I think it would be useful to begin with correlation of diaphragm movements and intra thoracic pressure.

The diaphragm plays a crucial role in respiration by contracting and flattening to increase thoracic cavity volume during inhalation, thereby reducing intrathoracic pressure and allowing air to flow into the lungs. While there isn't a single equation that exclusively describes diaphragm movement, its function is integral to the respiratory system's mechanics, which can be represented by the Equation of Motion in respiratory physiology.

Equation of Motion:

Pmus+Pvent=(V×E)+(Q×R)Pmus​+Pvent​=(V×E)+(Q×R)

Where:

• PmusPmus​: Pressure generated by respiratory muscles (including the diaphragm)
• PventPvent​: Pressure applied by the ventilator (if applicable)
• VV: Volume of air in the lungs
• EE: Elastance of the respiratory system (inverse of compliance)
• QQ: Airflow rate
• RR: Airway resistance

In spontaneous breathing, PmusPmus​ is primarily generated by the diaphragm's contraction. This equation illustrates how the pressure from muscle activity and any applied ventilatory pressure are balanced by the elastic recoil of the lungs and the resistance to airflow. Understanding this relationship is essential for analyzing respiratory mechanics and the diaphragm's role in ventilation.

Hi Theodore,

mathematical modelling of the respiratory systems is very obstacle because of the dynamic character of the respiratory meamics during Inspiration vs. expiration. These dynamic characteristics differ significantly between physiological versus pathological circumstances and change constantly depending on the extent of the pathological processes (e.g. the extent of lung stiffness - elastance -, or airway obstruction - resistance -, which in turn is influenced by various factors (such as bronchoconstriction, airwas collaps, mucosal oedema, secretion retention). These latter factors also vary and combine). All these components influence muscle strength in addition to neuromuscular function. On the other hand, this is additionally modified by the resistance of the abdominal organs (obesity, ascites).

Beyond the above mentioned parameters you should also take the hysteresis effect of the airways during expiration in to account. This is an important factor that fundamentally distinguishes the lung from a balloon. Therefore, linear models do not accurately describe lung function, whether in a physiological or pathological state.

Furthermore, a distinction should be made between spontaneous breathing and breathing under mechanical ventilation (controlled vs supported ventilation).

The corresponding equations for the respiratory functions in normal or diseased lungs can be found in the various respiratory physiology books (among others):

1. John B.West, Andrew M. Lucs: West's Respiratory Physiology: The Essentials (Lippincott Connect)

2. John B.West, Andrew M. Lucs: West's Pulmonary Pathophysiology: The Essentials (Lippincott Connect)

3. Fishman’s Pulmonary Diseases and Disorders. PART 2: Scientific Basis of Lung Function in Health and Disease

4. Tammeling G.J., Quanjer Ph.H.: Contours of breathing 1-2. 1985 Boehringer Ingelheim

I admit that I have no experience with mathematical models and don't currently know of any.

In the later 1990s, we completed respiratory mechanics measurements under different forms of non-invasive ventilation using the so-called ‘Michigan lung’ model. With minimal modifications (to model the hysteresis effect), this mechanical lung model proved to be very helpful and usable (https://www.michiganinstruments.com/lung-simulators/).

Ref. (unfortunately, for internal reasons, this work was only published in the form of a poster):

1. Juhász J, Gröschel A, Hormann W, Sybrecht G.W. The impact of diverse mask pressure stability on the lung mechanics during CPAP in an experimental Model. Am J Respir Crit Care Med 1999;159(Suppl. 3):A427

2. Gröschel A, Juhász J, Hormann W, Sybrecht G.W. A comparison of the work of breathing in different CPAP-machines. Am J Respir Crit Care Med 1999;159(Suppl. 3):A428

I hope, this is some help for you. Best regards, János