PLoS One. 2018 Jun 8;13(6):e0198210. doi: 10.1371/journal.pone.0198210.

Hemoglobin state-flux: A finite-state model representation of the hemoglobin signal for evaluation of the resting state and the influence of disease.

Barbour RL1,2, Graber HL2, Barbour SS2.
1 Department of Pathology, SUNY Downstate Medical Center, Brooklyn, NY, United States of America.
2 Photon Migration Technologies Corp., Brooklyn, NY, United States of America.



In this report we introduce a weak-model approach for examination of the intrinsic time-varying properties of the hemoglobin signal, with the aim of advancing the application of functional near infrared spectroscopy (fNIRS) for the detection of breast cancer, among other potential uses. The developed methodology integrates concepts from stochastic network theory with known modulatory features of the vascular bed, and in doing so provides access to a previously unrecognized dense feature space that is shown to have promising diagnostic potential. Notable features of the methodology include access to this information solely from measures acquired in the resting state, and analysis of these by treating the various components of the hemoglobin (Hb) signal as a co-varying interacting system.


The principal data-transform kernel projects Hb state-space trajectories onto a coordinate system that constitutes a finite-state representation of covariations among the principal elements of the Hb signal (i.e., its oxygenated (ΔoxyHb) and deoxygenated (ΔdeoxyHb) forms and the associated dependent quantities: total hemoglobin (ΔtotalHb = ΔoxyHb + ΔdeoxyHb), hemoglobin oxygen saturation (ΔHbO2Sat = 100Δ(oxyHb/totalHb)), and tissue-hemoglobin oxygen exchange (ΔHbO2Exc = ΔdeoxyHb-ΔoxyHb)). The resulting ten-state representation treats the evolution of this signal as a one-space, spatiotemporal network that undergoes transitions from one state to another. States of the network are defined by the algebraic signs of the amplitudes of the time-varying components of the Hb signal relative to their temporal mean values. This assignment produces several classes of coefficient arrays, most with a dimension of 10×10.


Motivating our approach is the understanding that effector mechanisms that modulate blood delivery to tissue operate on macroscopic scales, in a spatially and temporally varying manner. Also recognized is that this behavior is sensitive to nonlinear actions of these effectors, which include the binding properties of hemoglobin. Accessible phenomenology includes measures of the kinetics and probabilities of network dynamics, which we treat as surrogates for the actions of feedback mechanisms that modulate tissue-vascular coupling.


Qualitative and quantitative features of this space, and their potential to serve as markers of disease, have been explored by examining continuous-wave fNIRS 3D tomographic time series obtained from the breasts of women who do and do not have breast cancer. Inspection of the coefficient arrays reveals that they are governed predominantly by first-order rate processes, and that each array class exhibits preferred structure that is mainly independent of the others. Discussed are strategies that may serve to extend evaluation of the accessible feature space and how the character of this information holds potential for development of novel clinical and preclinical uses.

PMID: 29883456



Background: Cancer, like many forms of chronic disease, is characterized by phenotypes that are expressions of altered homeostatic mechanisms. Among these, commonly seen is the occurrence of a sustained inflammatory response [1], which is tightly coupled to upregulation of nitric oxide (NO) formation. This reactive species affects a variety of cellular and systems-level behaviors [2,3], and while serum biomarkers sensitive to inflammation are available [4], noninvasive measures that can be easily applied and are expectedly sensitive to other known actions NO are not easily determined.

Among the latter is the homeostatic control of blood delivery to tissue, which strikes a dynamic balance of tissue oxygen supply-demand needs. Having a feedforward effect is the impact of NO on vascular tone. Simultaneously, NO is known to exert a feedback effect on oxygen utilization by inhibiting the activity of mitochondrial cytochrome c-oxidase, the enzyme responsible for coupling oxygen utilization with metabolic energy formation in the form of ATP. This balance, thought to occur in all tissue types, is also impacted by cancer, fostering the widely recognized feature known as the Warburg Effect, wherein aerobic glycolysis is favored over mitochondrial oxidative activities in the presence of oxygen, leading to enhanced lactate production. In fact, because NO levels are often elevated in cancer, the feedback effect on mitochondrial function may be a causative contributor in driving this behavior.

While these features are widely understood, mainly lacking has been access to noninvasive resources that support a composite measure of factors sensitive to cancer-induced disturbances in NO metabolism. As described in [5], our approach has been to employ near infrared spectroscopy (NIRS), with the understanding that mentioned cancer phenotypes are likely to impact vascular tone and oxygen supply-demand balance, and hence features of the hemoglobin signal.

Also considered have been practical aims that can serve to maximize subject comfort and compliance, as well as to deploy analysis strategies that leverage other widely appreciated features of biology. The former was achieved by considering measures made in the resting state, using a custom-built instrument that provides for simultaneous, dual-breast volumetric NIRS imaging measures, among other capabilities [6].  Explored were 18 subjects with defined breast cancer, and 45 control subjects that included women with various forms of benign breast disease.

The mentioned analysis leveraged the understanding that many features of biology arise from adherence to generative processes that follow a limited set of rules. A good example of this is the transcription of DNA. Understood is that its information content is derived from a limited number of nucleotide base pairings that are read in defined ways. This mechanism, though limited in scope, provides for a near infinite variety of information. Similarly recognized is that this dependence holds for features common to basic understandings from the field of linguistics: i.e., languages in all their forms, including less recognized forms such as music and dance, are derived from adherence to generative rule sets.

Accordingly, a key focus of our report has been to identify methodology that leverages principal elements of language syntax, such as the hemodynamic equivalent of alphabet, words and sentences, that can serve as a basis for identifying intrinsic rule-based behaviors linked to the mentioned NO dependencies, which we believe will prove disease sensitive. While formally known as a grammar induction problem, our focus in this first report has been to invoke simplified rules that define the mentioned syntax, thereby allowing for the description of behaviors whose time dependencies can be quantified.

Invoked was a methodology similar to ones used to characterize finite Markov chains. To accomplish this, we needed to first define a set of finite states. Considered was the full complement of the hemoglobin signal that includes both its principal independent (relative changes in oxygenated (OxyHb) and deoxygenated (DeoxyHb) forms) and dependent components (relative changes in total hemoglobin (TotalHb = OxyHb + DeoxyHb), oxygen saturation ([OxyHb/TotalHb]*100) and efficiency of oxygen exchange (HbO2Exc = DeoxyHb – OxyHb). State definitions were based on the algebraic sign (+, -) of the amplitude of these five components relative to their temporal mean, thus yielding ten distinct states. Figure 1 illustrates how time evolution of the mentioned components leads to these state definitions. Accordingly, rules for defining syntax are: alphabet, algebraic sign of Hb components; words, five letter composites of Hb state definitions; sentences, time evolution of the Hb signal.



Figure 1. Inter-relationships between time-varying levels of hemoglobin signal components (upper panel) and their associated Hb States (lower panel). The Sentence depicted in this example is “‘-++–’ ‘-+++-’ ‘++++-’ ‘++-+-’ ‘++-++’ ‘+–++’ ‘+—+’ ‘—-+’ ‘–+-+’ ‘–+–’” or, alternatively, “‘oDEts’ ‘oDETs’ ‘ODETs’ ‘ODeTs’ ‘ODeTS’ ‘OdeTS’ ‘OdetS’ ‘odetS’ ‘odEtS’ ‘odEts’”.


To characterize behaviors linked the identified syntax, we invoked the commonly considered probabilistic description of the State transition matrix considered in Markov chain analysis. This produces three distinct feature spaces (state-dependent transition probabilities and rates, and Hb-component fluxes), each having dimensions of 10 × 10, and several classes of dependent spaces having similar dimensions. Of considerable interest was the finding that among these varied spaces, comparisons showed that they are largely independent of each other and were shown to have promising disease sensitivity (AUC values approaching 0.9).

Also recognized are alternative analysis strategies that appear capable of identifying hidden drivers of hemoglobin dynamics, and by inference are in some way linked to actions of nitric oxide. This consideration opens a yet broader view that divides principal elements of biology into three main categories: macromolecular elements, features of metabolism, and the actions of chemical signaling agents.  Comprising a diverse group of substances, of which many forms are themselves affected by nitric oxide and related compounds, the third category often take on the role of “traffic policeman,” impacting both the composition and level of the molecular machinery in cells and the outcomes [2] of their actions.

Of note here is the consideration that whereas there is a wealth of strategies available for exploring molecular machinery of cells and its effects, much less available are tools sensitive to influences of chemical signaling, especially using noninvasive methods. Thus, we hold that whereas the NIRS signal has been correctly identified as a window into actions of oxidative metabolism, less recognized is that behind these are the actions of the gaseous signaling agents, whose impacts are linked to many forms of chronic disease.

In summary, this report brings together two independent considerations: practical noninvasive sensing capabilities of near infrared spectroscopy (NIRS), and elementary understandings governing information transduction in biology, to provide for novel capabilities sensitive to prominent features of tumor biology that are recognized as key drivers of the tumorigenic state. Being a first demonstration, our focus has been to explore capabilities on a coarser level than needed, and under conditions that are easily implemented. The outcome is a methodological approach that allows access to new classes of previously unrecognized information accessible from whole organ-level measures, that appear likely to have utility for the detection and management of a range of chronic diseases, among other capabilities.



[1] M. Murata, “Inflammation and Cancer”, Environ Health Prev. Med. 23:50 (2018).

[2] B.A. Rizi, A. Achreja, and D.Nagrath, “Nitric Oxide – the forgotten child of tumor metabolism,” Trends Cancer 3, 659-672 (2017).

[3] D.D. Thomas, “Breathing new life into nitric oxide signaling: A brief overview of the interplay between oxygen and nitric oxide”, Redox Biology 5, 225-233, (2015).

[4] J.L. Sylman et al., “The Predictive Value of Inflammation-Related Peripheral Blood Measurements in Cancer Staging and Prognosis”, Front. Oncol. 8:78 (2018).

[5] R.L. Barbour, H.L. Graber and S.L. Barbour, “Hemodynamic state-flux: A finite-state model representation of the hemoglobin signal for evaluation of the resting state and the influence of disease,” PLOS One 13(6): e0198210 (2018). https://doi. org/10.1371/journal.pone.0198210.

[6] R. Al abdi, H.L. Graber, Y. Xu, and R.L. Barbour, “Optomechanical imaging system for breast cancer detection,” J. Optical Society of America A 28, pp. 2473-2493 (2011).