Scaling and regression.

<p>Throughout the figure, results for Image 1, Image 2 and Image 3 are denoted by blue dots, green triangles and red squares, respectively. (A,B) Scaled oxygen transfer rate (A) <i>NR</i> and (B) <i>N</i>/(<i>L</i>²/<i>R</i>)^1/3 versus pressure drop Δ<i>P</i>, on log-log axes. Solid lines denote numerical results and black triangles denote the gradients predicted by (A) scaling <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0165369#pone.0165369.e031" target="_blank">Eq (13)</a> and (B) scaling <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0165369#pone.0165369.e028" target="_blank">Eq (12)</a>. (C) Oxygen transfer rate <i>N</i> versus Δ<i>P</i> with the linearized oxygen–hemoglobin dissociation law. Solid lines denote numerical results and dashed lines denote results generated by the regression <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0165369#pone.0165369.e040" target="_blank">Eq (19)</a>. The upper bounds <i>N</i>_max on <i>N</i> are indicated, which have been calculated by solving the diffusion equation in the surrounding villous volume with the capillary assumed to be deoxygenated. (D) Oxygen transfer rate <i>N</i> with the nonlinear oxygen–hemoglobin dissociation law (solid lines) and average advection enhancement in each capillary (dashed lines) versus Δ<i>P</i>. Also shown on the figure is the oxygen advection enhancement parameter <i>B</i> (black dashed line), derived by linearizing the oxygen–hemoglobin dissociation law.</p

Methods.

<p>(A) An example of how 3D images are partitioned with a plane (shown in blue) to provide a surface for boundary conditions to be applied. The red surface represents the capillary surface (endothelium, Γ_cap) and the grey surface represents the villous surface Γ_vil. In this example the capillary bifurcates within the villous branch. (B) Example of the skeletonization of a capillary from a 3D image. The skeletonization line, representing the centreline of the lumen, is shown in black. (C) The 3D images of fetal capillaries and villous surfaces used in the simulations (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0165369#pone.0165369.s001" target="_blank">S1 Images</a>). The images shown have been partitioned by a plane on which boundary conditions are applied, for inflow (Γ_in), outflow (Γ_out) and villous tissue (Γ₀). (D) Polynomial fetal oxygen–hemoglobin dissociation law from [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0165369#pone.0165369.ref027" target="_blank">27</a>] (solid line) and the linear approximation between 0 and 60 mmHg, passing through the origin, found using a least-squares fit (dashed line). The gradient of the linear approximation is <i>K</i> = 0.019 mmHg⁻¹. (E) Schematic diagram of the idealized axisymmetric model of a fetal capillary dilation, showing an axisymmetric tube with a localized dilation. Blood flows into the capillary through <i>γ</i>_in and leaves through <i>γ</i>_out. Oxygen is provided along the dilated section of the capillary <i>γ</i>_<i>d</i>, denoted by the thicker black line. The undilated sections of the capillary are labeled <i>γ</i>_<i>u</i>.</p

Summary of oxygen transfer scaling results for <i>Pe</i>_eff = <i>Bu</i>₀<i>R</i>₀/<i>D</i> ≫ 1.

<p>Summary of oxygen transfer scaling results for <i>Pe</i>_eff = <i>Bu</i>₀<i>R</i>₀/<i>D</i> ≫ 1.</p

Geometric and calculated properties of the three 3D images used in the paper.

<p>Geometric and calculated properties of the three 3D images used in the paper.</p

Literature-based and estimated parameter values used in the ASM growth model.

<p>(^* indicates the values estimated from <i>in vitro</i> experiments.)</p

The role of individual inflammation history in the case of slow inflammation resolution (IR <<1).

<p>(a–c) ASM population size dynamics (<i>c</i>-cells, red; <i>p</i>-cells, blue; total population <i>s</i>, black) and (d–f) the corresponding inflammatory status evolution (μ, solid black; inflammatory thresholds μ₁ and μ₂, dashed), characterized by the same inflammation resolution rate, magnitude and average stimulus frequency (λ_d/λ_p  = 0.08, <i>a</i>/μ₁  = 0.5, ω/λ_p  = 0.25). (d) Regular series of inflammatory events; (e–f) two realisations of a series of inflammatory events at random times for the same mean frequency (about once a fortnight) as in (d). (g) Distribution of fold-increase in ASM mass after 300 days for a random sequence of inflammatory events with the same characteristics as in panels (b, c); arrows indicate the fold-increase corresponding to (a–c). (h) The distribution of outcomes with an increase of 25% in the inflammation resolution rate (λ_d/λ_p  = 0.1). (The outcome histograms (g,h) are computed for <i>N</i> = 1000 instances).</p

Survey of ASM growth scenarios, showing fold-increase in ASM population size after 300 days (colour scale) as a function of the inflammation resolution rate IR = λ_d/λ_p and (a) inflammation magnitude <i>a</i>/μ₁ (for fixed frequency ω/λ_p = 0.25) or (b) inflammation frequency ω/λ_p (for fixed magnitude <i>a</i>/μ₁ = 5).

<p>White dots indicate the growth regimes shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0090162#pone-0090162-g002" target="_blank">Fig. 2</a>. Solid black lines are the computed isolines of the 2- and 8-fold ASM growth, which agree with the theoretically predicted dependence λ_d ∼ ω log <i>a</i>/μ₂ (dashed white lines; see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0090162#pone.0090162.s001" target="_blank">Materials S1</a>).</p

A schematic of the model design.

<p>(a) Schematic representation of the model with <i>p</i> being the amount of ASM cells in proliferating state, <i>c</i> the amount of non-proliferative cells and μ the inflammatory status; λ_p is the proliferation rate, λ_a is the apoptosis rate, λ_cp and λ_pc are the switching rates between non-proliferative and proliferative states, λ_d is the inflammation clearance rate, and <i>f</i>(<i>t</i>) is a time-dependent external inflammatory stimulus. (b) Dependence of the model parameters on the inflammatory status μ (three levels of inflammation are characterised by the thresholds μ₁ and μ₂; see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0090162#pone-0090162-t001" target="_blank">Table 1</a> for reference values). Rates are plotted on a logarithmic vertical scale. (c) An illustration of the inflammatory status dynamics induced by a series of environmental stimuli such as shown in (a), illustrating graphically the parameters λ_d, <i>a</i>, and ω. (d) A simulation of the ASM cell population response (<i>p</i>, blue dash-dotted; <i>c</i>, red dashed; <i>s</i> = <i>p</i>+<i>c</i>, thick black solid) to a stepwise variation in the inflammation status (thin solid); the arrows show the direction of change in the ASM subpopulations. Although the inflammatory status returns to its initial state at the end of the simulation, the total ASM cell population has irreversibly increased, showing thereby “effective” hysteresis. Only the time spent in “severe” regime (μ>μ₂) contributes to substantial growth (over weeks); however, the “moderate” regime (μ₁<μ<μ₂) can also give rise to substantial growth over a longer timescale (months). Note that the proportion of proliferative cells (blue dash-dotted) is significant only during the “severe inflammation” regime (3).</p

Possible pathways of chronic ASM mass accumulation in asthma (solid lines indicate the mechanisms included in the mathematical model).

<p>Possible pathways of chronic ASM mass accumulation in asthma (solid lines indicate the mechanisms included in the mathematical model).</p

Representative ASM cell population growth dynamics over a period of 300 days, starting from a “healthy” state (<i>c</i> = 0.1, <i>p</i> = 0.01), for different values of normalised inflammation magnitude <i>a</i>/μ₁, frequency ω/λ_p and inflammation resolution rate λ_d/λ_p: <i>a</i>/μ₁ = {2 (a), 5 (b,c), 8 (d), 1.5 (e)}, ω/λ_p = {0.25 (a,c–e), 0.125 (b)} and λ_d/λ_p = {7 (a), 5 (b,c), 2 (d), 0.22 (e)}.

<p>(a,b) show no substantial ASM accumulation, (c,e) show “moderate” and (d) “severe” ASM hyperplasia. The amount of <i>c</i>-cells is depicted by the red line, <i>p</i>-cells (blue) and total population size <i>s</i> = <i>p</i>+<i>c</i> (thick black). Insets plot the corresponding inflammatory status μ (solid black) and the inflammation level thresholds (horizontal dashed lines).</p