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Complexity and Emergent Phenomena

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Abstract

Complex biological systems operate under non‐equilibrium conditions and exhibit emergent properties associated with correlated spatial and temporal structures. These properties may be individually unpredictable, but tend to be governed by power‐law probability distributions and/or correlation. This article reviews the concepts that are invoked in the treatment of complex systems through a wide range of respiratory‐related examples. Following a brief historical overview, some of the tools to characterize structural variabilities and temporal fluctuations associated with complex systems are introduced. By invoking the concept of percolation, the notion of multiscale behavior and related modeling issues are discussed. Spatial complexity is then examined in the airway and parenchymal structures with implications for gas exchange followed by a short glimpse of complexity at the cellular and subcellular network levels. Variability and complexity in the time domain are then reviewed in relation to temporal fluctuations in airway function. Next, an attempt is given to link spatial and temporal complexities through examples of airway opening and lung tissue viscoelasticity. Specific examples of possible and more direct clinical implications are also offered through examples of optimal future treatment of fibrosis, exacerbation risk prediction in asthma, and a novel method in mechanical ventilation. Finally, the potential role of the science of complexity in the future of physiology, biology, and medicine is discussed. © 2011 American Physiological Society. Compr Physiol 1:995‐1029, 2011.

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Figure 1. Figure 1.

Breathing irregularities in a preterm infant. Inter‐breath intervals (IBI) as a function of breath number at post‐conceptional ages of 39 (A) and 61 weeks (B). (C) Probability density distributions estimated from A and B and straight line fits on a log‐log graph; with permission from Reference .

Figure 2. Figure 2.

Color‐coded low attenuation areas (LAA) from a normal subject (left) and a patient with early emphysema (right). Different colors denote different contiguous but non‐overlapping clusters of LAA; with permission from Reference .

Figure 3. Figure 3.

Two sets of circles with different areas (A and B). The distribution of areas plotted on a double logarithmic graph in C demonstrates the difference between a Gaussian and a power law corresponding to the circles in A and B, respectively. The set of circles in B also shows schematically that as the scale (mean area corresponding to the rows) changes logarithmically, the ratio of the number of circles is preserved between scales.

Figure 4. Figure 4.

The scale (L) dependence of the number (N) of units in 1D (A) and in 2D (B).

Figure 5. Figure 5.

The scale dependence of the number of units in the Koch curve.

Figure 6. Figure 6.

Bond percolation on a square lattice (thin lines) for different values of the probability P. Thick line segments are occupied with probability P. The red curve marks the shortest percolating pathway at P = Pc = 0.5.

Figure 7. Figure 7.

Simulation of the progression of pulmonary fibrosis. The solid line shows the bulk modulus B of the elastic network versus the fraction of springs c randomly stiffened by a factor of 100. If all of the spring constants were uniformly stiffened in a gradual manner from the baseline value of 1 to 100, B would follow the dashed diagonal line. Shown at the top are the network configurations obtained when c = 0, c = 0.5, and c = 0.67 with thick line showing stiff springs; with permission from Reference .

Figure 8. Figure 8.

(A) A symmetrically bifurcating Weibel airway tree with each branch given a generation number starting at the trachea. (B) An asymmetrically branching Horsfield tree with each airway given an order number starting with the terminal bronchioles.

Figure 9. Figure 9.

A three‐dimensional branching model of the airway tree; with permission from .

Figure 10. Figure 10.

Laser scanning confocal microscopic image of the alveolar structure of a mouse lung fixed at 30 cmH2O airway pressure; with permission from .

Figure 11. Figure 11.

CT image of the lung of an emphysematous patient. Color clusters represent contiguous low attenuation areas; with permission from Reference .

Figure 12. Figure 12.

Exponent D of the cumulative distributions of cluster sizes as a function of the LAA% in normal (open symbols) and emphysematous (filled symbols) subjects. Solid line is model simulation. Based on Reference , with permission

Figure 13. Figure 13.

Network models mimicking emphysema. (A) Chemical breakdown of lung tissue. (B) Force‐based breakdown of lung tissue. Note that high forces (red) occur around the perimeter of the clusters especially in regions where a “thin wall” separates two clusters (arrow). (C) Size distributions of defect clusters from A and B as well as LAA cluster distribution from Fig. ; with permission from Reference .

Figure 14. Figure 14.

Inter‐cell and intra‐cell networks. Metabolic function reflects the behavior of a dynamic cell‐cell network. The nodes of this network are the various different cell phenotypes that communicate via ligand‐receptor interactions involving both short‐range adhesion molecules and long‐range mediator molecules. The cell phenotypes themselves can be viewed as dynamic attractors of the gene regulatory network that determines which proteins are going to be produced within the cell nucleus. The cell‐cell and gene regulatory networks interact via the network of intra‐cellular signaling molecules that operates similarly to a multi‐layer perceptron.

Figure 15. Figure 15.

A representative time series of twice‐daily peak expiratory flow (PEF) during 6 months, showing self‐similar fluctuations at different time scales. The inset shows a shorter time scale in which the statistical properties of the PEF series are similar to those of the entire series. Despite the random‐looking appearance, the fluctuations are not random but ordered, which means that any particular value is dependent on previous values; with permission from Reference .

Figure 16. Figure 16.

(A) A four‐generation tree model of the airways. The numbers in brackets are segments opening in an avalanche (thick lines) when pressure P = 0.5. Thin lines are segments that have not opened. The numbers to the right of the segments are the normalized opening pressures. (B) Parallel model with the same segments arranged in an increasing order of their opening pressures. (C) Crackle events as a function of time from the tree model. Groups of crackles come from avalanches. (D) Crackle events from the parallel model. (E) Distribution of inter‐crackle intervals in the two models; with permission from Reference .

Figure 17. Figure 17.

Power‐law stress relaxation in response to a 10% step change in uniaxial stretching of an isolated lung tissue strip on a double logarithmic graph. The exponent β is the negative slope of a straight line fit to the data (not shown). Black line is control, and blue curve is following digestion with elastase, while the sample was held at the same length as before digestion. Red curve is obtained after adjusting the length of the sample so that the mean force was the same as before digestion.

Figure 18. Figure 18.

Immunofluorescently labeled collagen in alveolar walls at 20% (left) and at 40% (right) uniaxial stretch in the vertical direction. Notice that after 20% stretch, the same fibers become less wavy (black arrows). Bar denotes 5 μm.

Figure 19. Figure 19.

Modeling the repair of the fibrotic lung. The colors are related to the force the elements carry (red high force). The thick and thin lines denote stiff and soft springs, respectively. (A) The initial configuration with the concentration of stiff springs c = 0.649 and the bulk modulus B = 26.6. (B) The configuration of the network following random repair with c = 0.611 and B = 22.6. (C) The configuration of the network following targeted repair with c = 0.63 and B = 16; with permission from Reference .

Figure 20. Figure 20.

A possible approach to monitoring and treating chronic diseases aided by the tools of complexity analysis; with permission from Reference .

Figure 21. Figure 21.

Conditional probability π as a function of the long‐range correlation exponent α that given the current value of the peak expiratory flow (PEF), in this case 95% of the predicted value, PEF will drop to below 80% of the predicted value within a month. Notice that the risk is small, only 13%, if the time series is highly correlated (α = 0.8) compared to a nearly uncorrelated time series (α = 0.55) when the risk is 84%; with permission from Reference .

Figure 22. Figure 22.

Normalized pressure‐volume curve in a model of an atelectatic region of the lung. With conventional mechanical ventilation, pressure is cycled between P1 = 0.3 and P2 = 0.7, and the recruited volume is V2. During variable ventilation, end‐inspiratory pressure is varied from inflation to inflation sampling the Gaussian around P2. In one inflation, pressure increases to 0.65 losing volume, whereas for the next inflation, pressure increases to 0.75 gaining recruited volume. Because of the nonlinearity of the pressure‐volume curve, the “gain” is far greater than the “loss,” and the recruited volume available for gas exchange increases by ΔV from V2 to V3; with permission from Reference .

Figure 23. Figure 23.

Secretion phosphatidylcholine as a function of percent variability in stretch‐by‐stretch amplitude superimposed on 50% mean stretch amplitude. *denote P < 0.05 increase over cells exposed to monotonous stretch; with permission from Reference .



Figure 1.

Breathing irregularities in a preterm infant. Inter‐breath intervals (IBI) as a function of breath number at post‐conceptional ages of 39 (A) and 61 weeks (B). (C) Probability density distributions estimated from A and B and straight line fits on a log‐log graph; with permission from Reference .



Figure 2.

Color‐coded low attenuation areas (LAA) from a normal subject (left) and a patient with early emphysema (right). Different colors denote different contiguous but non‐overlapping clusters of LAA; with permission from Reference .



Figure 3.

Two sets of circles with different areas (A and B). The distribution of areas plotted on a double logarithmic graph in C demonstrates the difference between a Gaussian and a power law corresponding to the circles in A and B, respectively. The set of circles in B also shows schematically that as the scale (mean area corresponding to the rows) changes logarithmically, the ratio of the number of circles is preserved between scales.



Figure 4.

The scale (L) dependence of the number (N) of units in 1D (A) and in 2D (B).



Figure 5.

The scale dependence of the number of units in the Koch curve.



Figure 6.

Bond percolation on a square lattice (thin lines) for different values of the probability P. Thick line segments are occupied with probability P. The red curve marks the shortest percolating pathway at P = Pc = 0.5.



Figure 7.

Simulation of the progression of pulmonary fibrosis. The solid line shows the bulk modulus B of the elastic network versus the fraction of springs c randomly stiffened by a factor of 100. If all of the spring constants were uniformly stiffened in a gradual manner from the baseline value of 1 to 100, B would follow the dashed diagonal line. Shown at the top are the network configurations obtained when c = 0, c = 0.5, and c = 0.67 with thick line showing stiff springs; with permission from Reference .



Figure 8.

(A) A symmetrically bifurcating Weibel airway tree with each branch given a generation number starting at the trachea. (B) An asymmetrically branching Horsfield tree with each airway given an order number starting with the terminal bronchioles.



Figure 9.

A three‐dimensional branching model of the airway tree; with permission from .



Figure 10.

Laser scanning confocal microscopic image of the alveolar structure of a mouse lung fixed at 30 cmH2O airway pressure; with permission from .



Figure 11.

CT image of the lung of an emphysematous patient. Color clusters represent contiguous low attenuation areas; with permission from Reference .



Figure 12.

Exponent D of the cumulative distributions of cluster sizes as a function of the LAA% in normal (open symbols) and emphysematous (filled symbols) subjects. Solid line is model simulation. Based on Reference , with permission



Figure 13.

Network models mimicking emphysema. (A) Chemical breakdown of lung tissue. (B) Force‐based breakdown of lung tissue. Note that high forces (red) occur around the perimeter of the clusters especially in regions where a “thin wall” separates two clusters (arrow). (C) Size distributions of defect clusters from A and B as well as LAA cluster distribution from Fig. ; with permission from Reference .



Figure 14.

Inter‐cell and intra‐cell networks. Metabolic function reflects the behavior of a dynamic cell‐cell network. The nodes of this network are the various different cell phenotypes that communicate via ligand‐receptor interactions involving both short‐range adhesion molecules and long‐range mediator molecules. The cell phenotypes themselves can be viewed as dynamic attractors of the gene regulatory network that determines which proteins are going to be produced within the cell nucleus. The cell‐cell and gene regulatory networks interact via the network of intra‐cellular signaling molecules that operates similarly to a multi‐layer perceptron.



Figure 15.

A representative time series of twice‐daily peak expiratory flow (PEF) during 6 months, showing self‐similar fluctuations at different time scales. The inset shows a shorter time scale in which the statistical properties of the PEF series are similar to those of the entire series. Despite the random‐looking appearance, the fluctuations are not random but ordered, which means that any particular value is dependent on previous values; with permission from Reference .



Figure 16.

(A) A four‐generation tree model of the airways. The numbers in brackets are segments opening in an avalanche (thick lines) when pressure P = 0.5. Thin lines are segments that have not opened. The numbers to the right of the segments are the normalized opening pressures. (B) Parallel model with the same segments arranged in an increasing order of their opening pressures. (C) Crackle events as a function of time from the tree model. Groups of crackles come from avalanches. (D) Crackle events from the parallel model. (E) Distribution of inter‐crackle intervals in the two models; with permission from Reference .



Figure 17.

Power‐law stress relaxation in response to a 10% step change in uniaxial stretching of an isolated lung tissue strip on a double logarithmic graph. The exponent β is the negative slope of a straight line fit to the data (not shown). Black line is control, and blue curve is following digestion with elastase, while the sample was held at the same length as before digestion. Red curve is obtained after adjusting the length of the sample so that the mean force was the same as before digestion.



Figure 18.

Immunofluorescently labeled collagen in alveolar walls at 20% (left) and at 40% (right) uniaxial stretch in the vertical direction. Notice that after 20% stretch, the same fibers become less wavy (black arrows). Bar denotes 5 μm.



Figure 19.

Modeling the repair of the fibrotic lung. The colors are related to the force the elements carry (red high force). The thick and thin lines denote stiff and soft springs, respectively. (A) The initial configuration with the concentration of stiff springs c = 0.649 and the bulk modulus B = 26.6. (B) The configuration of the network following random repair with c = 0.611 and B = 22.6. (C) The configuration of the network following targeted repair with c = 0.63 and B = 16; with permission from Reference .



Figure 20.

A possible approach to monitoring and treating chronic diseases aided by the tools of complexity analysis; with permission from Reference .



Figure 21.

Conditional probability π as a function of the long‐range correlation exponent α that given the current value of the peak expiratory flow (PEF), in this case 95% of the predicted value, PEF will drop to below 80% of the predicted value within a month. Notice that the risk is small, only 13%, if the time series is highly correlated (α = 0.8) compared to a nearly uncorrelated time series (α = 0.55) when the risk is 84%; with permission from Reference .



Figure 22.

Normalized pressure‐volume curve in a model of an atelectatic region of the lung. With conventional mechanical ventilation, pressure is cycled between P1 = 0.3 and P2 = 0.7, and the recruited volume is V2. During variable ventilation, end‐inspiratory pressure is varied from inflation to inflation sampling the Gaussian around P2. In one inflation, pressure increases to 0.65 losing volume, whereas for the next inflation, pressure increases to 0.75 gaining recruited volume. Because of the nonlinearity of the pressure‐volume curve, the “gain” is far greater than the “loss,” and the recruited volume available for gas exchange increases by ΔV from V2 to V3; with permission from Reference .



Figure 23.

Secretion phosphatidylcholine as a function of percent variability in stretch‐by‐stretch amplitude superimposed on 50% mean stretch amplitude. *denote P < 0.05 increase over cells exposed to monotonous stretch; with permission from Reference .

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Béla Suki, Jason H.T. Bates, Urs Frey. Complexity and Emergent Phenomena. Compr Physiol 2011, 1: 995-1029. doi: 10.1002/cphy.c100022