Table of notations.

<p>Table of notations.</p

Evolution of hepatitis B infection homogenous case.

<p>With <i>n</i> = 5 simulations of agent-based stochastic model, we averaged results and present for sinusoidal viral particles (Panel A) and intracellular particles for hepatocytes (Panel B). Panel C plots percentage of infection across all hepatocytes where we define infected as at least one HBV or cccDNA copy. Results presented include extra-hepatic HBV replication and no cccDNA persistence in new replenishment hepatocytes. Note, scaling translates from the model values of, say, 7.5 HBV copies in our simulated liver to an entire liver at 10⁶ HBV copies/mL on the graph. Simulated inoculum of 2.10⁹ HBV copies/mL at t = 0, Δ<i>t</i> = <i>τ</i>/30 = 1min, <i>N</i> = 150 cells, <i>κ</i> = 0.3. Immune response activation at day 40 with CYL strength <i>δ</i> = 0.6 and Non-CYL strength <i>u</i> = 0.25. Note negative logarithmic values occur (Panels A and B) since the average number of copies/cell fall under 1, e.g., sums of HBV particles across all cells in our model translates into 1 particle per the population of 150 or log(1/150).</p

NTCP spatial heterogeneities effects on HBV infection and DoC.

<p>(Panel A) Results plotted are in percentage of cells; averaged over <i>n</i> = 5 simulations of agent-based stochastic model. Inoculum of 2.10⁹ HBV copies per mL at day 0 and immune responses activation at day 40 (<i>δ</i> = 0.6, <i>u</i> = 0.25) Plain lines represent percentage of cells in the liver with at least 1 HBV copy, Dotted lines represent percentage of cells in the liver with at least 1 cccDNA copy. cccDNA survival probability to replenishment hepatocyes set to zero. (blue trace) Gradient-based heterogeneities of HBV replication cycle efficiency, (red trace) Gradient-based heterogeneities of HBV replication cycle efficiency excluding the HBV uptake from sinusoid to cell (parameters Γ), (black trace) Gradient-based heterogeneity of the HBV uptake (parameter <i>κ</i> linked to NTCP presence), (green trace) Homogeneous base line case with activation of NTCP blocking (homogenous strength <i>u</i>₂ = 0.5) at day 40. Heterogeneities linear distributions with variances set to 0.8 (see text). (Panels B & C) Ribbon plots showing progression for cccDNA counts over all 30 hepatocytes (sinusoid 5) with spatially homogeneous (Panel B) and heterogeneous Γ (Panel C) up to <i>t</i> = 250 days. Note, homogeneous distribution at these immune activity levels leads to clearance at around 100 days.</p

Table of parameters.

<p>Table of parameters.</p

Spatial heterogeneities effects on HBV infection and clearance.

<p>(Panel A) Results are in percentage of cells; averaged over <i>n</i> = 5 simulations of agent-based stochastic model. Inoculum of 2.10⁹ HBV copies per mL at day 0 and immune responses activation at day 40 (<i>δ</i> = 0.3, <i>u</i> = 0.75). Legend: Solid lines represent percentage of cells in the liver with at least 1 HBV copy (<i>V</i>_Φ<i>k</i>+<i>R</i>_Φ<i>k</i>) and dotted lines percentage of cells in the liver with at least 1 cccDNA copy (<i>C</i>_Φ<i>k</i>). cccDNA survival to replenishment hepatocytes was set to zero probability. (1) Homogeneous base line case; (2) Gradients of HBV replication cycle efficiency (parameters Γ and <i>κ</i>) (3) Gradients of <i>δ</i>, (4) Gradients of both (Γ, <i>κ</i>) and <i>δ</i>. (Panels B & C) Ribbon plots before immune activation (<i>t</i> = 1–30 days) showing intracellular viral particles (<i>V</i>_Φ) for each hepatocyte (sinusoid 5). Hepatocyte number order reflects proximity to sinusoidal periportal entrance, i.e., #1 at entry point, and #30 at pericentral exit. Panel B shows particle counts over sinusoid with homogeneous spatial distribution of HBV replication rates (Γ) and uptake (<i>κ</i>) whereas Panel C shows distribution with linear gradient of same parameters. Peak parameter values situated at periportal hepatocyte #1 and minimal at pericentral #30. Note linear distribution of <i>V</i>_Φ corresponding to distribution of parameters with variance of 0.8; see section (3.2) for details.</p

Reinfection of transected or transplanted liver with solo cccDNA copy.

<p>Results here with increased hepatocyte population than in other results: 50 sinusoids of 300 hepatocytes or 15,000 total cells, all homogeneously distributed (no spatial gradients of any parameters). This corresponds to 0.006% of cell population with 1 cccDNA. cccDNA survival probability to replenishment hepatocytes here set to zero. (1) Evolution of HBV from only 0.006% cells infected with 1 cccDNA without any immune response (2)(1) Evolution of HBV from only 0.006% cells infected with 1 cccDNA with activation of a weak immune response at day 10 (<i>δ</i> = 0.3, <i>u</i> = 0.25) Legend: Plain lines represent percentage of cells in the liver with at least 1 HBV copy (<i>V</i>_Φ<i>k</i>+<i>R</i>_Φ<i>k</i>). Dotted lines represent percentage of cells in the liver with at least 1 cccDNA copy.</p

Diagram of the multiscale modeling framework.

<p>Assembly of the multi-scaled model including <i>k</i> = 1…<i>n</i> intracellular hepatocyte HBV models (Φ_<i>k</i>) coupled to a sinusoidal lumen (Ψ) and extra-hepatic blood compartments (Ψ_<i>b</i>). The original Murray-Goyal model was for a single hepatocyte; we here expand their single-hepatocyte model across <i>n</i> unique hepatocytes with <i>n</i> intracellular HBV models (Φ_<i>k</i>). Thus, the <i>k</i>^<i>th</i> hepatocyte expresses a unique number of viral particles as determined by its own respective <i>k</i>^<i>th</i> intracellular compartmental model. Within the <i>k</i> = 1…<i>n</i> hepatocytes, the HBV model functions as follows. Complete viral particles (<i>V</i>_Ψ) taken into the cell, denoted <i>V</i>_Φ<i>k</i>, then convert to cccDNA in the nucleus (<i>C</i>_Φ<i>k</i>) which in turn trigger production of viral proteins p36 (<i>P</i>_Φ<i>k</i>) and ‘single-strand’ viral DNA (<i>S</i>_Φ<i>k</i>). <i>S</i>_Φ<i>k</i> is in turn converted to ‘relaxed’ viral DNA (<i>R</i>_Φ<i>k</i>) that either returns to the nucleus reinforcing the cccDNA population (<i>C</i>_Φ<i>k</i>) or is released from the cell to <i>V</i>_Ψ. The amount of <i>P</i>_Φ<i>k</i> in the cell determines which route <i>R</i>_Φ<i>k</i> will take, and <i>P</i>_Φ<i>k</i> itself is also released into Ψ as <i>P</i>_Ψ. Note release of complete viral particles from the <i>k</i>^<i>th</i> hepatocyte (e.g., <i>V</i>_Φ<i>k</i> and <i>P</i>_Φ<i>k</i>) is added to the total pool of sinusoidal lumen viral particles (e.g., <i>V</i>_Ψ and <i>P</i>_Ψ), at the respective spatial location for the <i>k</i>^<i>th</i> hepatocyte. Hence, the suite of <i>n</i> hepatocytes are effectively coupled by viral transport to and from the sinusoid lumen (Ψ). No other coupling between the hepatocytes occurs in our model (e.g., gap-junctions, etc.) An additional compartment representing the extra-hepatic blood, Ψ_<i>b</i>, includes dynamics of infection for red blood cells as noted in the text.</p

Spatial plots showing viral particles levels over cells within a single sinusoid per row (‘unit model’ #1) in three different parameter sets.

<p>Plots are averages of <i>n</i> = 5 simulations that are evolution of only one sinusoid for 3 cases of agent-based stochastic model: (a) Homogeneous baseline case, (b) linear gradient-based heterogeneities of Γ and <i>κ</i> combined, and (c) linear gradient-based heterogeneity of <i>κ</i> alone. Columns correspond to the suite of either sinus particles (<i>V</i>_<i>s</i>, <i>P</i>_Ψ, two leftmost columns) or the full cohort of intracellular particles (<i>V</i>_Φ<i>k</i> to <i>R</i>_Φ<i>k</i>, 3^rd column to last). For each plot, x-axis represents the time in days, y-axis represents spatial location so that <i>y</i> = 1 is the first hepatocyte cell at the periportal entry of the sinusoid and <i>y</i> = 30 is the 30^th hepatocyte cell at the pericentral end of the sinusoid. Colour bars above each plot show levels of each particle for each respective column across the rows. Inoculation at day <i>t</i> = 0 and activation of immune responses (CYL and non-CYL) at day <i>t</i> = 40 (<i>δ</i> = 0.9, <i>u</i> = 0.25) similar to prior results shown, and cccDNA survival probability to daughter hepatocytes set to zero again. Variance for gradient heterogeneities: 0.8. (See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0188209#pone.0188209.g003" target="_blank">Fig 3</a> for depiction of these gradient-based distributions.)</p

Table1_Modeling and experimental approaches for elucidating multi-scale uterine smooth muscle electro- and mechano-physiology: A review.docx

The uterus provides protection and nourishment (via its blood supply) to a developing fetus, and contracts to deliver the baby at an appropriate time, thereby having a critical contribution to the life of every human. However, despite this vital role, it is an under-investigated organ, and gaps remain in our understanding of how contractions are initiated or coordinated. The uterus is a smooth muscle organ that undergoes variations in its contractile function in response to hormonal fluctuations, the extreme instance of this being during pregnancy and labor. Researchers typically use various approaches to studying this organ, such as experiments on uterine muscle cells, tissue samples, or the intact organ, or the employment of mathematical models to simulate the electrical, mechanical and ionic activity. The complexity exhibited in the coordinated contractions of the uterus remains a challenge to understand, requiring coordinated solutions from different research fields. This review investigates differences in the underlying physiology between human and common animal models utilized in experiments, and the experimental interventions and computational models used to assess uterine function. We look to a future of hybrid experimental interventions and modeling techniques that could be employed to improve the understanding of the mechanisms enabling the healthy function of the uterus.</p