Study the rate of spread of Hemolytic Anemia then finding the effect of natural Drug- Curcumin on this disease using rat model through Terahertz & Math-modeling

PROJECT SUMMARY

The proposed work is to study “Study the rate of spread of Hemolytic Anemia then finding the effect of natural Drug- Curcumin on this disease using rat model through Terahertz & Math-modeling,”. The absorption coefficients of erythrocytes of diseased rat blood are compared to that of normal rat blood. The Mathematical model is proposed to commensurate theory with experimental results. Literature survey makes it well known that phenyl hydrazine induces hemolytic anemia. This is understood as result from the reaction of phenyl hydrazine with hemoglobin.

The two healthy Wistar rats are treated with phenyl hydrazine that induces hemolytic anemia in their blood. The blood is extracted from each and its erythrocytes are studied using terahertz spectroscopy. The change of in the number of RBC with time will define the rate of hemolytic anemia. One of them is treated with antioxidant, “Curcumin” the derivative of Turmeric. Drug Curcumin is checked to adverse the effect of phenyl hydrazine. The RBC count of third blood sample will lead into the information of the effect of this natural herb on blood.

The motivation for this research comes from the fact that there is difficulty in understanding the exact cause of the spread of this disease and its rate of spread in the body. This study will lead to its better understanding in future.

Experimentally induced auto-immune hemolytic anemia (AIHA) in rats is characterized either by constant depressed erythrocyte numbers, or by oscillatory erythrocyte numbers about a depressed level (periodic auto-immune hemolytic anemia). We will rely on the reduction in the rate of erythrocytes that will be counted through terahertz. It is previously shown that terahertz (THz) time-domain spectroscopy (TDS) can be used to portray the blood. The complex optical constants of erythrocytes were obtained in the THz frequency region.

In this we count the volume percentage of erythrocytes in extracted blood and compare it with the conventional RBC counter results. Recently, Curcumin has shown to counteract the development of disease-induced anemia. The blood is extracted from rats thrice, firstly without inducing any drug, secondly 3 days after inducing double dosage of phenyl hydrazine and 3 days after inducing double dosage of “Curcumin ” the bioactive compound of Turmeric in one of the rats.

Immediate therapy with Curcumin may significantly reversed these deteriorating effects of PHZ on RBCs. The literature survey has confirmed that The THz absorption constants are shown to vary linearly with the RBC concentration. The pervious study also demonstrate that THz-TDS imaging can facilitate the quantitative analysis of blood, using these facts we will study AIHA (auto-immune hemolytic anemia) and try to proportionate the experimental results with for Math Models.

The mathematical Modelling is based on the fact that the induction of auto-immune hemolytic anemia (AIHA) in rats is sometimes marked by a steady depression of hemoglobin levels, and at other times by sustained oscillations with the occurrence of a Hopf bifurcation in hemoglobin concentration and reticulocyte numbers with a period of 16 to 17 days. The in vivo erythrocyte production rate in rats has the form of a Hill function, showing saturation at low hemoglobin levels. The effect of drug is induced in the model by considering the fact RBC reacts if its production is changed under anemia conditions where RBC has no feedback to its production. This knowledge can be used when extending the disease model to a drug-disease model for a drug acting as an erythropoietin stimulating agent (ESA).

Drug concentrations at the target (bone marrow) are determined by the PK of the drug, and effects on RBC (or Hb) production by its Pharmacodynamics. In this model we will connect drug concentrations to Hb production via an Emax model with parameters Smax and SC50. As there is no feedback control of Hb, any drug dosage regimen needs to be carefully evaluated with respect to avoiding excessive Hb values.

PROJECT DESCRIPTION

The terahertz region lies between the microwave and infrared regions of the electromagnetic spectrum such that it is strongly attenuated by water and very sensitive to water content. Terahertz radiation has very low photon energy and thus it does not pose any ionization hazard for biological tissues.

Inspiration:

This study is enthused by the rigorous work has been done in Terahertz (THz) detection of the signature of Bacillus thuringiensis Spores in DiPel using THz frequency domain spectroscopy and study the imaging of different biomaterials using THz. The attenuation from transmission measurements was measured in THz wavelengths (~750-250 μm) and absorbance signature was observed at 917 GHz. Biomolecules and bacterial spores have been demonstrated to have characteristic signatures in the absorption patterns. On the similar lines, I propose to study the erythrocyte content of human blood. This will lead to better understanding of hemolytic anemia that can lead to the effective cure of the rare disease. The previous work in imaging is done by developing a reflective THz imaging system that operates at a center frequency of 0.525 THz with a bandwidth of 0.125 THz.

Background and Literature Survey:

Red blood cells (RBCs), also called erythrocytes are hemoglobin containing molecules, making an almost quarter of blood cell and the vertebrate organism’s principal means of delivering oxygen (O₂) to the body tissues via the blood flow through the system. They take up oxygen in the lungs and release it into tissues while squeezing through the body’s capillaries.

The cell membrane of RBC is composed of proteins and lipids, and this structure provides properties essential for physiological cell function such as deformability and stability while traversing the circulatory system and specifically the capillary network. In humans, mature red blood cells are flexible and oval biconcave disks. They lack a cell nucleus and most organelles, in order to accommodate maximum space for hemoglobin. Approximately 2.4 million new erythrocytes are produced per second. The cells develop in the bone marrow and circulate for about 100-120 days in the body before their components are recycled by macrophages. Each circulation takes about 20 seconds.

Autoimmune hemolytic anemia is a rare disorder caused by auto reactive red blood cell (RBC) antibodies that destroy RBCs.

Autoimmune hemolytic anemia (AIHA), caused by auto reactive red blood cell (RBC) antibodies along with clinical and laboratory evidence of hemolysis, is estimated to occur in approximately 1 in 80,000 patients annually.

The RBC autoantibodies are generally classified as either warm RBC autoantibodies, if optimum reactivity with RBCs occurs at 37⁰C, or cold RBC autoantibodies, if optimum reactivity with RBCs occurs at less than 30⁰C.

Erythrocytes age predominantly by damage to their cell membrane as they pass through the capillaries. Since erythrocytes have no nucleus to produce proteins that effect repair of the cell membrane, the membrane slowly loses flexibility and the cells can no longer efficiently deliver O₂ to the tissues. The immune system recognizes this membrane breakdown and tags the membrane with special markers. These markers then signal macro­phages or white blood cells to actively degrade or phagocytize the oldest erythrocytes after a period of time that is about 120 days in humans.

The process by which Pluripotent stem cells give rise to erythrocytes is called erythropoiesis. The stem cells have the capability of infinite division that the mature cells cannot do. The five stages from the most immature to the most mature are the proerythroblast, the basophilic norm oblast (early erythroblast), polychromatophilic norm oblast (intermediate erythroblast), ortho chromatophilic norm oblast (late erythroblast) and reticulocyte.

Further scope of this study can acknowledge more about RBC autoantibodies that are encountered frequently and can complicate transfusion workups, delay distribution of compatible units, have variable clinical significance that ranges from benign to life-threatening, and may signal an underlying disease or disorder.

The diagram shows the regulation of red blood cells (RBCs). Upon erythropoietin stimulus, the bone marrow releases RBCs into the blood. RBCs contain hemoglobin (Hb) which binds oxygen (O₂) and carries it in the blood to peripheral tissues. If the tissue oxygen supplies is insufficient, cells in the kidney increase production of erythropoietin.

Math Model of AIHA:

Delay Differential Equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times.

An age-structured mode is developed for erythropoiesis and is reduced to a system of threshold-type differential delay equations.

Since literature articulates that hemolytic anemia is a periodic disease that arises due to abnormalities in feedback mechanism of blood and is caused by the disruption of a steady state behavior of immune system of body, or due to the replacement by a new periodic regime by the normal periodic one. Mathematical Model relies on the oscillations in this feedback mechanism that is used as a basis to study the onset conditions of disease.

Assumptions for Math Model: The dynamics of undifferentiated stem cells and the committed stem cells are regulated by two types of feedback mechanisms:

1. Colony stimulating factor renal erythropoietin for the erythrocytes. Colony-stimulating factors (CSFs) are secreted glycoproteins that bind to receptor proteins on the surfaces of hemopoietic stem cells; thereby activating intracellular signaling pathways that can cause the cells to proliferate and differentiate into a specific kind of blood cell colony-stimulating factors is present.
2. An intrinsic property of these feedback mechanisms is the presence of time delays which arise because of finite nonzero cell maturation times and cell-replication times.
3. Presence of control parameters are those quantities which in comparison to blood-cell number either do not change with time, or change very little or hence are regarded constant. Examples of control parameters in the regulation of hematopoiesis are the maturation times and the destruction rate.

Proposed Math Model for the kinetics of circulating blood for numerical simulations:

The math model is divided into 2 parts

1. Normal Blood
2. Diseased Blood

Normal Blood:

The math modeling is done on the cells in a compartment that die randomly as time proceeds.

The idea that older cells are actively degraded has several applications to the study of population dynamics of blood cells.

The above hypothesis is based on the fact that Blood cells from humans decay exponentially, and t_{1/2 is} approximately 6.9 hours for humans   γ =ln2/ 6.9 hr⁻¹ = 0.1hr⁻¹

The various parameters involved are:  V (t) be the number of circulating cells per unit volume at a given time t, I (t) be the input flux, Cells are being lost at a random rate γ,

Efflux/ death rate is D (t) = γ V (t)

V (t) = V₀e^{− γ} (t−t₀) +
γ (days⁻¹) = loss rate of red blood cells in the circulationBalance equation stating that the rate of change of erythrocyte numbers is a balance between their production and their destruction:

dE(t)dt

= production-destruction

Delay Time for rats is approximately 5.7 days

The next step in our model construction is to define some appropriate form for the production function F. In vivo measurements of erythrocyte production rates F in rats and other mammals including humans indicate that the feedback function saturates at low erythrocyte numbers and is a decreasing function of increasing red blood cell levels (i.e., negative feedback).

A convenient function that captures this behavior, and which has sufficient flexibility to be able to fit the data, as well as easily handled analytic properties, is given by

where F₀ (units of cells/kg-day) is the maximal red blood cell production rate that the body can approach at very low circulating red blood cell numbers, n is a positive exponent, and θ (units of cells/kg) is a shape parameter. These three parameters have to be determined from experimental data related to red blood cell production rates.

Negative Feedback:

The peripheral erythrocyte destruction rate is γ (day^-1) while F (cells/kg/day) is the cellular production rate in the early erythroid series cells. The total average time between the entrance of a cell into the erythroid series and the release of a mature erythrocyte into the blood is τ (days). The circulating density of erythrocytes is E (cells/kg), so their dynamics are described by:

dEdt=-γE+F(Eτ)Where E (τ) = E (t-τ). The erythrocyte production rate is a function of the circulating erythrocyte numbers at a time τ previously because of the finite time needed for cell maturation.

The in vivo erythrocyte production rate in rats has the form of a Hill function, showing saturation at low hemoglobin levels.

FEτ=F0θnEτn+θn

where F₀ (units of cells/kg-day) is the maximal red blood cell production rate that the body can approach at very low circulating red blood cell numbers, n is a positive exponent, and θ (units of cells/kg) is a shape parameter. These three parameters are determined from experimental data related to red blood cell production rates.

The Age-structured model with Constant Flux:

The moving boundary condition has the advantage of better capturing the physiological reality of apoptosis in circulating cells. Moreover, the model is sufficiently general to characterize other hematopoietic lines.

The erythrocyte precursor cells begin from fact that has differentiated into a self-sustaining population which eventually leads to the production of mature erythrocytes. The model considers two populations of cells: the precursor cells, denoted by p (t, µ), and the mature non-proliferative cells, denoted by m (t, v).

Let p (t, µ) denote the population of precursor cells at time t and age µ, and let V (E) be the velocity of maturation, which may depend on the hormone concentration, E. If S₀ (E) is the number of cells recruited into the proliferating precursor population, then the entry of new precursor cells into the age-structured model will satisfy the boundary condition suggesting that EPO mediates apoptosis and, by interrupting the programmed cell death, controls the number of cells that mature.

V (E) p (t, 0) = S₀ (E).

Let the birth rate for proliferating precursor cells be β (µ, E) and death rate through apoptosis α (µ, E). Let h (µ −

µ¯) be the density of the distribution of maturity levels of the cells when released into the circulating blood, where

µ¯represents the mean age of mature precursor cells and

∫0µFh(µ-µ¯)dµ

=1

The disappearance rate function is given by

Hµ=hµ-µ¯∫µµ¯hs-µ¯dsWith these conditions the age-structured model for the population of precursor cells with t > 0 and 0 < µ < µ_F, satisfies:

dpdt+VEdpdµ=VE[βµ,Ep-αµ,Ep-Hµp]Let m (t, v) be the population of mature non-prolifer­ating cells at time I and age v. Assume that the mature cells age at a rate W, which is considered to be a constant for erythropoiesis since the aging process appears to depend only on the number of times that an erythrocyte passes through the capillaries. From the disappearance rate function, the boundary given condition by the for cells following entering expression: the mature population is given by the following expression age, where for a cell maturity reaching level µ_F is the maximum age for a cell reaching the maturity.

This results in a moving boundary condition with the age of boundary the oldest condition, erythrocyte, derived v (t) varying in t.

The boundary condition is:

W-dvFdtmt,vF=QIf γ (v) is death rate of mature cells (depending only on age where the maximum age, v_F (t), is determined by:

dmdt+Wdmdv=-Wγvm

The hormone EPO exerts control in the model through the boundary conditions, the birth and death of precursor cells, and the velocity of aging. Using the method of characteristics and the techniques one can reduce this system of equations to a system of threshold delay equations. The following system of delay differential equations with a fixed delay T and a state dependent delay occurring in the equation governing the age at which mature cells die is obtained.

The hormone level E is governed by a differential equation with a negative feedback.

Let V(t ), the total population of mature cells, be given by

where k is the decay constant for the hormone and f(M) is a monotone decreasing function of M (negative feedback).

dEdt=fV-γEfV=a1+KV

The above is Differential delay equation (DDE) given by Hill function

Model for the effect of drug:

The effect of the drug is induced in the model by considering the fact RBC reacts if its production is changed under anemia conditions where RBC has no feedback to its production. This knowledge can be used when extending the disease model to a drug-disease model for a drug acting as an erythropoietin stimulating agent (ESA). Drug concentrations at the target (bone marrow) are determined by the PK of the drug, and effects on RBC (or Hb) production by its PD. In this model we linked drug concentrations to Hb production via an Emax model with parameters Smax and SC50. As there is no feedback control of RBC, any drug dosage regimen needs to be carefully evaluated with respect to avoiding excessive Hb values.

dVdt=rint-rint-LS;

rint=rin0.DE;

DE=1+Smax .c(t)SC50+c(t);

ct=PK(DR,PKpar,t)

V=Intial Value of RBC in blood

LS=Life Span of RBC

rin0=RBC production rate at start of treatment

DE is the drug effect

Smax=Maximum realtive increase in erthropoiesis stimulation.Generally taken as 0.5

S50=Drug concentration producing half maximum stimulation

DR=dosage regime

PK=purely kinetic parameter

t=time after treatment of start

ct=drug concentration

Parameters for numerical simulations:

For normal situation (i.e., not autoimmune hemolytic anemia):

γ = 2.31×10⁻² day⁻¹,

F₀ = 7.62×10¹⁰ cells/kg-day,

n = 7.6,

θ = 2.47×10¹¹ cells/kg,

τ = 5.7 days.

These parameters correspond to a steady-state circulating red blood cell mass of E* = 3.3 × 10¹¹ cells/kg, and from the linear analysis of the previous section, it is predicted that this steady state is stable. Red blood cell destruction rate γ is increased through the action of cell damage (lysis) by the injected antibody.

Numerical simulations of with the parameters show that the linear analysis results quoted above give a very accurate picture of the full nonlinear behavior, including the values of γ at which stability of E* is lost and regained, and the period of the solutions at these stability boundaries The computer simulation of the model is seen as a function of the peripheral destruction rate (γ). Plot the solution of this equation with

γ=0.05,0.08 and 0.01

The Terahertz (THz) or T-rays region lies between the microwave and infrared regions of the electromagnetic spectrum. Thus THz is situated in the frequency regime between optical and electronic techniques. This regime is typically defined as 0.1-10 THz. Applications investigated hitherto range from using THz spectroscopic features to identify explosives, to checking for defects in airplanes and to imaging diseases.

Terahertz has many interesting properties that make it very useful for technical and biological applications.

1. It is strongly attenuated by water and very sensitive to water content.
2. Compared to other radiations, THz waves can realize a non-invasive and non-ionizing imaging for biological tissues due to the low photon energy.

In THz-Time Domain Spectroscopy techniques coherent detection is used to record the THz wave’s temporal electric fields leads to obtaining of both the amplitude and phase of the THz wave simultaneously. These temporal waveforms can be treated with Fourier analysis to give the spectra. This allows precise measurements of the refractive index and absorption coefficient of samples without resorting to the Kramers- Kronig relations. The energy of rotational and vibrational transitions of molecules lies in the THz region and intermolecular vibrations such as hydrogen bonds exhibit different spectral characteristics in the THz range.

Experimentally it is verified that exposure bio fluids to EM radiations offer significant indication towards the mechanism of living cells and associated diseases.

Terahertz (THz) spectroscopy can be used to get the qualitative analysis of Hemolytic anemia. The data of concentration of different blood constituents and their rate of change can be obtained in the THz frequency. This is done by comparing the volume percentage of RBCs in diseased blood with the normal blood.

The observations are based on various properties of blood like refractive index and absorption coefficients in THz regime will be altered when RBC concentration levels are varied.

Since THz waves have a scattering rate lower than that of infrared and visible light and hence may be used as a feasible probe for analyzing blood inside vessels. We measured the optical constants of blood and its components using THz time-domain spectroscopy (TDS) and determined the correlation between the RBC concentration and the THz absorbance.

The first sample was collected into a tube contained disodium salt of Ethylene Diamine Tetra Acetic Acid (EDTA) anticoagulant and used for assessment of the following erythrocyte indices: Red Blood Cell (RBC) count, Hemoglobin (HGB) concentration.

For a pure material, multiple Debye type relaxation processes are possible where the complex dielectric coefficient,, is described, ε∞ is the real part of the dielectric constant at the high frequency limit, ∆ε = ε_j-ε_j+1,

The ∆ε term can be considered to be an ‘amplitude’, indicating the ‘quantity’ of that particular relaxation in the material under investigation.

The complex dielectric coefficient of a material is simply related to the complex refractive index, as described by equation, where are the real and imaginary parts of the complex dielectric coefficient. The complex dielectric coefficient of a material can be determined simply from the measured values for absorption coefficient, µa, and refractive index, n,

Where n () and  are the real and imaginary parts of the refractive index of blood. T () _a-b, which is the transmission coefficient from medium a to medium b, has the form of:

where is the complex refractive index.

RESULTS AND DISCUSSION

Hematological data is expected to show that there was significant drop in Hemoglobin (HGB) concentrations, RBCs counts and Hematocrit (HCT) values in rats at day 3 post-injection of PHZ and these drops in RBCs.

The experimental part of this study is based on measuring the amplitude of the pulse signal passing through the different RBC of blood in the frequency domain. As the RBC concentration decreases, the THz absorption should increase.

(THz) time-domain spectroscopy (TDS) can also be used to characterize the blood using complex optical constants of blood and its constituents in THz frequency range.

Absorbance of blood can be represented by a simple linear equation

α_blood=V⋅α_RBCs + (1−V) ⋅α_plasma, where α_blood, α_RBCs, and α_plasma are the absorption coefficients of blood, RBCs, and plasma and V indicates the volume fraction of the RBCs.

Schematic diagram of a photoconductive antenna device operating as a THz emitter and as a detector.

The microwatt-level THz pulse train, having the same repetition frequency as the excitation laser can span the range, 0.1 to 5 THz (3.3 cm^-1 to 165 cm^-1) with a central frequency at about 1 THz.

The THz detector, which can be another photoconductive antenna switch or an electro-optic crystal, is fed with an optical pulse trigger coming from the same excitation laser source. The temporal shape of the broadband THz wave is then measured in femtosecond time-resolution by an optical delay line provided by a motorized stage. By Fourier-transformation of the time-domain data, the terahertz spectra (usually between 0.1 to 5 THz) is obtained.

The schematic diagram of a dipole photoconductive antenna is shown below. The metal transmission lines are typically deposited on either a semi-insulating GaAs substrate or a low temperature-grown GaAs film on semi-insulating GaAs. The femtosecond laser pulse focused on the biased antenna gap (shown in the diagram) causes photo generated electrons to rapidly traverse the gap, creating a transient photocurrent that produces a THz electromagnetic pulse.

This THz radiation is then collected by the Si lens. When used as a detector, a time-varying bias in the photoconductive antenna is provided by the time-dependent induced electric field due to the THz pulse. A part of the same excitation laser pulse is then made incident on the detector to create photo-electrons and the measured current at the electrodes relative to the timing of the laser pulse and the incident THz wave reproduces the THz waveform in the time domain. 

Electro-optic detection is carried out via the Pockels effect where a crystal (e.g. ZnTe) that becomes birefringent in the presence of an electric field is utilized.

The transient field induced by the THz pulse leads to a polarization change in an incident laser probe pulse which is proportional to the field strength. This is then measured using polarization-sensitive detection. 

Schematic diagram of a THz-TDS system.

Duration of the project                                                                                                    

The project will start in August 2017. The duration of the project will be two years (until 2019).

Human Subjects

My project involves experimentation on 2 Wistar Rats animals. If approved by your side will obtain approval for your project through your institution’s office of Institutional Compliance.

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 Major Publications

Bhalla Chunesh D. (2001) Causality in Chaotic Processes. Cambridge: Cambridge University, Press.

Bhalla Chunesh D. (2002) “Physics and Reality,” in Uskali Mäki (ed.) Fact and Fiction in Physics. Cambridge: Cambridge University Press.

Bhalla Chunesh D. (2003) “Nonstationary Time Series Equations” British Journal for the Philosophy of Science, 2003.