in $1\times 10^{-7}$ M C5a to a maximum of $1.9\times 10^{-8} \text{ cm }^2/\text{ sec }$ in $1\times 10^{-11}$ M C5a”. We thus take $\mu _M$ to be the average of these two values, that is $\mu _M = 1.38\times 10^{-8}\text{ cm }^2/\text{ s }\approx 0.12 \text{ mm }^2/\text{ day }$.

1.4 Advection Parameters $\lambda _1$ and $\lambda _2$

We will take $\lambda _2^\mathrm{chem}$ to be equal to $v_{IF}$ calculated in “Appendix of Sizes, Weights, Equilibria and Velocities”; thus, $\lambda _2^\mathrm{chem}$ = 864 mm/day. For $\lambda _1^\mathrm{chem}$, it is more complicated, but we would say that if $C_\mathrm{op}$ reaches the maximum possible value $C_{\max }$ calculated in “Appendix of Maximum Capillary Density $C_{\max }$”, then $\lambda _1^\mathrm{chem}\cdot C_\mathrm{op} = v_\mathrm{lymph}$, which was calculated in “Appendix of Sizes, Weights, Equilibria and Velocities”. That is, we assume that if the skin is “packed” with open capillaries, then the resulting flow will be the same as the usual lymph flow in the skin lymphatics). Hence, $\lambda _1^\mathrm{chem} = v_\mathrm{lymph}/C_{\max } = 0.0135 \text{ mm } \text{ day }^{-1}\text{ cell }^{-1}$. For cells, we assume smaller values due the higher friction that cells encounter in the tissue. In the absence of relevant empirical data, we take ${\lambda _1^\mathrm{cell}={1}/{10}\cdot \lambda _1^\mathrm{chem}}$ and ${\lambda _2^\mathrm{cell}={1}/{10}\cdot \lambda _2^\mathrm{chem}}$.

1.5 Rate at Which TGF-$\beta$ is Internalised by Macrophages $\gamma _1$

At equilibrium, $C=C^\mathrm{eq}$ and thus $p(C)=0$. Therefore, the equation for T at equilibrium becomes

$$\begin{aligned} a_M M^\mathrm{eq} (T_L+r_1M^\mathrm{eq}) - d_1 T^\mathrm{eq} - \gamma _1 T^\mathrm{eq} M^\mathrm{eq} = 0 , \end{aligned}$$

which leads to

$$\begin{aligned} \gamma _1 = \frac{a_M M^\mathrm{eq} (T_L+r_1M^\mathrm{eq}) - d_1 T^\mathrm{eq}}{T^\mathrm{eq} M^\mathrm{eq}} \approx 0.0042 \, \frac{\text{ mm }^3}{\text{ cells }\cdot \text{ day }}. \end{aligned}$$

1.6 Chemotaxis Parameters

1.6.1 Macrophage Chemotactic Sensitivity Towards TGF-$\beta$ $\chi _1$

In Li Jeon et al. (2002, Table 1), the chemotaxis coefficients of neutrophils for different gradients of interleukin-8 are listed (ranging from $0.6\times 10^{-7}$ to $12\times 10^{-7}$ mm$^2\cdot$mL$\cdot$ng$^{-1}\cdot$s$^{-1}$). We take the intermediate value $\chi _1 = 5\times 10^{-7}\text{ mm }^2\text{ mL } \text{ ng }^{-1}\text{ s }^{-1}\approx 4\times 10^{-2}\text{ mm }^2(\text{ pg/mm }^3)^{-1}\text{ day }^{-1}$. To compare this value with one from another source, we consider Tranquillo et al. (1988, Figure 8): although the chemotaxis coefficient is shown to depend on the attractant concentration, an average value is $\chi = 150 \text{ cm }^2\text{ sec }^{-1}\text{ M }^{-1}\approx 5.18\times 10^{-2}\text{ mm }^2(\text{ pg/mm }^3)^{-1}\text{ day }^{-1}$ (using the TGF-$\beta$ molecular weight found in “Appendix of TGF-$\beta$ Molecular Weight and Equilibrium $T^\mathrm{eq}$”). This result is encouraging because it is of the same order of magnitude as the previous estimate.

1.6.2 LEC Chemotactic Sensitivity Towards VEGF $\chi _2$

In Barkefors et al. (2008), a quantification is made of the effects of FGF2 and VEGF165 on HUVEC and HUAEC chemotaxis. In Barkefors et al. (2008, Figure 6A), it is reported that the total distance migrated per HUVEC in response to a 50 ng/mL gradient of VEGFA165 was about 150 $\mu$m. Considering that the analysed area of the cell migration chamber was 800 $\mu$m long and that the experiment lasted 200 minutes, we can estimate the endothelial cell velocity to be 150/200 = 0.75 $\mu$m/min = 1.08 mm/day and the VEGF gradient to be 50 ng/mL / 800 $\mu$m = 62.50 (pg/mm$^3$)/mm. Now, the flux $\mathcal {J}$ in our equation is given by $\mathcal {J} = \chi _2 L \frac{\partial V}{\partial x}$; however, $\mathcal {J}$ can also be seen as the product of the mass density and the velocity of the flowing mass (Douglas et al. 2005). Therefore, with L being our mass density, we have

$$\begin{aligned} \text{ cell } \text{ velocity } = \chi _2 \frac{\partial V}{\partial x} \end{aligned}$$

and then we can use the previous calculations to estimate

$$\begin{aligned} \chi _2 = \frac{\text{ cell } \text{ velocity }}{\text{ VEGF } \text{ gradient }} = \frac{1.08 \text{ mm/day }}{62.50 \text{(pg/mm }^3\text{)/mm }} = 0.0173 \, \frac{\text{ mm }^2}{\text{ day }} \frac{\text{ mm }^3}{\text{ pg }}. \end{aligned}$$

In order to have realistic cell movement dynamics, $\chi _2$ is taken to be 10 times bigger. This can be justified by the fact that the aforementioned data refer to HUVECs, and LECs might be faster than these cell types. A more suitable data set for this parameter would be very useful to better inform this estimate, but we are not aware of such data. Also, chemical gradients created in vitro are usually different between those observed in vivo and they are known to highly affect cell velocity.

1.6.3 Density Dependence of the Macrophage Chemotactic Sensitivity $\omega$

The cell density dependence of the macrophage velocity is given by the factor $1/(1+\omega M)$. This velocity is maximal when M is close to zero, and we assume that it is halved when M reaches its carrying capacity $k_1^{old}$ (that is, the parameter $k_1$ in Bianchi et al. (2015)). We therefore take $\omega$ to be the inverse of the macrophage-carrying capacity $k_1^{old}$.

1.7 Macrophage Inflow $\phi _1$

We expect $\phi _1$ to be proportional to the lymph flow (estimated in “Appendix of Sizes, Weights, Equilibria and Velocities” as $v_\mathrm{lymph} = 864 \text{ mm } \text{ day }^{-1}$) and macrophage presence in the lymph. In the same source Fischer et al. (1996) that we used to estimate $v_\mathrm{lymph}$, it is reported that the mean capillary diameter is 55 $\mu$m. Thus, about $2.05 \text{ mm }^3$ of lymph pass through a capillary bi-dimensional section in 1 day.

In Cao et al. (2005), we find that a mouse leucocyte count in the blood is approximately 3 to $8\times 10^6$ cells/mL and that of these about $2\times 10^6$ are macrophages coming from the lymph nodes; so we have a macrophage density of $2\times 10^3 \text{ cells/mm }^3$ in the lymph. Therefore, each day about $2.05 \text{ mm }^3 \times 2\times 10^3 \text{ cells/mm }^3 = 4.11\times 10^3$ macrophages pass in one capillary. Converting capillaries into cell density as was done in “Appendix of Maximum Capillary Density $C_{\max }$”, we have an influx equal to $\frac{4.11}{20}\times 10^3\text{ day }^{-1}=0.205\times 10^3 \text{ day }^{-1}$. However, the macrophage density reported in Cao et al. (2005) refers to blood; we assume that this quantity in lymph (especially during inflammation) will be about 10 times bigger. Therefore, we will take $\phi _1 = 2.05\times 10^3 \text{ day }^{-1}$.