A Novel Mathematical Equation for a Noninvasive Determination of the Number of Alveoli of the Human Lung

Main Article Content

Ikechukwu I. Udema

Abstract

Background: There had always been a spirited effort in understanding the transport of air or molecular oxygen plus other gases from alveolar air space into the pulmonary capillaries and from the latter back into the former using mathematical models; the determination of the number of alveoli using cadaver and invasive and partially noninvasive methods have been made. There is a need for a noninvasive method of mathematical nature, with evaluative, diagnostic, and prognostic application.

Objectives: The objectives of this research were to derive a mathematical equation for the noninvasive determination of the number of alveoli during rest and physical activity and elucidate the usefulness and advantage of the model over known methods.

Methods: Theoretical and computational (calculational) methods; data in the literature were substituted into the model mathematical equation for the computation of the number of alveoli in the human lungs.

Results and Discussion: The computed number (Nalv) of alveoli differed from one country or subcontinental region to another. The Nalv for the male were expectedly larger than for the female subjects.

Conclusion: The mathematical equation for totally noninvasive determination by computation is derivable and was derived. The total number (Nalv) of alveoli mobilised for function is a function of the width (d) of the nares (d 22/15), rate (Rv) of gas flow , and radius (ralv) of a functional alveolus . The equation has the potential to be of diagnostic, evaluative and prognostic value in medical practice. This new computational approach could be faster than other known approaches for the determination of the Nalv. A noninvasive approach by computation, relying on other noninvasively determined respiratory parameters, can eliminate the possibility of tissue damage.

Keywords:
Derivation of mathematical equation, computation, number of alveolar at rest, exercise, width of nares, air, molecular oxygen

Article Details

How to Cite
Udema, I. I. (2020). A Novel Mathematical Equation for a Noninvasive Determination of the Number of Alveoli of the Human Lung. Asian Journal of Biology, 10(1), 38-47. https://doi.org/10.9734/ajob/2020/v10i130098
Section
Short Research Article

References

Leong AFT, Buckley GA, Paganin DM, Hooper SB, Wallace MJ, Kitchen MJ. Real-time measurement of alveolar size and population using phase contrast x-ray imaging. Biomedical Optics Express. 2014; 5(11):1-15.

Ben-Tal A. Simplified models for gas exchange in the human lungs. J Theor Biol. 2006; 238: 474–495.

Angus GE, Thurlbeck WM. Number of alveolar in human lung. J. Appl. Physiol. 1972; 32(4): 483–485.

Iorungwa IS. A mathematical model of airflow into the lungs and its subsequent diffusion into the blood stream Thesis. 2011; 1-77.

Wagner PD. The physiological basis of pulmonary gas exchange: Implications for clinical interpretation of arterial blood gases. Eur. Respir. J. 2015; 45: 227-243.

Ochs M, Nyengaard JR, Jung A, Knudsen L, Voigt M, Wahlers T, et al. The number of alveoli in the human lung. Am J. Respir. Crit care Med. 2004; 169: 120-124.

Hyde DM, Tyler NK, Putney LF, Singh P, Gundersen HJG. Total number and mean size of alveoli in mammalian lung estimated using fractionators sampling and unbiased estimates of the Euler characteristic of alveolar openings. The Anatomical Record Part A. 2004; 274A: 216–226.

Mathison S, Nagilla R, Kompella UB. Nasal route for direct delivery of solutes to the central nervous system: Fact or fiction? J Drug Target. 1998; 5: 413–441.

Namati E, Thiesse T, de Ryk J, McLennan G. Alveolar dynamics during respiration: Are the pores of Kohn. A pathway to recruitment? Am. J. Respir. Cell Mol. Biol. 2008; 38(5): 572–578.

Schwenninger D, Runck H, Schumann S, Haberstroh J, Meissner S, Koch E, Guttmann J. Intravital microscopy of subpleural alveoli via transthoracic endoscopy. J. Biomed. Opt. 2011; 16(4):1-6.

Liu Y, Ohnson MR, Matida EA, Kherani S, Marsan J, Creation of a standardized geometry of the human nasal cavity. J. Appl. Physiol. 2009; 106: 784–795.

Udema II. Reason for higher rate of gas flow per unit cross-sectional area of smaller pore aperture. Asian J. Phys. Chem. Sci. 2018; 6(3): 1–11.

Zaidi AA, Mattern BC, Claes P, McEcoy B, Hughes C, Shriver MD. Investigating the case of human nose shape and climate adaptation. PLoS Genet. 2017; 13(3):1–31.

Baláshsházyl I, Hofmann W, Farkas A, Madas BG. Three-dimensional model for aerosol transport and deposition in expanding and contracting alveoli. Inhalation Toxicology. 2008; 20: 611-621.

Schriever VA, Hummel T, Johan N. Lundström JN, Freiherr J. Size of nostril opening as a measure of intranasal volume. Physiol. Behav. 2013; 110-111:1-3.

Hajari AJ, Yablonskiy DA, Suksstanskii AL, Quirk JD, Conradi MS, Woods JC. Morphometric changes in the human pulmonary acinus during inflation. J. Appl. Physiol. 2012; 112: 937–943.

Burton DA, Stokes K, Hall GM. Physiological effect of exercise. CEACCP. 2004; 4(6): 185–188.

George SC, Babb AL, Hlastala MP. Dynamics of soluble gas exchange in the airways iii. Single-exhalation breathing maneuvers. The American Physiological Society. 1993; 2439-2449.

Hlastala MP, Powell FL, Anderson JC. Airway exchange of highly soluble gases. Appl Physiol. 2013; 114: 675–680.

Reynolds A, Ermentrout GB, Clermont G. A mathematical model of pulmonary gas exchange under inflammatory stress. J Theor Biol. 2010; 264(2): 161–173.

Farha S, Laskowski D, George D, Park MM, Tang WHW, Dweik RA, et al. Loss of alveolar membrane diffusing capacity and pulmonary capillary blood volume in pulmonary arterial hypertension. Respir. Physiol. 2013; 14(6):1-8.

Roughton FJ, Forster RE. Relative importance of diffusion and chemical reaction rates in determining rate of exchange of gases in the human lung, with special reference to true diffusing capacity of pulmonary membrane and volume of blood in the lung capillaries. J. Appl. Physiol. 1957; 11(2): 290–302.

Cheng YS. Aerosol deposition in the extra-thoracic region. Aerosol Sci Technol. 2003; 37: 659–671.

Illum L. Transport of drugs from the nasal cavity to the central nervous system. Eur J Pharm Sci. 2000; 11:1–18.

Kelly JT, Prasad AK, Wexler AS. Detailed flow patterns in the nasal cavity. J Appl Physiol. 2000; 89: 323–337.

Kim SK, Chung SK. An investigation on airflow in disordered nasal cavity and its corrected models by tomographic PIV. Measurement Sci Technol. 2004; 15: 1090–1096.

Mathison S, Nagilla R, Kompella UB. Nasal route for direct delivery of solutes to the central nervous system: Fact or fiction? J Drug Target.1998; 5: 413–441.

Piiper J, Meyer M, Scheid P. Diffusion limitation in alveolar-capillary CO2 transfer in human lungs: Experimental evidence from re-breathing equilibration. Bauer C et al., (eds), Biophysics and Physiology of Carbon dioxide Springer-Verlag Berlin Heidelberg. 1980; 359- 360.Leong AFT, Buckley GA, Paganin DM, Hooper SB, Wallace MJ, Kitchen MJ. Real-time measurement of alveolar size and population using phase contrast x-ray imaging. Biomedical Optics Express. 2014; 5(11):1-15.

Ben-Tal A. Simplified models for gas exchange in the human lungs. J Theor Biol. 2006; 238: 474–495.

Angus GE, Thurlbeck WM. Number of alveolar in human lung. J. Appl. Physiol. 1972; 32(4): 483–485.

Iorungwa IS. A mathematical model of airflow into the lungs and its subsequent diffusion into the blood stream Thesis. 2011; 1-77.

Wagner PD. The physiological basis of pulmonary gas exchange: Implications for clinical interpretation of arterial blood gases. Eur. Respir. J. 2015; 45: 227-243.

Ochs M, Nyengaard JR, Jung A, Knudsen L, Voigt M, Wahlers T, et al. The number of alveoli in the human lung. Am J. Respir. Crit care Med. 2004; 169: 120-124.

Hyde DM, Tyler NK, Putney LF, Singh P, Gundersen HJG. Total number and mean size of alveoli in mammalian lung estimated using fractionators sampling and unbiased estimates of the Euler characteristic of alveolar openings. The Anatomical Record Part A. 2004; 274A: 216–226.

Mathison S, Nagilla R, Kompella UB. Nasal route for direct delivery of solutes to the central nervous system: Fact or fiction? J Drug Target. 1998; 5: 413–441.

Namati E, Thiesse T, de Ryk J, McLennan G. Alveolar dynamics during respiration: Are the pores of Kohn. A pathway to recruitment? Am. J. Respir. Cell Mol. Biol. 2008; 38(5): 572–578.

Schwenninger D, Runck H, Schumann S, Haberstroh J, Meissner S, Koch E, Guttmann J. Intravital microscopy of subpleural alveoli via transthoracic endoscopy. J. Biomed. Opt. 2011; 16(4):1-6.

Liu Y, Ohnson MR, Matida EA, Kherani S, Marsan J, Creation of a standardized geometry of the human nasal cavity. J. Appl. Physiol. 2009; 106: 784–795.

Udema II. Reason for higher rate of gas flow per unit cross-sectional area of smaller pore aperture. Asian J. Phys. Chem. Sci. 2018; 6(3): 1–11.

Zaidi AA, Mattern BC, Claes P, McEcoy B, Hughes C, Shriver MD. Investigating the case of human nose shape and climate adaptation. PLoS Genet. 2017; 13(3):1–31.

Baláshsházyl I, Hofmann W, Farkas A, Madas BG. Three-dimensional model for aerosol transport and deposition in expanding and contracting alveoli. Inhalation Toxicology. 2008; 20: 611-621.

Schriever VA, Hummel T, Johan N. Lundström JN, Freiherr J. Size of nostril opening as a measure of intranasal volume. Physiol. Behav. 2013; 110-111:1-3.

Hajari AJ, Yablonskiy DA, Suksstanskii AL, Quirk JD, Conradi MS, Woods JC. Morphometric changes in the human pulmonary acinus during inflation. J. Appl. Physiol. 2012; 112: 937–943.

Burton DA, Stokes K, Hall GM. Physiological effect of exercise. CEACCP. 2004; 4(6): 185–188.

George SC, Babb AL, Hlastala MP. Dynamics of soluble gas exchange in the airways iii. Single-exhalation breathing maneuvers. The American Physiological Society. 1993; 2439-2449.

Hlastala MP, Powell FL, Anderson JC. Airway exchange of highly soluble gases. Appl Physiol. 2013; 114: 675–680.

Reynolds A, Ermentrout GB, Clermont G. A mathematical model of pulmonary gas exchange under inflammatory stress. J Theor Biol. 2010; 264(2): 161–173.

Farha S, Laskowski D, George D, Park MM, Tang WHW, Dweik RA, et al. Loss of alveolar membrane diffusing capacity and pulmonary capillary blood volume in pulmonary arterial hypertension. Respir. Physiol. 2013; 14(6):1-8.

Roughton FJ, Forster RE. Relative importance of diffusion and chemical reaction rates in determining rate of exchange of gases in the human lung, with special reference to true diffusing capacity of pulmonary membrane and volume of blood in the lung capillaries. J. Appl. Physiol. 1957; 11(2): 290–302.

Cheng YS. Aerosol deposition in the extra-thoracic region. Aerosol Sci Technol. 2003; 37: 659–671.

Illum L. Transport of drugs from the nasal cavity to the central nervous system. Eur J Pharm Sci. 2000; 11:1–18.

Kelly JT, Prasad AK, Wexler AS. Detailed flow patterns in the nasal cavity. J Appl Physiol. 2000; 89: 323–337.

Kim SK, Chung SK. An investigation on airflow in disordered nasal cavity and its corrected models by tomographic PIV. Measurement Sci Technol. 2004; 15: 1090–1096.

Mathison S, Nagilla R, Kompella UB. Nasal route for direct delivery of solutes to the central nervous system: Fact or fiction? J Drug Target.1998; 5: 413–441.

Piiper J, Meyer M, Scheid P. Diffusion limitation in alveolar-capillary CO2 transfer in human lungs: Experimental evidence from re-breathing equilibration. Bauer C et al., (eds), Biophysics and Physiology of Carbon dioxide Springer-Verlag Berlin Heidelberg. 1980; 359- 360.Leong AFT, Buckley GA, Paganin DM, Hooper SB, Wallace MJ, Kitchen MJ. Real-time measurement of alveolar size and population using phase contrast x-ray imaging. Biomedical Optics Express. 2014; 5(11):1-15.

Ben-Tal A. Simplified models for gas exchange in the human lungs. J Theor Biol. 2006; 238: 474–495.

Angus GE, Thurlbeck WM. Number of alveolar in human lung. J. Appl. Physiol. 1972; 32(4): 483–485.

Iorungwa IS. A mathematical model of airflow into the lungs and its subsequent diffusion into the blood stream Thesis. 2011; 1-77.

Wagner PD. The physiological basis of pulmonary gas exchange: Implications for clinical interpretation of arterial blood gases. Eur. Respir. J. 2015; 45: 227-243.

Ochs M, Nyengaard JR, Jung A, Knudsen L, Voigt M, Wahlers T, et al. The number of alveoli in the human lung. Am J. Respir. Crit care Med. 2004; 169: 120-124.

Hyde DM, Tyler NK, Putney LF, Singh P, Gundersen HJG. Total number and mean size of alveoli in mammalian lung estimated using fractionators sampling and unbiased estimates of the Euler characteristic of alveolar openings. The Anatomical Record Part A. 2004; 274A: 216–226.

Mathison S, Nagilla R, Kompella UB. Nasal route for direct delivery of solutes to the central nervous system: Fact or fiction? J Drug Target. 1998; 5: 413–441.

Namati E, Thiesse T, de Ryk J, McLennan G. Alveolar dynamics during respiration: Are the pores of Kohn. A pathway to recruitment? Am. J. Respir. Cell Mol. Biol. 2008; 38(5): 572–578.

Schwenninger D, Runck H, Schumann S, Haberstroh J, Meissner S, Koch E, Guttmann J. Intravital microscopy of subpleural alveoli via transthoracic endoscopy. J. Biomed. Opt. 2011; 16(4):1-6.

Liu Y, Ohnson MR, Matida EA, Kherani S, Marsan J, Creation of a standardized geometry of the human nasal cavity. J. Appl. Physiol. 2009; 106: 784–795.

Udema II. Reason for higher rate of gas flow per unit cross-sectional area of smaller pore aperture. Asian J. Phys. Chem. Sci. 2018; 6(3): 1–11.

Zaidi AA, Mattern BC, Claes P, McEcoy B, Hughes C, Shriver MD. Investigating the case of human nose shape and climate adaptation. PLoS Genet. 2017; 13(3):1–31.

Baláshsházyl I, Hofmann W, Farkas A, Madas BG. Three-dimensional model for aerosol transport and deposition in expanding and contracting alveoli. Inhalation Toxicology. 2008; 20: 611-621.

Schriever VA, Hummel T, Johan N. Lundström JN, Freiherr J. Size of nostril opening as a measure of intranasal volume. Physiol. Behav. 2013; 110-111:1-3.

Hajari AJ, Yablonskiy DA, Suksstanskii AL, Quirk JD, Conradi MS, Woods JC. Morphometric changes in the human pulmonary acinus during inflation. J. Appl. Physiol. 2012; 112: 937–943.

Burton DA, Stokes K, Hall GM. Physiological effect of exercise. CEACCP. 2004; 4(6): 185–188.

George SC, Babb AL, Hlastala MP. Dynamics of soluble gas exchange in the airways iii. Single-exhalation breathing maneuvers. The American Physiological Society. 1993; 2439-2449.

Hlastala MP, Powell FL, Anderson JC. Airway exchange of highly soluble gases. Appl Physiol. 2013; 114: 675–680.

Reynolds A, Ermentrout GB, Clermont G. A mathematical model of pulmonary gas exchange under inflammatory stress. J Theor Biol. 2010; 264(2): 161–173.

Farha S, Laskowski D, George D, Park MM, Tang WHW, Dweik RA, et al. Loss of alveolar membrane diffusing capacity and pulmonary capillary blood volume in pulmonary arterial hypertension. Respir. Physiol. 2013; 14(6):1-8.

Roughton FJ, Forster RE. Relative importance of diffusion and chemical reaction rates in determining rate of exchange of gases in the human lung, with special reference to true diffusing capacity of pulmonary membrane and volume of blood in the lung capillaries. J. Appl. Physiol. 1957; 11(2): 290–302.

Cheng YS. Aerosol deposition in the extra-thoracic region. Aerosol Sci Technol. 2003; 37: 659–671.

Illum L. Transport of drugs from the nasal cavity to the central nervous system. Eur J Pharm Sci. 2000; 11:1–18.

Kelly JT, Prasad AK, Wexler AS. Detailed flow patterns in the nasal cavity. J Appl Physiol. 2000; 89: 323–337.

Kim SK, Chung SK. An investigation on airflow in disordered nasal cavity and its corrected models by tomographic PIV. Measurement Sci Technol. 2004; 15: 1090–1096.

Mathison S, Nagilla R, Kompella UB. Nasal route for direct delivery of solutes to the central nervous system: Fact or fiction? J Drug Target.1998; 5: 413–441.

Piiper J, Meyer M, Scheid P. Diffusion limitation in alveolar-capillary CO2 transfer in human lungs: Experimental evidence from re-breathing equilibration. Bauer C et al., (eds), Biophysics and Physiology of Carbon dioxide Springer-Verlag Berlin Heidelberg. 1980; 359- 360.