Biomedical Engineering - Life Support Systems

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Life Support Systems – A.A. 2017/2018 Balance equations – diffusive phenomena propagation in time Maria Laura Costantino, Giustina Casagrande Pag. 1/7 1 – Balance equations1 – Definitions and use Control volume The control volume (vc) is a portion of space delimited by a boundary-closed surface that represents the border with the outer space. This surface can be “virtual”, if it is only geometrically defined, or “real”, when it corresponds to a separation surface between two different means (for example the internal surface of the wall of a channel). If necessary, the boundary line can be considered as the connection of several partial surfaces that behave in different modes in relation with the external space. From a time evolution point of view, the control volume can be fixed or movable in relation with the chosen reference system; it can also be deformable or not. Interactions between the control volume and the outer space The control volume can interact with the external space. The interactions occurring across the boundaries concern extensive quantities (additive quantities such as mass and volume, while temperature and pressure are intensive quantities). We usually deal with mass (molecular diffusion, described by Fick’s law) energy (heat conduction, described by Fourier’s law) and momentum transport (viscosity, described by Newton’s law). It is useful remind the definitions regarding thermodynamic systems: • Open system: the boundary allows both mass and energy transfer • Closed system: the boundary allows energy transfer but it is not permeable to mass; in this case talking about control mass instead of control volume is more precise. • Isolated: the boundary is not permeable to mass and energy. Control volume, can be generally considered as an open thermodynamic system. Phenomena occurring inside the control volume Some of the quantities that the system exchanges with the outer space, can go through transformations inside the control volume, resulting in a overall variation of the quantity included in the volume: i.e. it is possible to have production or consumption of substances (i.e. oxygenated haemoglobin). Physical phenomena that bring to consumption or production of different extensive quantities inside the control volume can be very different according to the quantities. For example, to produce or consume mass, relatively to a specific chemical substance, a chemical reaction is required. Similarly, the thermal energy is a product of human metabolism. 1 Note: some paragraphs are taken from: Ferruccio Miglietta, Appunti di Fisica Tecnica, Pitagora Editore. Life Support Systems – A.A. 2017/2018 Balance equations – diffusive phenomena propagation in time Maria Laura Costantino, Giustina Casagrande Pag. 2/7 Definition of a general variation equation Variation equations come from the application, to a specific control volume, of a balance equation that represents the time variation of a specific extensive quantity A. The general form of a variation equation is: !"#$%&%' !!"#$"%$&'!"=!"#$%&%' !!"#$%!"&!"−!"#$%&%' !!"#$!%&'(!"+!"#$%&%' !!"#$%&'$!"−!"#$%&%' !!"#$%&'(!" A variation equation is generally a partial differential equation to be time integrated with suitable initial conditions and in a specific spatial domain with suitable boundary conditions. If the term on the left side of the equation is null, the balance represents the conservation of quantity A inside the control volume: this is a conservation equation in the steady state. According to the considered quantity A, the equation can be represented in a scalar, vectorial or tensor form. A scheme of blood oxygenation process, for example, beyond the indication of blood mass flow rate incoming and out-coming across the two base sections of a cylindrical capillary (for simplicity), requires to consider the O2 and CO2 passage across the whole lateral surface of the capillary (diffusion across a membrane). Mass variation The variation of the mass included in an open thermodynamic system (identified by their control volume) is given by: !"#$%"$%"&'(#!"##!"#$"%$&'!"=!"#$%&"'!"## !"#$ !"#$!"−!"#$%&'()!"## !"#$ !"#$!"+!"##!"#$%&'$!" !ℎ! !"#$!"!"#$!"−!"##!"#$%&'(!" !ℎ! !"#$!"#$%&!" d m d t =  m i n ∑ −  m o u t ∑ +  m p r −  m c (1.1) where  m = d m d t . The term  m p r refers to the mass of any substance potentially produced and can be also included in the term  m i n , while  m c refers to the mass of any substance potentially consumed and can be also included in the term  m o u t . Mass conservation For a thermodynamic system in a steady state, the conditions do not vary in the time domain, therefore the first member of the equation (1.1) is null and becomes:  m i n ∑ +  m p r −  m c =  m o u t ∑ (1.2) If no production or consumption of mass occurs, we obtain the continuity equation:  m i n ∑ =  m o u t ∑ (1.3) Life Support Systems – A.A. 2017/2018 Balance equations – diffusive phenomena propagation in time Maria Laura Costantino, Giustina Casagrande Pag. 3/7 Energy variation In the most general case, the energy variation of the fluid inside the control volume, considering positive the incoming energy, is given by: !"#$%"$%"&'(#!"!#$%!"#$"%$&'!"=!"#$% !""#$%!&'(!" !ℎ! !"#$%!"&!"##!"−!"#$% !""#$%!&'(!" !ℎ! !"#$!%&'(!"##!"+!"#$ !"#$%&'$!" !ℎ! !"#$ !"#!"#!" +!"#$ !"#ℎ!"#$%!" !ℎ! !"#$ !"#$%&!"+!"#$ !"#ℎ!"#$%!" !ℎ! !"#$ !"#$%&!" where the heat exchanged in the time domain (thermal power) is identified by:  Q =  Q i n −  Q o u t and the work exchanged in the time domain (mechanical power) is identified by  L =  L i n −  L o u t . The heat produced in the time domain (  Q p r ) is sometimes included in the thermal power exchanged. In differential terms: d E d t ( ) v c =  m e ( ) i n −  m e ( ) o u t +  Q p r +  Q +  L ( ) v c (1.4) In the equation (1.4) the terms e i n = u + P v + β w 2 2 + g z ⎛⎝⎜ ⎞⎠⎟ i n ed e o u t = u + P v + β w 2 2 + g z ⎛⎝⎜ ⎞⎠⎟ o u t are the mean energy quantities per unit of incoming and outcoming mass: there is a term of internal energy (u), a term representing pressure energy (Pv), a kinetic term ( )2w, a term of potential energy (gz). Remind:  Q  m = q is the heat exchanged per unit of mass and  L  m = l is the work exchanged per unit of mass. The pressure work (work and energy have the same unit of measure) is the work performed by the external forces on the fluid to insert it in the control volume and the work performed by the fluid on the external forces to get out of the control volume. In the closed thermodynamic systems the mass transport does not occur across the boundary and the pressure work is null. The multiplicative coefficient β that appears in the kinetic terms, is a form factor depending on the velocity profile on the section: β = v 3 d A A ∫ A w 3 , c o n 1 < β ≤ 2 where A is the section area of fluid passage, w the mean velocity and v the local velocity of the fluid in the section. In case of fully developed motion, the extreme values of β are respectively in turbulent and laminar motion conditions: β=1 for the fully developed turbulent motion β=2 for the fully developed laminar motion However, β can be equal to infinite values included between the two extremities. Remind: a fluid in motion inside a duct is in laminar flow when Re2300. For values of Reynolds number between 2000 and 2300 the fluid is in transition flow. Finally, the energy variation equation in complete form for a fluid that crosses the control volume per time unit can be written as: d E d t ( ) v c =  m ⋅ u + P v + β w 2 2 + g z ⎛⎝⎜ ⎞⎠⎟ ⎡⎣⎢⎢ ⎤⎦⎥⎥ i n −  m u + P v + β w 2 2 + g z ⎛⎝⎜ ⎞⎠⎟ ⎡⎣⎢⎢ ⎤⎦⎥⎥ o u t +  Q p r +  Q +  L ⎛⎝⎜⎜ ⎞⎠⎟⎟ v c Considering the transport in the biological systems, the term expressing the mechanical power is often negligible, apart from the case of cardiac work. The thermal power in the majority of the cases refers to the produced thermal power (for example the metabolic heat). The variations in potential energy are often negligible. Energy conservation Here we consider the steady state flow. The energy equation (first law of thermodynamics), for open systems in steady state, can be expressed as: ( u + P v + 12 β w 2 + g z ) i n +  Q  m +  L  m = ( u + P v + 12 β w 2 + g z ) o u t (1.5) The equation (1.5) can be explained as follow: the energy of the fluid at the outlet of the control volume is equal to the sum of the energy of the fluid at the inlet, accounting for the supplied or subtracted energy, in the form of heat or work, during the cross of the volume. In terms of powers:  m ⋅ ( u + P v + 12 β w 2 + g z ) i n +  Q +  L =  m ⋅ ( u + P v + 12 β w 2 + g z ) o u t (1.6) Reminding that h=u+Pv is the mass enthalpy, the equation (1.6) can be written as:  m ⋅ ( h + 12 β w 2 + g z ) i n +  Q +  L =  m ⋅ ( h + 12 β w 2 + g z ) o u t (1.7)  Q +  L =  m ⋅ ( h + 12 β w 2 + g z ) o u t − ( h + 12 β w 2 + g z ) i n ⎡⎣⎢ ⎤⎦⎥ (1.8) The enthalpy term takes into account the energy degradation and, consequently, the reality of the processes. Thus, in differential form: dh Tds vdP =+ The first term of the second member contains the entropy variation per unit of flowing mass, which, in steady state, is the sum of two terms: ∆!= !!"−!!"#=!!+!!"" Where sq is the entropy variation associated to the thermal energy exchange between the control volume and the external space and sirr is the entropy variation due to irreversible processes and internal irreversible nature. Life Support Systems – A.A. 2017/2018 Balance equations – diffusive phenomena propagation in time Maria Laura Costantino, Giustina Casagrande Pag. 5/7 Diffusive phenomena propagation in the time domain Semi-infinite slab Ex. 1 A Becker containing venous blood treated with heparin is placed on a desk without any cover. The blood inside the Becker is homogenously oxygenated with pO2=40mmHg. Consider thermal equilibrium between air and blood at a T=20° C and the diffusion coefficient of oxygen in the blood equal to D=1.75*10-9 m2/s for blood with haematocrit = 45%. The pO2 in the air (21% O2, 78% N2, 1% other gases) is 21656 Pa. Determine: 1. The oxygen flux at the interface in the initial conditions, 2. The oxygen quantity in the blood at a depth of 5 mm, after 2 hours of air exposure, 3. How many hours have to pass in order to obtain a blood with pO2=100 mmHg, knowing that the Becker is 5 cm deep? 4. How much time would be required to reach the same oxygenation condition if we had water instead of blood? (Consider for water D=1.97*10-9m2/s and the same oxygen partial pressures) Solution: 1. The O2 concentration in the air can be considered a constant value; it is thus possible to determine the O2 concentration profile trend in the blood and to determine the O2 flux at the interface, assuming it with a s thickness. In the air: ca= pO2a RTa = 21656⋅Pa 260 J kg⋅K ⋅293.15⋅K =0.28kg m3 This concentration refers to a specific content. In the blood: cb= pO2b RTa = 40⋅133.32⋅Pa 260 J kg⋅K ⋅293.15⋅K =0.07kg m3 Assuming the existence of a film of blood 1 mm thick (s) at the interface, the oxygen flux across the film is: j= D s ca−cb ( )= 1.75⋅10−9m2 s 1⋅10−3m ⋅(0.28kg m3−0.07kg m3)=3.7⋅10−7 kg m2s ⎛ ⎝⎜ ⎞ ⎠⎟ 2. Using the theory of the semi-infinite slab, it is possible to calculate the oxygen concentration in the blood at a certain distance x (5 mm), after a time t (2 hours). cx=ca+(cb−ca)⋅erf x 4Dt ⎛ ⎝⎜ ⎞ ⎠⎟=0.28kg m3+(0.07−0.28)kg m3⋅erf 0.005m 4⋅1.75⋅10−9m2 s ⋅2⋅3600s ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ =0.138kg m3 3. The O2 concentration in the blood, when pO2=100mmHg, is: co2=pO2f RO2T = 100⋅133.32Pa 260 J kgK⋅293.15K =0.175kg m3 X defines the argument of the function erf, thus: Life Support Systems – A.A. 2017/2018 Balance equations – diffusive phenomena propagation in time Maria Laura Costantino, Giustina Casagrande Pag. 6/7 X= x 4Dt= 0.05m 4⋅1.75⋅10−9m2 s⋅t =597.6⋅t−12 From which erf X( )=cf−ca cb−ca =0.175−0.28 0.07−0.28=0.5 From the table of the error function: X=0.48 and, consequently: tblood= 597.6 X ⎛ ⎝⎜ ⎞ ⎠⎟ 2 = 597.6 0.48 ⎛ ⎝⎜ ⎞ ⎠⎟ 2 =1550025s=430h33'45" 4. In case of water X= x 4Dt= 0.05m 4⋅1.97⋅10−9m2 s⋅t =560⋅t−12 From which, if erf(X)=0.5, X=0.48 and: twater= 560 X ⎛ ⎝⎜ ⎞ ⎠⎟ 2 = 560 0.48 ⎛ ⎝⎜ ⎞ ⎠⎟ 2 =1361111s=378h5'11"Vcapsule Microporous membrane permeance The presence of a microporous membrane, where diffusion of some molecules takes place only through the pores which are filled by the liquid that is present at both sides of the membrane, slightly modifies the diffusion equation however, without altering the overall concept. A “partition coefficient” H A can be defined to account for the mass flow rate reduction, which is allowed only through the membrane pores instead of through the whole membrane: 2 2 A membrane the of surface totalpo re s the of surface total Hr n Ar N p p = × = = Where: N number of pores r average radius of the pores A total membrane surface area n density of the pores in the membrane The diffusing molecules have not a point-like structure so, assuming their shape as spherical with radius R, they can cross the pores only if the centre of the spherical molecule is at a distance from the centre of the pore which is smaller than (r-R). The effective area of each pore is thus smaller than pr 2: p(r-R) 2 = pr 2(1-R/r) 2 In addition, the free diffusion coefficient D through the pores is reduced by the bouncing of the molecules on the pores walls which increases the diffusion distance. Fig. 1 Fig. 2 Diffusing molecule Pore of the membrane To account for this phenomenon it is necessary to introduce the “hindrance” coefficient e < 1. This coefficient takes into account the effects due to both the geometry of the pores and the bouncing effects. The expression of e, which is experimentally determined, is: e = (1-R/r) 2[1 – 2.1 (R/r) + 2.09 (R/r) 3 – 0.95 (R/r) 5] The course of e as a function of the ratio R/r is shown in Fig. 3. The solid line refers to the pure geometrical effects which is related to the relative dimensions of the molecule and the pore, (p(r-R) 2 = pr 2(1-R/r) 2), while the dotted curve refers to the global action exerted by both the geometrical effects and the bouncing effects. The experimental points refer to measurements performed on a porous membrane with 66 Ǻ effective radius of the pores. The diffusion coefficient D p through the pores of the microporous membrane can thus be written as D p= eD Fig. 3 The permeance P L of the microporous membrane, which refers to the capacity of the membrane to allow the diffusion of molecules, can thus be written as: D x H D xr n P A L e e p D = D = 2 P L is a function of the geometrical characteristics of the microporous membrane, of the relative dimensions of the diffusing molecules and of the pores, and of the free diffusion coefficient of the molecules in the solvent. Convective and diffusive mass flow through porous membranes If we apply a hydraulic pressure difference Dp at the two sides of a porous membrane, where mass flow is a function of the hindrance coefficient, partition coefficient, etc., a convective flow rate of the solvent, which is determined by this pressure difference, takes place through the pores in addition to the solute diffusive mass flow that has been calculated in the previous section. The solvent flow rate through each pore (Q pore ) can be written as: p x 8r Q 4 po ro D × D × × = µ p where: r is the pore radius, Dx is the pore length. If n is the pore density over the membrane, the convective flow of the solvent through the pores of the membrane can be written as: 4 convporopr jnQn pLp 8x p µ × =×=× ×D=×D ×D where L p is the so-called filtration coefficient per square unit of the membrane. Life Support Systems – A.A. 2017/2018 Membrane for Hemodialysis Maria Laura Costantino, Giustina Casagrande Pag. 1/3 2 – Sizing a Membrane for Hemodialysis A porous membrane for hemodialysis is 50 µm thick. The pores have a radius of 50 Å and the urea molecule (considered spherical) has a radius of 2.04 Å. Determine the partition coefficient HA (or coefficient of free area) and the permeance of the membrane PL, assuming the pores disposed in parallel rows, with 1.5 diameters distances between their centres both in longitudinal and transverse direction. The free diffusion coefficient is D = 12 · 10-6 cm2/s for urea and η=0,9 is the effective porosity (number of effective pores with respect to the total porosity). Determine: - the trend of urea concentration in the patient during dialysis; - the size of the membrane (area), knowing that: - during dialysis, the patient production rate of urea is G = 0.095 mMol/min; - the total body volume of fluids to be purified (to be considered constant for this exercise, assuming ultrafiltration=0) is V = 43.5 l; - the concentration of urea at the beginning of dialysis is 11.5 mMol / l; - the dialysis filter manufacturer states that the device clearance (urea removal capability) of the membrane is constant and equal to: != (!!!)!"!(!!!)!"#!!"=225 !"!"# Where Qb = 300 cc / min is the blood flow rate and c the urea concentration. The indices in and out respectively refer to the blood inlet and outlet of the dialysis filter. The urea concentration at the dialysis filter inlet can be assumed equal to the urea concentration at the outlet of the patient. The reference treatment is 4 hours long. Life Support Systems – A.A. 2017/2018 Membrane for Hemodialysis Maria Laura Costantino, Giustina Casagrande Pag. 2/3 Solution It is useful to report all the data in International System units: Membrane thickness: dx = 5·10-5 m Pores radius: r = 5·10-9 m Radius of the urea molecule: R = 2,04·10-10 m Diffusion coefficient of urea: D = 1,2·10-9 m2/s Membrane effectiveness: η = 0,9 Initial concentration of urea: C0 = 11,5·103 mMol/m3 Urea generation rate: G = 15,83·10-4 mMol/s Urea clearance: K = 3,75·10-6 m3/s Total volume of fluids into the body (average): V = 43,5·10-3 m3 Since the pores are displaced in parallel rows, with distances of 1.5 diameters, the number of pores per unit of membrane area can be determined considering each pore in the centre of a square whose side is L = 3r . Therefore, the superficial density of pores in the membrane is: ( ) ( ) 15 2 22 9 11 4, 444 10 m 3r 3510 m n − − == =⋅ ⋅⋅ The partition coefficient HA is the ratio between the total surface of the pores and the total surface of the membrane: H A = n ⋅ P o r e A r e a = n ⋅ π r 2 = 1 ( 3 r ) 2 ⋅ π r 2 = π 9 = 0 , 3 4 9 The passage of the urea molecules through the pores of the membrane has to be studied taking into account that the available surface of the pores is the one of the circle of radius r-R and that the path of the molecules through the pores is characterized by bumps against the walls of the pore. It is, therefore, necessary to calculate the hindrance factor to account for these effects: 235 112,12,09 0,95 0,841364 RRR R rrr r ε ⎛⎞ ⎛⎞ ⎛⎞ ⎛⎞=− −+−= ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎝⎠ ⎝⎠ It is then possible to calculate the membrane permeance: 2 96 A L 5 H 0, 349 m m P= εηD 0, 841364 0, 9 1, 2 10 6, 34 10 dx s s 510 m −− − =⋅⋅⋅⋅=⋅ ⋅ In order to evaluate the trend of urea concentration in the patient during dialysis, it is possible to write a variation equation as: v a r i a t i o n o f t h e q u a n t i t y o f u r e a ⎛⎝⎜⎜⎜⎜ ⎞⎠⎟⎟⎟⎟ p a t i e n t = f l o w r a t e o f t h e i n c o m i n g u r e a ⎛⎝⎜⎜⎜⎜ ⎞⎠⎟⎟⎟⎟ p a t i e n t - f l o w r a t e o f t h e o u t g o i n g u r e a ⎛⎝⎜⎜⎜⎜ ⎞⎠⎟⎟⎟⎟ p a t i e n t + u r e a g e n e r a t i o n r a t e ⎛⎝⎜⎜⎜ ⎞⎠⎟⎟⎟ p a t i e n t V d c ( t ) d t = Q b c ( ) i - Q b c ( ) o + G where the subscripts i and o respectively refer to the inlet and outlet from the patient point of view. From the dialysis filter point of view, inlet (in) and outlet (out) are reversed. Disregarding the ultrafiltration rate of the blood through the dialyzer, it is possible to consider ci = co = cin and cout, namely: Life Support Systems – A.A. 2017/2018 Membrane for Hemodialysis Maria Laura Costantino, Giustina Casagrande Pag. 3/3 Q b c ( ) i = Q b c ( ) out Q b c ( ) o = Q b c ( ) i n ⎫ ⎬ ⎪ ⎭ ⎪ Q b c ( ) i − Q b c ( ) o = Q b c ( ) out − Q b c ( ) i n = − K ⋅ c i n ( t ) It can be assumed cin = cu = c, considering the blood urea concentration at the dialyzer inlet coincident with that of the patient. The equation of variation can be then wrote as V dc ( t ) dt + K ⋅ c ( t ) =G The integral of the differential equation is: c ( t ) = GK + B e − K V t Imposing the boundary condition at the initial time: at t = 0, c0 = 11,5·103 mMol/m3, it is possible to calculate the value of the constant K: B = 1 1 , 0 7 7 8 ⋅ 1 0 3 m M o l m 3 Graphically: The membrane surface area can be determined writing the expression for the urea flow rate as a function of the membrane area and of the clearance. Q u r e a = J u r e a ⋅ A = K ⋅ c ( t ) The urea flux trough the membrane can be calculated from the membrane permeance, determined in the first part of the practical, remembering that the urea concentration is equal to zero in the dialysis fluid side of the filter. L L sangue fluido fluido Lsangue L J=P Δc=P (c (t)-c (t)) c (t)=0 t J =P c (t)=P c(t) urea urea ⋅ ∀ ⋅⋅ Imposing the equivalence of the equations, it is possible to determine the surface membrane area, which results to be not dependent from the variation of the urea concentration during time. P L ⋅ c ( t ) ⋅ A = K ⋅ c ( t ) A = K P L = 3 , 7 5 ⋅ 1 0 - 6 m 3 s 6 , 3 4 ⋅ 1 0 - 6 m s = 0 , 5 9 m 2 Life Support Systems Hemodialysis Treatment Maria Laura Costantino, Giustina Casagrande Pag. 1/8 3 – Hemodialysis Treatment Mr Pissanò suffers from renal failure with no residual renal functionality. He undergoes hemodialysis treatment three times a week: in Tab. 1 the patient and the treatment data are summarized. Tab. 1 – Patient and treatment data Body mass 75 kg Mean blood volume 7.5 l Mean hematocrit 43 % Urea production rate 150 g/week Treatment frequency 3 treat./week Urea Concentration required at the end of the session 0.4 g/l Ultrafiltration volume to extract 10 l/week In the operating manual of the hemodialyzer used in the treatment, the chart of the urea clearance as a function of the blood flow rate is reported (Fig.1). We assume the dialysis fluid flow rate as constant. The chart in Fig. 2 represents the ultrafiltration flow rate as a function of the trans-membrane pressure (TMP, the hydraulic pressure difference between blood and dialysate compartments). Q [ml/min] CL [ml/min] 0 50 100 150 200 250 300 350 400 450 0 200 400 600 800 10 0 0 12 0 0 14 0 0 16 0 0 B Fig. 1 – Characteristic curve of the hemodialyzer urea clearance Fig. 2 - Characteristic curve of the hemodialyzer ultrafiltration Life Support Systems Hemodialysis Treatment Maria Laura Costantino, Giustina Casagrande Pag. 2/8 Requests: 1. Calculate the mean urea concentration at the beginning of a dialysis session and the ultrafiltration volume to be drained during each dialysis session. 2. Based on the urea clearance, evaluate the mean duration of the treatment (TD), in order to reach the target urea concentration at the end of the session, that is specified in Tab. 1. Make the calculation for each of the following blood flow rates: 250, 350, 500, 1000, 1500 ml/min. 3. For each pair of blood flow rate-dialysis time (QB, TD) determined at the point 2, evaluate the TMP to apply, in order to drain the ultrafiltration volume calculated at the point 1. Assumptions • Model the patient as a reservoir containing a solution with a uniform urea concentration (single pool model – Fig.3). Assume this solution volume composed by the total body fluids (equal to about 57% of the body mass). • Disregard the urea patient production during the dialysis session. • Neglect the ultrafiltration flow rate when evaluating urea mass transfer. Dializzatore V, C (t) Q ,C(t)B Q ,C (t)B, r r CL QUF Fig. 3 – Single pool patient and hemodialysis circuit scheme. Life Support Systems Hemodialysis Treatment Maria Laura Costantino, Giustina Casagrande Pag. 3/8 Solution 1) Mean urea concentration at the beginning of the treatment and ultrafiltration volume Under the hypothesis of a uniform subdivision of the treatments over the week, with NT = 3 (number of treatments per week), the mass of urea stored between two consecutive sessions can be evaluated as: AU = P N U T = 150 g / we e k 3 t r e at m e nt s / we e k =50 g This mass can be considered as homogeneously dissolved in the fluid volume V, that represents the 57% of the patient body mass. Considering the fluid density equal to the one of the water: V kg kg l = ⋅ 0 57 75 1 . = 42.75 l=42,75*10-3m3 The mean urea concentration at the beginning of the treatment is given by the sum of urea concentration at the end of the previous treatment and the accumulated mass of urea during the inter-dialysis period in the volume V: CI = CF + AU / V =0,4 [g/l] +50[g]/42,75 [l]= 1.57 g/l=1.57 *103 g/m3 This computation is related to the worst case. In real conditions we should calculate the concentration related to the stored mass using the volume that also involves the ultrafiltration volume, since this is the volume gained by the patient between one treatment and the next one, as follows: CI = CF + AU / V =0,4 [g/l] +50[g]/46,08 [l]= 1.48 g/l=1.48 *103 g/m3 The ultrafiltration volume to be extracted per each treatment is: VUF = VUFtot/NT=10 [l/week]/3 [treat/week]=3.33 l/treat=3.33*10-3m3/treat 2) Mean duration of the treatment Assuming that the urea concentration C(t) in the patient is uniform (single-pool model), to obtain C(t) during the treatment we need to write a variation equation on the “patient system”: u r e a a c c u m u l a t i o n d u r i n g t h e t r e a t m e n t ⎛⎝⎜⎜⎜⎜⎜⎜⎜ ⎞⎠⎟⎟⎟⎟⎟⎟⎟ = u r e a f l o w r a t e ⎛⎝⎜⎜⎜ ⎞⎠⎟⎟⎟ i n l e t − u r e a f l o w r a t e ⎛⎝⎜⎜⎜ ⎞⎠⎟⎟⎟ o u t l e t + u r e a p r o d u c t i o n d u r i n g t h e t r e a t m e n t ⎛⎝⎜⎜⎜⎜⎜⎜⎜ ⎞⎠⎟⎟⎟⎟⎟⎟⎟ − r e s i d u a l r e n a l r e m o v a l ⎛⎝⎜⎜⎜ ⎞⎠⎟⎟⎟ Since the patient has no residual renal functionality and neglecting the urea production during the treatment, the last two terms can be put equal to zero. The hypothesis to neglect the ultrafiltration flow rate, brings to say that: QB,r = QB ; V = cost The volume here considered should be the one involving the total volume of body fluids plus the ultrafiltration volume accumulated between a treatment and the following one. The mass balance equation becomes: QB Cr(t) − QB C(t) = V dC t dt () QB(Cr(t)-C(t))V 1 = dC t dt () (1) Life Support Systems Hemodialysis Treatment Maria Laura Costantino, Giustina Casagrande Pag. 4/8 Under the same hypothesis, the clearance of the dialyzer can be written as: K = QB C t C t C t r () () () − (2) Matching the equations (1) and (2) in a system, we can obtain a linear 1st order differential equation with constant coefficients: !"(!)!"+!!!!=0 Solving this equation with the initial condition C(0)=CI=1,48g/l we obtain a negative exponential trend: !!=!!!!!!! At the end of the treatment we want an urea concentration of CF = 0.4 g/l, therefore for t = TD, it has to be C(TD) = CF , and the duration needed for a treatment is: !!=!!!"!!!! (3) Based on the chart in fig. 1 and equation (3) it is possible to calculate different durations for the treatment (Tab.3: a: C(0)=CI =1,48 g/l; b:C(0)=CI =1,57 g/l (worst case)) Tab. 3: treatment duration as a function of the blood flow rate, when a. b. QB [ml/min] K [ml/min] TD [min] TD [h : min] 250 200 301 05:01 350 250 241 04:01 500 300 201 03:21 1000 400 151 02:31 1500 450 134 02:14 3) Transmembrane pressure Assuming a constant ultrafiltration flow rate along the treatment, we can calculate it as: QUF=VUF /TD. From the chart in Fig. 2 it is possible to determine the necessary transmembrane pressure. Tab. 4: transmembrane pressure drop as a function of the blood flow rate QB [ml/min] 250 350 500 1000 1500 QUF [ml/min] 11,0 13,8 16,6 22,1 24,9 QUF [ml/h] 663 829 994 1326 1491 Δp [mmHg] 175 210 260 380 >500 QB [ml/min] K [ml/min] TD [min] TD [h : min] 250 200 315 05:15 350 250 252 04:12 500 300 210 03:30 1000 400 158 02:38 1500 450 140 02:20 Life Support Systems Hemodialysis Treatment Maria Laura Costantino, Giustina Casagrande Pag. 5/8 Additional observations: • Haemolysis determination in a single treatment Two different 15 G-diameter needles (outlet and inlet accesses) are used to connect the patient to the dialysis circuit. The geometrical characteristics of the needles are reported in the table: Geometrical characteristics of a 15 G diameter needle The internal diameter of the needles is much smaller than the dimensions of any other component of the circuit. For this reason an overall gross evaluation of the haemolysis induced by the dialysis circuit, allows to calculate haemolysis only in the needles as they are for sure the most haemolytic elements of the whole circuit. To determine the degree of haemolysis risk for the treatment, we have to evaluate for each blood flow rate, the shear stress intensity and the related time duration, at each time blood flows into the needles. These values have to be compared, in the τ−t plane, with the Tillmann limit curve. The wall shear stress in a cylindrical channel can be determined as: τ = 2 λ ρ Q b 2 π 2 D 4 (4), with λ = 64 Re f or l a m i na r f l ow λ = 0.316 Re 0. 25 f or t ur bul e nt f l ow ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ where D is the needle diameter and the Reynolds Number Re = 4 ρ Q b π µ D . The blood characteristics are assumed as: density ρ = 1060 kg/m3, and viscosity µ = 2.652 cP, calculated by the Bull’s equation, at T = 37°C and Ht = 43%. Assuming that the patient blood volume (Vb = 7.5 l) is processed by the machine n times during the treatment, the processed total volume is equal to: n ⋅ Vb = Qb ⋅ Td (5) Since in the circuit two identical 15G diameter needles are present, the blood crosses a needle 2n times, during each treatment. The transit time taken to cross each needle is: tp = π L D 2 4 Q b (6) External diameter 1.9 mm Internal diameter 1.5 mm Length 40 mm Life Support Systems Hemodialysis Treatment Maria Laura Costantino, Giustina Casagrande Pag. 6/8 The total transit time during the treatment is Tp = 2 ⋅ n ⋅ tp , and using the equations (5) and (6): Tp = π L D 2 T d 2 V b (7) In Tab.5 shear stress intensity and time durations (calculated by using equations (4) and (7)) are summarized, for each considered blood flow rate. Note that, at a blood flow rate of 500 ml/min, the Reynolds Number gives transition flow conditions in the needle. Therefore, the calculations are performed both for laminar and turbulent conditions (the real value of τ will be an average value between them). Tab. 5: Study of the haemolysis of the system Qb [ml/min] 250 350 500 1000 1500 Re 1414 1979 2827 5655 8482 λ (laminar) 0.04527 0.03234 0.02264 λ (turbulent) 0.04333 0.03644 0.03293 τ [Pa] (laminar) 33.3 46.7 66.7 τ [Pa] (turbulent) 127.7 429.5 873.2 Tp [s] 0.330 0.265 0.221 0.165 0.147 The pairs of the calculated τ and Tp, are reported in Fig.4 together with the Tillmann limit curve. Observing the position of these points, it is possible to point out that flow rates higher than 500 ml/min would induce unacceptable haemolysis. Fig. 4: Tilmann chart - Plot of the studied working conditions in the plane τ−t Life Support Systems Hemodialysis Treatment Maria Laura Costantino, Giustina Casagrande Pag. 7/8 Blood flow rate optimization The identification of the optimal blood flow rate to be set during a dialysis session, implies the following considerations. 1. Social recovery of the patient: it increases with the decrease of the treatment duration; as shown in Tab.2, the duration decreases with the increase of the blood flow rate. However, at very high flow rates (high-flux dialysis: QB = 1000 ÷ 1500 ml/min), the decrease becomes less and less significant, because the clearance has a non-linear trend as a function of QB. 2. Cardiovascular impairments: the working point of the heart is very susceptible to the blood flow rate through an external circuit. As the blood flow rate increases, the heart work increases more and more, even if for a shorter duration. Therefore, the evaluation of the cardiac behaviour of the patient cannot be avoided: it’s quite common for the dialysis patients to develop cardiovascular dialysis related cardiovascular diseases. 3. Haemolysis: the most haemolytic component of the extracorporeal dialysis circuit is the needle. Haemolysis increases with QB. Moreover, an increase in QB would make also the roller pump more haemolytic, due to the fact that during dialysis treatments the pump is set as totally volumetric. The haemolysis evaluation in the needles shows haemolysis to become harmful already at QB = 500 ml/min. 4. Technological considerations: with the increase of the blood flow rate, that would imply a reduction in the duration of the treatment, also the TMP has to increase (Tab. 3) to allow a proper ultrafiltration. In these conditions blood proteins undergo a polarization phenomenon which makes them to adhere to the membrane, increasing the local oncotic pressure. For this reason a further QB [ml/min] TD [min] Life Support Systems Hemodialysis Treatment Maria Laura Costantino, Giustina Casagrande Pag. 8/8 higher TMP would be required. The technological problems to consider are related to the mechanical resistance of the membranes and of the potting. Conclusions: In conclusion, with the exception of Factor 1, all the other factors suggest the choice of low blood flow rates. Specifically Factor 3 has the highest weight in the choice of the QB with a maximum acceptable value of 350 ÷ 400 ml/min. QB [ml/min] Δp [mmHg] Life Support Systems Membrane oxygenator Maria Laura Costantino, Giustina Casagrande Pag. 1/8 4- Flat membrane oxygenator Design criteria and evaluation of blood-gas exchange Breathing, also called ventilation, is the process that moves air in and out of the lungs. Breathing is part of physio-logical respiration required to sustain life, the body cells in fact require oxygen to release energy via cellular respi-ration and produce carbon dioxide. Venous blood, interacting with a gas, changes its chemical-physical composition and becomes arterial. This gas exchange process occurs by passive diffusion between the blood in the lung capillaries and the gas (air at atmos-pheric pressure) in the pulmonary alveoli. If, for some reasons, the lungs cannot allow blood oxygenation (for example, during operations under extracorpo-real circulation), the problem to replace the oxygenating function of the lung with suitable devices arises. The oxy-genators are the devices that are used to this purpose. The use of oxygenating devices can guarantee survival not only during surgery under extracorporeal circulation, but also during temporary respiratory assistance, either total or partial, of patients who suffer from severe dysfunc-tions of the pulmonary system (e.g. due to the inhalation of toxic substances or lung disease or consequent to a chest trauma, etc. In the blood oxygenators the respiratory gases O2 and CO2 are exchanged by diffusion and convection; they move from blood through the membranes towards the pulmonary alveoli and vice versa, respectively. In the lungs the oxygenating gas is constituted by air, in the artificial membrane oxygenators, however, the gas is a mixture of air and oxygen. The amount of oxygen may vary from 50% up to 100% depending on the gas transfer to be ob-tained. The design criteria for an oxygenating device can be appropriately defined remembering that the laws governing the exchange of respiratory gases may be equally applied to a natural or an artificial system. Appropriate coeffi-cients have to be used to take into account both the type of membrane, and the peculiar fluid dynamics of the exchanger. The equations of variations allow both to determine the efficiency of the device, and to determine how the system should or could be designed to optimize its performances. The resolution of the equations requires the knowledge of the geometric dimensions, the exchanger geometry, the flow regime, the artificial membrane characteristics, and of the boundary conditions for the involved gas species. In a mass exchanger it will also be necessary to study the trend of the mass transfer also as a function of the mu-tual direction of motion of the fluids (co-current, counter-current or cross current). Design of the device You have to design a membrane oxygenator, containing a solid (not porous) silicone membrane. The device has to be used during Extra-Corporeal Membrane Oxygenation (ECMO) procedures. Fick law explains why, to increase the gas exchange, the diffusion distance has to be limited. At the same time the devices assembling requires the presence of an appropriate thickness of sealer. It is then possible to consider the distance of diffusion to the blood side equal to the amplitude of the interspace within which blood flows (s1 = 0.2 mm). The thickness of the membrane is equal to 0.001 mm (0.18 with silicone + Dacron, 0.06 only silicone, those in polysulfone are thick 0.0005 mm). The inlet section is 12 cm large. Design the device so as to obtain laminar flow in the blood side, to minimize haemolytic effects. From the gas side the flow regime does not affect the exchange, since the diffusion coefficients in the gaseous phase are some or-ders of magnitude higher than those in the liquid phase. Assume a very high gas flow rate, so as that the mass transfer that takes place through the membrane will not affect oxygen concentration in the gas side of the device. Assume that the blood enters the device in venous physiological standard conditions (pO2 = 40 mmHg, pCO2 = 46 mmHg) and leaves it fully oxygenated (pO2 = 100 mmHg, pCO2 = 40 mmHg). Also assume counter-current direction of the blood and gas flow inside the device , then it will be possible to solve the problem by using a bi-dimensional approach, neglecting the transverse components of the mass exchange, into the equations. Define in first approximation, the size of the exchange surface area of the device, both when in the gas side flows pure oxygen, and with percentage of pure oxygen FiO2 = 70% and 40%. Life Support Systems Membrane oxygenator Maria Laura Costantino, Giustina Casagrande Pag. 2/8 Specifically, optimize the length of the oxygenator given a inlet blood flow rate equal to 4 l/min and by imposing the achievement of the arterial conditions at the outlet. Please consider that the reduction of the length of the ox-ygenator implies at least two advantages in terms of biocompatibility of the system: the reduction of the foreign surfaces in contact with blood and the reduction of the head losses, that can induce haemolysis. y"z"x"Blood"Flow"Gas"Flow"Gas"Flow"Con0nuum"membrane" Life Support Systems Membrane oxygenator Maria Laura Costantino, Giustina Casagrande Pag. 3/8 Solution To deal with a similar problem it is first of all necessary to properly formulate it, choosing the most appropriate approach. The step after is the formulation of the equations of variations and the identification of the necessary features, so as to carry out any necessary experimental tests if some of the parameters are not known. Final stage is therefore the resolution itself. In the most general case for oxygen and carbon dioxide it is possible to wrote: where the terms on the left were null under the hypothesis of device working under steady conditions. What re-mains to consider were the diffusive and convective terms. Applying to the studied problem the transport and exchange equations for the oxygen and the carbon dioxide as above reported, it is possible to obtain different formalisations depending on the mass exchanger configuration. In particular, if the reciprocal motion of the blood and the gas is co-current or counter-current, a bi-dimensional ap-proach is sufficient to study the problem, neglecting the transversal components of the transport. When instead a cross-currents mass exchanger is studied, the three-dimensional approach becomes necessary and in the transport equations the diffusive terms in all the direction must be considered. The considered mass exchanger is counter-current and is pointed out that the gas flow rate is very high: it is then possible to adopt a bi-dimensional approach. Wishing to analytically solve the problem, it is to remember that it is not possible to find an analytical first approx-imation approach allowing taking into account the real phenomena that take place in the blood, already since the release of oxygen bound by the red blood cells. Analytical approach tends to simplify the problem taking account the dissolved oxygen in the plasma, which is transferred to the tissues surrounding the capillaries. The calculated concentration of O2, however, does not disregard the presence of bounded oxygen. A first approach to the problem, certainly too simplistic and not realistic, although of simple application could in-clude the study of the gas-membrane-blood system in analogy with the transmission of thermal power between solids. Basically, there would be a model of only diffusive transport of oxygen, both in the membrane, where the transport of O2 is actually diffusive, both in the blood where instead is the convective transport to prevail. As al-ready said a similar modelling approach leads to realistic results only to study the mass exchange into the mem-brane. Remembering that the adopted schematization also allows to study the single elementary exchange unit, as repre-sentative of the process, it is possible at this point to separately study the trend of the oxygen concentration into the membrane and in the blood side of the oxygenator. ( ) ( ) ( ) ∂ ∂ t c Ns c v c Ns O O O O O O 2 2 2 2 2 2 + = ∇ ⋅ ∇ − ⋅ ∇ + D (accumulation) = (diffusion) + (convection) ( ) ( ) ( ) ∂ ∂ t c c c c c v c c c CO HCO HbCO CO CO HCO HCO CO HCO HbCO 2 3 2 2 2 3 3 2 3 2 + + = ∇ ⋅ ∇ + ∇ − ⋅ ∇ + + − − − − D D Life Support Systems Membrane oxygenator Maria Laura Costantino, Giustina Casagrande Pag. 4/8 Trend of oxygen concentration into the blood Available data that refers to the blood: • Haematocrit: Ht = 35% The value take into account the dilution of the blood typical of extracorporeal cir-culation treatments. This parameter usually ranges between 30 and 40% • Temperature: T = 37C, since we are operating in normo-thermic conditions. Sometime, it could happen to work in slight hypothermia, in such a case the different saturation has to be taken into account. • Diffusivity of the oxygen in the blood: D b = 1 , 7 5 ⋅ 1 0 − 5 c m 2 s , • Solubility of oxygen in the blood: α O 2 b = 3 ⋅ 10 − 5 m l O 2 m l b m m H g , • The ability of O2 to bind the haemoglobin: C A P = 0 , 2 m l O 2 m l b . • Partial pressure of oxygen in arterial blood: pO2a = 100 mmHg • Partial pressure of oxygen in venous blood: pO2v = 40 mmHg • Blood flow rate to be treated Qs=4 l/min=66,67ml/s=6,67*10-5 m3/s Hypothesis: • Steady state conditions, • Oxygen concentration homogeneous in the y direction (perpendicular to the motion) in the blood, • Equal oxygen concentration at the interface among the membrane and the blood interface. This means to con-sider all the oxygen contained in the blood, also the linked part, as determinant of the diffusion phenomena in the membrane. First of all, the laminarity of the flow inside the layer has to be verified. Remembering the definition of the Reynold’s number, the values of dynamic viscosity, density and blood speed were needed. From the literature, in the considered working conditions, it is known that: μ is around 3cP. To be more precise it is possible to take into account the haematocrit with the equation μb=μp (1+2,5 Ht)=2,42Ÿ10-3 PaŸs, calculated with μp=1,8 μH2O, among 20 and 37°C. The problem has been solved using μ=3cP=3Ÿ10-3 PaŸs. ρ b=(1-Ht)Ÿ ρ p+HtŸ ρRBC=1054,25 kg/m3 with ρ p=1035 kg/m3 e ρ RBC=1090 kg/m3 To evaluate the blood speed, the dimension of the input section of this oxygenator has to be calculated: A=s1Ÿf Where f is the width of the inlet. Using the international system units: s1=0,2mm=2Ÿ10-4m f=12cm=0,12m Where A=0,24Ÿ10-5m2 The speed is then vb=Qb/A=2,78m/s. The Reynolds number, calculated using the hydraulic diameter, is 390, the motion regimen is therefore laminar. The general equation for the oxygen transport into the blood is obtained by formalizing a mass balance of the oxy-gen in an established control volume. In Cartesian coordinates, the equation of continuity, combined with Fick's law appears to be as: ∂ C ∂ t = D b _ O 2 ∂ 2 C 0 2 ∂ x 2 + ∂ 2 C 0 2 ∂ y 2 + ∂ 2 C 0 2 ∂ z 2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ − v b ∂ C 0 2 ∂ x ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ Where D b _ 0 2 is the diffusion coefficient for the oxygen into the blood, vb is the local velocity vector for the blood. In particular, the first three terms on the right of the equal sign are the diffusive contribution in the three main directions and the fourth term represents the convective contribution due to the motion of the blood in the z direc-tion. It is in this phase possible to neglect the reaction term; the amount of oxygen bound to haemoglobin will be Life Support Systems Membrane oxygenator Maria Laura Costantino, Giustina Casagrande Pag. 5/8 instead considered when determining the value of the oxygen concentration into the blood. The total concentration of oxygen in the blood can be calculated as the sum of the concentrations of the O2 bound to the haemoglobin (oxy-haemoglobin), carried by red blood cells (cb), and the concentration of O2 dissolved in the plasma. Cb= cb+ cd = CAP·sat+αO2_b·pO2_b For the venous blood that enters the oxygenator cb_0=0,15105mlO2/mlb, while for the arterial blood we have cb_L=0,19285mlO2/mls. It may also be noticed that in this case the hypothesis of steady state result verified and then: 0= ∂ ∂ t C Since we are considering a counter-current flat membrane oxygenator, in the y direction, there is neither convec-tion nor diffusion, so it is possible to write: 2 2 0 2 C 0 y ∂ = ∂ Transport by convection in the blood side and oxygen inlet from the membrane ==> realistic Considering, in steady state conditions, a volume element fixed in the space (Euler approach) of infinitesimal length dx and writing a balance between the inlet flow rates of oxygen and the outlet one, under the hypothesis that there is no oxygen production or consumption, we have: 0 = v b ⋅ f ⋅ s 1 2 ⋅ c b ( x ) − c b ( x + d x ) ( ) + J z ( x ) ⋅ f ⋅ d x Where the first term at the second member express the convective transport (axial) and the second one the diffu-sive transport of oxygen in the z direction. It is possible to explicit it as: v b ⋅ f ⋅ s 1 2 ⋅ ∂ c b ∂ x = J z ( x ) ⋅ f v b 2 ⋅ ∂ c O 2 _ b ∂ x = J z ( x ) s 1 Where Jz (x) is the oxygen flux trough the membrane into the blood. Under the hypothesis of constant flow, however realistic, a linear trend could be obtained, but this would not cor-respond with what actually happens on the two sides of the membrane. The flow in the z direction has to be ex-pressed as a function of x, then studying what indeed takes place in the membrane. Trend of the oxygen concentration in the membrane: diffusion ==> realistic Considering a oxygen flow variable as a function of the x coordinate, the driving force of the exchange decreases as x increases, while the diffusion coefficient in the membrane remains unchanged. The membrane is characterized by: D O 2 m = 1 , 7 ⋅ 1 0 − 7 c m 2 s , α O 2 m = 6 , 5 8 ⋅ 1 0 − 4 m l O 2 m l m m m H g . When considering the same concentrations of oxygen at the blood-membrane interface, we will have: c O 2 _ b 0 , s 2 ( ) = c O 2 _ b ( 0 ) = 0 , 15105 m l O 2 m l m c O 2 _ b L , s 2 ( ) = c O 2 _ b ( L ) = 0 , 1 9 2 8 5 m l O 2 m l m In this case it is assumed that all the oxygen available in the blood is also available to the diffusion in the mem-brane, including the part bounded to haemoglobin. Even more realistic hypothesis might be that the oxygen partial pressures will be equal at the blood membrane interface (pO2s = pO2m). The equation that allows determining the O2 flux trough the membrane, along the z-axis, is: J z ( x ) = − D O 2 _ m ⋅ ∂ c O 2 _ m z ( ) ∂ z = D O 2 _ m z m ⋅ ( c O 2 _ g − c O 2 _ b ( x ) ) Life Support Systems Membrane oxygenator Maria Laura Costantino, Giustina Casagrande Pag. 6/8 Integrating what written for the blood and the membrane it is possible to obtain: v b ⋅ s 1 2 ⋅ ∂ c O 2 _ b ( x ) ∂ x = D O 2 _ m z O 2 _ m ⋅ ( c O 2 _ g − c O 2 _ b ( x ) ) ∂ c O 2 _ b ( x ) ∂ x + D O 2 _ m ⋅ 2 z O 2 _ m ⋅ s 1 ⋅ v b ⋅ c O 2 _ b ( x ) = D O 2 _ m ⋅ 2 z O 2 _ m ⋅ s 1 ⋅ v b c O 2 _ g Defining: D 1 = D O 2 _ m ⋅ 2 z O 2 _ m ⋅ s 1 ⋅ v b it is possible to write ∂ c O 2 _ b ( x ) ∂ x + D 1 ⋅ c O 2 _ b ( x ) = D 1 ⋅ c O 2 _ g The general solution of a not homogeneous linear differential equation of the first order could be written: c O 2 _ b ( x ) = e − D 1 x ⋅ K + D 1 ⋅ c O 2 _ g ( ) ∫ ⋅ e D 1 x d x ( ) c O 2 _ b ( x ) = K ⋅ e − D 1 x + c O 2 _ g The value of K can be determined imposing the boundary condition for which cO2_x(0)=cb_0=0,13105mlO2/mls, from which K=cb_0-cO2_g. A variation in FiO2 implies changes in the gas oxygen concentration. Natural air includes about 20% oxygen. Oxy-gen-enriched air has a higher FiO2 up to 100% oxygen. In the considered cases oxygen concentration into the gas is: pO2_g(100%oxygen)=760mmHg pO2_g(FiO2=70%)=70*760 mmHg+(1-70)*760*0,2 mmHg=577,6 mmHg pO2_g(FiO2=40%)=40*760 mmHg+(1-40)*760*0,2 mmHg=395,2 mmHg the corresponding concentrations, determined as pO2*aO2 were: cO2_g(100%oxygen)=0,5 mlO2/mlb cO2_g(FiO2=70%)=0,38 mlO2/mlb cO2_g(FiO2=40%)=0,26 mlO2/mlb Along the oxy the oxygenation in the different considered situations is: 1,40E-01 1,50E-01 1,60E-01 1,70E-01 1,80E-01 1,90E-01 2,00E-01 0 100 200 300 400 500 600 700 800 900 Blodd O2 concentra.on [mlO 2/mlb] Oxy-Membrane Lenght [cm] 40 70 100 Fi0 2 Arterial Blood Venous Blood Figure 1: Variation of the Blood O2 concentration along the oxygenator, when the FiO2 is varied. Life Support Systems Membrane oxygenator Maria Laura Costantino, Giustina Casagrande Pag. 7/8 To analytically determine the length of the oxygenator (inferable also from the figure 1), the output concentration has to be imposed equal to the desired one, allowing finding the value of L for which the equation is satisfied. c O 2 _ b ( L ) = c O 2 _ b L = c O 2 _ b _ 0 − c O 2 _ g ( ) ⋅ e − D 1 x + c O 2 _ g L = − z O 2 _ m ⋅ s 1 ⋅ v b D O 2 _ m ⋅ 2 ⋅ l n c O 2 _ b _ L − c O 2 _ g c O 2 _ b _ 0 − c O 2 _ g ⎛⎝⎜ ⎞⎠⎟ On the considered cases: L100%=2,09 m L70%=3,3 m L40%=7,92 m The resulting lengths are very high, and de facto a similar oxygenator would not be achievable. To solve this problem, using the same silicone membrane, it is possible to act on the blood velocity, reducing it. To this extent it is possible to construct a device with a number n of overlapping cavities, so that the flow rate in each of them is equal to Qi=Qb/n. The velocity in each cavity (vi) is vi=vb/n Considering 30 layers, the length of the device results to be: L100%=0,07 m = 7 cm L70%=0,11 m = 11 cm L40%=0,26 m = 26 cm If a more realistic membrane characterized by the same diffusive properties, but thick 0,06 mm is considered, the results change as follows. When a single layer is hypothesis: L100%=125 m L70%=198 m L40%=475 m Using 30 layers, the length of the device (membrane 0,06 mm thick) results to be: L100%=4,17 m L70%=6,59 m L40%=15,8 m The device reaches manageable dimensions only considering at least 1000 layers, reaching: L100%=0,13 m =13 cm L70%=0,20 m=20 cm L40%=0,48 m=48 cm In this last situation the device will be 26cm deep. Life Support Systems Membrane oxygenator Maria Laura Costantino, Giustina Casagrande Pag. 8/8 Additional considerations: Evaluation of the device performances Once defined the dimensions of the device, choosing one of the working conditions (i.e. FiO2 =40%), it is possible to determine the variations in the device performances when the inlet blood flow rate is varied. It is therefore pos-sible to drawn diagrams representing the specific /total exchange vs blood flow rate. The total exchange diagram, as a function of the blood flow rate, is: VO [mlO /min] 2 2 Q [l/min]s . 240245250255260265270275 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 0 1 2 3 4 5 6 7 8 mlO 2/s Blood flow rate [l/min] Total exchange On the left the ideal diagram that would be obtained with experimental test on the device. On the righ the diagram obtained analytically solving the problem, with the current data, for the considered membrane. The diagram shows the tendency of the exchange in reaching a plateau. The initial linear trend is nearly not appreciable in the analytical solution, due to the applied simplificative hypothesis. Looking to the specific exchange vs blood flow rate diagram: vO [mlO /100 ml] 2 2 Q [l/min]s . 5 Q* 240245250255260265270275280 0,00E+00 2,00E-02 4,00E-02 6,00E-02 8,00E-02 1,00E-01 0 1 2 3 4 5 6 7 8 mlO 2/s Blood flow rate [l/min] Specific exchange Q s* On the left the ideal diagram that would be obtained with experimental test on the device. On the right the diagram obtained analytically solving the problem, with the current data, for the considered membrane. It is possible to observe that the simplified approach, neglecting fluid dynamic phenomena, as the development of the boundary layer, does not allow correctly describing all the curve, but only the second part. Life Support Systems – A. A. 2017/2018 Pumps for biomedical applications Maria Laura Costantino, Giustina Casagrande Pag. 1/13 Pompa centrifuga Tubo arterioso Blocco ossigenatore - scambiatore Tubo venoso Cannula venosa Cannula arteriosa Riserva venosa 5a – How to identify the external characteristic of a pump Choice of a hydraulic pump for biomedical applications The choice of the pump to be used is determined by several factors as: 1) The desired hydraulic head; 2) The needed flow rate; 3) The properties of the fluid to be pumped. For specific fluids, the pumps have to be built with proper materials; 4) The available energy sources (useful to choose the proper motor to be matched with the pump). Every commercial pump has to be provided by the builder with the operating diagram together with the efficiency diagram, in order to allow a correct sizing of the system. Indeed, in the choice of the pump, we have to consider not only that at a preset flow rate the hydraulic head of the pump has to overcome the pressure drops along the circuit, but even that the pump has to work with a suitable efficiency. In the following table the qualitative characteristics of different kinds of pumps are summarized. Purpose Machine to be chosen Low hydraulic head (60-100 m); medium-low flow rate Centrifugal pump High hydraulic head (over 100 m); low flow rate Volumetric pump Low hydraulic head (up to 10 m); huge flow rates Axial pump When on equal conditions of flow rate and reliability of the system, I want to increase the hydraulic head Several pumps in series When on equal conditions of hydraulic head, I want to increase the flow rate and the reliability of the system Several pumps in parallel How to trace the characteristic curve of the circuit and identify the operating point Choice of the pump. We have to properly size the pump to be used in a cardiorespiratory assistance circuit, on the blood side. The pump has to drive the blood when its temperature ranges between 30 and 37°C. The blood is withdrawn from a venous reservoir, placed 50 cm below the operating table. The configuration of the circuit is shown in Fig.1 Fig.1 Configuration of the cardiorespiratory assistance circuit. Range of flow rates. The patients undergoing surgery in the hospital that is buying the pump can have age ranging from 20 to 90 years, with a body surface area (BSA) included in a range between 1.10 and 2.00 m2 and cardiac index (CI) approximatively of 2.4 l/(min*m2). Knowing that the flow rate to be used is set for each patient as equal to: Q=BSA*CI; determine the range of working flow rates for the pump. Identification of the external characteristic of the pump. This curve will be a function of the flow rate that goes through the pump and it will be given by the sum of the Life Support Systems – A. A. 2017/2018 Pumps for biomedical applications Maria Laura Costantino, Giustina Casagrande Pag. 2/13 geodetic quote (ρg (zA − zR)) and of distributed and concentrated pressure drops (ΔPd+ΣΔPc) along the inflow and outflow lines. Yext = ρg (zA − zR)+ MAP + ΔPd+ΣΔPc Where MAP is the mean arterial pressure. The distributed pressure drops can be evaluated as: ∆!= ! !"!!!!!!! [Pa] The term λ can be evaluated as: in the hypothesis of turbulent flow (worst case). The length of the tubes can be considered equal to 2.5 m (reference mean value). The diameter of the tubes is standardized and equal to 9.5 mm (3/8"). Blood properties are: viscosity µs = 2.57 cP and density ρs = 1049.5 kg/m3. Further pressure drops are related to the presence of the oxygenator-heat exchanger block and the arterial cannula and can be avaluated through the diagram in Figure 2 and table 1. Take also into account that at the outlet of the arterial cannula, a MAP of 70 mmHg is required. Δp = a Q2 + b Q [mmHg] (1 Fr = 0.33 mm) Cardiac Output [l/min] ∅ cannula [Fr] a b < 2.00 13.5 12.3 −3.95 2.00 - 4.00 15.5 5.75 −0.08 >4.00 19.5 3.11 −5.93 Fig.2 Pressure drops as a function of the flow rate for the oxygenator-heat exchanger block (Dideco® D703 compactflow). Data provided by the company Tab.1 Pressure drops in the arterial cannula, as a function of the flow rate After the evaluation of the total pressure drops along the circuit, represent in a chart the system characteristic curve (pump external characteristic curve) in the range of the flow rates provided for the pump. Knowing that the pump internal characteristic curve is the one shown in Fig.3, determine the operating point of the pump, once this is coupled with a ci