? ?? In medicine, mechanical ventilation is a procedure that assists or replaces spontaneous breath- ing for
Fantastic news! We've Found the answer you've been seeking!
Question:
???
Transcribed Image Text:
In medicine, mechanical ventilation is a procedure that assists or replaces spontaneous breath- ing for critically ill patients, using a medical device called a ventilator. Some people attribute the first mechanical ventilation to Andreas Vesalius in 1555. Negative pressure ventilators (iron lungs) came into use in the 1940s-1950s in response to poliomyelitis (polio) epidemics. Philip Drinker and Louis Shaw are credited with its invention. Modern ventilators use positive pressure to inflate the lungs of the patient. In the ICU (intensive care unit), common indications for the ini- tiation of mechanical ventilation are acute respiratory failure, acute exacerbation of chronic obstructive pulmonary disease, coma, and neuromuscular disorders. The goals of mechanical ventilation are to provide oxygen to the lungs and to remove carbon dioxide. In this project, we model the mechanical process performed by the ventilator. We make the following assumptions about this process of filling the lungs with air and then letting them deflate to some rest volume (see Figure 2.13). (i) The length (in seconds) of each breath is fixed (fot) and is set by the clinician, with each breath being identical to the previous breath. (ii) Each breath is divided into two parts: inspiration (air flowing into the patient) and expiration (air flowing out of the patient). We assume that inspiration takes place over the interval [0, t;] and expiration over the time interval [ti, fot]. The time t; is called the inspiratory time. [1] Papp during inspiration airway-resistance pressure drop, P, lung elastic pressure, Pe and residual pressure, Pex (iii) During inspiration the ventilator applies a constant pressure Papp to the patient s air- way, and during expiration this pressure is zero, relative to atmospheric pressure. This is called pressure-controlled ventilation. (iv) We assume that the pulmonary system (lung) is modeled by a single compartment. Hence, the action of the ventilator is similar to inflating a balloon and then releasing the pressure. (v) At the airway there is a pressure balance: (1) P₁+ P₂ + Pex = P + where P, denotes pressure losses due to resistance to flow into and out of the lung, P, is the elastic pressure due to changes in volume of the lung, Pex is a residual pressure that remains in the lung at the completion of a breath, and Paw denotes the pressure applied to the airway. (Paw = Papp during inspiration and Paw = 0 during expiration.) The residual pressure P is called the end-expiratory pressure. (vi) Let V(t) denote the volume of the lung at time t, with V.(t), 0 ≤ t ≤t;, denoting its volume during inspiration and V.(t), t; sts for its volume during expiration. We assume that V;(0) = V. (f) = 0. The number V;(1) = Vr is called the tidal volume of the breath. (vii) We assume that the resistive pressure P, is proportional to the flows into and out of the lung such that P, = R(dv/dt), and we assume that the proportionality constant R is the same for inspiration and expiration. (viii) Furthermore, we assume that the elastic pressure is proportional to the instantaneous volume of the lung. That is, P = (1/C)V, where the constant C is called the compliance of the lung. (2) R Using the pressure equation in (1) together with the above assumptions, a mathematical model for the instantaneous volume in the single compartment lung is given by the following pair of first-order linear differential equations: (3) R Group Projects for Chapter 2 dt dv dt + 81 + V; + P = P = Papp 0≤t≤t;, Ve + Pex=0, 1; ≤1 ≤ hot. The initial conditions, as indicated in assumption (vi), are V;(0) = 0 and V₂(t) = V(t) = VT. The constant Per is not known a priori but is determined from the end condition on the expiratory volume: V₂(1) = 0. This will make each breath identical to the previous breath. To obtain a for- mula for Pex, complete the following steps. The initial conditions, as indicated in assumption (vi), are V;(0) = 0 and V₂(1) = V₂ (1) = VT. The constant Per is not known a priori but is determined from the end condition on the expiratory volume: V(c) = 0. This will make each breath identical to the previous breath. To obtain a for- mula for Pex, complete the following steps. (a) Solve equation (2) for V;(1) with the initial condition V;(0) = 0. (b) Solve equation (3) for V₂ (1) with the initial condition V₂ (1;) = V₁. (c) Using the fact that V;(1) = V₁, show that Pes 1) Pupp Au/RC 1 (eV/RC ] (d) For R = 10 cm (H₂O)/L/sec, C = 0.02 L/cm (H₂O), Papp = 20 cm (H₂O), t; = 1 sec and tot = 3 sec, plot the graphs of V;(1) and V₂(1) over the interval [0, 1]. Compute Pex for In medicine, mechanical ventilation is a procedure that assists or replaces spontaneous breath- ing for critically ill patients, using a medical device called a ventilator. Some people attribute the first mechanical ventilation to Andreas Vesalius in 1555. Negative pressure ventilators (iron lungs) came into use in the 1940s-1950s in response to poliomyelitis (polio) epidemics. Philip Drinker and Louis Shaw are credited with its invention. Modern ventilators use positive pressure to inflate the lungs of the patient. In the ICU (intensive care unit), common indications for the ini- tiation of mechanical ventilation are acute respiratory failure, acute exacerbation of chronic obstructive pulmonary disease, coma, and neuromuscular disorders. The goals of mechanical ventilation are to provide oxygen to the lungs and to remove carbon dioxide. In this project, we model the mechanical process performed by the ventilator. We make the following assumptions about this process of filling the lungs with air and then letting them deflate to some rest volume (see Figure 2.13). (i) The length (in seconds) of each breath is fixed (fot) and is set by the clinician, with each breath being identical to the previous breath. (ii) Each breath is divided into two parts: inspiration (air flowing into the patient) and expiration (air flowing out of the patient). We assume that inspiration takes place over the interval [0, t;] and expiration over the time interval [ti, fot]. The time t; is called the inspiratory time. [1] Papp during inspiration airway-resistance pressure drop, P, lung elastic pressure, Pe and residual pressure, Pex (iii) During inspiration the ventilator applies a constant pressure Papp to the patient s air- way, and during expiration this pressure is zero, relative to atmospheric pressure. This is called pressure-controlled ventilation. (iv) We assume that the pulmonary system (lung) is modeled by a single compartment. Hence, the action of the ventilator is similar to inflating a balloon and then releasing the pressure. (v) At the airway there is a pressure balance: (1) P₁+ P₂ + Pex = P + where P, denotes pressure losses due to resistance to flow into and out of the lung, P, is the elastic pressure due to changes in volume of the lung, Pex is a residual pressure that remains in the lung at the completion of a breath, and Paw denotes the pressure applied to the airway. (Paw = Papp during inspiration and Paw = 0 during expiration.) The residual pressure P is called the end-expiratory pressure. (vi) Let V(t) denote the volume of the lung at time t, with V.(t), 0 ≤ t ≤t;, denoting its volume during inspiration and V.(t), t; sts for its volume during expiration. We assume that V;(0) = V. (f) = 0. The number V;(1) = Vr is called the tidal volume of the breath. (vii) We assume that the resistive pressure P, is proportional to the flows into and out of the lung such that P, = R(dv/dt), and we assume that the proportionality constant R is the same for inspiration and expiration. (viii) Furthermore, we assume that the elastic pressure is proportional to the instantaneous volume of the lung. That is, P = (1/C)V, where the constant C is called the compliance of the lung. (2) R Using the pressure equation in (1) together with the above assumptions, a mathematical model for the instantaneous volume in the single compartment lung is given by the following pair of first-order linear differential equations: (3) R Group Projects for Chapter 2 dt dv dt + 81 + V; + P = P = Papp 0≤t≤t;, Ve + Pex=0, 1; ≤1 ≤ hot. The initial conditions, as indicated in assumption (vi), are V;(0) = 0 and V₂(t) = V(t) = VT. The constant Per is not known a priori but is determined from the end condition on the expiratory volume: V₂(1) = 0. This will make each breath identical to the previous breath. To obtain a for- mula for Pex, complete the following steps. The initial conditions, as indicated in assumption (vi), are V;(0) = 0 and V₂(1) = V₂ (1) = VT. The constant Per is not known a priori but is determined from the end condition on the expiratory volume: V(c) = 0. This will make each breath identical to the previous breath. To obtain a for- mula for Pex, complete the following steps. (a) Solve equation (2) for V;(1) with the initial condition V;(0) = 0. (b) Solve equation (3) for V₂ (1) with the initial condition V₂ (1;) = V₁. (c) Using the fact that V;(1) = V₁, show that Pes 1) Pupp Au/RC 1 (eV/RC ] (d) For R = 10 cm (H₂O)/L/sec, C = 0.02 L/cm (H₂O), Papp = 20 cm (H₂O), t; = 1 sec and tot = 3 sec, plot the graphs of V;(1) and V₂(1) over the interval [0, 1]. Compute Pex for
Expert Answer:
Related Book For
Applied Statistics in Business and Economics
ISBN: 978-0073521480
4th edition
Authors: David Doane, Lori Seward
Posted Date:
Students also viewed these mathematics questions
-
Make the following assumptions about Gwinn Company and Harris Company for the purpose of this problem. 1. The two companies ended 2010 with the exact same balances in all of the accounts reported on...
-
Patients suffering from acute respiratory failure were randomly assigned to either be placed in a prone (face down) position or a supine (face up) position. In the prone group, 21 out of 152 patients...
-
By filling carbon nanotubes with miniscule wires made of iron and iron carbide, incredibly thin nanowires can be extruded by blasting the carbon nanotubes with an electron beam. If Gentech...
-
A car has a sticker price of $69,000. The car has a 100 hp engine and can accelerate from 0 to 60 mph in 15.8 seconds. The lease rate is 4.6%. The term of the lease is three years. The buyout is...
-
1. Analyze Meredith's marketing information system. What are its strengths and weaknesses? 2. Can impersonal data points really result in meaningful relationships? Explain. 3. Does Meredith's...
-
An article in the Wall Street Journal contained the following observation: "Every month, millions of workers leave the job market because of retirement, to care for children or aging parents, to...
-
If a well-behaved investment alternative's internal rate of return (IRR) is equal to MARR, which of the following statements about the other measures of worth for this alternative must be true? 1....
-
On January 3, 2012, Trusty Delivery Service purchased a truck at a cost of $90,000. Before placing the truck in service, Trusty spent $3,000 painting it, $1,500 replacing tires, and $4,500...
-
Structured query language (SQL) is divided into two categories: data definition language (DDL) commands and data manipulation language (DML) commands. Data manipulation language (DML) is a set of...
-
Natura Cosmeticos SA, Brazils all natural, ethically-sourced, sustainable beauty products company, recently purchased Avon Products Inc. as a way to enter the U.S., Europe, and Chinese cosmetics...
-
Based on the facts below, prepare for Shelli Thomson her US (federal) individual income tax return (Form 1040), her Schedule A (Itemized Deductions), Schedule 3 (Additional Credits and Payments)....
-
calculate Company's Total Value by follow table . C. Specia. Funds grants be 20% Devity 20% 20% Petared Earnings 20% 20% Company's Total Value
-
Janet is running a painting business.She receives $200,000 payment from buyers at the end of year, at which time she also pays $20,000 fixed cost and 15% of the revenues as a variable cost. (1) An...
-
The equity sections for Atticus Group at the beginning of the year (January 1) and end of the year (December 31) follow. Stockholders' Equity (January 1) Common stock-$5 par value, 100,000 shares...
-
A car and a bus set out at 2 pm from the same spot, headed in the same direction. The average speed of the car is twice the average speed of the bus. After 2 hours, the car is 68 miles ahead of the...
-
Consider the linear system: = x-3y dt = 3x-5y a) Find the general solution of the system. b) Describe the critical point (0,0) of the system using all the applicable words from the Word List below.
-
Reword the following Selling a property is one of the biggest financial transactions you'll make, at OBrien Real Estate we aim to take the stress out of the equation and deliver a result over and...
-
You are standing on the top of a building and throw a ball vertically upward. After 2 seconds, the ball passes you on the way down, and 2 seconds after that, it hits the ground below. a. What is the...
-
The data set below shows a sample of salaries for 39 engineers employed by the Solnar Company along with each engineer's years of experience. (a) Construct a scatter plot using Salary as the response...
-
A sample of size n 5 70 showed a skewness coefficient of 0.773 and a kurtosis coefficient of 1.277. What is the distribution's shape?
-
Consider the following data on 20 chemical reactions, with Y = chromatographic retention time (seconds) and X = molecular weight (gm/mole). (a) Make a scatter plot. (b) Use Excel to t the regression,...
-
A big challenge for the GM managers at the auto plant making the Chevrolet Sonic will be maintaining positive employee morale. They will be managing a workforce of 1,550 employees that is composed of...
-
Assume you are an employee in a situation similar to the one described in this case, a situation in which you believe your union has not represented your interests fairly and made a deal with...
-
A British Workplace Behaviour Survey explored prevalence of a wide range of behaviors that comprise ill treatment in the UK workplace. Ill treatment included 47 percent reported unreasonable...
Study smarter with the SolutionInn App