Introductory Chapter: Mathematical Modeling as Part of a Collaborative Effort to Improve COPD Treatment

The current state of COPD treatment, in particular the inability to do much more than ease symptoms and slow the progression of the disease [1], underscores the need for extensive research into the condition and development of new strategies. This need cannot be fulfilled through a single discipline, but will require collaboration in fields of pathology, anatomy, genetics, engineering, physics, biochemistry, cell biology, informatics, microbiology, material science, clinical medicine, and numerous other areas. These fields have already contributed substantially to the current state of the art, and all continue to develop and use modern tools to continue their investigations. The relevance of mathematical modeling as a component in the development of new strategies may not be obvious to the general public, but modeling is supported by and supports more directly biologically related components. It is likely to be an important area of study as researchers investigate treatments that can not only halt progression of COPD but also reverse the disease’s course.

Models are only as valid as the data that support them. They, along with the overall future directions in COPD therapy, will benefit from a greater understanding of fundamental pulmonary physiology and the complex interactions that occur among the gases, tissues, blood, antigens, pathways, genetic transcription, membrane permeabilities, and reaction kinetics. Biological and biochemical studies provide a wealth of information about the individual components of this physiology. Mathematical models can quantify these components to identify subtle changes in, for example, concentrations that may lead to substantial changes, therapeutic or pathological.

Multiple basic mechanisms remain unexplained related to both gas exchange and inflammation despite excellent relevant work that spans back over decades. It may not be surprising that the mechanism for matching perfusion with ventilation is unclear [2], given that the mechanism through which hypoxia increases blood flow in the peripheral circulation, a well-known and long-studied effect, is also unknown. Still, elucidation of the mechanism may have clinical applications. Similarly, the mechanism for hyperoxic hypercapnia is unknown, despite multiple studies and hypotheses [3, 4]. This phenomenon describes the increase in blood CO₂ that occurs when COPD patients are given supplementary oxygen that raises their resting blood oxygen saturation to the 96 to 99% levels that are considered normal. To avoid hypercapnia, physicians generally recommend that patients’ saturation be kept between 88 and 92% [5].

The tendency for COPD to lead to worse COPD raises another pervasive theme, which is one of the positive feedback mechanisms in the disease. Some of these mechanisms are beneficial, such as the matching of blood flow with airflow. The matching can be considered as positive feedback because it implies that increased blood flow leads to increased airflow which in turn further increases blood flow. Of course, the effect is ultimately limited in normal physiology. COPD appears to disrupt this mechanism [6], leading to poor matching between ventilation and perfusion. Positive feedback is also present in the monocyte recruitment and macrophage activation processes of normal inflammation [7, 8, 9, 10]. In contrast, COPD can trigger pathological positive feedback mechanisms, such as chronic inflammation that cannot resolve itself.

1. Clinical importance of modeling

Mathematical models can add to the overall understanding of pulmonary physiology. Because the behavior of the lung depends on excitatory and inhibitory pathways that interact with one another, the overall result of a change in chemical concentration, cellular mobility, timing delays, transport barriers, receptor densities, and other parameters can be unpredictable through qualitative analysis. It seems obvious that a regional increase in, for example, a vasodilator will lead to increased blood flow, but paradoxical effects can arise if the increase triggers a pathway that counteracts the effect. A good quantitative model can reveal these types of effects and can indicate which pathways are responsible for them. In vivo whole-animal models can similarly reveal these integrated phenomena but do not provide as much data. In instances where the model does not match the in vivo behavior, the discrepancy can be used to improve estimates of the time delays, production rates, receptor densities, and other parameters involved, and it may demonstrate the presence of a currently undiscovered component of the system. Thus, one fundamental role of modeling is to identify mathematical discrepancies in our current understanding of pulmonary physiology and to suggest additional experiments that should be done. The results of those experiments, combined with the consequent revised models, can suggest new potential clinical targets for treatment.

2. Systems that can be modeled

No model can describe a biological organ completely. To do so would require the model to account for every cell, cellular function, chemical agent, receptor, agonist, and subsystem in the organ and adjacent organs. COPD models therefore address specific aspects of the disease and pulmonary physiology. These aspects include flow in the airways, blood flow regulation, gas exchange, tissue remodeling, neural control, inflammation, mechanical stress, infection, and clearing of debris.

2.1 Airflow

Flow in the airways must consider the forces for inspiration and expiration, mainly those caused by the diaphragm and the intercostal muscles, the configuration of the lung lobes and the branching patterns of the airways, smooth muscle tone, mucus thickness, mucus viscosity, hydrostatic pressure, the elasticity of alveoli, bronchi, and bronchioles, and concentrations of oxygen, carbon dioxide, and nitric oxide. The models can, in theory, be specific to a given patient, but the vast amount of data that would be necessary to model a specific lung and the consequent computational requirements would be prohibitory. Thus, models tend to rely on general descriptions of a generic lung.

Specific rules are devised to describe the relationships between parent and daughter airways at branching locations. Much of our current understanding of this branching network comes from the meticulous studies of Weibel [11, 12, 13]. Islam et al. [14] reviewed models of the pulmonary airway tree and emphasized the importance of geometric configuration to both airflow and particle deposition. These models can help predict the amount of each gas species that enters the alveoli, the regions of the lung that are more susceptible to deposition of inflammatory particles, and optimal particle size for aerosol treatments.

For COPD, it is important to know how airflow is distributed to different regions of the lung. This distribution changes when airways become obstructed with mucus and narrowed by inflammation. In addition, the effort required for the patient to exhale increases as the alveoli lose elasticity. The increased pressure required to expel gas from the alveoli compresses the bronchi, tending to further narrow them and increase the difficulty of exhalation [15]. The patient compensates for this effect through the use of pursed-lip breathing, where the mouth is used to apply a back pressure on the airways to keep them open [16]. These mechanical issues have been reviewed by Bhana and Magan [17]. The ability of the alveoli to take in and expel gas, and hence the distribution of alveolar ventilation, is also affected by the surfactant. Ventzislava et al. demonstrated that the bronchoalveolar lavage fluid from COPD subjects had less surfactant lipids and proteins than that from non-COPD subjects [18]. Albert [19] proposed that when mechanical ventilation is used, it can decrease surfactant levels, which in turn increases damage to the alveoli.

2.2 Blood flow

Blood flow is the second half of the fluid mechanical aspects of lung function. As in airflow, blood flow occurs at length scales that vary over three orders of magnitude, and the considerations that arise at the length scales of the pulmonary artery and vein differ from those at the alveolar capillary level. Of particular importance is the regulation of blood flow to the arterioles and capillaries, as this determines the ability of the lung to match blood perfusion with alveolar ventilation [20]. Blood flow modeling at the arteriolar level requires not only that mechanical flow resistance be considered, but that the feedback mechanisms among shear stress, release of vasodilators and vasoconstrictors, endothelial and epithelial function, smooth muscle cell physiology, blood oxygen saturation, and transport barriers be considered [21, 22, 23, 24].

2.3 Feedback/neural control

Neural control of breath rate and airflow volume is intriguing because it occurs both consciously and unconsciously. It is interesting from an engineering standpoint as both a feedback control problem and a transport problem that involves blood pH and blood concentrations of CO₂ and O₂. Although CO₂ is the primary driver of blood pH, chemoreceptors sense the two parameters separately [25]. An early model was published by Grodins et al. [26]. The component of respiratory drive that is governed by CO₂ arises from both peripheral chemoreceptors at the carotid bodies and a medullary chemoreceptor [15]. The carotid bodies also sense O₂. Overall, the neural respiratory drive is increased in patients with COPD [27] and the respiratory discomfort that these patients endure can arise from both excess CO₂ and from insufficient O₂.

Neural feedback control of pulmonary gas exchange is further complicated because the blood gas concentrations also affect pulmonary blood flow resistance, heart rate, cardiac output, and other parameters. Because the feedback mechanisms are both positive and negative [28], the effect of a single parameter becomes difficult to predict without a mathematical model that looks quantitatively at the complete system.

2.4 Gas exchange

Gas exchange is the most fundamental aspect of lung function. A quantitative study of gas exchange involves gas concentrations in the alveoli and blood, the transport barrier between the alveoli and the capillaries, the transport barrier between blood plasma and hemoglobin in the red blood cell, capillary blood flow, airflow, the kinetics of the interaction between O₂ and hemoglobin, the reactions involved in transport of CO₂ by plasma and the red blood cell, and the role of vasodilators such as nitric oxide in the control of both airflow and blood flow. One fundamental aspect of the interaction of CO₂ and O₂ with hemoglobin is the Bohr effect, which tends to offload more oxygen in a high CO₂ environment [29, 30] and the Haldane effect, which tends to offload more CO₂ in an oxygen-rich environment [29, 31].

Because overall gas exchange depends on blood flow and airflow, which in turn depend on neuronal control, studies tend to combine these effects into an integrated model [24]. Such models have been presented by Ursino’s group at the University of Bologna [23, 32, 33] and by Tehrani [34]. These studies did not incorporate the effects of nitric oxide in their models. Buess et al. [35] specifically modeled the exchange of nitric oxide in the airways. They focused less on the vascular side of the pulmonary system, but rather considered blood to be a sink for nitric oxide.

2.5 Tissue remodeling

The ultimate treatment for COPD is to have the lung repair itself, which could mean the ambitious goal of reconstructing lost alveoli, or the more modest, but still ambitious goal of restoring the alveolar elasticity [36]. This therapeutic strategy requires an understanding of lung tissue remodeling that far exceeds current knowledge. It is known that lung tissue can remodel itself, but the remodeling can be pathological as much as therapeutic. Some insights can be gained from the changes in lung structure following lung reduction surgery, a procedure used in cases of severe hyperinflation [37]. In lung volume reduction surgery, a portion of the lung is removed to reduce lung volume, thus allowing the diaphragm and chest wall to operate at a volume that is more efficient [38].

Guarniere et al. [39] reviewed recent advances in the use of mesenchymal stem cells to promote tissue regenerating for COPD patients. They identified several major questions that must be answered before a viable method can be applied clinically. Among these are the cell culture environment and number of passages for the donor cells, cell dose, timing of the dose, route of administration, target tissue microenvironment, and localization of the cells to the target tissue (homing). Furthermore, the ability of mesenchymal stem cells to differentiate depends in a complicated manner on the chemical environment within the host. For example, while TGF-β promotes stem cell differentiation into smooth muscle cells [40], it also promotes the differentiation of epithelial cells to a mesenchymal phenotype [41]. This growth factor is one of the host of agents that must be present in the correct amount at the correct location to have the desired effect. These considerations illustrate how modeling of mass transport issues can support and in turn be supported by both in vivo and in vitro studies.

2.6 Inflammation

Chronic and recurrent inflammation is a hallmark of COPD [42] that leads to increased difficulty in breathing through production of excessive mucus [43] and to tissue damage [8]. Inflammatory cells, such as macrophages and lymphocytes, produce proteases [44] that promote degradation of alveolar connective tissue and cellular apoptosis [45]. Inflammation is initiated by triggers such as cigarette smoking, but it can persist years after smoking cessation [46]. Smoking appears to induce a change from an inflammatory process that is self-limiting to one that is self-perpetuating [42]. Such irreversible changes are characteristic of the branching solutions of nonlinear equations that can help to elucidate the underlying inflammation-perpetuating mechanisms.

2.7 Mechanical stress

Mechanical stresses are important to lung physiology on large and small scales, ranging from the muscular effort involved in respiration, to the elastic properties of the parenchyma that are relevant to ventilation efficiency [17], to the forces of mechanical ventilation that can lead to tissue damage [10, 47], and to the forces that lead to functional [48] and phenotypic changes in the numerous involved cell types [49]. The mechanical environment also affects the cycle of inflammation [49].

2.8 Infection

COPD exacerbations are particularly problematic in terms of morbidity, mortality, quality of life, COPD progression, and health care costs [50, 51]. Approximately half of all exacerbations are caused by infections, while irritants, allergic reactions, and other inflammatory triggers are also involved [50]. Common viral infections include rhinoviruses, influenza, parainfluenza, coronavirus, and adenovirus. Infections have been addressed mathematically in the case of acute respiratory distress syndrome [52].

Moxnes and Hausken [53] describe a viral infection model that describes the subsequent immune response. They compare the infections of H1N1 to H3N2 and noted a higher viral load and duration for the former. Dobrovolny et al. [54] reviewed eight mathematical models of influenza and compared quantifiable results to patient outcomes. Specifically, they examined those models where the authors had disabled cytotoxic T lymphocytes, antibodies, of interferon. None of the models were able to match all experiments, and the authors emphasized that additional work was needed both experimentally, to determine the important input parameters, and mathematically, to capture all important pathways.

2.9 Clearing of debris

Another aspect of pulmonary physiology that is relevant to COPD is the clearing of particulates from the airways. Mathematical models of this function were reviewed by Xu and Jiang [55]. The models must consider properties of the mucus (viscosity and viscoelasticity), the motion and coordination of the cilia, and additional forces such as those exerted by coughing.

3. Further questions to address

Several major questions need to be answered before significant progress can be made in treating, arresting, or reversing COPD. Some of these questions are clinical and require new methods to evaluate individual patients, map, and quantify regions of emphysema and bronchitis, characterize individual immune response, localize dead space, and quantify the spatial distributions of air and blood flow. With such measurements, it may be possible to treat each patient in a manner that is optimized to the individual’s physiology. Other questions relate more to the specific mechanisms by which respiratory physiology adapts over short and long time spans.

3.1 Ventilation/perfusion matching

The mechanism through which the lung matches ventilation with blood perfusion is intriguing from the scientific standpoint in that it appears to operate in the opposite direction from the vascular control in the peripheral circulation. There, hypoxia increases blood flow, whereas in the lung, hypoxia decreases blood flow. The difference may result from a yet unidentified agent in the lung or it may be a result of phenotypic differences in the vascular cells (endothelial and smooth muscle) or an effect of the epithelial cells. More studies on the behavior of these cells, alone and in co-culture, may be helpful.

3.2 Hyperoxic hypercapnia

A general guideline for COPD patients on supplemental oxygen is to maintain oxygen saturation between 88 and 92% to avoid the tendency for hypercapnia that increases mortality risk [5, 56]. A better understanding of the mechanism by which high oxygen saturation leads to excess carbon dioxide in COPD patients might allow these patients to enjoy fully saturated blood without the dangers of hypercarbia. Multiple mechanisms have been suggested [4]. A computational study by Hanson et al. [57] suggests that the increase in dead space, coupled with the Haldane effect, is sufficient to explain this phenomenon. This conclusion is consistent with clinical measurements by Sassoon et al. [58]. It suggests that hyperoxic hypercapnia should vary with a patient’s specific COPD physiology (e.g., degree of dead space) and that the 88–92% rule may not be appropriate for all COPD patients.

3.3 Inflammation and exacerbation

A third major question relates to the mechanisms through which inflammation is perpetuated even after smoking cessation and the causes of inflammation-induced exacerbations. While it seems intuitive that exacerbations must be triggered by an external stimulus, rates of exacerbations and their time courses vary from patient to patient [50], and variations in immune responses are likely responsible for some of this variability. A detailed analysis of an individual’s immune response may help to predict the time course of an exacerbation and inform treatment. It is well known that positive feedback systems can lead to oscillatory behaviors [59], and given that inflammation includes positive feedback aspects, it is reasonable to propose that some episodes of exacerbation are not immediately triggered events, but are the consequence of a natural, though pathological, cycle in certain COPD patients.

3.4 Remodeling

Promotion of lung tissue remodeling through stem cell therapy, tissue engineering, genetic engineering, or other manipulations of cell growth and differentiation is a goal that is probably years from being reached. Achievement of this goal will require meticulous mechanistic experiments on the molecular, cellular, and tissue level. The problem thus lends itself to multiscale modeling [60] and can benefit from advancements in those techniques.

4. Conclusion

Further advancements in the treatment of COPD will require a concerted effort by researchers from a wide variety of fields. The associated problems are challenging, but well worth addressing for the health, well-being, and quality of life of the patients. Mathematical modeling can contribute significantly to an understanding of the mechanisms involved in COPD and to the development of modern treatments of the disease.