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The pressure in the aorta changes throughout the cardiac cycle. During systole, as the heart contracts, the outflux of blood into the aorta causes an increase in pressure, whereas during diastole the pressure decreases as the heart relaxes. A simple model for the aortic pressure waveform is given by the Windkessel effect described by the image below. In this model, the heart is considered a pressure generating pump which is directly connected to an elastic compartment (the aorta), which in turn is connected to a rigid set of peripheral vessels (the hose of the firefighter). ● Pump ● Heart Air Windkessel In order to find the aortic pressure waveform from the Windkessel model, a mass balance formulation around the aorta must be formulated. Coming into the aorta from the heart we have the flowrate Q(t). According to conservation of mass, this inflow rate Q(t) must be equal to the outflow rate into the peripheral vessels and the change in volume of the aorta. To find these two relationships, the following assumptions must be made: Elastic arteries Assume the blood pressure in the aorta is given by p Assume that the change in volume (and therefore mass) in the aorta with respect to time is proportional to the change in pressure, p, with respect to time up to a proportionality constant K Assume that the outflow rate into the peripheral vessels is dependent on the resistance of those vessels R and the blood pressure in the aorta p. This outflow relationship follows a form like Ohm's Law, where the pressure is similar to the voltage and the flow similar to the current. Based on the assumptions described before, write the first order differential equation that describes the mass balance of the Windkessel model.

The pressure in the aorta changes throughout the cardiac cycle. During systole, as the heart contracts, the outflux of blood into the aorta causes an increase in pressure, whereas during diastole the pressure decreases as the heart relaxes. A simple model for the aortic pressure waveform is given by the Windkessel effect described by the image below. In this model, the heart is considered a pressure generating pump which is directly connected to an elastic compartment (the aorta), which in turn is connected to a rigid set of peripheral vessels (the hose of the firefighter). ● Pump ● Heart Air Windkessel In order to find the aortic pressure waveform from the Windkessel model, a mass balance formulation around the aorta must be formulated. Coming into the aorta from the heart we have the flowrate Q(t). According to conservation of mass, this inflow rate Q(t) must be equal to the outflow rate into the peripheral vessels and the change in volume of the aorta. To find these two relationships, the following assumptions must be made: Elastic arteries Assume the blood pressure in the aorta is given by p Assume that the change in volume (and therefore mass) in the aorta with respect to time is proportional to the change in pressure, p, with respect to time up to a proportionality constant K Assume that the outflow rate into the peripheral vessels is dependent on the resistance of those vessels R and the blood pressure in the aorta p. This outflow relationship follows a form like Ohm's Law, where the pressure is similar to the voltage and the flow similar to the current. Based on the assumptions described before, write the first order differential equation that describes the mass balance of the Windkessel model.

Human Anatomy & Physiology (11th Edition)
Human Anatomy & Physiology (11th Edition)
11th Edition
ISBN:9780134580999
Author:Elaine N. Marieb, Katja N. Hoehn
Publisher:Elaine N. Marieb, Katja N. Hoehn
Chapter1: The Human Body: An Orientation
Section: Chapter Questions
Problem 1RQ: The correct sequence of levels forming the structural hierarchy is A. (a) organ, organ system,...
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The pressure in the aorta changes throughout the cardiac cycle. During systole, as the heart contracts, the outflux of blood into the aorta causes an increase in pressure, whereas during diastole the pressure decreases as the heart relaxes. A simple model for the aortic pressure waveform is given by the Windkessel effect described by the image below. In this model, the heart is considered a pressure generating pump which is directly connected to an elastic compartment (the aorta), which in turn is connected to a rigid set of peripheral vessels (the hose of the firefighter). 5 Pump Heart Air Windkessel Elastic arteries In order to find the aortic pressure waveform from the Windkessel model, a mass balance formulation around the aorta must be formulated. Coming into the aorta from the heart we have the flowrate Q(t). According to conservation of mass, this inflow rate Q(t) must be equal to the outflow rate into the peripheral vessels and the change in volume of the aorta. To find these two relationships, the following assumptions must be made: Assume the blood pressure in the aorta is given by p Assume that the change in volume (and therefore mass) in the aorta with respect to time is proportional to the change in pressure, p, with respect to time up to a proportionality constant K Assume that the outflow rate into the peripheral vessels is dependent on the resistance of those vessels R and the blood pressure in the aorta p. This outflow relationship follows a form like Ohm's Law, where the pressure is similar to the voltage and the flow similar to the current. Based on the assumptions described before, write the first order differential equation that describes the mass balance of the Windkessel model.
Transcribed Image Text:The pressure in the aorta changes throughout the cardiac cycle. During systole, as the heart contracts, the outflux of blood into the aorta causes an increase in pressure, whereas during diastole the pressure decreases as the heart relaxes. A simple model for the aortic pressure waveform is given by the Windkessel effect described by the image below. In this model, the heart is considered a pressure generating pump which is directly connected to an elastic compartment (the aorta), which in turn is connected to a rigid set of peripheral vessels (the hose of the firefighter). 5 Pump Heart Air Windkessel Elastic arteries In order to find the aortic pressure waveform from the Windkessel model, a mass balance formulation around the aorta must be formulated. Coming into the aorta from the heart we have the flowrate Q(t). According to conservation of mass, this inflow rate Q(t) must be equal to the outflow rate into the peripheral vessels and the change in volume of the aorta. To find these two relationships, the following assumptions must be made: Assume the blood pressure in the aorta is given by p Assume that the change in volume (and therefore mass) in the aorta with respect to time is proportional to the change in pressure, p, with respect to time up to a proportionality constant K Assume that the outflow rate into the peripheral vessels is dependent on the resistance of those vessels R and the blood pressure in the aorta p. This outflow relationship follows a form like Ohm's Law, where the pressure is similar to the voltage and the flow similar to the current. Based on the assumptions described before, write the first order differential equation that describes the mass balance of the Windkessel model.
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