The use of the modified Beer-Lambert law to improve biomolecular detection: diagnosing hemoglobin disorders and identifying pathogens

The use of the modified Beer-Lambert law to improve biomolecular detection: diagnosing hemoglobin disorders and identifying pathogens | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article The use of the modified Beer-Lambert law to improve biomolecular detection: diagnosing hemoglobin disorders and identifying pathogens Mohamed Ilyes Habia, Ghania Habia, Aissa Manallah This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6231215/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Biomolecule detection techniques are still a top priority in modern research because they are essential for clinical point-of-care diagnostics. This is especially true for pathogens like viruses and bacteria. Hemoglobin disorders, including thalassemia and sickle cell disease, substantially hinder oxygen transport and overall health. Hemoglobinopathies like methemoglobin (MetHb), carboxyhemoglobin (COHb), and sulfhemoglobin (SulHb) are difficult to diagnose because blood samples break down quickly after being collected, making it difficult to do an accurate and quick analysis. The Beer-Lambert Law (BLL), which is the basis of absorbance spectroscopy, is used a lot in analytical methods because it gives accurate quantitative data with little sample preparation. The Modified Beer-Lambert Law (MBLL) builds on the basic idea by adding scattering effects and uneven mediums. This versatility makes it especially useful for biological systems that are very complicated. Its integration simplifies the process of obtaining precise concentration readings and monitoring metabolic activities in real-time, particularly in environments such as cloudy or scattering blood. The MBLL is used to think about hemoglobin diseases, and Matlab is used to look at the optical properties of MetHb, COHb, and SulHb. The MBLL correlates light absorbance with the concentration of absorbing species, facilitating a more profound comprehension of the spectroscopic characteristics of various hemoglobin derivatives. This method makes diagnosis more accurate and shows how advanced spectroscopic modeling can be used to solve clinical problems related to hemoglobinopathies. Modified Beer-Lambert Absorbance hemoglobin disorder blood disease Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 1. Introduction Hemoglobin is a protein found within red blood cells that transports oxygen. When hemoglobin is absent, faulty, or impaired, the decreased oxygen levels in the blood cause dizziness, fatigue, shortness of breath, and an abnormal heart rate. Anemia is defined as a low hemoglobin concentration caused by more than seventeen different diseases, the most common of which are iron deficiency, hemoglobinopathy, and malaria [ 1 ]. Iron (Fe³⁺) in hemoglobin's (Hb) heme group changes into ferric iron (Fe³⁺). MetHb undergoes this transformation. MetHb's ferric iron is incapable of binding oxygen (O 2 ) [ 2 ]. As a result, the O 2 dissociation curve is shifted to the left, making it more difficult to release O 2 and provide adequate tissue oxygenation [ 3 ]. MetHb formation also reduces the amount of Hb available for O 2 binding and transport [ 4 ]. One way to find out how much oxygen is bound to hemoglobin molecules is to compare the total amount of oxygen that can bind to them to the average amount of oxygen that is already bound. When one gram of O2 of functional hemoglobin unites with 1.34 ml of O 2 , the oxygen capacity of normal blood is (150 g Hb/liter) (1.34 ml O 2 g Hb) = 200 ml O 2 /liter [ 5 ]. Carboxyhemoglobin (COHb) is molded when Hb and carbon monoxide (CO) interact; CO binds to heme molecules 240 times more than O 2 [ 6 ]. The resulting CO Hb limits the blood’s O 2 -carrying capacity. In healthy, non-smoking individuals, Met Hb is in the range of 0.67 ± 0.33% and CO Hb in the range of 0.5–1.5%. Sulfhaemoglobin can be caused by exposure to any substance that contains a sulfur atom capable of binding to hemoglobin. SulHb, also called sulfhaemoglobin, is a rare disorder that can happen when people are exposed to chemicals that have sulfur atoms that can attach to hemoglobin [ 7 ]. This binding process hinders the ability of hemoglobin to carry oxygen. This is in contrast to MetHb, which has a known antidote in the form of methylene blue. SulHb is irreversible and has no known antidote [ 7 ]. Early detection, diagnosis, and intervention are required to prevent end-organ damage, especially when sulfhemoglobin levels are high. SPR can be applied specifically in healthcare systems and biotechnology labs to contribute to the diagnosis and prognosis of hemoglobin anomalies. [ 8 ]. 2. The Essential Concepts In the near-infrared (NIR) spectrum (NIR "window": about 650–900 nm), biological tissues don't absorb much light. Because of this, NIR light can be used to get physiological information from deeper tissue layers, often without cutting them. Continuous wave near-infrared spectroscopy (cwNIRS) is often used to measure changes in the amounts of tissue chromophores, especially oxyhemoglobin (HbO₂) and deoxyhemoglobin (Hb). The BLL law describes how a beam of visible light is absorbed by a medium that loses energy and how the intensity of the radiation is related to the optical path inside the medium [ 9 ]. The BLL applies to all electromagnetic radiation and light-absorbing substances, such as gases, solids, liquids, molecules, atoms, and ions [ 10 ]. And the rule serves as a quantitative foundation for absorbance photometry, colorimetric analysis, and other applications. It means that the amount of light absorbed by a clear substance doesn't change depending on how bright the light is hitting it. Eq. ( 4 ) [ 11 ] can be used to find the absorbance. $$A=\log \left( {\frac{1}{T}} \right)=\varepsilon \left( \lambda \right)LC$$ 1 . $$T=\frac{{{I_{}}}}{{{I_0}}}$$ 2 $$I={I_0}\exp ( - {\mu _a}L)$$ 3 $${\mu _a}=\varepsilon \left( \lambda \right)C$$ 4 Where A is the absorbance, T is the transmittance, C is the concentration of a light-absorbing substance, L is the optical path or the thickness of the substance layer, and ε (λ) is the molar extinction coefficient depending on wavelength. I denotes the transmitted light intensity, whereas I 0 represents the incident light intensity. µ a is the absorption coefficient (cm⁻¹) of the medium [ 12 ]. In a homogeneous medium, absorbance A and light path length b are directly proportional. As the thickness of the absorbent medium increases, the intensity of transmitted radiation falls. The Beer-Lambert law establishes a direct link between absorbance and concentration, enabling precise measurement of analytes in a solution [ 13 ]. This method is very useful for assessing the concentrations of things like medicines, biomolecules, and contaminants. Real-time Monitoring The law enables real-time monitoring of reactions and processes by measuring how absorbance varies over time. This feature is critical for kinetic research and quality control applications. This feature enables non-destructive testing [ 14 ]. The Beer-Lambert law lets tests be done without damaging the material, since measuring absorbance doesn't change it much [ 15 ]. The MBLL relies on two assumptions: (1) the medium's absorption varies uniformly, and (2) the scattering loss is constant [ 16 ]. The method is especially useful in biological applications, where sample integrity is critical. It may fail at high absorber or scattered concentrations due to increasing molecular interactions (e.g., self-quenching effects). As concentration grows, light absorption becomes nonlinear, and departures from linearity might occur, confounding measurement attempts [ 17 ]. Optical Inhomogeneity: Biological tissues are often heterogeneous, with different parts having different optical properties [ 18 ]. For example, blood vessels, fat, and connective tissue are all heterogeneous. The resultant non-homogeneity may hamper the use of BLL, which presupposes a homogenous medium. Variations in tissue composition can lead to variable absorption and scattering, making it impossible to use a straightforward linear connection between absorbance and concentration [ 12 ]. The usual way to do cwNIRS uses one or more light source–detector pairs, thinks of the lit tissue as optically homogeneous, and uses a changed version of Beer–Lambert's law [ 16 ]. $$A=\log \left( {\frac{1}{T}} \right)=\varepsilon \left( \lambda \right)LC+S$$ 5 The component S, which is independent of absorption and represents intensity loss due to scattering, is a geometry-dependent component. Sassaroli and Fantini showed that the above equation is wrong (L should be changed to its mean over the absorption coefficient range of 0–±a) [ 19 ]. However, the differential form of the MBLL is still valid for small changes in attenuation as long as S stays the same and absorption changes evenly within the illuminated tissue volume [ 16 ]. $$\Delta A=L\Delta {\mu _a}$$ 6 The difference in attenuation can be found by comparing the intensity values of two different states of the tissue. This change is directly related to the change in absorption. The latter represents the weighted sum of the variation in the concentration of tissue chromophores [ 20 ]. $$\Delta {\mu _a}=\sum {{\varepsilon _i}(\lambda )\Delta {C_i}}$$ 7 We can write the above equation as [ 16 ] for oxyhemoglobin and deoxyhemoglobin. $$\Delta {\mu _a}={\varepsilon _{Hb}}(\lambda )\Delta {C_{Hb}}+{\varepsilon _{HbO}}(\lambda )\Delta {C_{HbO}}$$ 8 An expression representing the variations in scattering must be incorporated into the MBLL. Attenuation, being a function of the absorption coefficient and the transport scattering coefficient, can be expressed as [ 16 ]. $$\Delta A=L\Delta {\mu _a}+P\Delta \mu {'_s}$$ 9 µ's is the reduced scattering coefficients [ 18 ]. Which values of L and K are used depend on the tissue's initial optical properties (µa and µs') at the given wavelengths and the exact shape of the measurement. The outbound intensity relation can be written as: $$I={I_0}\exp - (L\Delta {\mu _a}+P\Delta \mu {'_s})$$ 10 The refractive index of the analyte medium (blood) in the visual region is defined by [ 21 ]: $${n_s}={n_{Hb}}=n+i \times k$$ 11 Friebel defines n as the real component of hemoglobin's refractive index, which changes as a function of wavelength and concentration [ 21 ][ 22 ]. $${n_{Hb}}=n(\lambda ,{C_{Hb}})={n_{{H_2}O}}(\lambda ) \times \left( {\beta (\lambda ) \times {C_{Hb}}+1} \right)$$ 12 In this equation, n H₂O represents the refractive index of water as a function of wavelength, β(λ) is the specific refractive increase based on wavelength, and C Hb represents the concentration of hemoglobin. Using Prahl's molar extinction coefficient data [ 21 ][ 23 ], you can figure out the imaginary part (k) of the refractive index that changes with the amount of hemoglobin. The calculation can be done at any wavelength. $$k=(2.303 \times e \times {C_{Hb}} \times \lambda )/(4\pi \times {M_{Hb}})$$ 13 In this relation, e represents the molar extinction coefficient, and MHb represents the molar mass of hemoglobin. 3. Results and discussion In this work model and source, code under Matlab program developed under the rule of MBLL to simulate the outbound intensity and the absorbance for each species. The extinction molar coefficients (in mM⁻³•cm⁻³) of five types of hemoglobin are shown in Fig. 3 . These are Hb (deoxyhemoglobin), HbO₂ (oxyhemoglobin), COHb (carboxyhemoglobin), SulfHb (sulfhemoglobin), and MetHb (methemoglobin). The wavelengths are shown in nm. The extinction coefficient tells us how well each type of hemoglobin can absorb light at different wavelengths. This is very important for spectroscopic analysis and diagnostic uses. The blue curve representing Hb has a pronounced absorption peak about 560 nm, followed by a subsequent rise in the lower wavelength region of 400–430 nm. This particular absorption profile shows how iron molecules are arranged when they are deoxygenated. When oxygen is not bound, it changes the electronic transitions inside the porphyrin ring. HbO₂ (red curve) exhibits a distinct double-peak pattern in the 540–580 nm spectrum, with maxima at around 542 nm and 578 nm. In pulse oximetry and spectroscopic analysis, the standard "double-hump" pattern is often used to tell the difference between hemoglobin that is oxygenated and hemoglobin that is not oxygenated. The SulfHb (magenta curve) has a distinct absorption peak near 620 nm, making it spectrally unique compared to other hemoglobin types. This one-of-a-kind peak marks SulfHb as an optical aberration, which makes it easier to find even in very small amounts in clinical samples. On the other hand, MetHb (the cyan curve) has a complicated absorption spectrum with clear peaks at around 500 nm and 630–650 nm. This clear pattern shows how ferrous (Fe²⁺) iron changes into ferric (Fe³⁺) iron, which greatly changes the electrical properties of the heme group. In the 550–600 nm range, where Hb, HbO₂, and COHb all absorb strongly, the graph shows a lot of spectral overlap. This evidence supports the claim that simple single-wavelength analysis is not enough to accurately tell the difference between species in clinical specimens. To get accurate concentration information from complicated mixtures, you need to use multivariate analytical methods like principal component analysis or partial least squares regression. The strength of the argument is shown by looking at the extinction coefficients: at about 570 nm, both Hb and COHb have strong absorption values, making it harder to tell them apart directly. Using observations at different wavelengths to build overdetermined systems of equations is the scientific way to solve this overlap problem. This makes it possible to get an accurate concentration reading even when measurement noise is present. Isolated absorption peaks serve as diagnostically significant markers. The research presents compelling evidence that specific wavelengths have distinct diagnostic significance. The single peak of SulfHb at 620 nm and the unique absorption of MetHb at 630 nm are the best spectroscopic clues for these medical situations. This claim is based on scientific evidence that certain wavelengths make it easier to find abnormal hemoglobin species, even when they are present in small amounts or with a lot of normal hemoglobin variants. The fact that long wavelengths (700 nm) are not absorbed has a big effect on the design of spectroscopic instruments and the choice of measurement parameters in clinical and research settings. This supports the claim that near-infrared spectroscopy is useful for in vivo applications where tissue penetration is important. This careful study of spectroscopic evidence and use of well-known biophysical principles proves beyond a reasonable doubt that the different extinction coefficient profiles of different hemoglobin species provide a solid basis for their quantitative differentiation and clinical evaluation. Figure 4 shows four sets of spectroscopic data that strongly support the idea that different types of hemoglobin interact with light in complex ways in the visible and near-infrared ranges. Figure 4 .a: Total Absorption Coefficient: Fig. 4 .a illustrates the total absorption coefficient as a function of wavelength (in nm). Figure 4 .a displays the overall absorption coefficient for the wavelength range of 400 to 900 nm. An evident absorption peak is observed in the range of 550to580 nm. The range of 450–500 nm reduces the absorption peak, indicating robust absorption at this wavelength. The absorption coefficient markedly diminishes on both sides of this peak, indicating that the material absorbs light most specifically in this area. The values shown in Fig. 4 .b are the overall absorption coefficients. These values show the complex spectrum signature that comes from mixing different types of hemoglobin (HbO₂, Hb, COHb, MetHb, and SulfHb). The cumulative effect signifies more than the simple aggregation of separate components; it embodies the weighted contributions according to their various concentrations. I contend that this wavelength dependency is fundamental to spectroscThe visible spectrum (450–650 nm) reveals significant changes in absorption characteristics among hemoglobin species, while the near-infrared spectrum (700–900 nm) reveals very minor variances. very minor variances. This observation undermines the oversimplified notion that hemoglobin derivatives may be uniformly differentiated throughout the whole spectrum. Consequently, the selection of critical wavelengths is crucial for particular diagnostic applications. The contributions to absorption data elucidate how each hemoglobin variant disproportionately affects overall absorption at specific wavelengths. Oxyhemoglobin predominates with its unique double-peak pattern at around 542 and 578 nm, whereas deoxyhemoglobin displays its characteristic offer supplementary spectrum characteristics: COHb exhibits a notable peak at approximately 570 nm, MetHb displays peaks at 500 and 630 nm, and SulfHb is characterized by its unique peak around 556 nm absorption at 620 nm. In Fig. 4 .c The exponential behavior has significant consequences for Traditional pulse oximetry uses the red (660 nm) and infrared (940 nm) wavelengths to tell the difference between HbO₂ and Hb, but I believe this method is flawed in the present situation of deoxyhemoglobin. The transmitted intensity patterns show that deoxyhemoglobin make absorption profiles that aren't accurate, and standard two-wavelength methods can't tell the difference. The evidence strongly indicates that multi-wavelength spectroscopy is crucial for precise hemoglobin differentiation. Using measuring wavelengths between 600 and 630 nm will make it much easier to find MetHb, and looking at things between 570 and 580 nm will help figure out how much COHb is present. The absorption patterns. This depiction elucidates the spectral signatures of several hemoglobin species, especially in the visible spectrum where unique absorption peaks arise. Still, I argue that interpreting absorbance spectra is very hard when they are used for measurements that take place inside living things. According to Mie theory, tissue scattering changes with wavelength. This makes a sloping baseline that can hide fine spectral features. When other chromophores like melanin, bilirubin, and water are added, their absorption spectra overlap, which makes analysis more difficult. 4. Effect in increasing hemoglobin’s species concentrations Effect of Increasing COHb Levels Figure 5 .a shows how rising amounts of carboxyhemoglobin (COHb) alter light intensity throughout wavelengths. As COHb levels grow (from 0–5%), the intensity curve shifts slightly, particularly in the visible band (400–700 nm). The most substantial changes occur in specific wavelength ranges, likely correlating to COHb absorption peaks. This suggests that COHb has distinct absorption characteristics that influence light transmission. Non-invasive spectroscopic techniques could use the data to estimate COHb levels in blood, a crucial step in diagnosing carbon monoxide poisoning. Effect of Increasing MetHb Levels The effect of increasing methemoglobin (MetHb) levels (0–5%) on light intensity in Fig. 5 .b shows that. There are notable fluctuations in intensity, notably in the 500–700 nm region, with distinct peaks and troughs changing as MetHb levels grow. MetHb has unique absorption capabilities that affect light transmission in specific wavelength bands. This information is valuable for detecting methemoglobinemia, a condition where MetHb levels are abnormally high, impairing oxygen delivery. Effect of Increasing SulfHb Levels The impact of sulfhemoglobin (SulfHb) levels (0–5%) on light intensity (Fig. 5 .c).The intensity curves reveal considerable changes in the visible spectrum, with shifts in absorption peaks as SulfHb levels increase. SulfHb's absorption properties can be utilised to measure its levels in blood.This is significant for diagnosing sulfhemoglobinemia, a rare illness caused by sulfur attaching to hemoglobin, which can hamper oxygen transport. Effect of Oxygen Saturation (SO2) Figure 5 .d shows how varying oxygen saturation levels (0–100%) affect light intensity. There are big changes in the intensity curves, especially in the red and near-infrared (600–900 nm) ranges, which are often used in pulse oximetry. The link between oxygen saturation and light intensity is the basis for pulse oximetry, a non-invasive way to test blood. The data emphasizes the importance of specific wavelengths for accurate oxygen saturation measurements. These graphs illustrate the capabilities of spectroscopic methods in medical diagnostics. By looking at how much light is absorbed and transmitted at different wavelengths based on MBLL, it is possible to figure out the levels of oxygen saturation and different types of hemoglobin in the blood. The ability to measure COHb, MetHb, SulfHb, and SO 2 levels non-invasively is critical for diagnosing and monitoring conditions such as carbon monoxide poisoning, methemoglobinemia, sulfhemoglobinemia, and hypoxemia. Further studies could focus on refining the wavelength ranges and boosting the sensitivity of spectroscopic equipment. The incorporation of these discoveries into wearable or portable devices could increase real-time monitoring of blood parameters in clinical and emergency scenarios. In the red and near-infrared (600–900 nm) spectrums, which are used in pulse oximetry, HbO₂ and Hb have different absorption maxima. Conversely, COHb predominantly influences the visible spectrum. COHb lowers the amount of oxygen that blood can carry by competing with oxygen for binding sites on hemoglobin. This can happen even when oxygen saturation levels are normal. In contrast to HbO₂ and Hb, MetHb is incapable of binding oxygen, hence hindering oxygen transport and delivery. This results in methemoglobinemia, a disorder marked by cyanosis and hypoxia. Spectroscopic methods can tell the difference between MetHb concentrations because MetHb changes the spectrum in a way that HbO₂ and Hb do not. The spectral signature of SulfHb is different from those of HbO₂ and Hb, which makes it easier to identify and measure. SulfHb, in contrast to HbO₂ and Hb, is incapable of efficiently transporting oxygen, resulting in compromised oxygen delivery. This property is different from HbO₂ and Hb, which have wider effects in the red and near-infrared ranges. SulfHb changes things more in the visible range. 5. Conclusion This study shows that the Modified Beer-Lambert Law (MBLL) has a lot of potential for using it to look at hemoglobin abnormalities using spectroscopy. This is a big step forward in non-invasive diagnostic methods. This research uses a lot of mathematical modelling and spectroscopic analysis of different types of hemoglobin to build a solid base for telling the difference between normal hemoglobin and deoxyhemoglobin, such as methemoglobin (MetHb), carboxyhemoglobin (COHb), and sulfhemoglobin (SulfHb). Because each type of hemoglobin has its own unique spectral signature, it is possible to accurately find and measure them. The absorption peaks at certain wavelengths—HbO₂ has a double peak at 542/578 nm, MetHb has peaks at 500/630 nm, and SulfHb has a noticeable absorption at 620 nm—are important biomarkers for the diseases they represent. The differences in the spectrum make it easier to make custom diagnostic tools that can find abnormal hemoglobin variants even when they are present in very small amounts within complex biological matrices. Our study clearly shows that multi-wavelength spectroscopic techniques are necessary for accurately telling the difference between hemoglobin species. This is because two-wavelength techniques, like those used in standard pulse oximetry, are limited when deoxyhemoglobin are present. A careful study of how different concentrations change the transmission of light gives us a way to use math to figure out threshold values and diagnostic algorithms that work in the real world. Because the MBLL can include both absorption and scattering effects, it is very useful for biological applications. It gets around the problems that the standard Beer-Lambert Law has with mediums that aren't all the same, like blood and tissue. This better modelling method makes it easier to get more accurate concentration readings when the liquid is cloudy, bringing laboratory accuracy into line with point-of-care uses. This study is a big step forward in biomedical optics and hemoglobin research. It shows that using the Modified Beer-Lambert Law for spectroscopic analysis is a good way to diagnose and keep an eye on hemoglobin disorders without having to cut them open. This has big implications for clinical practice and the development of new medical devices. Declarations Author Contribution Habia mohamed ilyes :wrote the main manuscript, Simulation & interpretation Habia Ghania : interpretation & Correction Manallah Aissa: interpretation, reviewing & validation References N. Taparia, K. C. Platten, K. B. Anderson, and N. J. Sniadecki, “A microfluidic approach for hemoglobin detection in whole blood,” AIP Adv. , vol. 7, no. 10, 2017, doi: 10.1063/1.4997185. L. Gharahbaghian, B. Massoudian, and G. 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Meinke, “Model function to calculate the refractive index of native hemoglobin in the wavelength range of 250-1100 nm dependent on concentration,” Appl. Opt. , vol. 45, no. 12, pp. 2838–2842, 2006, doi: 10.1364/AO.45.002838. S. Prahl, “Optical absorption of hemoglobin,” Oregon Medical Laser Center , 1999. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6231215","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":432170263,"identity":"a8c6226c-bd88-4776-a59c-6b40f23701ad","order_by":0,"name":"Mohamed Ilyes Habia","email":"","orcid":"","institution":"University Ferhat Abbas of Setif","correspondingAuthor":false,"prefix":"","firstName":"Mohamed","middleName":"Ilyes","lastName":"Habia","suffix":""},{"id":432170264,"identity":"f09314fa-e859-41e8-bb3a-041baed64ddd","order_by":1,"name":"Ghania Habia","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Ghania","middleName":"","lastName":"Habia","suffix":""},{"id":432170265,"identity":"21b82235-3356-4d95-a4e6-c2ae69a11530","order_by":2,"name":"Aissa Manallah","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABBklEQVRIie2PIWvEMBiGvxC7+br+hajAWOmJ/YwzLYGeCkxWnMhRSOXZuf6CwanplA+mwmoLNTvuD0ROnFhyc4P2NjdYHvNFvA9vXoBI5I9iapYBUKrA1f4BZGeWBQrGPlb+EkWebBUUdVUB6/CSpLcav3qXhLTd98YwXKct2bkbO+TPLfqWbbaeU5hF8MpGHpA0SVJP4sWWXnmtpJpTEgHo2L08+C0Js5PgxitE4aySdqfQQmXXkOaj1G+CD8dlBUYalAepkOik1ybn45UWZgXzShW26DtlRcFH31IsbEnb/uTMOZPdHnE61/mKD5vju9tm8x/7TnlJFj+NB1a/CUcikcj/4BMOxnB6OzzHOwAAAABJRU5ErkJggg==","orcid":"","institution":"University Ferhat Abbas of Setif","correspondingAuthor":true,"prefix":"","firstName":"Aissa","middleName":"","lastName":"Manallah","suffix":""}],"badges":[],"createdAt":"2025-03-15 07:38:12","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6231215/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6231215/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":79377562,"identity":"4944bcf0-d191-4ab0-9bd9-d6bb1ef56021","added_by":"auto","created_at":"2025-03-27 15:36:50","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":109475,"visible":true,"origin":"","legend":"\u003cp\u003eIllustration of the Beer-Lambert-Bouguer law\u003c/p\u003e\n\u003cp\u003eLight Absorbed, 2) Light Directly Transmitted Detected\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-6231215/v1/dd279981063065925a7d73ba.png"},{"id":79378335,"identity":"0a047024-9724-44b5-8f1a-39814af2b18f","added_by":"auto","created_at":"2025-03-27 15:44:50","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":192350,"visible":true,"origin":"","legend":"\u003cp\u003eThe principal of the modified Beer-Lambert law\u003c/p\u003e\n\u003cp\u003e1)Light absorbed, 2) light directly transmitted detected, 3) light scattered detected, 4) light scattered not detected\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-6231215/v1/1f09c0d9f951f746293933e6.png"},{"id":79377566,"identity":"bafc7336-a4cb-4889-b6ca-bda0d498e926","added_by":"auto","created_at":"2025-03-27 15:36:50","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":164490,"visible":true,"origin":"","legend":"\u003cp\u003eExtinction molar coefficient of hemoglobin species\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-6231215/v1/70d1401bd1b1c917f17b75aa.png"},{"id":79378336,"identity":"f17ff0ac-416e-4d11-8552-1cb9ba5e588b","added_by":"auto","created_at":"2025-03-27 15:44:50","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":162568,"visible":true,"origin":"","legend":"\u003cp\u003ePhotonic interactions of haemoglobin variants over the visible and near-infrared spectra\u003c/p\u003e\n\u003cp\u003ea) Total Absorbance coefficient, b) Absorption contribution of hemoglobin species, c) transmitted intensity, d) absorbance spectra\u003c/p\u003e","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-6231215/v1/eb6f0f023ff053c9e03d14bf.png"},{"id":79377574,"identity":"8999dd3f-8729-4ac2-80ff-7160be7d584e","added_by":"auto","created_at":"2025-03-27 15:36:50","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":188260,"visible":true,"origin":"","legend":"\u003cp\u003eEffect in increasing hemoglobin’s species concentrations\u003c/p\u003e\n\u003cp\u003ea) Carboxyhemoglobin, b) Methemoglobin, c) Sulfhemoglobin, d)\u003cstrong\u003e \u003c/strong\u003eOxygen Saturation\u003c/p\u003e","description":"","filename":"floatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-6231215/v1/a2639b14964c1d2126783f6e.png"},{"id":85506291,"identity":"eeb086de-5f0c-4823-927b-8d5bdd2687de","added_by":"auto","created_at":"2025-06-26 15:31:48","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1306196,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6231215/v1/669274f4-8393-40fa-8d44-f3440ee45262.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"The use of the modified Beer-Lambert law to improve biomolecular detection: diagnosing hemoglobin disorders and identifying pathogens","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eHemoglobin is a protein found within red blood cells that transports oxygen. When hemoglobin is absent, faulty, or impaired, the decreased oxygen levels in the blood cause dizziness, fatigue, shortness of breath, and an abnormal heart rate. Anemia is defined as a low hemoglobin concentration caused by more than seventeen different diseases, the most common of which are iron deficiency, hemoglobinopathy, and malaria [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. Iron (Fe\u0026sup3;⁺) in hemoglobin's (Hb) heme group changes into ferric iron (Fe\u0026sup3;⁺). MetHb undergoes this transformation. MetHb's ferric iron is incapable of binding oxygen (O\u003csub\u003e2\u003c/sub\u003e) [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. As a result, the O\u003csub\u003e2\u003c/sub\u003e dissociation curve is shifted to the left, making it more difficult to release O\u003csub\u003e2\u003c/sub\u003e and provide adequate tissue oxygenation [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. MetHb formation also reduces the amount of Hb available for O\u003csub\u003e2\u003c/sub\u003e binding and transport [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. One way to find out how much oxygen is bound to hemoglobin molecules is to compare the total amount of oxygen that can bind to them to the average amount of oxygen that is already bound. When one gram of O2 of functional hemoglobin unites with 1.34 ml of O\u003csub\u003e2\u003c/sub\u003e, the oxygen capacity of normal blood is (150 g Hb/liter) (1.34 ml O\u003csub\u003e2\u003c/sub\u003e g Hb)\u0026thinsp;=\u0026thinsp;200 ml O\u003csub\u003e2\u003c/sub\u003e/liter [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. Carboxyhemoglobin (COHb) is molded when Hb and carbon monoxide (CO) interact; CO binds to heme molecules 240 times more than O\u003csub\u003e2\u003c/sub\u003e[\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. The resulting CO\u003csub\u003eHb\u003c/sub\u003e limits the blood\u0026rsquo;s O\u003csub\u003e2\u003c/sub\u003e-carrying capacity. In healthy, non-smoking individuals, Met\u003csub\u003eHb\u003c/sub\u003e is in the range of 0.67\u0026thinsp;\u003cem\u003e\u0026plusmn;\u003c/em\u003e\u0026thinsp;0.33% and CO\u003csub\u003eHb\u003c/sub\u003e in the range of 0.5\u0026ndash;1.5%. Sulfhaemoglobin can be caused by exposure to any substance that contains a sulfur atom capable of binding to hemoglobin. SulHb, also called sulfhaemoglobin, is a rare disorder that can happen when people are exposed to chemicals that have sulfur atoms that can attach to hemoglobin [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e]. This binding process hinders the ability of hemoglobin to carry oxygen. This is in contrast to MetHb, which has a known antidote in the form of methylene blue. SulHb is irreversible and has no known antidote [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e]. Early detection, diagnosis, and intervention are required to prevent end-organ damage, especially when sulfhemoglobin levels are high. SPR can be applied specifically in healthcare systems and biotechnology labs to contribute to the diagnosis and prognosis of hemoglobin anomalies. [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e].\u003c/p\u003e"},{"header":"2. The Essential Concepts","content":"\u003cp\u003eIn the near-infrared (NIR) spectrum (NIR \"window\": about 650\u0026ndash;900 nm), biological tissues don't absorb much light. Because of this, NIR light can be used to get physiological information from deeper tissue layers, often without cutting them. Continuous wave near-infrared spectroscopy (cwNIRS) is often used to measure changes in the amounts of tissue chromophores, especially oxyhemoglobin (HbO₂) and deoxyhemoglobin (Hb). The BLL law describes how a beam of visible light is absorbed by a medium that loses energy and how the intensity of the radiation is related to the optical path inside the medium [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. The BLL applies to all electromagnetic radiation and light-absorbing substances, such as gases, solids, liquids, molecules, atoms, and ions [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. And the rule serves as a quantitative foundation for absorbance photometry, colorimetric analysis, and other applications. It means that the amount of light absorbed by a clear substance doesn't change depending on how bright the light is hitting it. Eq.\u0026nbsp;(\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e4\u003c/span\u003e) [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e] can be used to find the absorbance.\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$A=\\log \\left( {\\frac{1}{T}} \\right)=\\varepsilon \\left( \\lambda \\right)LC$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e.\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$T=\\frac{{{I_{}}}}{{{I_0}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$I={I_0}\\exp ( - {\\mu _a}L)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$${\\mu _a}=\\varepsilon \\left( \\lambda \\right)C$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere A is the absorbance, T is the transmittance, C is the concentration of a light-absorbing substance, L is the optical path or the thickness of the substance layer, and ε (λ) is the molar extinction coefficient depending on wavelength. \u003cem\u003eI\u003c/em\u003e denotes the transmitted light intensity, whereas \u003cem\u003eI\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e represents the incident light intensity. \u0026micro;\u003csub\u003ea\u003c/sub\u003e is the absorption coefficient (cm⁻\u0026sup1;) of the medium [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eIn a homogeneous medium, absorbance A and light path length b are directly proportional. As the thickness of the absorbent medium increases, the intensity of transmitted radiation falls. The Beer-Lambert law establishes a direct link between absorbance and concentration, enabling precise measurement of analytes in a solution [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]. This method is very useful for assessing the concentrations of things like medicines, biomolecules, and contaminants. Real-time Monitoring The law enables real-time monitoring of reactions and processes by measuring how absorbance varies over time. This feature is critical for kinetic research and quality control applications. This feature enables non-destructive testing [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]. The Beer-Lambert law lets tests be done without damaging the material, since measuring absorbance doesn't change it much [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]. The MBLL relies on two assumptions: (1) the medium's absorption varies uniformly, and (2) the scattering loss is constant [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. The method is especially useful in biological applications, where sample integrity is critical. It may fail at high absorber or scattered concentrations due to increasing molecular interactions (e.g., self-quenching effects). As concentration grows, light absorption becomes nonlinear, and departures from linearity might occur, confounding measurement attempts [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e]. Optical Inhomogeneity: Biological tissues are often heterogeneous, with different parts having different optical properties [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]. For example, blood vessels, fat, and connective tissue are all heterogeneous. The resultant non-homogeneity may hamper the use of BLL, which presupposes a homogenous medium. Variations in tissue composition can lead to variable absorption and scattering, making it impossible to use a straightforward linear connection between absorbance and concentration [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]. The usual way to do cwNIRS uses one or more light source\u0026ndash;detector pairs, thinks of the lit tissue as optically homogeneous, and uses a changed version of Beer\u0026ndash;Lambert's law [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e].\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$A=\\log \\left( {\\frac{1}{T}} \\right)=\\varepsilon \\left( \\lambda \\right)LC+S$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe component S, which is independent of absorption and represents intensity loss due to scattering, is a geometry-dependent component. Sassaroli and Fantini showed that the above equation is wrong (L should be changed to its mean over the absorption coefficient range of 0\u0026ndash;\u0026plusmn;a) [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]. However, the differential form of the MBLL is still valid for small changes in attenuation as long as S stays the same and absorption changes evenly within the illuminated tissue volume [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e].\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\Delta A=L\\Delta {\\mu _a}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe difference in attenuation can be found by comparing the intensity values of two different states of the tissue. This change is directly related to the change in absorption. The latter represents the weighted sum of the variation in the concentration of tissue chromophores [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e].\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$\\Delta {\\mu _a}=\\sum {{\\varepsilon _i}(\\lambda )\\Delta {C_i}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWe can write the above equation as [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e] for oxyhemoglobin and deoxyhemoglobin.\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$$\\Delta {\\mu _a}={\\varepsilon _{Hb}}(\\lambda )\\Delta {C_{Hb}}+{\\varepsilon _{HbO}}(\\lambda )\\Delta {C_{HbO}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAn expression representing the variations in scattering must be incorporated into the MBLL. Attenuation, being a function of the absorption coefficient and the transport scattering coefficient, can be expressed as [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e].\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$$\\Delta A=L\\Delta {\\mu _a}+P\\Delta \\mu {\u0026#039;_s}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e\u0026micro;'s is the reduced scattering coefficients [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]. Which values of L and K are used depend on the tissue's initial optical properties (\u0026micro;a and \u0026micro;s') at the given wavelengths and the exact shape of the measurement. The outbound intensity relation can be written as:\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$$I={I_0}\\exp - (L\\Delta {\\mu _a}+P\\Delta \\mu {\u0026#039;_s})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003c/p\u003e\u003cp\u003eThe refractive index of the analyte medium (blood) in the visual region is defined by [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e]:\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e\n$${n_s}={n_{Hb}}=n+i \\times k$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eFriebel defines n as the real component of hemoglobin's refractive index, which changes as a function of wavelength and concentration [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e][\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e].\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ12\" name=\"EquationSource\"\u003e\n$${n_{Hb}}=n(\\lambda ,{C_{Hb}})={n_{{H_2}O}}(\\lambda ) \\times \\left( {\\beta (\\lambda ) \\times {C_{Hb}}+1} \\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn this equation, n\u003csub\u003eH₂O\u003c/sub\u003e represents the refractive index of water as a function of wavelength, β(λ) is the specific refractive increase based on wavelength, and C\u003csub\u003eHb\u003c/sub\u003e represents the concentration of hemoglobin.\u003c/p\u003e \u003cp\u003eUsing Prahl's molar extinction coefficient data [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e][\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e], you can figure out the imaginary part (k) of the refractive index that changes with the amount of hemoglobin. The calculation can be done at any wavelength.\u003cdiv id=\"Equ13\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ13\" name=\"EquationSource\"\u003e\n$$k=(2.303 \\times e \\times {C_{Hb}} \\times \\lambda )/(4\\pi \\times {M_{Hb}})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e13\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn this relation, e represents the molar extinction coefficient, and MHb represents the molar mass of hemoglobin.\u003c/p\u003e"},{"header":"3. Results and discussion","content":"\u003cp\u003eIn this work model and source, code under Matlab program developed under the rule of MBLL to simulate the outbound intensity and the absorbance for each species. The extinction molar coefficients (in mM⁻\u0026sup3;\u0026bull;cm⁻\u0026sup3;) of five types of hemoglobin are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. These are Hb (deoxyhemoglobin), HbO₂ (oxyhemoglobin), COHb (carboxyhemoglobin), SulfHb (sulfhemoglobin), and MetHb (methemoglobin). The wavelengths are shown in nm. The extinction coefficient tells us how well each type of hemoglobin can absorb light at different wavelengths. This is very important for spectroscopic analysis and diagnostic uses. The blue curve representing Hb has a pronounced absorption peak about 560 nm, followed by a subsequent rise in the lower wavelength region of 400\u0026ndash;430 nm. This particular absorption profile shows how iron molecules are arranged when they are deoxygenated. When oxygen is not bound, it changes the electronic transitions inside the porphyrin ring. HbO₂ (red curve) exhibits a distinct double-peak pattern in the 540\u0026ndash;580 nm spectrum, with maxima at around 542 nm and 578 nm. In pulse oximetry and spectroscopic analysis, the standard \"double-hump\" pattern is often used to tell the difference between hemoglobin that is oxygenated and hemoglobin that is not oxygenated.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe SulfHb (magenta curve) has a distinct absorption peak near 620 nm, making it spectrally unique compared to other hemoglobin types. This one-of-a-kind peak marks SulfHb as an optical aberration, which makes it easier to find even in very small amounts in clinical samples. On the other hand, MetHb (the cyan curve) has a complicated absorption spectrum with clear peaks at around 500 nm and 630\u0026ndash;650 nm. This clear pattern shows how ferrous (Fe\u0026sup2;⁺) iron changes into ferric (Fe\u0026sup3;⁺) iron, which greatly changes the electrical properties of the heme group. In the 550\u0026ndash;600 nm range, where Hb, HbO₂, and COHb all absorb strongly, the graph shows a lot of spectral overlap. This evidence supports the claim that simple single-wavelength analysis is not enough to accurately tell the difference between species in clinical specimens. To get accurate concentration information from complicated mixtures, you need to use multivariate analytical methods like principal component analysis or partial least squares regression. The strength of the argument is shown by looking at the extinction coefficients: at about 570 nm, both Hb and COHb have strong absorption values, making it harder to tell them apart directly. Using observations at different wavelengths to build overdetermined systems of equations is the scientific way to solve this overlap problem. This makes it possible to get an accurate concentration reading even when measurement noise is present. Isolated absorption peaks serve as diagnostically significant markers.\u003c/p\u003e \u003cp\u003eThe research presents compelling evidence that specific wavelengths have distinct diagnostic significance. The single peak of SulfHb at 620 nm and the unique absorption of MetHb at 630 nm are the best spectroscopic clues for these medical situations. This claim is based on scientific evidence that certain wavelengths make it easier to find abnormal hemoglobin species, even when they are present in small amounts or with a lot of normal hemoglobin variants. The fact that long wavelengths (700 nm) are not absorbed has a big effect on the design of spectroscopic instruments and the choice of measurement parameters in clinical and research settings. This supports the claim that near-infrared spectroscopy is useful for in vivo applications where tissue penetration is important. This careful study of spectroscopic evidence and use of well-known biophysical principles proves beyond a reasonable doubt that the different extinction coefficient profiles of different hemoglobin species provide a solid basis for their quantitative differentiation and clinical evaluation.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e shows four sets of spectroscopic data that strongly support the idea that different types of hemoglobin interact with light in complex ways in the visible and near-infrared ranges. Figure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.a: Total Absorption Coefficient: Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.a illustrates the total absorption coefficient as a function of wavelength (in nm). Figure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.a displays the overall absorption coefficient for the wavelength range of 400 to 900 nm. An evident absorption peak is observed in the range of 550to580 nm. The range of 450\u0026ndash;500 nm reduces the absorption peak, indicating robust absorption at this wavelength. The absorption coefficient markedly diminishes on both sides of this peak, indicating that the material absorbs light most specifically in this area. The values shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.b are the overall absorption coefficients. These values show the complex spectrum signature that comes from mixing different types of hemoglobin (HbO₂, Hb, COHb, MetHb, and SulfHb). The cumulative effect signifies more than the simple aggregation of separate components; it embodies the weighted contributions according to their various concentrations.\u003c/p\u003e \u003cp\u003eI contend that this wavelength dependency is fundamental to spectroscThe visible spectrum (450\u0026ndash;650 nm) reveals significant changes in absorption characteristics among hemoglobin species, while the near-infrared spectrum (700\u0026ndash;900 nm) reveals very minor variances. very minor variances. This observation undermines the oversimplified notion that hemoglobin derivatives may be uniformly differentiated throughout the whole spectrum. Consequently, the selection of critical wavelengths is crucial for particular diagnostic applications.\u003c/p\u003e \u003cp\u003eThe contributions to absorption data elucidate how each hemoglobin variant disproportionately affects overall absorption at specific wavelengths. Oxyhemoglobin predominates with its unique double-peak pattern at around 542 and 578 nm, whereas deoxyhemoglobin displays its characteristic offer supplementary spectrum characteristics: COHb exhibits a notable peak at approximately 570 nm, MetHb displays peaks at 500 and 630 nm, and SulfHb is characterized by its unique peak around 556 nm absorption at 620 nm.\u003c/p\u003e \u003cp\u003eIn Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.c The exponential behavior has significant consequences for Traditional pulse oximetry uses the red (660 nm) and infrared (940 nm) wavelengths to tell the difference between HbO₂ and Hb, but I believe this method is flawed in the present situation of deoxyhemoglobin. The transmitted intensity patterns show that deoxyhemoglobin make absorption profiles that aren't accurate, and standard two-wavelength methods can't tell the difference.\u003c/p\u003e \u003cp\u003eThe evidence strongly indicates that multi-wavelength spectroscopy is crucial for precise hemoglobin differentiation. Using measuring wavelengths between 600 and 630 nm will make it much easier to find MetHb, and looking at things between 570 and 580 nm will help figure out how much COHb is present. The absorption patterns. This depiction elucidates the spectral signatures of several hemoglobin species, especially in the visible spectrum where unique absorption peaks arise.\u003c/p\u003e \u003cp\u003eStill, I argue that interpreting absorbance spectra is very hard when they are used for measurements that take place inside living things. According to Mie theory, tissue scattering changes with wavelength. This makes a sloping baseline that can hide fine spectral features. When other chromophores like melanin, bilirubin, and water are added, their absorption spectra overlap, which makes analysis more difficult.\u003c/p\u003e "},{"header":"4. Effect in increasing hemoglobin’s species concentrations","content":"\u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003e \u003cb\u003eEffect of Increasing COHb Levels\u003c/b\u003e \u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.a shows how rising amounts of carboxyhemoglobin (COHb) alter light intensity throughout wavelengths. As COHb levels grow (from 0\u0026ndash;5%), the intensity curve shifts slightly, particularly in the visible band (400\u0026ndash;700 nm). The most substantial changes occur in specific wavelength ranges, likely correlating to COHb absorption peaks.\u003c/p\u003e \u003cp\u003eThis suggests that COHb has distinct absorption characteristics that influence light transmission. Non-invasive spectroscopic techniques could use the data to estimate COHb levels in blood, a crucial step in diagnosing carbon monoxide poisoning.\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003e \u003cb\u003eEffect of Increasing MetHb Levels\u003c/b\u003e \u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eThe effect of increasing methemoglobin (MetHb) levels (0\u0026ndash;5%) on light intensity in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.b shows that. There are notable fluctuations in intensity, notably in the 500\u0026ndash;700 nm region, with distinct peaks and troughs changing as MetHb levels grow.\u003c/p\u003e \u003cp\u003eMetHb has unique absorption capabilities that affect light transmission in specific wavelength bands.\u003c/p\u003e \u003cp\u003eThis information is valuable for detecting methemoglobinemia, a condition where MetHb levels are abnormally high, impairing oxygen delivery.\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003e \u003cb\u003eEffect of Increasing SulfHb Levels\u003c/b\u003e \u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eThe impact of sulfhemoglobin (SulfHb) levels (0\u0026ndash;5%) on light intensity (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.c).The intensity curves reveal considerable changes in the visible spectrum, with shifts in absorption peaks as SulfHb levels increase. SulfHb's absorption properties can be utilised to measure its levels in blood.This is significant for diagnosing sulfhemoglobinemia, a rare illness caused by sulfur attaching to hemoglobin, which can hamper oxygen transport.\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003e \u003cb\u003eEffect of Oxygen Saturation (SO2)\u003c/b\u003e \u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.d shows how varying oxygen saturation levels (0\u0026ndash;100%) affect light intensity.\u003c/p\u003e \u003cp\u003eThere are big changes in the intensity curves, especially in the red and near-infrared (600\u0026ndash;900 nm) ranges, which are often used in pulse oximetry. The link between oxygen saturation and light intensity is the basis for pulse oximetry, a non-invasive way to test blood. The data emphasizes the importance of specific wavelengths for accurate oxygen saturation measurements.\u003c/p\u003e \u003cp\u003eThese graphs illustrate the capabilities of spectroscopic methods in medical diagnostics. By looking at how much light is absorbed and transmitted at different wavelengths based on MBLL, it is possible to figure out the levels of oxygen saturation and different types of hemoglobin in the blood.\u003c/p\u003e \u003cp\u003eThe ability to measure COHb, MetHb, SulfHb, and SO\u003csub\u003e2\u003c/sub\u003e levels non-invasively is critical for diagnosing and monitoring conditions such as carbon monoxide poisoning, methemoglobinemia, sulfhemoglobinemia, and hypoxemia. Further studies could focus on refining the wavelength ranges and boosting the sensitivity of spectroscopic equipment. The incorporation of these discoveries into wearable or portable devices could increase real-time monitoring of blood parameters in clinical and emergency scenarios. In the red and near-infrared (600\u0026ndash;900 nm) spectrums, which are used in pulse oximetry, HbO₂ and Hb have different absorption maxima. Conversely, COHb predominantly influences the visible spectrum. COHb lowers the amount of oxygen that blood can carry by competing with oxygen for binding sites on hemoglobin. This can happen even when oxygen saturation levels are normal. In contrast to HbO₂ and Hb, MetHb is incapable of binding oxygen, hence hindering oxygen transport and delivery. This results in methemoglobinemia, a disorder marked by cyanosis and hypoxia.\u003c/p\u003e \u003cp\u003eSpectroscopic methods can tell the difference between MetHb concentrations because MetHb changes the spectrum in a way that HbO₂ and Hb do not. The spectral signature of SulfHb is different from those of HbO₂ and Hb, which makes it easier to identify and measure. SulfHb, in contrast to HbO₂ and Hb, is incapable of efficiently transporting oxygen, resulting in compromised oxygen delivery. This property is different from HbO₂ and Hb, which have wider effects in the red and near-infrared ranges. SulfHb changes things more in the visible range.\u003c/p\u003e"},{"header":"5. Conclusion","content":"\u003cp\u003eThis study shows that the Modified Beer-Lambert Law (MBLL) has a lot of potential for using it to look at hemoglobin abnormalities using spectroscopy. This is a big step forward in non-invasive diagnostic methods. This research uses a lot of mathematical modelling and spectroscopic analysis of different types of hemoglobin to build a solid base for telling the difference between normal hemoglobin and deoxyhemoglobin, such as methemoglobin (MetHb), carboxyhemoglobin (COHb), and sulfhemoglobin (SulfHb). Because each type of hemoglobin has its own unique spectral signature, it is possible to accurately find and measure them. The absorption peaks at certain wavelengths\u0026mdash;HbO₂ has a double peak at 542/578 nm, MetHb has peaks at 500/630 nm, and SulfHb has a noticeable absorption at 620 nm\u0026mdash;are important biomarkers for the diseases they represent. The differences in the spectrum make it easier to make custom diagnostic tools that can find abnormal hemoglobin variants even when they are present in very small amounts within complex biological matrices. Our study clearly shows that multi-wavelength spectroscopic techniques are necessary for accurately telling the difference between hemoglobin species. This is because two-wavelength techniques, like those used in standard pulse oximetry, are limited when deoxyhemoglobin are present. A careful study of how different concentrations change the transmission of light gives us a way to use math to figure out threshold values and diagnostic algorithms that work in the real world. Because the MBLL can include both absorption and scattering effects, it is very useful for biological applications. It gets around the problems that the standard Beer-Lambert Law has with mediums that aren't all the same, like blood and tissue. This better modelling method makes it easier to get more accurate concentration readings when the liquid is cloudy, bringing laboratory accuracy into line with point-of-care uses. This study is a big step forward in biomedical optics and hemoglobin research. It shows that using the Modified Beer-Lambert Law for spectroscopic analysis is a good way to diagnose and keep an eye on hemoglobin disorders without having to cut them open. This has big implications for clinical practice and the development of new medical devices.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eHabia mohamed ilyes :wrote the main manuscript, Simulation \u0026amp; interpretation Habia Ghania : interpretation \u0026amp; Correction Manallah Aissa: interpretation, reviewing \u0026amp; validation\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eN. Taparia, K. C. Platten, K. B. Anderson, and N. J. Sniadecki, \u0026ldquo;A microfluidic approach for hemoglobin detection in whole blood,\u0026rdquo; \u003cem\u003eAIP Adv.\u003c/em\u003e, vol. 7, no. 10, 2017, doi: 10.1063/1.4997185.\u003c/li\u003e\n\u003cli\u003eL. Gharahbaghian, B. Massoudian, and G. Dimassa, \u0026ldquo;Methemoglobinemia and sulfhemoglobinemia in two pediatric patients after ingestion of hydroxylamine sulfate.,\u0026rdquo; \u003cem\u003eWest. J. Emerg. 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Booth, \u0026ldquo;Hematology and Oncology,\u0026rdquo; in \u003cem\u003eManaging Emergencies in the Outpatient Setting: Pearls for Primary Care\u003c/em\u003e, G. M. Booth and S. Frattali, Eds. Cham: Springer International Publishing, 2023, pp. 211\u0026ndash;222. doi: 10.1007/978-3-031-15270-2_11.\u003c/li\u003e\n\u003cli\u003eR. Di Capua, F. Offi, and F. Fontana, \u0026ldquo;Check the Lambert-Beer-Bouguer law: A simple trick to boost the confidence of students toward both exponential laws and the discrete approach to experimental physics,\u0026rdquo; \u003cem\u003eEur. J. Phys.\u003c/em\u003e, vol. 35, no. 4, 2014, doi: 10.1088/0143-0807/35/4/045025.\u003c/li\u003e\n\u003cli\u003eA. C. Judge, J. S. Brownless, N. A. R. Bhat, J. E. Sipe, M. J. Steel, and C. Martijn De Sterke, \u0026ldquo;Effective photons in weakly absorptive dielectric media and the Beer-Lambert-Bouguer law,\u0026rdquo; \u003cem\u003eNew J. 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Pirovano \u003cem\u003eet al.\u003c/em\u003e, \u0026ldquo;Effect of adipose tissue thickness and tissue optical properties on the differential pathlength factor estimation for NIRS studies on human skeletal muscle,\u0026rdquo; \u003cem\u003eBiomed. Opt. Express\u003c/em\u003e, vol. 12, no. 1, p. 571, 2021, doi: 10.1364/boe.412447.\u003c/li\u003e\n\u003cli\u003eM. I. Habia, A. Manallah, and K. Ayadi, \u0026ldquo;Plasmonic biosensor for the study of blood diseases by analysis of hemoglobin concentration,\u0026rdquo; \u003cem\u003eOpt. Quantum Electron.\u003c/em\u003e, vol. 55, no. 3, p. 234, Mar. 2023, doi: 10.1007/s11082-022-04503-z.\u003c/li\u003e\n\u003cli\u003eM. Friebel and M. Meinke, \u0026ldquo;Model function to calculate the refractive index of native hemoglobin in the wavelength range of 250-1100 nm dependent on concentration,\u0026rdquo; \u003cem\u003eAppl. Opt.\u003c/em\u003e, vol. 45, no. 12, pp. 2838\u0026ndash;2842, 2006, doi: 10.1364/AO.45.002838.\u003c/li\u003e\n\u003cli\u003eS. Prahl, \u0026ldquo;Optical absorption of hemoglobin,\u0026rdquo; \u003cem\u003eOregon Medical Laser Center\u003c/em\u003e, 1999.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Modified Beer-Lambert, Absorbance, hemoglobin disorder, blood disease","lastPublishedDoi":"10.21203/rs.3.rs-6231215/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6231215/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eBiomolecule detection techniques are still a top priority in modern research because they are essential for clinical point-of-care diagnostics. This is especially true for pathogens like viruses and bacteria. Hemoglobin disorders, including thalassemia and sickle cell disease, substantially hinder oxygen transport and overall health. Hemoglobinopathies like methemoglobin (MetHb), carboxyhemoglobin (COHb), and sulfhemoglobin (SulHb) are difficult to diagnose because blood samples break down quickly after being collected, making it difficult to do an accurate and quick analysis. The Beer-Lambert Law (BLL), which is the basis of absorbance spectroscopy, is used a lot in analytical methods because it gives accurate quantitative data with little sample preparation. The Modified Beer-Lambert Law (MBLL) builds on the basic idea by adding scattering effects and uneven mediums. This versatility makes it especially useful for biological systems that are very complicated. Its integration simplifies the process of obtaining precise concentration readings and monitoring metabolic activities in real-time, particularly in environments such as cloudy or scattering blood. The MBLL is used to think about hemoglobin diseases, and Matlab is used to look at the optical properties of MetHb, COHb, and SulHb. The MBLL correlates light absorbance with the concentration of absorbing species, facilitating a more profound comprehension of the spectroscopic characteristics of various hemoglobin derivatives. This method makes diagnosis more accurate and shows how advanced spectroscopic modeling can be used to solve clinical problems related to hemoglobinopathies.\u003c/p\u003e","manuscriptTitle":"The use of the modified Beer-Lambert law to improve biomolecular detection: diagnosing hemoglobin disorders and identifying pathogens","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-03-27 15:36:45","doi":"10.21203/rs.3.rs-6231215/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"5a6b11de-8c61-4db4-936f-6041e1c164ff","owner":[],"postedDate":"March 27th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-06-26T15:23:39+00:00","versionOfRecord":[],"versionCreatedAt":"2025-03-27 15:36:45","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-6231215","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6231215","identity":"rs-6231215","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}