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The main aim of this paper is to investigate the application of Calculus in blood flow and pharmacology. Although it may seem unlikely, biology heavily relies on the application of calculus, giving rise to a specialized field known as biocalculus.
In this article, we delve into how calculus is employed to calculate the velocity of blood flow in arteries and veins, as well as to determine drug sensitivity.
Poiseuille's law, velocity, pressure, cardiac output, dosage of drugs.
Contrary to common perception, biology extensively incorporates calculus in its various applications, leading to the emergence of a specialized field known as biocalculus. In this section, we explore how calculus plays a crucial role in several biology and medicine domains.
Calculus is instrumental in deriving Poiseuille's law, which allows us to calculate the velocity of blood flow in arteries or veins at specific points in time, as well as the volume of blood traversing through arteries. Poiseuille's law is represented as:
ΔP/l = (8ηlπr⁴)/(πR⁴)
Where:
The flow rate of blood can be determined by integrating the velocity function across the artery's cross-section, which simplifies to:
ΔP/l = (4ηlπr⁴)/(πR⁴)
Cardiac output is calculated using the dye dilution method, wherein blood is pumped into the right atrium and flows into the aorta.
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A device placed in the aorta measures the dye concentration leaving the heart at equal time intervals until the dye dissipates.
This measurement can be expressed using the equation:
C(t) = A / (T∫0 c(t) dt)
Where:
Calculus is also utilized to determine high and low blood pressure points during the cardiac cycle through optimization techniques.
In the field of oncology, calculus comes into play to assess the rate of tumor growth or shrinkage and the number of cells constituting the tumor. This is achieved using a differential equation known as the Gompertz Equation, given by:
dV/dt = aV * ln(b/V)
Where:
In pharmacology, calculus plays a vital role in determining drug sensitivity, where a drug's sensitivity is the derivative of its strength. Optimization techniques are employed to find the dosage that maximizes both sensitivity and strength of a drug. Integration methods can also be utilized to calculate potential side effects of drugs, such as temperature changes in the body.
Microbiologists employ various mathematical models, including logistic, exponential, and differential equations, to calculate the rate at which bacteria grow.
Physiologists use calculus to find the rate of change of shortening velocity concerning the load when modeling muscle contractions.
Integration techniques are applied in neurology to calculate the voltage of a neuron at a specific point in time. Differential equations are utilized to determine the change in voltage of a neuron with respect to time, as shown in the equation below:
dV/dt = ΔV/Δt
The Nicholson-Bailey model, employing partial fractions, can effectively model the dynamics of a host-parasitoid system.
Entomologists find partial derivatives useful in modeling the crawling speed of larvae, particularly in the field of forensic entomology.
Exploring the application of Fractional Calculus (FC), we initially examine traditional integer differentiation for a power function of time:
f(t) = tm
f'(t) = m * tm-1
f''(t) = m(m-1) * tm-2
f'''(t) = m(m-1)(m-2) * tm-3
...
f(n)(t) = m(m-1)(m-2)...(m-n+1) * tm-n
where m ≠ n
Similarly, the nth repetitive integer integration process can be examined for a power function of time:
∫tm dt = tm+1 / (m+1) + C
∫∫tm dt = tm+2 / ((m+1)(m+2)) + C
∫∫∫tm dt = tm+3 / ((m+1)(m+2)(m+3)) + C
...
∫∫...∫tm dt = tm+n / ((m+1)(m+2)...(m+n)) + C
where m ≠ -1, -2, -3, ...
The gamma function, denoted as Γ(x), can be defined as:
Γ(x) = ∫0∞ tx-1 * e-t dt
It is important to note that the gamma function is not defined for x equal to zero or negative integer values. However, for positive integer values of x, the gamma function connects smoothly with the factorial function, making it suitable for defining non-integer factorial values.
Fractional Calculus can also be applied to the Taylor series of an exponential function:
et = 1 + t + (t2 / 2!) + (t3 / 3!) + (t4 / 4!) + ...
For sufficiently large N, an exponential function can be accurately approximated as a summation of power functions. Using this methodology, the Taylor series for an exponential can be term-by-term fractionally differentiated or integrated:
et = 1 + (tq / q!) + (t2q / (2q)!) + (t3q / (3q)!) + ...
It's crucial to exercise caution when selecting integer values of q to avoid undefined values of the gamma function.
The esophageal Doppler monitor (EDM) is a common tool used to assess aortic blood flow velocity during systole. Clinicians rely on EDM to accurately measure cardiac output and stroke volume in anesthesia and critical care settings. Figure 2 depicts this waveform. The velocity, v(t), can be modeled as:
v(t) = a * ebt (15)
Where 'a' represents an acceleration term, and 'b' is a dimensionless gain. The duration of systole is known as flow time, FT, and 'g' can be determined as:
g = ∫0FT v(t) dt (16)
The Systolic Pressure – Flow Relationship in the Aorta
A simplified model of aortic blood pressure, P(t), as a function of aortic blood flow velocity during systole is:
P(t) = r * [a∫0t v(t) dt - b∫0t v(t) dt] * Za + k (16)
Where 'r' represents the radius of the aorta, and 'a' and 'b' are velocity-based differential integrals of orders -0.7 and 0.1, respectively. The terms 'Za' and 'Zb' are analogous to combinations of elastance, resistance, inertia, and resistance. Additionally, 'C' is a constant of integration, and 'k' converts units from Pascals to mmHg. Furthermore:
k = a∫0T v(t) dt - b∫0T v(t) dt (17)
For one of the smaller human arteries with η = 0.027, R = 0.008 cm, L = 2 cm, and p = 4000 dyne (1 dyne = 1 g.cm / s² = 10⁻⁵ N), calculate the blood flow velocity at the centerline and at r = 0.002 cm. Also, determine the velocity gradient at r = 0.002 cm.
V = (4p / ηR²) * (1 - (r / R)²) (18)
VCL = 1.185 cm/s
Vr=0.002 = 1.111 cm/s < VCL
Velocity gradient at r = 0.002 cm:
V' = -74 cm/s / cm
A patient is prescribed 500 mg of metformin hydrochloride, and the available tablets are 1000 mg each. How many tablets should be administered?
Amount = 500 mg / 1000 mg
Amount = 1 / 2 tablet
A client weighing 60 kg needs to be given a drug at a rate of 2 mg/kg. The stock strength of the drug is 40 mg/2 ml. Calculate the volume of the drug to be administered.
Stock required = 60 kg * 2 mg/kg
Amount = 120 mg
Amount = 6 ml
A doctor prescribes 10,000 mg of a drug for a patient, and the available tablets are 4 mg each. How many tablets should be administered?
Doctor's order × Conversion factor = 10,000 mg / 4 mg/tablet
Amount = 2,500 tablets
In this study, we have conducted an analysis of blood flow and pharmacology based on Poiseuille's law in calculus. Beginning with a brief historical overview, we have presented the equations for our model. We have approached them from an abstract mathematical perspective to derive insights into blood flow in arteries and veins. Additionally, we have provided a concise analysis of pharmacology. "The calculus of utility aims at supplying the ordinary needs of humanity at the least cost in labor."
A Study on Application of Calculus in Blood Flow and Pharmacology. (2024, Jan 24). Retrieved from https://studymoose.com/document/a-study-on-application-of-calculus-in-blood-flow-and-pharmacology
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