Biorheology, 1976, Vol. 13, pp. 201-210. Pergamon Press. Printed in Great Britain
A CONSTITUTIVE EQUATION FOR WHOLE HUMAN BLOOD FREDERICK J. WALBURN* and DANIEL J. SCHNECKt Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A.
(Received 19 April 1975; in revised/orm 27 September 1975) Abstract-A constitutive equation for whole human blood was developed using a power law functional form containing two parameters, a consistency index and a non-Newtonian index. These two parameters were determined by a multiple regression technique performed on viscometric data obtained from anticoagulated blood samples of known hematocrit and chemical composition. An initial constitutive model depending only on shear rate was found to be lacking any substantial degree of significance. When hematocrit was included as an indepe~dent variable, there was a considerable increase in the fit of experimental data with the analytic model. This fit became even better when fibrinogen and globulin concentrations were further taken into account. The best constitutive model ultimately included a dependence of shear stress on shear rate hematocrit and the sum of fibrinogen and globulin concentrations. Plasma lipids and the protein albumin wer~ found to contribute little to the rheologic behavior of blood. INTRODUCTION
Through the ages many noted scholars have studied the flow of blood [1], among them Hippocrates (460--375 B.C.), Aristotle (384-322 B.C.), Leonardo da Vinci (1452-1519), Sir WilWilliam Harvey (1587-1657), Reverend Steven Hales (1733), and Jean L. M. Poiseuille (1828). This interest has continued, and today many investigators are studying the complex role of blood in the cardiovascular system of man and animals. Yet, even with the modern sophisticated research methods available today, there is still much that is not known about this extremely complicated fluid. In order to obtain more information concerning blood flow, there are two approaches which could be utilized. The first is experimental, where either an in vitro model of the in vivo flow situation is constructed, and relevant information is obtained from the model, or where the information is obtained directly from an animal. The second approach is to develop a mathematical model of the in vivo flow situation. Such a model for an incompressible fluid consists of an equation of continuity (conservation of mass), three generalized momentum equations, and a constitutive equation. When the constitutive equation is introduced into the generalized momentum equations, they become specific for the particular fluid involved in the flow situation. With appropriate simplifications and boundary conditions, the equation of continuity and the specific momentum equations can be solved to yield information about the flow: local velocities, shear rates, shear stresses, etc. In accordance with the second approach, this paper offers an empirical constitutive equation for use in describing the behavior of blood. The dependence of the viscosity of blood upon the shear rate has been thoroughly investigated, and it is universally agreed that the viscosity of whole blood decreases as the rate of shear increases. A fluid which exhibits this type of behavior is called pseudoplastic [2]. In addition to its pseudoplastic behavior, whole blood also exhibits a yield stress-that is, a certain minimum force is necessary in order to initiate flow [3]. A fluid exhibiting such a yield stress is called a Bingham Plastic [2]. Several functional forms for constitutive equations have been used to model the non-Newtonian behavior of blood [3,4]. An equation which attempts to describe both Bingham Plastic and pseudoplastic behavior was proposed in 1959 by Casson [5] who at the time was working with pigment-oil suspensions of printing ink. The equation has the form: T 1/2
where
=
kcit 1/2 + T~/2,
shear stress, it == shear rate, kc == Casson viscosity, and Ty == yield stress. T==
*Research Assistant, Department of Engineering Science and Mechanics. t Assistant Professor, Department of Engineering Science and Mechanics. 201
(1)
F. J.
202
WALBURN
and D. J.
SCHNECK
Merrill et al. [6], and other investigators [3,7-10] have used this equation with a reasonable degree of success to model the behavior of whole blood. However, it should be pointed out that a yield stress for whole human blood has been observed and measured only under static loading conditions. Investigators have attributed this behavior, among other things, to rouleaux formation. There is some doubt, and probably rightly so, as to whether the yield stress is at all manifest in a dynamic situation. That is, even though fluid velocities and velocity gradients pass routinely through zero in an oscillating flow situation, there is probably not enough time for static phenomena to appear. Thus, the constitutive form chosen for this study was a general power law: T
where
=
ky",
(2)
shear stress (dyn/cm 2), y shear rate (sec-I), k consistency index (P-sec"-\ n =non-Newtonian index, and the validity of this expression was examined for shear rates in excess of 23.28 reciprocal seconds. The parameters k and n are assumed to be constant for a given hematocrit level and a given chemical composition. In addition to the comments made earlier, it may further be noted that the yield stress for blood is extremely small, virtually constant, and is due mainly to interactions between the protein fibrinogen and the erythrocytes. In many cases the addition of an anticoagulant either reduces these interactions or eliminates them completely [11]. Thus, the power law equation is considered valid even though whole blood does exhibit a slight yield stress under static conditions. Note that the power law equation can easily be modified to allow for the presence of a yield stress, i.e. it can be written as follows: T=
= =
T
where
Ty
=
k'Y" + Ty
(3)
is the yield stress. METHODS
Materials Two hundred human blood samples of 10 ml each were obtained from admissions processed through the hematology laboratory of the Montgomery County Hospital, Montgomery County, VA. These blood samples, anticoagulated with EDT A, were carefully mixed by holding a test tube in one hand and gently rotating the wrist in a back and forth motion. Two 1.2 ml aliquots were then drawn from each 10 ml test tube and viscosity measurements were obtained at 37°C on a Wells-Brookfield cone and plate viscometer with accompanying temperature control bath. Each aliquot was examined at shear rates of 23.28, 46.56, 116.40, and 232.80 reciprocal seconds and the results at each shear rate were averaged for both aliquots. This reduced the chance for an error due to incomplete mixing, inasmuch as the pairs of viscosity values were compared and if large discrepancies were noted, more viscosity measurements were made. The chemical analysis (S.M.A. 10) of each blood sample was obtained from the same hospital and total lipid determinations were performed according to the method of Zollner and Kirsch[12].
Variables The dependence of the viscosity of blood upon shear rate and hematocrit is well documented and was included in the constitutive model. The chemical variables investigated included the total lipids, albumin, and total protein minus albumin (TPMA). TPMA is composed of fibrinogen and the globulins. These chemical variables were chosen because they are composed of long chain asymmetric molecules which exist in blood in large numbers and interact more than do symmetric particles. Thus, it is probable that they contribute most to the rheologic properties of the fluid. In the analysis of the viscometric data, albumin, TPMA, total lipid, hematocrit, and the shear rate were considered to be independent variables and the apparent viscosity was considered to be the dependent variable. A multiple regression computer procedure was then used to determine the variables of greatest significance. From the point of view of this investigation, it was not of any consequence whether the blood sample was drawn from a patient that had a pathologic condition
Aconstitutive equation for whole human blood
203
or from one who was perfectly normal. Since the intent of the study was to examine the mechanical properties of the fluid solely as a function of its actual physical and chemical composition, it was not immediately relevant to know what factors might be responsible for these compositions.
Statistical analysis Equation (2) was the basic functional form that was selected for developing the constitutive equation. Since the viscosity of blood varies for different shear rates, it is convenient to introduce a new quantity called apparent viscosity (/J-a), which is defined as the viscosity of the fluid at a given shear rate:
(4) When equation (4) is substituted into equation (2), an expression relating the apparent viscosity to the shear rate results: (5)
Equation (5) is nonlinear, making it extremely complicated from the point of view of least squares regression analysis. Therefore, it is more convenient to linearize the equation by taking logarithms to the base e: loge /J-a = log. k + (n -1) log. 1'.
(6)
Equation (6) is of the form: Z=mX+b
where Z = dependent variable = log. /J-a, X = independent variable = loge it, m = slope of the line = (n - 1), and b = the Z intercept = loge k. The parameters (n - 1), loge k, and loge /La were examined in the regression analysis as functions of loge it, albumin, TPMA, total lipid, hematocrit, their respective squares, inverses, the squares of their inverses, and all interaction terms (albumin x TPMA, albumin x loge 1', albumin x l/hematocrit, and so on). Loge /La was the basic dependent variable. A maximum R -square improvement regression procedure [13] was used to analyze the data. This procedure finds, first, the "best" one variable model that produces the highest R -square statistic (see Appendix). That is, of the independent variables chosen for analysis, the computer program selects one and uses the linear least squares method to determine an equation of the form:
(7) It then computes an associated R -square statistic for this equation. Going through this procedure separately for each independent variable, the computer finally chooses the model with the largest R -square statistic as the best one variable model. To determine the best two variable model, the statistical program now adds one of the remaining variables to the best one variable model. The linear least squares regression analysis is used to determine an equation of the form: (8)
and the associated R -square statistic for this equation is computed. A different variable is then added to the best one variable model, and again the R -square statistic is computed. This procedure is followed for all remaining variables to determine which one, when added to XI, will produce the greatest increase in R -square for equation (8).
204
F. J. WALBURN and D. J. SCHNECK
At this point, it is realized that the XI variable which produced the best one variable model may not necessarily be one of the two which produces the best two variable model. Thus, the program goes back to check each of the variables, both XI and X 2 in the model it has just obtained. That is, it determines whether there will be a further increase in the R -square value if either of the presently included variables is replaced by one which was excluded. After all possible comparisons are made, the combination of two variables which produces the greatest increase in R -square over the previous two variable model is isolated. The procedure continues cycling until no further increase in R -square is found. The result is the best two variable model. Essentially, this procedure determines whether or not the variable in the best one variable model is also significant in the best two variable model. For higher order models, the entire procedure is repeated. Since the goal of this study was to seek a practical as well as realistic constitutive equation, the best three variable model was the highest order model investigated. Furthermore, the model was restricted to the normal physiologic range of hematocrit, i.e. 35-50%. RESULTS AND DISCUSSION
The best one variable model Since the behavior of whole blood is primarily pseudoplastic, that is, the apparent viscosity decreases as the shear rate increases, it was not surprising to find that the best one variable rheologic model shows the shear rate to be the single most significant independent variable. This equation is: /-La
where k
=
(9)
ky"-I,
= 0.134 P(sec)"-I, and
n = 0.785, both constant.
Equation (9) has an R -square value of 0.6187 and a M.S.E. of 0.0218, so the fit of this equation to the experimental data is poor. Observe in particular that the non-Newtonian index is constant for this model. Physically this is unrealistic, since the non-Newtonian behavior of blood arises primarily from the interactions of the red blood cells with each other and so one would expect hematocrit to enter the analysis at a significant level. The poor representation of the real behavior of blood by this model can clearly be seen in Fig. 1, where the best one variable model curve is shown along with two experimental curves. 9·
8
tl. u
7
+c: ~
2l.
0.
4
';; t-
o:
~
0
0.
n.
0
A CONSTITUTIVE EQUATION FOR WHOLE HUMAN BLOOD FREDERICK J. WALBURN* and DANIEL J. SCHNECKt Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A.
(Received 19 April 1975; in revised/orm 27 September 1975) Abstract-A constitutive equation for whole human blood was developed using a power law functional form containing two parameters, a consistency index and a non-Newtonian index. These two parameters were determined by a multiple regression technique performed on viscometric data obtained from anticoagulated blood samples of known hematocrit and chemical composition. An initial constitutive model depending only on shear rate was found to be lacking any substantial degree of significance. When hematocrit was included as an indepe~dent variable, there was a considerable increase in the fit of experimental data with the analytic model. This fit became even better when fibrinogen and globulin concentrations were further taken into account. The best constitutive model ultimately included a dependence of shear stress on shear rate hematocrit and the sum of fibrinogen and globulin concentrations. Plasma lipids and the protein albumin wer~ found to contribute little to the rheologic behavior of blood. INTRODUCTION
Through the ages many noted scholars have studied the flow of blood [1], among them Hippocrates (460--375 B.C.), Aristotle (384-322 B.C.), Leonardo da Vinci (1452-1519), Sir WilWilliam Harvey (1587-1657), Reverend Steven Hales (1733), and Jean L. M. Poiseuille (1828). This interest has continued, and today many investigators are studying the complex role of blood in the cardiovascular system of man and animals. Yet, even with the modern sophisticated research methods available today, there is still much that is not known about this extremely complicated fluid. In order to obtain more information concerning blood flow, there are two approaches which could be utilized. The first is experimental, where either an in vitro model of the in vivo flow situation is constructed, and relevant information is obtained from the model, or where the information is obtained directly from an animal. The second approach is to develop a mathematical model of the in vivo flow situation. Such a model for an incompressible fluid consists of an equation of continuity (conservation of mass), three generalized momentum equations, and a constitutive equation. When the constitutive equation is introduced into the generalized momentum equations, they become specific for the particular fluid involved in the flow situation. With appropriate simplifications and boundary conditions, the equation of continuity and the specific momentum equations can be solved to yield information about the flow: local velocities, shear rates, shear stresses, etc. In accordance with the second approach, this paper offers an empirical constitutive equation for use in describing the behavior of blood. The dependence of the viscosity of blood upon the shear rate has been thoroughly investigated, and it is universally agreed that the viscosity of whole blood decreases as the rate of shear increases. A fluid which exhibits this type of behavior is called pseudoplastic [2]. In addition to its pseudoplastic behavior, whole blood also exhibits a yield stress-that is, a certain minimum force is necessary in order to initiate flow [3]. A fluid exhibiting such a yield stress is called a Bingham Plastic [2]. Several functional forms for constitutive equations have been used to model the non-Newtonian behavior of blood [3,4]. An equation which attempts to describe both Bingham Plastic and pseudoplastic behavior was proposed in 1959 by Casson [5] who at the time was working with pigment-oil suspensions of printing ink. The equation has the form: T 1/2
where
=
kcit 1/2 + T~/2,
shear stress, it == shear rate, kc == Casson viscosity, and Ty == yield stress. T==
*Research Assistant, Department of Engineering Science and Mechanics. t Assistant Professor, Department of Engineering Science and Mechanics. 201
(1)
F. J.
202
WALBURN
and D. J.
SCHNECK
Merrill et al. [6], and other investigators [3,7-10] have used this equation with a reasonable degree of success to model the behavior of whole blood. However, it should be pointed out that a yield stress for whole human blood has been observed and measured only under static loading conditions. Investigators have attributed this behavior, among other things, to rouleaux formation. There is some doubt, and probably rightly so, as to whether the yield stress is at all manifest in a dynamic situation. That is, even though fluid velocities and velocity gradients pass routinely through zero in an oscillating flow situation, there is probably not enough time for static phenomena to appear. Thus, the constitutive form chosen for this study was a general power law: T
where
=
ky",
(2)
shear stress (dyn/cm 2), y shear rate (sec-I), k consistency index (P-sec"-\ n =non-Newtonian index, and the validity of this expression was examined for shear rates in excess of 23.28 reciprocal seconds. The parameters k and n are assumed to be constant for a given hematocrit level and a given chemical composition. In addition to the comments made earlier, it may further be noted that the yield stress for blood is extremely small, virtually constant, and is due mainly to interactions between the protein fibrinogen and the erythrocytes. In many cases the addition of an anticoagulant either reduces these interactions or eliminates them completely [11]. Thus, the power law equation is considered valid even though whole blood does exhibit a slight yield stress under static conditions. Note that the power law equation can easily be modified to allow for the presence of a yield stress, i.e. it can be written as follows: T=
= =
T
where
Ty
=
k'Y" + Ty
(3)
is the yield stress. METHODS
Materials Two hundred human blood samples of 10 ml each were obtained from admissions processed through the hematology laboratory of the Montgomery County Hospital, Montgomery County, VA. These blood samples, anticoagulated with EDT A, were carefully mixed by holding a test tube in one hand and gently rotating the wrist in a back and forth motion. Two 1.2 ml aliquots were then drawn from each 10 ml test tube and viscosity measurements were obtained at 37°C on a Wells-Brookfield cone and plate viscometer with accompanying temperature control bath. Each aliquot was examined at shear rates of 23.28, 46.56, 116.40, and 232.80 reciprocal seconds and the results at each shear rate were averaged for both aliquots. This reduced the chance for an error due to incomplete mixing, inasmuch as the pairs of viscosity values were compared and if large discrepancies were noted, more viscosity measurements were made. The chemical analysis (S.M.A. 10) of each blood sample was obtained from the same hospital and total lipid determinations were performed according to the method of Zollner and Kirsch[12].
Variables The dependence of the viscosity of blood upon shear rate and hematocrit is well documented and was included in the constitutive model. The chemical variables investigated included the total lipids, albumin, and total protein minus albumin (TPMA). TPMA is composed of fibrinogen and the globulins. These chemical variables were chosen because they are composed of long chain asymmetric molecules which exist in blood in large numbers and interact more than do symmetric particles. Thus, it is probable that they contribute most to the rheologic properties of the fluid. In the analysis of the viscometric data, albumin, TPMA, total lipid, hematocrit, and the shear rate were considered to be independent variables and the apparent viscosity was considered to be the dependent variable. A multiple regression computer procedure was then used to determine the variables of greatest significance. From the point of view of this investigation, it was not of any consequence whether the blood sample was drawn from a patient that had a pathologic condition
Aconstitutive equation for whole human blood
203
or from one who was perfectly normal. Since the intent of the study was to examine the mechanical properties of the fluid solely as a function of its actual physical and chemical composition, it was not immediately relevant to know what factors might be responsible for these compositions.
Statistical analysis Equation (2) was the basic functional form that was selected for developing the constitutive equation. Since the viscosity of blood varies for different shear rates, it is convenient to introduce a new quantity called apparent viscosity (/J-a), which is defined as the viscosity of the fluid at a given shear rate:
(4) When equation (4) is substituted into equation (2), an expression relating the apparent viscosity to the shear rate results: (5)
Equation (5) is nonlinear, making it extremely complicated from the point of view of least squares regression analysis. Therefore, it is more convenient to linearize the equation by taking logarithms to the base e: loge /J-a = log. k + (n -1) log. 1'.
(6)
Equation (6) is of the form: Z=mX+b
where Z = dependent variable = log. /J-a, X = independent variable = loge it, m = slope of the line = (n - 1), and b = the Z intercept = loge k. The parameters (n - 1), loge k, and loge /La were examined in the regression analysis as functions of loge it, albumin, TPMA, total lipid, hematocrit, their respective squares, inverses, the squares of their inverses, and all interaction terms (albumin x TPMA, albumin x loge 1', albumin x l/hematocrit, and so on). Loge /La was the basic dependent variable. A maximum R -square improvement regression procedure [13] was used to analyze the data. This procedure finds, first, the "best" one variable model that produces the highest R -square statistic (see Appendix). That is, of the independent variables chosen for analysis, the computer program selects one and uses the linear least squares method to determine an equation of the form:
(7) It then computes an associated R -square statistic for this equation. Going through this procedure separately for each independent variable, the computer finally chooses the model with the largest R -square statistic as the best one variable model. To determine the best two variable model, the statistical program now adds one of the remaining variables to the best one variable model. The linear least squares regression analysis is used to determine an equation of the form: (8)
and the associated R -square statistic for this equation is computed. A different variable is then added to the best one variable model, and again the R -square statistic is computed. This procedure is followed for all remaining variables to determine which one, when added to XI, will produce the greatest increase in R -square for equation (8).
204
F. J. WALBURN and D. J. SCHNECK
At this point, it is realized that the XI variable which produced the best one variable model may not necessarily be one of the two which produces the best two variable model. Thus, the program goes back to check each of the variables, both XI and X 2 in the model it has just obtained. That is, it determines whether there will be a further increase in the R -square value if either of the presently included variables is replaced by one which was excluded. After all possible comparisons are made, the combination of two variables which produces the greatest increase in R -square over the previous two variable model is isolated. The procedure continues cycling until no further increase in R -square is found. The result is the best two variable model. Essentially, this procedure determines whether or not the variable in the best one variable model is also significant in the best two variable model. For higher order models, the entire procedure is repeated. Since the goal of this study was to seek a practical as well as realistic constitutive equation, the best three variable model was the highest order model investigated. Furthermore, the model was restricted to the normal physiologic range of hematocrit, i.e. 35-50%. RESULTS AND DISCUSSION
The best one variable model Since the behavior of whole blood is primarily pseudoplastic, that is, the apparent viscosity decreases as the shear rate increases, it was not surprising to find that the best one variable rheologic model shows the shear rate to be the single most significant independent variable. This equation is: /-La
where k
=
(9)
ky"-I,
= 0.134 P(sec)"-I, and
n = 0.785, both constant.
Equation (9) has an R -square value of 0.6187 and a M.S.E. of 0.0218, so the fit of this equation to the experimental data is poor. Observe in particular that the non-Newtonian index is constant for this model. Physically this is unrealistic, since the non-Newtonian behavior of blood arises primarily from the interactions of the red blood cells with each other and so one would expect hematocrit to enter the analysis at a significant level. The poor representation of the real behavior of blood by this model can clearly be seen in Fig. 1, where the best one variable model curve is shown along with two experimental curves. 9·
8
tl. u
7
+c: ~
2l.
0.
4
';; t-
o:
~
0
0.
n.
0
