An apparatus for monitoring a vascular access of a patient

<u>Background of the Invention.</u>

0001The invention relates to a system for monitoring vascular access in a patient undergoing extracorporeal blood treatment.

0002Specifically, though not exclusively, the invention can be usefully applied in the field of extracorporeal treatment for kidney failure.

0003Setting up an extracorporeal blood treatment, such as for example hemodialysis therapy, requires blood circulation in an extracorporeal circuit connected to the cardiovascular circuit of the patient through a vascular access.

0004The blood, taken from the patient and sent through an extracorporeal circuit, is subjected to a treatment, generally passing through a treatment unit (for example a dialyzer filter) and thereafter returned to the patient. The vascular access is where the blood is removed from the cardio-vascular system of the patient and returned to the system.

0005One of the vascular accesses most commonly used in hemodialysis therapy is the Cimino-Brescia artero-venous fistula access. Other vascular access types are known, however. For reasons of simplicity the present description will make reference to the artero-venous fistula as an example of vascular access, without excluding other types of vascular access from the claimed field of protection.

0006In an extracorporeal treatment the blood is usually taken from the vascular access by an arterial needle fluidly connected to the extracorporeal circuit. After having passed through the treatment unit, the blood is sent back to the vascular access through a venous needle. Generally blood circulation in the extracorporeal circuit is performed by a positive displacement pump, generally peristaltic.

0007One of the problems of extracorporeal blood treatment is monitoring the efficiency of the vascular access.

0008A parameter indicative of this efficiency is the blood flow rate which the vascular access can supply. This flow rate is usually greater than the blood flow rate through the extracorporeal circuit. For example, in normal conditions the blood flow rate at the vascular access is about 800÷1200 ml/minute, while the blood flow rate in the extracorporeal circuit varies between 100 and 700 ml/minute.

0009The flow rate at the vascular access can diminish due to a vascular pathology, such as, for example, a stenosis, i.e. a narrowing of the blood passage section, or for example due to a drop in cardiac output. The presence and location of a stenosis at the vascular access should be determined as soon as possible in order to prevent the stenosis degenerating into a thrombosis (occlusion of the blood vessels).

0010A reduced-efficiency vascular access can lead to the undesirable phenomenon of recirculation of blood during treatment. Recirculation consists in the presence during treatment of blood flow proceeding in an opposite to the desired direction, i.e. from the return zone of the treated blood (venous needle) to the supply zone of the blood to be treated (arterial needle). Flow recirculation therefore consists in return (in the extracorporeal circuit) of blood which has already been subjected to treatment, with a consequent diminution in treatment efficiency.

0011Various systems have been proposed for monitoring vascular access and, more generally, the cardiovascular system of a patient subjected to extracorporeal blood treatment.

0012<patcit id="pcit0001" dnum="EP1044695A2"><text>EP 1 044 695 A2</text></patcit> teaches a method for determining the blood flow rate in a vascular access during hemodialysis treatment. The method varies the blood flow rate of the extracorporeal circuit and measures the arterial and venous pressures in the extracorporeal circuit during the above-cited flow rate variations. The operations are carried out in two different conditions: first with the vascular access open, in which a part of the blood flow passes through the vascular access between the withdrawal needle and the return needle, and then when the vascular access is closed, in which the vascular access flow between withdrawal needle and return needle is zero. According to the method of <patcit id="pcit0002" dnum="EP1044695A2"><text>EP 1 044 695 A2</text></patcit>, vascular access blood flow rate, with the vascular access open, is judged to be equal to the blood pump flow rate at which the difference of arterial pressure (or venous pressure) in the two different situations is zero.

0013This method has the drawback that it is necessary to intervene mechanically on the fistula to interrupt blood flow.

0014<patcit id="pcit0003" dnum="WO0018451A"><text>WO 00/18451</text></patcit> teaches a method for determining the flow in a fistula of a patient using an extracorporeal blood flow circuit, such as for example a hemodialysis circuit, in which the blood flows from a withdrawal point in the fistula to a return point in the fistula. The method varies the blood flow rate in the extracorporeal circuit and takes a reading of a signal which can be correlated with the fistula flow rate downstream of the withdrawal point. The blood flow rate upstream of the withdrawal point is evaluated at equal the blood flow rate obtaining in the extracorporeal circuit when the fistula blood flow rate downstream, read with the above-described signal, is zero. <patcit id="pcit0004" dnum="WO0018451A"><text>WO 00/18451</text></patcit> includes an embodiment in which the signal which can be correlated to the fistula blood flow downstream of the withdrawal point is generated by an ultrasonic sensor which operates directly on the patient's vascular access.

0015The use of a sensor to directly measure the blood flow rate in the fistula tract comprised between the withdrawal needle and the return needle leads to a certain constructional complication, as well as some discomfort for the patient.

0016<patcit id="pcit0005" dnum="EP1020199A2"><text>EP 1 020 199 A2</text></patcit> teaches a method for detecting the presence of a stenosis in a vascular access during extracorporeal blood treatment. The method includes the use of at least one pressure sensor predisposed in the extracorporeal circuit along the arterial line upstream of the blood pump. A stenosis can be calculated from the entity of the pressure pulse measured by the pressure sensor.

0017A pressure sensor can be placed on the arterial line too downstream of the blood pump and upstream of a dialyzer, and a further pressure sensor can be placed on the venous line downstream of the dialyzer. The method also includes a reading of the pressure pulse frequency and use of that frequency as a signal entity correction factor. The pressure pulse frequency signal can be corrected by means of a function depending on the blood pump flow rate.

0018The data deductible from the method described in <patcit id="pcit0006" dnum="EP1020199A2"><text>EP 1 020 199 A2</text></patcit> is however limited: in particular, the method provides only a general indication of the hemodynamic state of the fistula, signalling the presence of a stenosis, but it cannot gather more detailed data, such as for example the vascular access blood flow rate or the location of any stenoses found.

0019<patcit id="pcit0007" dnum="US5830365A"><text>US 5,830,365</text></patcit> teaches a method for determining some hemodynamic parameters, among which the blood flow rate in a fistula during an extracorporeal blood treatment. The method involves the alteration of at least one chemical-physical characteristic of the blood in the venous line of the extracorporeal circuit, and recording the change which occurs in the arterial line following this alteration. The alteration can be a change in the concentration of an indicator, or a change in the temperature or pressure. In a specific embodiment use is made of a hemodialysis machine provided with a dialyzer where a dialysis solution containing an indicator flows and the concentration change of the indicator in the venous and arterial lines of the extracorporeal circuit connected to the dialyzer is registered. In the venous line the concentration of the indicator increases by effect of back-filtration through the dialyser. In the arterial line the concentration of the indicator increases by effect of recirculation in the fistula. The change of concentration in the arterial and venous lines is read by ultrasonic sensors. Alteration (in this case the change in concentration) is performed in two stages: first when the blood flows in the normal direction through the extracorporeal circuit, then when the blood flows in the opposite direction. The method includes the use of a device for inverting the blood flow direction in the extracorporeal circuit. According to the method taught in <patcit id="pcit0008" dnum="US5830365A"><text>US 5,830,365</text></patcit> the change in concentration measured in the first stage enables calculation of recirculation at normal flow rate, while the change in concentration measured in the second stage enables calculation of recirculation when the flow is inverted. The two calculated values thus enable a calculation of various hemodynamic parameters among which the blood flow rate in the fistula.

0020However the alteration of the chemical-physical properties of the blood and the inversion of the flow during the course of extracorporeal treatment lead to various drawbacks: a constructional complication, a delay in carrying out the treatment, an invasive intervention on the blood, quite removed from the course of normal treatment.

0021<patcit id="pcit0009" dnum="WO0204044A"><text>WO 02/04044</text></patcit> teaches a method for identifying problems in arterial flow during an extracorporeal blood treatment in which the blood is transferred, by means of a positive displacement pump, from the vascular access of a patient to a blood treatment device through an arterial line and then sent by the treatment device to the vascular access through a venous line of the extracorporeal circuit. The method consists in measuring the amplitude of the periodic variations in pressure in the venous line induced by the rotation of the blood pump, by comparing the variations with a threshold value and generating a control signal if the threshold value is exceeded. <patcit id="pcit0010" dnum="WO0204044A"><text>WO 02/04044</text></patcit> further describes another method according to which, during a dialysis treatment, the amplitude of the periodical variations of the pressure of the dialysis fluid (and not the venous line) is measured. The result is compared with a threshold value and if the threshold value is exceeded a control signal is generated.

0022The methods described in <patcit id="pcit0011" dnum="WO0204044A"><text>WO 02/04044</text></patcit> are not however able to provide data relating to the blood flow rate at the vascular access.

0023<patcit id="pcit0012" dnum="US6221040B"><text>US 6,221,040</text></patcit> discloses a system for monitoring a vascular access during a dialysis treatment in which the pressures in both the arterial and venous branches of the extracorporeal blood system are monitored by pressure sensors. A computer unit generates first and second characteristic values for the integrity of the vascular access from measured arterial and venous pressures. An analyser unit analyses the integrity of the vascular access by comparing the first and second characteristic values to first and second ranges of predetermined values. Calculating a sum of the venous and arterial pressure generates the first characteristic value, and calculating a difference between the venous and the arterial pressure generates the second characteristic value.

0024The object of <patcit id="pcit0013" dnum="US6221040B"><text>US 6,221,040</text></patcit> is to provide a monitoring system that allows detection of the venous cannula slipping out of the vascular access as well as detection of a blood leak in the venous branch of the extracorporeal circuit. It is not directed to determination of fistula flow.

0025<patcit id="pcit0014" dnum="US5866015A"><text>US 5,866,015</text></patcit> and <patcit id="pcit0015" dnum="EP0773035A"><text>EP 0 773 035</text></patcit> disclose a method for determining hemodynamic parameters during an extracorporeal hemotherapy, including the steps of measuring the blood temperature in the arterial branch of the extracorporeal circuit, varying the blood flow in the extracorporeal circuit, storing the values of the extracorporeal blood flow and the measured values of the blood temperature, and determining a value of the blood flow from the stored sequence of value pairs of blood temperature and of extracorporeal blood flow, at which value, after it is exceeded, the amount of the change in the blood temperature within a specific blood flow interval is greater than a predetermined limiting value. The fistula flow is inferred from the determined blood flow value.

0026The method is based on the fact that the measuring curve existing in discrete measured values is able to be represented by two subfunctions, the first subfunction indicating the blood temperature as a function of the extracorporeal blood flow for blood flow values smaller than the fistula flow or equal to the fistula flow, and the second subfunction indicating the blood temperature as a function of the blood flow for blood flow values greater than or equal to the fistula flow. The intersection of the two subfunctions indicates the point where the extracorporeal blood flow equals fistula flow. Thus, from the "break point" of the characteristic function curve, i.e., from the discontinuity in the rise of the curve, the point is able to be defined where fistula recirculation begins, i.e., where blood flow equals fistula flow.

0027In addition to measuring temperature, the concentration of a blood constituent (hematocrit) can also be measured, as can the density, speed of sonic propagation, optical density, and conductivity or viscosity.

0028The blood characteristic to be measured must have a different value in the venous branch of the extracorporeal circuit than it does in the blood flowing to fistula. It is assumed that the blood characteristic, preferably the temperature, is kept constant in the venous branch of the extracorporeal circuit while the measured values are recorded. If this characteristic is not constant, a regulating device to keep the characteristic in the venous branch constant must be provided. In the case of a temperature measurement, for example, this can be realised as a temperature controller.

0029Another drawback of this method is that the delivery rate of blood pump, which predetermines the extracorporeal blood flow, is increased starting from a lower value to an upper limiting value which must be greater than the fistula flow to be expected. Fistula flows can only be determined within the adjustable blood flow range. Therefore the fistula flow is not determinable if it is equal to or greater than the upper limiting value of the adjustable blood flow range.

0030The prior art comprises the scientific publication entitled: "<nplcit id="ncit0001" npl-type="s"><text>On-line dynamic measurement of fistula pressure during hemodialysis for detection of access stenosis and bad needle placement", Abstract from the 24th EDTNA-ERCA Conference, Prague, 5-8 July 1997, page 23, authors Polaschegg, Techert and Wizemann</text></nplcit>.

0031According to this publication it is possible to calculate the pressure of a vascular access by measuring the pressure in an extracorporeal blood circuit connected to the vascular access, with the aim of detecting any stenoses in the access itself.

0032In a scientific publication entitled "<nplcit id="ncit0002" npl-type="s"><text>Dynamic pressure measurement for detection of blood access stenosis", published in the EDTNA-ERCA Journal, 1998, XXIV, 4, on pages 39-44, authors Polaschegg, Techert and Wizemann</text></nplcit>, more detail is given on monitoring problems in a patient's vascular access. The method is based on the determination of the venous and arterial pressures (upstream of the blood pump) in an extracorporeal blood circuit connected to the vascular access to be monitored. The method comprises a preliminary stage in which, through <i>in vitro</i> tests in which the extracorporeal circuit is not connected-to a real vascular access, fluid resistances in the arterial and venous lines of the extracorporeal circuit are calculated. During a second stage the extracorporeal circuit is connected to the real vascular access of the patient in order to initiate an extracorporeal treatment. During the extracorporeal treatment the venous and arterial pressures are calculated in the extracorporeal circuit. As the venous and arterial pressures in the extracorporeal circuit are known, as are the fluid resistances in the arterial and venous lines of the extracorporeal circuit, the pressures in the vascular access can be calculated. The dynamic measurement at different flow rates and the comparison with static measures enables stenoses at the vascular access to be identified.

0033The scientific publication entitled "<nplcit id="ncit0003" npl-type="s"><text>Pressure drops in cannulas for hemodialysis", author H.D. Polaschegg, published in The International Journal of Artificial Organs, Vol. 24, No. 9, 2001, pp. 614-623</text></nplcit>, relates to a method for determining a fall in pressure in an arterial or venous line in hemodialysis, with which the vascular access pressures can be determined starting from the pressures measured in the extracorporeal circuit of the hemodialysis machine.

0034The scientific publication entitled "<nplcit id="ncit0004" npl-type="s"><text>Extracorporeal pressure monitoring and the detection of vascular access stenosis", authors Kleinekofort, Kraemer, Rode and Wizemann, published in The International Journal of Artificial Organs, Vol. 25, No. 1, 2002, pp. 45-50</text></nplcit>, presents a method for identifying the presence of stenoses in a vascular access, even where the stenosis is located between the withdrawal needle and the return needle. The method comprises measuring the static pressures in the arterial and venous lines of an extracorporeal circuit and in calculating the pressures at the vascular access at the withdrawal point and the return point..These pressures, which correspond to the pressures which would be measured by two pressure sensors directly connected to the withdrawal and return needles, are used in order to identify the presence of a stenosis. A knowledge of the pressures both at the point of withdrawal and at the point of return of the vascular access provides more accurate indications and enables a first approximate localization of the stenosis, especially enabling to detect if the stenosis is in venous tract or is located between the needles.

0035The method described here is not however able to determine the blood flow rate in the vascular access.

0036The publication entitled "<nplcit id="ncit0005" npl-type="s"><text>Utility of intra-access pressure monitoring in detecting and correcting venous outlet stenoses prior to thrombosis", in Kidney International, Vol. 47 (1995), pages 1364-1373, authors Besarab, Sullivan, Ross, Moritz</text></nplcit>, teaches a method for deriving the pressure internally of the vascular access (intra-access pressure) from the pressure measured in the hemodialysis machine, as a function of the type of needle used, the blood flow rate of the hemodialysis machine, and the hematocrit of the blood. Other methods for determining the pressure at the vascular access are cited or described in the following publications: <ul id="ul0001" list-style="dash"><li>"<nplcit id="ncit0006" npl-type="s"><text>Detection of access strictures and outlet stenoses in vascular accesses. Which test is best?", in ASAIO Journal, 1997, Vol. 43: pages M543-M547, authors Besarab, Lubkowski, Frinak, Ramanathan, Escobar</text></nplcit>;</li><li>"<nplcit id="ncit0007" npl-type="s"><text>Simplified measurement of intra-access pressure", in Journal of the American Society of Nephrology, 1998, Vol. 9, pages 284-289, authors Besarab, Frinak, Sherman, Goldman, Dumpler, Devita, Kapoian, Al-Saghir, Lubkowski</text></nplcit>;</li><li>"<nplcit id="ncit0008" npl-type="s"><text>Effect of systemic hemodynamics on flow within vascular accesses used for hemodialysis", in ASAIO Journal 2001, Vol. 47, pages 501-506, authors Besarab, Lubkowski, Vu, Aslam, Frinak</text></nplcit>;</li><li>"<nplcit id="ncit0009" npl-type="s"><text>Dynamic venous access pressure ratio test for hemodialysis access monitoring", in American Journal of Kidney Disease, Vol. 40, N° 4 (October), pages 760-768, 2002, authors Frinak, Zasuwa, Dunfee, Besarab, Yee</text></nplcit>.</li></ul>

0037An abstract entitled "<nplcit id="ncit0010" npl-type="s"><text>A novel model-based method for monitoring the hemodialysis vascular access", published in the Journal of the American Society of Nephrology, 2001, Vol. 12, N. A1513, pages 294A-295A, authors Lodi, Monari, Fava, Paolini, Grandi, Galato, Cavalcanti</text></nplcit>, cites a mathematical model based on the hemodynamic description of the vascular access which enables the arterial and venous pressures at the vascular access to be calculated and also the flow in vascular access starting from extracorporeal arterial and venous pressures. The model, which includes three parameters (resistance to flow of the anastomosis, resistance between arterial and venous access, the resistance which expresses the efficiency of venous circulation drainage), was used to analyse the data gathered during a normal hemodialysis therapy operation. The abstract states that the extracorporeal venous and arterial pressures were measured after having set four different flow rates on the blood pump and that the above-cited parameters included in the mathematical model were calculated using the mathematical model.

<u>Summary of the Invention.</u>

0038The present invention provides a system for controlling vascular access adequacy during an extracorporeal blood treatment.

0039An aim of the invention is to enable calculation of some hemodynamic parameters at the vascular access. Knowledge of these parameters enables both regulation of the blood pump flow rate operation in the extracorporeal circuit and intervention in case of detection of a pathological situation in the vascular access.

0040A further aim of the invention is to enable evaluation of the blood flow circulating in the vascular access of a patient during an extracorporeal blood treatment.

0041A further aim of the invention is to make available a system for evaluating vascular hydraulic resistance in various tracts of the patient's vascular system. In particular, an aim of the invention is to evaluate vascular resistance upstream of the blood withdrawal zone from the vascular access, downstream of the blood return zone, and in the tract of vascular access comprised between the withdrawal zone and the return zone.

0042An advantage of the invention is that it provides indicative values of the efficiency of the vascular access simply, automatically, using devices (such as for example pressure transducers, blood pump, drainage pump) which are normally already present in machines for extracorporeal blood treatment. A further advantage is that the invention enables monitoring of the vascular access at any time during the extracorporeal blood treatment.

0043A further advantage of the invention is that the monitoring procedure does not cause extra stress to the patient. The procedure can be carried out by means of variations in the blood pump or the drainage pump flow rates, or both, within flow rate intervals which are normally compatible with the extracorporeal treatment the patient undergoes. The intervals can be those normally used during the course of therapy.

0044These aims and others besides are all attained by the invention as it is characterised in one or more of the appended claims.

0045In a special function of the invention, a mathematical model is used which contains at least two parameters in which a first parameter relates to the hemodynamics of the vascular access, and a second parameter relates to the blood flow rate in the extracorporeal circuit.

0046The mathematical model comprises a third parameter relating to at least one blood characteristic: this characteristic can be any physical, chemical or physical-chemical property thereof which characterises the blood in a vessel and which can be related to the blood flow rate in that vessel. A peculiarity of the invention is that the mathematical model used describes the relationship between the selected blood property (physical, chemical or physical-chemical) and the blood flow rate in the vessel. In particular the mathematical model describes the relationship in the vascular access. For example, the mathematical model can describe the fluid-dynamic situation of the vascular access; the model can describe a relationship between the difference of pressure at two points of the vascular access and the flow rate crossing the points. Apart from the pressure it is also possible to select other properties (physical, chemical or physical-chemical) of the blood which are influenced by the flow rate, such as, for example: the difference in induced potential, speed of sound, optical characteristics, temperature, concentration of an indicator, and so on.

0047According to the invention, the monitoring of the vascular access is performed by varying the flow rate of at least one fluid (for example blood or the product of ultrafiltration), which runs either in the extracorporeal circuit or in at least one hydraulic line (for example an ultrafiltration line) connected to the extracorporeal circuit.

0048The monitoring can be carried out by varying both the above-cited flows.

0049The monitoring determines the values of at least one characteristic of the blood, in at least one zone of the blood circulation path, and at at least two different values of the flow rate of the fluid.

0050As mentioned above, the cited characteristic of the blood can be a physical, chemical or chemical-physical one. In an embodiment of the invention, among the various characteristics of the blood that depend on blood flow, the selected characteristic to be used is the pressure.

0051The monitoring procedure involves calculating one or more of the hemodynamic parameters of the vascular access contained in the mathematical model, using the values of the blood characteristic determined previously during the course of the procedure.

0052In an embodiment of the invention, a multiplicity of values of the blood characteristic is determined; then the said hemodynamic parameters are calculated, by means of the mathematical model, using approximation algorithms (of known type). The algorithms can be chosen, for example, from those which enable determination of the value of the hemodynamic parameter, by virtue of which the blood characteristic values calculated using the mathematical model, at different flow rate values, are those which are closest to the blood characteristic values which were previously determined during the course of the procedure, at the same flow rate values.

0053In an embodiment of the invention, the mathematical model used is descriptive of the pressure variation at the vascular access: it comprises at least one hemodynamic parameter relative to at least one characteristic of the vascular access; at least one parameter relative to the blood characteristic; and at least one parameter relative to the blood flow rate in the extracorporeal circuit.

0054The hemodynamic parameter can be relative to at least one of the following characteristics of the vascular access: the blood flow rate upstream of a withdrawal zone of the blood.from the access, the blood flow rate between the withdrawal zone and a blood return zone at the access, the blood flow rate downstream of the blood return zone, the vascular hydraulic resistance upstream of the blood withdrawal zone from the access, the vascular hydraulic resistance between the blood withdrawal zone and the blood return zone, and the vascular hydraulic resistance downstream of the blood return zone.

0055In a further embodiment of the invention, the monitoring procedure includes determining the values assumed by the blood characteristic in at least two zones of the blood circulation path (where the blood circulation path comprises both the intracorporeal circuit and the extracorporeal circuit) and at at least two different flow rate values of one fluid (blood or the product of ultrafiltration).

0056In a further embodiment of the invention, the monitoring procedure includes determining the values assumed by the blood characteristic in at least one zone of the blood circulation path and at at least two different flow rate values of two fluids (blood and the product of ultrafiltration).

0057In a further embodiment of the invention, the monitoring comprises a measuring stage of a blood characteristic, in a zone of the extracorporeal circuit arranged downstream of the blood withdrawal zone, or in a zone arranged upstream of the blood return zone, or in both above zones. The monitoring includes determining the blood characteristic in the vascular access, in the withdrawal zone, or in the return zone, or in both zones, by means of one or more mathematical models describing the variation of the said blood characteristic between the zones of withdrawal and return in the vascular access and the measuring zones in the extracorporeal circuit. The mathematical models can be, in particular, models descriptive of the variation of the said blood characteristic in the passage through the arterial or venous needles. In an embodiment of the invention, these mathematical models comprise at least one parameter which is relative to the blood flow, or at least one parameter relative to the hematocrit of the blood, or both said parameters. In particular the mathematical models can be represented by one or more interpolating formulas of experimental data; the formulas can be, for example, second-order polynomials with one or more parameters chosen between the flow rate and the hematocrit of the blood.

0058In a special operation of the invention, at regular time intervals the monitoring procedure determines the values assumed by the blood characteristic in at least one zone of the blood circulation path during the flow rate change, evaluates the variation of the blood characteristic, selects the values assumed by the blood characteristic when the variation has exceeded a threshold limit value, and uses the selected values to calculate the value of the blood characteristic at the vascular access.

0059In a further special operation of the invention, at regular time intervals the monitoring procedure determines the values of the blood characteristic in two different zones of the blood circulation path during flow rate change, compares the variation of the blood characteristic detected in a first zone of the blood circulation path and the variation of the characteristic detected in a second zone thereof, selects the values of the blood characteristic when the difference between the variations has exceeded a threshold limit value, and uses the selected values in calculating the value of the blood characteristic at the vascular access.

0060In another characteristic of the invention, in calculating the value of the characteristic of the vascular access, the monitoring procedure considers the values of the blood characteristic in a stationary blood flow situation, i.e. after having kept the flow rate constant for a determined period of time.

0061The monitoring procedure is applied by means of a machine for blood treatment in an extracorporeal circuit, in particular for a machine for treatment of kidney failure, predisposed to perform one or more of the following therapies: hemodialysis, hemofiltration, hemodiafiltration, pure ultrafiltration, plasmapheresis.

0062The machine is provided with a timer for carrying out the monitoring procedure at least once during the extracorporeal treatment.

0063The monitoring procedure can be initiated on command of an operator, or automatically at a predetermined moment during the treatment.

0064The extracorporeal circuit can be included in the complex of fluid distribution lines, of the disposable type, normally removably associated and used in a machine for treatment of renal failure.

0065The machine is normally equipped with pressure transducers operating in the blood withdrawal line, before the blood pump, and in the blood return line, after the blood treatment unit.

0066Further characteristics and advantages of the present invention will better emerge from the detailed description that follows, of a specific embodiment of the invention, illustrated purely in the form of a non-limiting example in the figures of the drawings.

<u>Brief Description of the Drawings.</u>

0067The description will be made herein below with reference to the appended figures of the drawings, here given by way of non-limiting illustration, in which: <ul id="ul0002" list-style="dash"><li><figref idref="f0001">figure 1</figref> is a diagram of a machine for an extracorporeal blood treatment provided with a monitoring device of the vascular access according to the invention;</li><li><figref idref="f0003">figure 2</figref> is a diagram of blood flow in a patient connected up to the machine of <figref idref="f0001">figure 1</figref>;</li><li><figref idref="f0002">figure 3</figref> is an electrical diagram which describes by analogy the circulation of extracorporeal and intracorporeal blood of the patient subjected to the extracorporeal treatment with the machine of <figref idref="f0001">figure 1</figref>;</li><li><figref idref="f0002">figure 4</figref> shows a diagram of the relation between ΔP_f and q_b, where ΔP_f = P_af - P_vf (difference between arterial pressure in the vascular access P_af and venous pressure in the vascular access Pv_f) and q_b is the extracorporeal flow rate of the blood;</li><li><figref idref="f0003">figure 5</figref> is a diagram of the relation between (P_vf - P_v) e q_uf, where (P_vf - P_v) is the difference between the venous pressure in the vascular access P_vf and the systemic venous pressure P_v, and q_uf is the ultrafiltration flow rate.</li></ul>

<u>Detailed Description.</u>

0068The machine illustrated in <figref idref="f0001">figure 1</figref> is a machine for hemodiafiltration comprising a unit for an extracorporeal blood treatment (a filter for hemodiafiltration 1) having two chambers 2, 3 separated by a semipermeable membrane 4. A first chamber 2 has an inlet which is connected to an arterial line 5 (blood withdrawal line from the patient) of an extracorporeal blood circuit. The arterial line 5 is connectable with a vascular access 6 of a patient by means of an access tool constituted in the example by an arterial needle N_A. The arterial line 5 is provided with a pressure sensor 8 and a positive displacement pump 9 for blood circulation along the extracorporeal circuit in the direction of the arrow 7.

0069The first chamber 2 has an outlet connected to a venous line 10 (blood return line to the patient) of the extracorporeal blood circuit. The venous line 10 is connectable to the vascular access 6 of the patient by means of an access tool constituted in the illustrated embodiment by a venous needle N_v. The venous line 10 is provided with a pressure sensor 12.

0070The second chamber 3 of the filter 1 has an inlet connected to a supply line 14 of a fresh treatment fluid (dialysis liquid) and an outlet connected to a discharge line 15 of a discharge fluid (the dialysis liquid and the ultrafiltered liquid). The supply line 14 is provided with a supply pump 13 of the fresh treatment fluid. The discharge line 15 is provided with a drainage pump 16 for the circulation of the discharge fluid in the direction of the arrow 11.

0071The dialysis machine further comprises a control and calculation unit 17 connected to a screen and also to a keyboard through which the user communicates to the control and calculation unit the setting values for machine operation. One of the setting values which the control and calculation unit 17 receives from the user is the blood flow rate q_b in the arterial blood withdrawal line 5. The control and calculation unit 17 can control the speed of the blood pump 9 in order to have the predetermined value of flow rate q_b. The control and calculation unit 17 can be connected to at least one measuring device, able to provide information relating to the effective blood flow rate in the arterial line. The measuring device can comprise, for example, a flowmeter, or an encoder connected to the rotor of a blood pump. The control and calculation unit 17 is further connected to the pressure sensors 8 and 12 and receives therefrom the signals indicating the detected pressure.

0072The control and calculation unit 17 controls the operation of the various motor devices of the machine, in particular the blood pump 9 and drainage pump 16, according to the instructions received from the user and the programmed algorithms contained in its memory.

0073The machine can further comprise sensors (of known type and not illustrated) for detecting the blood viscosity upstream and downstream of the treatment unit 1. The sensors can comprise, for example, measuring devices for the blood hematocrit level.

0074The control and calculation unit is programmed to carry out, automatically or by request of the user, a series of operations which enable the vascular access to be monitored.

0075<figref idref="f0003">Figure 2</figref> shows the patient's blood circulation subjected to extracorporeal treatment with the machine of <figref idref="f0001">figure 1</figref>. The vascular access 6, through which the extracorporeal blood circuit is connected to the cardio-vascular circuit of the patient is, in the embodiment, a fistula of the Cimino-Brescia type. In <figref idref="f0003">figure 2</figref> H indicates the patient's heart, P denotes the pulmonary circuit, V denotes the vascular system (or systemic circuit, or intravascular circuit or intracorporeal circuit). The arterial line 5 and the venous line 10 are connected at one end to the vascular access 6 and at the other end to the dialysis filter 1.

0076<figref idref="f0002">Figure 3</figref> shows an electrical diagram which, by analogy, describes the blood circulation of the patient subjected to the extracorporeal blood treatment.

0077The legend to <figref idref="f0002">figure 3</figref> is as follows.

0078<b>Quantities controller by the control unit 17:</b><ul id="ul0003" list-style="none"><li>q_b blood pump flow rate [ml/min]</li><li>q_uf ultrafiltration flow rate [ml/min]</li></ul>

0079<b>Known quantities (measurable directly or indirectly or determinable from indirect measurements using a mathematical model):</b><dl id="dl0001"><dt>P_am</dt><dd>extracorporeal arterial pressure [mmHg]</dd><dt>P_vm</dt><dd>extracorporeal venous pressure [mmHg]</dd><dt>E_art</dt><dd>hydrostatic pressure related to the height level difference between the pressure sensor 8 in the arterial line of the extracorporeal circuit and the arterial needle N_A [mmHg]</dd><dt>E_ven</dt><dd>hydrostatic pressure related to the height level difference between the pressure sensor 12 in the venous line of the extracorporeal circuit and the venous needle N_v [mmHg]</dd><dt>R_am</dt><dd>hydraulic resistance of the extracorporeal arterial line [mmHg•min/ml]</dd><dt>P_vm</dt><dd>hydraulic resistance of the extracorporeal venous line [mmHg•min/ml]</dd><dt>P_af</dt><dd>vascular access arterial pressure [mmHg]</dd><dt>P_vf</dt><dd>vascular access venous pressure [mmHg]</dd><dt>P_a</dt><dd>mean systemic arterial pressure (MAP) [mmHg]</dd><dt>P_v</dt><dd>venous pressure (venous return pressure) [mmHg]</dd></dl>

0080<b>Unknown quantities to be determined:</b><dl id="dl0002"><dt>q_a</dt><dd>blood flow rate at the vascular access, upstream of the arterial access [ml/min]</dd><dt>q_f</dt><dd>blood flow rate of artero-venous anastomosis in the vascular access tract comprised between the arterial access and the venous access, (q_f = q_a - q_b) [ml/min]</dd><dt>q_v</dt><dd>blood flow rate downstream of the venous access, (q_v = q_a - q_uf) [ml/min]</dd><dt>R_d</dt><dd>hydraulic resistance upstream of the vascular access [mmHg•min/ml]</dd><dt>R_f</dt><dd>hydraulic resistance between the arterial access and the venous access [mmHg•min/ml]</dd><dt>R_v</dt><dd>hydraulic resistance downstream of the vascular access [mmHg•min/ml]</dd></dl>

0081In the diagram of <figref idref="f0002">figure 3</figref> the extracorporeal blood circuit is traced in bold line, while the intracorporeal circulation in the vascular access is drawn in thin line.

0082The nodes where the extracorporeal circuit meets with the vascular access are the zones where pressures P_af e P_vf are determined (either directly measured or calculated).

0083Various methods are known, based on mathematical models, for calculating pressures P_af e P_vf from known pressures P_am e P_vm in the extracorporeal circuit. Some of these methods are described in the scientific publications cited in the present description. Herein below details will be given of a method founded on a new mathematical model based on the:electrical diagram represented in <figref idref="f0002">figure 3</figref>.

0084In the following a mathematical model is shown, also based on the electrical diagram of <figref idref="f0002">figure 3</figref>, representative of the hemodynamics of the vascular access of an extracorporeal blood circuit in which the blood is removed from the patient through an arterial needle, is made to circulate through the extracorporeal circuit and is returned through a venous needle.

0085The mathematical model describes the variation of pressure in the vascular access as a function of the blood flow rate.

0086The mathematical model is expressed in the following three equations which can be derived from the electrical diagram represented in <figref idref="f0002">figure 3</figref>. <maths id="math0001"><math display="block"><msub><mi>q</mi><mi>a</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mi>a</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub></mrow><msub><mi>R</mi><mi>d</mi></msub></mfrac></math><img file="EP1590018B1_D0001.tif" /></maths><maths id="math0002"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>=</mo><msub><mi>R</mi><mi>f</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mi>b</mi></msub></mfenced></math><img file="EP1590018B1_D0002.tif" /></maths><maths id="math0003"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">v</mi></msub><mo>=</mo><msub><mi>R</mi><mi>v</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mi mathvariant="italic">uf</mi></msub></mfenced></math><img file="EP1590018B1_D0003.tif" /></maths> where, as mentioned herein above, the symbols have the following meanings: <dl id="dl0003" compact="compact"><dt>q_a =</dt><dd>blood flow rate at the vascular access 6 (fistula), upstream of the withdrawal point of the arterial needle N_A</dd><dt>q_b =</dt><dd>blood flow rate in the arterial line 5 of the extracorporeal circuit</dd><dt>P_a =</dt><dd>mean systemic arterial pressure measured at patient's arm</dd><dt>P_af =</dt><dd>arterial pressure in the vascular access 6, i.e. the pressure in the vascular access (in the embodiment, with a Cimino-Brescia fistula, this is a tract of arterialized vein) at the point of withdrawal of the arterial needle N_A</dd><dt>R_d =</dt><dd>resistance of the tract of arterialised vein comprised between the anastomosis and the point of withdrawal of the arterial needle N_A</dd><dt>P_vf =</dt><dd>venous pressure in the vascular access 6, i.e. the pressure in the fistula at the return point of the venous needle N_v</dd><dt>R_f =</dt><dd>vascular resistance of the tract of fistula comprised between the two needles N_A and N_v and representing the resistance between the two points at which P_af and P_vf are determined</dd><dt>P_v =</dt><dd>venous pressure of the blood in the distal venous branch; the Pv value can be unknown during the extracorporeal treatment; in this case it can be placed at a constant physiological value (e.g. P_v = 0)</dd><dt>R_v =</dt><dd>vascular resistance in the venous branch of the blood return zone at the zone where venous pressure P_v is evaluated; where P_v = 0, the resistance R_v represents total venous resistance, i.e. the vascular resistance met by the blood in returning from the venous needle N_v to the heart H, which constitutes an indicative value of the drainage efficiency of the venous circulation</dd><dt>q_uf</dt><dd>= ultrafiltration flow rate (in case of hemodiafiltration, q_uf is the difference between the discharge fluid flow rate in the discharge line 15 and the fresh dialysis fluid flow rate in the supply line 14).</dd></dl>

0087The pressures in the above-indicated mathematical model relate to atmospheric pressure. The arterial and venous pressures P_af and P_vf in the vascular access are measurable directly, for example using pressure sensors operating directly on the vascular access 6 in proximity or internally of the arterial and venous needles N_A e N_v.

0088As previously mentioned, the pressures P_af and P_vf are also determinable indirectly using a mathematical model which includes, among its parameters, pressures P_am and P_vm (arterial and venous pressures) measured in the extracorporeal circuit by the pressure sensors 8 and 12. The prior art comprises various mathematical models usable for calculating pressures P_af and P_vf when pressures P_am and P_vm are known. Some of the above-cited prior art contains examples of so-usable mathematical models. There follows a further example of a mathematical model usable for determining the intravascular pressures of the blood starting from the easily-measurable values of the extracorporeal blood pressures.

<u>Determination of P</u>_<u>af</u><u>and P</u>_<u>vf</u><u>with P</u>_<u>am</u><u>and P</u>_<u>vm</u><u>known.</u>

0089The mathematical model used comprises the two equations which can be derived from the electrical diagram of <figref idref="f0002">figure 3</figref>: <maths id="math0004"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub><mo>=</mo><msub><mi>P</mi><mi mathvariant="italic">am</mi></msub><mo>+</mo><msub><mi>E</mi><mi mathvariant="italic">art</mi></msub><mo>+</mo><msub><mi>R</mi><mi mathvariant="italic">am</mi></msub><mo>⋅</mo><msub><mi>q</mi><mi>b</mi></msub></math><img file="EP1590018B1_D0004.tif" /></maths><maths id="math0005"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>=</mo><msub><mi>P</mi><mi mathvariant="italic">vm</mi></msub><mo>+</mo><msub><mi>E</mi><mi mathvariant="italic">ven</mi></msub><mo>-</mo><msub><mi>R</mi><mi mathvariant="italic">vm</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>b</mi></msub><mo>-</mo><msub><mi>q</mi><mi mathvariant="italic">uf</mi></msub></mfenced></math><img file="EP1590018B1_D0005.tif" /></maths>

0090Resistances R_am and R_vm can be considered equal, with satisfactory approximation, to the hydraulic resistance of the arterial needle N_A and, respectively, the venous needle N_v; it is therefore assumed for the sake of simplicity that the whole drop in pressure in the arterial and venous lines is concentrated at the respective needles.

0091To calculate the hydraulic resistance R of a needle, the following mathematical model is used: it makes use of an equation which connects the hydraulic resistance of the needle with the blood flow rate and the blood hematocrit. <maths id="math0006"><math display="block"><mi>R</mi><mo>=</mo><mfenced><msub><mi>A</mi><mn>2</mn></msub><mo>⋅</mo><msubsup><mi>q</mi><mi>b</mi><mn>2</mn></msubsup><mo>+</mo><msub><mi>A</mi><mn>1</mn></msub><mo>⋅</mo><msub><mi>q</mi><mi>b</mi></msub><mo>+</mo><msub><mi>B</mi><mn>2</mn></msub><mo>⋅</mo><msup><mi mathvariant="italic">Hct</mi><mn>2</mn></msup><mo>+</mo><msub><mi>B</mi><mn>1</mn></msub><mo>⋅</mo><mi mathvariant="italic">Hct</mi><mo>+</mo><msub><mi>B</mi><mn>0</mn></msub></mfenced><mo>⋅</mo><msub><mi>R</mi><mi mathvariant="italic">Poiseuille</mi></msub></math><img file="EP1590018B1_D0006.tif" /></maths> where <dl id="dl0004" compact="compact"><dt>q_b =</dt><dd>blood flow rate</dd><dt>Hct =</dt><dd>blood hematocrit</dd></dl><maths id="math0007"><math display="block"><msub><mi>R</mi><mi mathvariant="italic">Poiseuille</mi></msub><mo>=</mo><mfrac><mrow><mn>8</mn><mo>⋅</mo><mi>L</mi></mrow><mrow><mi>π</mi><mo>⋅</mo><msup><mi>r</mi><mn>4</mn></msup></mrow></mfrac></math><img file="EP1590018B1_D0007.tif" /></maths><dl id="dl0005" compact="compact"><dt>L =</dt><dd>length of needle</dd><dt>r =</dt><dd>radius of internal section of the needle</dd></dl>

0092R_poiseuille is the theoretical hydraulic resistance calculated using the Hagen-Poiseuille law for a liquid with viscosity equal to one.

0093A₂, A₁, B₂, B₁ and B₀ are coefficients characteristic of each needle, the value being obtained by means of experimental preliminary laboratory testing, by measuring the fall of pressure through the needle with different blood and hematocrit flow rates. In experimental tests the flow rate was varied within a range from 0 to 500 ml/minute, while the hematocrit was varied within a range from 30 to 45%. The coefficients differ for a same needle according to blood flow direction, that is whether the needle is used as an arterial needle or as a venous needle. These preliminary <i>in vitro</i> tests serve to experimentally characterise the needles which will then be used for the extracorporeal blood treatment. The tests include simulation of the extracorporeal treatment (for example dialysis) using a machine for performing the treatment (for example a dialysis machine) with an extracorporeal circuit lacking the device for effecting the treatment (for example lacking a dialyzer filter), causing bovine blood to circulate, exiting from a container and returning thereto. The blood is kept at a constant temperature of 37°C. The blood hematocrit is measured. The machine and the circuit used in the tests can be the same as those illustrated in <figref idref="f0001">figure 1</figref>.

0094At intervals of about 1 minute the blood pump flow rate q_b is changed, starting from a zero flow rate q_b0 = 0 ml/minute and increasing it by 50 ml/minute up to a maximum flow rate of 500 ml/minute (q_b1=50ml/min, q_b2=100ml/min, ..., q_bi=i • 50ml/min, ..., q_b10=500ml/min). In general, the flow rate q_b assumes N different values q_bi with i = 0, 1, 2, ..., N (N ≥ 3).

0095At each interval pressures P_ami and P_vmi are measured using the pressure sensors placed along the extracorporeal circuit. From each pressure value measured, P_ami and P_vmi, we subtract the hydrostatic pressure due to the different blood level in the container with respect to the point of measurement of the pressure on the machine. From pressures P_ami and P_vmi we can deduce the pressure falls of the corresponding needles ΔP_ai and ΔP_vi, with i = 0, 1, 2, ..., N (N ≥ 3).

0096The same operations are repeated, each time controlledly changing the value of the hematocrit in the bovine blood. The blood flow rate values q_b are the same each time, i.e. q_b = q_bi, with i = 0, 1, 2, ..., N.

0097The hematocrit can be varied by dilution with physiological solution (in this case the hematocrit diminishes each time). For each series of operations the value of the hematocrit is measured. Purely by way of an example, the operations can be performed with the following hematocrit values: about 44%, about 42%, about 40%, about 38%, about 36%, about 34%, about 32 %. In general the value of the hematocrit Hct assumes M different Hct_j values with j = 1, 2, ..., M (with M ≥ 2).

0098Thus, for each needle we obtain a number N•M of values ΔP_aij and ΔP_vij with i = 0, 1, 2, ..., N (with N ≥ 3) e j = 1, 2, ..., M (with M ≥ 2).

0099A processor calculates the hydraulic resistances of the needle, normalised with respect to the Poiseuille resistance, for one of the hematocrit values (for example Hct = H_ct1) according to the equation: <maths id="math0008"><math display="block"><msub><mi>R</mi><mrow><mi mathvariant="italic">ai</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi>P</mi><mrow><mi mathvariant="italic">ai</mi><mo>⁢</mo><mn>1</mn></mrow></msub></mrow><msub><mi>q</mi><mi mathvariant="italic">bi</mi></msub></mfrac><mo>⋅</mo><mfrac><mn>1</mn><msub><mi>R</mi><mi mathvariant="italic">Poiseuille</mi></msub></mfrac></math><img file="EP1590018B1_D0008.tif" /></maths><maths id="math0009"><math display="block"><msub><mi>R</mi><mrow><mi mathvariant="italic">vi</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi>P</mi><mrow><mi mathvariant="italic">vi</mi><mo>⁢</mo><mn>1</mn></mrow></msub></mrow><msub><mi>q</mi><mi mathvariant="italic">bi</mi></msub></mfrac><mo>⋅</mo><mfrac><mn>1</mn><msub><mi>R</mi><mi mathvariant="italic">Poiseuille</mi></msub></mfrac></math><img file="EP1590018B1_D0009.tif" /></maths> in which <ul id="ul0004" list-style="none"><li>R_ail = resistance of the arterial needle at flow rate q_b = q_bi and with hematocrit Hct = Hct₁</li><li>R_vil = resistance of the venous needle at flow rate q_b = q_bi and with hematocrit Hct = Hct₁</li><li>ΔP_ail = pressure drop on the arterial needle at flow rate q_b =q_bi and with hematocrit Hct = Hct₁.</li><li>ΔP_vil = pressure drop on the venous needle at flow rate q_b = q_bi and with hematocrit Hct = Hct₁.</li></ul>

0100Hence we obtain two series of values R_ail and R_vil of resistances (one arterial and the other venous) corresponding to a determined hematocrit value (in the example Hct = Hct₁), with i = 0, 1, 2, ..., N, with N = number of times we determine ΔP_ail and ΔP_vil at different flow rates q_bi.

0101Each of the two series of values (R_a and R_v) is interpolated by the processor using a second-order polynomial: <maths id="math0010"><math display="block"><mi>R</mi><mo>=</mo><msub><mi>A</mi><mn>2</mn></msub><mo>⁢</mo><msubsup><mi>q</mi><mi>b</mi><mn>2</mn></msubsup><mo>+</mo><msub><mi>A</mi><mn>1</mn></msub><mo>⁢</mo><msub><mi>q</mi><mi>b</mi></msub><mo>+</mo><msub><mi>b</mi><mn>1</mn></msub></math><img file="EP1590018B1_D0010.tif" /></maths> and we thus obtain, for each type of needle, a pair of coefficients A₂ and A₁ for each flow direction (i.e. we obtain a pair of coefficients which characterise the arterial needle and a pair of coefficients which characterise the venous needle). Coefficient b₁ depends on the blood hematocrit value.

0102Coefficients B₂, B₁ and B₀ are obtained as follows.

0103Let us for a moment consider only one blood flow direction through the needle: for example, the arterial needle.

0104The processor also calculates the resistances R_a of the arterial needle for the other hematocrit values Hct = Hct_j (j = 2,..., M), at different blood flow rates q_b=q_bi (i = 0, 1, 2,..., N), thus obtaining various series of values: <maths id="math0011"><math display="block"><msub><mi>R</mi><mi mathvariant="italic">aij</mi></msub><mo>=</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi>P</mi><mi mathvariant="italic">aij</mi></msub></mrow><msub><mi>q</mi><mi mathvariant="italic">bi</mi></msub></mfrac><mo>⋅</mo><mfrac><mn>1</mn><msub><mi>R</mi><mi mathvariant="italic">Poiseuille</mi></msub></mfrac></math><img file="EP1590018B1_D0011.tif" /></maths>

0105These values of R_a are interpolated, for each hematocrit value Hct, according to the blood flow rate q_b, using a second order polynomial: <maths id="math0012"><math display="block"><mtable columnalign="left"><mtr><mtd><msub><mi>R</mi><mrow><mi mathvariant="italic">ai</mi><mo>⁢</mo><mn>2</mn></mrow></msub><mo>=</mo><msub><mi>A</mi><mn>2</mn></msub><mo>⁢</mo><msubsup><mi>q</mi><mi>b</mi><mn>2</mn></msubsup><mo>+</mo><msub><mi>A</mi><mn>1</mn></msub><mo>⁢</mo><msub><mi>q</mi><mi>b</mi></msub><mo>+</mo><msub><mi>b</mi><mn>2</mn></msub><mspace width="2em" /><mi>for Hct</mi><mo>=</mo><msub><mi>Hct</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>R</mi><mrow><mi mathvariant="italic">ai</mi><mo>⁢</mo><mn>3</mn></mrow></msub><mo>=</mo><msub><mi>A</mi><mn>2</mn></msub><mo>⁢</mo><msubsup><mi>q</mi><mi>b</mi><mn>2</mn></msubsup><mo>+</mo><msub><mi>A</mi><mn>1</mn></msub><mo>⁢</mo><msub><mi>q</mi><mi>b</mi></msub><mo>+</mo><msub><mi>b</mi><mn>3</mn></msub><mspace width="2em" /><mi>for Hct</mi><mo>=</mo><msub><mi>Hct</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd><mo>⋯</mo></mtd></mtr><mtr><mtd><msub><mi>R</mi><mi mathvariant="italic">aiM</mi></msub><mo>=</mo><msub><mi>A</mi><mn>2</mn></msub><mo>⁢</mo><msubsup><mi>q</mi><mi>b</mi><mn>2</mn></msubsup><mo>+</mo><msub><mi>A</mi><mn>1</mn></msub><mo>⁢</mo><msub><mi>q</mi><mi>b</mi></msub><mo>+</mo><msub><mi>b</mi><mi>M</mi></msub><mspace width="2em" /><mi>for Hct</mi><mo>=</mo><msub><mi>Hct</mi><mi mathvariant="normal">M</mi></msub></mtd></mtr></mtable></math><img file="EP1590018B1_D0012.tif" /></maths> with i = 0, 1, 2, ..., N (with N ≥ 3), in order to obtain a series of values b_j (j = 1, 2,..., M).

0106In substance, exemplifying the above-mentioned process step by step, for j = 1 the processor interpolates values R_ail (for Hct = Hct₁) according to the equation <maths id="math0013"><math display="block"><msub><mi>R</mi><mrow><mi mathvariant="italic">ai</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>A</mi><mn>2</mn></msub><mo>⁢</mo><msubsup><mi>q</mi><mi>b</mi><mn>2</mn></msubsup><mo>+</mo><msub><mi>A</mi><mn>1</mn></msub><mo>⁢</mo><msub><mi>q</mi><mi>b</mi></msub><mo>+</mo><msub><mi>b</mi><mn>1</mn></msub></math><img file="EP1590018B1_D0013.tif" /></maths> and thus determines b₁.

0107Then it interpolates values R_ai2 for j = 2 (for Hct = Hct₂) following the equation <maths id="math0014"><math display="block"><msub><mi>R</mi><mrow><mi mathvariant="italic">ai</mi><mo>⁢</mo><mn>2</mn></mrow></msub><mo>=</mo><msub><mi>A</mi><mn>2</mn></msub><mo>⁢</mo><msubsup><mi>q</mi><mi>b</mi><mn>2</mn></msubsup><mo>+</mo><msub><mi>A</mi><mn>1</mn></msub><mo>⁢</mo><msub><mi>q</mi><mi>b</mi></msub><mo>+</mo><msub><mi>b</mi><mn>2</mn></msub></math><img file="EP1590018B1_D0014.tif" /></maths> and determines b₂, and so on up until j = M, thus obtaining M values of b_j.

0108At this point the processor makes a further interpolation, using the values of b_j according to the equation <maths id="math0015"><math display="block"><mi>b</mi><mo>=</mo><msub><mi>B</mi><mn>2</mn></msub><mo>⁢</mo><msup><mi mathvariant="italic">Hct</mi><mn>2</mn></msup><mo>+</mo><msub><mi>B</mi><mn>1</mn></msub><mo>⁢</mo><mi mathvariant="italic">Hct</mi><mo>+</mo><msub><mi>B</mi><mn>0</mn></msub></math><img file="EP1590018B1_D0015.tif" /></maths> and thus determines coefficients B₂, B₁ and B₀.

0109The same series of interpolations is effected using the data relating to the venous needle.

0110Hereafter we report some examples of values of the coefficients A₂, A₁, B₂, B₁ and B₀ experimentally obtained.

0111With a needle having the following characteristics: gauge = 15 (internal diameter = 1.6mm), length = 28mm, the following is obtained: <ul id="ul0005" list-style="none"><li>A₂ (arterial) = -0.00004, A₁(arterial) = 0.0351,</li><li>B₂ (arterial) = 0.0192, B₁(arterial) = -0.9398,</li><li>B₀(arterial) = 21.059, R_poiseuille = 0.022</li><li>A₂ (venous) -0.000026, A₁(venous) = 0.0266, B₂(venous) = 0.0403, B₁(venous) = -2.2937, B₀(venous) = 41.969, Rp_oiseuille = 0.022</li></ul>

0112With a needle having the following characteristics: gauge = 16 (internal diameter = 1.4mm), length = 33mm, the following is obtained: <ul id="ul0006" list-style="none"><li>A₂ (arterial) = -0.00004375, A₁(arterial) = 0.0309,</li><li>B₂ (arterial) = 0.0081, B₁(arterial) = -0.3226,</li><li>B₀(arterial) = 8.3882, R_Poiseuille = 0.0442</li><li>A₂ (venous) = -0.00002875, A₁(venous) = 0.0193,</li><li>B₂(venous) = 0.0037, B₁(venous) = 0.0487,</li><li>B₀(venous) = 1.4565, R_Poiseuille = 0.0442.</li></ul>

0113The control and calculation unit 17 memory is preloaded with the coefficient values A₂, A₁, B₂, B₁ and B₀ of the most commonly used needles (the memory contains two series of coefficients for each needle, one for each blood flow direction, i.e. a series relating to a needle's use as an arterial needle and as a venous needle). The control and calculation unit 17 recognises the needle used in the extracorporeal treatment time by time and consequently in the calculation of P_af and P_vf uses the coefficients relating to the needle being used. Recognition of the needle can be automatic (for example by means of an identification system associated to the needle) or can be user-guided.

0114Thus a mathematical model is defined, usable by the control and calculation unit 17 for determining the pressures in the vascular access by measuring the pressure in the extracorporeal circuit.

0115Herein below some operative methods are defined by means of which a processor in the control and calculation unit 17 of the machine can monitor the vascular access during an extracorporeal treatment.

<u>First monitoring procedure.</u>

0116In this first operative mode q_b is varied at q_uf = constant (= 0), while P_am and P_vm are measured.

0117The operative mode is now described step by step. <ol id="ol0001"><li>a. Determine values P_af1 and P_vf1 of the arterial pressure and, respectively, the venous pressure in the vascular access (fistula) at a known blood pump flow rate q_bl.</li><li>b. Save and store values q_bl, P_af1 and P_vf1 in a memory.</li><li>c. Change the blood pump f low rate to a known value q_b2. At the same time the <u>ultrafiltration</u> flow rate q_uf is kept constant.</li><li>d. Keep the blood pump flow rate at q_b2 for a determined period of time (for example about ten seconds) to let the system become stable.</li><li>e. Determine values P_af2 and P_vf2 of the arterial pressure and, respectively, of the venous pressure in the vascular access (fistula) at blood pump flow rate q_b2.</li><li>f. Save and store values q_b2, P_af2 and P_vf2.</li><li>g. Steps c-f can be repeated for a desired number of times so as to save and store a series of values q_bi, p_afi, P_vfi, with i = 1, 2, 3, ..., N, where N is an integer number greater than 1.</li><li>h. Calculate R_f and q_a using the values stored in the memory and the mathematical model expressed by the equation <maths id="math0016"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>=</mo><msub><mi>R</mi><mi>f</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mi>b</mi></msub></mfenced></math><img file="EP1590018B1_D0016.tif" /></maths></li><li>i. Save and store the values calculated for R_f and q_a.</li><li>j. Calculate R_v using at least a part of the stored values and the mathematical model expressed by the equation <maths id="math0017"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">v</mi></msub><mo>=</mo><msub><mi>R</mi><mi>v</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mi mathvariant="italic">uf</mi></msub></mfenced></math><img file="EP1590018B1_D0017.tif" /></maths></li><li>k. Save and store the calculated value for R_v.</li><li>l. Calculate R_d using at least a part of the stored values and the mathematical model expressed in the equation <maths id="math0018"><math display="block"><msub><mi>q</mi><mi>a</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mi>a</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub></mrow><msub><mi>R</mi><mi>d</mi></msub></mfrac></math><img file="EP1590018B1_D0018.tif" /></maths> where P_a (mean systemic arterial pressure or MAP) is measured at the patient's arm in known ways and the measured value of P_a is transmitted to the control and calculation unit 17.</li><li>m. Save and store the value calculated for R_d.</li></ol>

0118The calculation of R_f and q_a in point h can be done in the following way.

0119The stored values of q_bi, P_afi and P_vfi, with i = 1, 2, ..., N (with N ≥ 2), are introduced into the equation <maths id="math0019"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>=</mo><msub><mi>R</mi><mi>f</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mi>b</mi></msub></mfenced></math><img file="EP1590018B1_D0019.tif" /></maths> so as to obtain a system of N equations with 2 unknowns q_a and R_f. <maths id="math0020"><math display="block"><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi>P</mi><mrow><mi>f</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>R</mi><mi>f</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mrow><mi>b</mi><mo>⁢</mo><mn>1</mn></mrow></msub></mfenced></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi>P</mi><mrow><mi>f</mi><mo>⁢</mo><mn>2</mn></mrow></msub><mo>=</mo><msub><mi>R</mi><mi>f</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mrow><mi>b</mi><mo>⁢</mo><mn>2</mn></mrow></msub></mfenced></mtd></mtr><mtr><mtd><mo>⋯</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi>P</mi><mi mathvariant="italic">fN</mi></msub><mo>=</mo><msub><mi>R</mi><mi>f</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mi mathvariant="italic">bN</mi></msub></mfenced></mtd></mtr></mtable></math><img file="EP1590018B1_D0020.tif" /></maths> where ΔP_fi = P_afi - P_vfi with i = 1, 2, ..., N (N ≥ 2)

0120The unknown quantities q_a and R_f can be determined by calculating the optimal solution of the above-indicated equation system.

0121If N = 2 the system has an analytical solution.

0122If N > 2 the two unknowns q_a and R_f can be determined using an optimisation algorithm.

0123For example the processor calculates the two values, one q_a and the other R_f, for which the corresponding values of ΔP_f calculated by the above-indicated system of equations are the closest to the ΔP_fi values previously determined at point e.

0124The following calculation procedure can be used. Using the values stored in memory, q_bi, p_afi and P_vfi, by means of a mathematical interpolation algorithm previously stored in memory the processor determines a linear equation which approximates the relation between ΔP_f and q_b. Then the value of q_b at ΔP_f = 0 is calculated, using the above-indicated linear equation. The value of q_b at ΔP_f = 0 is assumed to be equal to the flow rate q_a of the vascular access. The value of q_a thus determined is stored in memory. Further, the processor calculates the value assumed by ΔP_f at q_b = 0, once more using the same linear equation. The value of ΔP_f at q_b = 0 is assumed to be equal to the product of R_f·q_a. At this point, using the previously-stored value of q_a the value of R_f can be calculated with a simple quotient.

0125Graph ΔP_f - q_b of <figref idref="f0003">figure 2</figref> illustrates this mode of procedure. The points in <figref idref="f0003">figure 2</figref> represent the determined values ΔP_fi of ΔP_f according to the blood pump flow rate q_b. The straight line interpolating the various points is the graphic representation of the linear mathematical relation which connects ΔP_f with q_b.

0126The interpolation method can be any known linear interpolation method. The straight line of interpolation intersects the horizontal axis (q_b) at q_a and the vertical axis (ΔP_f) at R_f·q_a.

0127Another way of calculating q_a and R_f is based on the description of the relation between q_b and ΔP_f using a non-linear mathematical relation (for example a polynomial of a degree greater than one), derived by the processor with an interpolation method using the values stored in the memory q_bi, P_afi e P_vfi. After having derived this non-linear relation, the value assumed by q_b at ΔP_f = 0 is assumed to be equal to the flow rate q_a of the vascular access. The value of q_a thus determined is stored in memory. Further, the processor calculates the value assumed by ΔP_f at q_b = 0, using the above-cited non-linear equation as well. The value of ΔP_f at q_b = 0 is assumed to be equal to the product of R_f·q_a. At this point, using the previously-stored value of q_a it is possible to calculate, by a simple division, the value of R_f. This value represents, in the embodiment, the value of hydraulic resistance R_f at point q_b = 0 (i.e. at zero blood flow rate in the extracorporeal circuit).

0128At point c., the blood pump flow rate is varied from q_b1 to q_b2 so that, in consequence of the change of flow rate q_b2 - q_b1, the pressure difference ΔP_f = P_af - P_vf varies significantly in absolute value and sufficiently to be appreciated (for example at least 2 mmHg), i.e. so that <maths id="math0021"><math display="block"><mfenced open="|" close="|"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">P</mi><mrow><mi mathvariant="normal">f</mi><mo>⁢</mo><mn mathvariant="normal">1</mn></mrow></msub><mo mathvariant="normal">-</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">P</mi><mrow><mi mathvariant="normal">f</mi><mo>⁢</mo><mn mathvariant="normal">2</mn></mrow></msub></mfenced><mo mathvariant="normal">≥</mo><mn mathvariant="normal">2</mn><mspace width="1em" /><mi>mmHg</mi><mo mathvariant="normal">,</mo></math><img file="EP1590018B1_D0021.tif" /></maths> where <maths id="math0022"><math display="block"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">P</mi><mrow><mi mathvariant="normal">f</mi><mo>⁢</mo><mn mathvariant="normal">1</mn></mrow></msub><mo mathvariant="normal">=</mo><msub><mi mathvariant="normal">P</mi><mrow><mi>af</mi><mo>⁢</mo><mn mathvariant="normal">1</mn></mrow></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="normal">P</mi><mrow><mi>vf</mi><mo>⁢</mo><mn mathvariant="normal">1</mn></mrow></msub></math><img file="EP1590018B1_D0022.tif" /></maths> and <maths id="math0023"><math display="block"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">P</mi><mrow><mi mathvariant="normal">f</mi><mo>⁢</mo><mn>2</mn></mrow></msub><mo mathvariant="normal">=</mo><msub><mi mathvariant="normal">P</mi><mrow><mi>af</mi><mo>⁢</mo><mn>2</mn></mrow></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="normal">P</mi><mrow><mi>vf</mi><mo>⁢</mo><mn>2</mn></mrow></msub></math><img file="EP1590018B1_D0023.tif" /></maths>

0129The same occurs for each flow rate change from q_bi to q_b(i+1). The values of q_bi are selected so that the difference between the minimum value and the maximum value of q_bi does not exceed a predefined value (for example about 600 ml/min) in order that q_a and R_f can be considered as constant in the calculation with good approximation.

0130At point c. the ultrafiltration flow rate q_uf is kept constant = 0.

0131At point j. the resistance R_v is calculated assuming q_uf = 0. The R_v stored in memory can be one of the estimated R_vi or the mean value of the estimated R_vi. <maths id="math0024"><math display="block"><msub><mi>R</mi><mi mathvariant="italic">vi</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mi mathvariant="italic">vfi</mi></msub><mo>-</mo><msub><mi>P</mi><mi>v</mi></msub></mrow><msub><mi>q</mi><mi>a</mi></msub></mfrac></math><img file="EP1590018B1_D0024.tif" /></maths>

0132At point 1. the resistance R_d stored in the memory can be one of the R_di calculated with equation (1) or the mean value of the calculated R_di. <maths id="math0025"><math display="block"><msub><mi>R</mi><mi mathvariant="italic">di</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mi>a</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">afi</mi></msub></mrow><msub><mi>q</mi><mi>a</mi></msub></mfrac></math><img file="EP1590018B1_D0025.tif" /></maths>

<u>Second monitoring procedure.</u>

0133In the second operative mode q_uf is changed to q_b = constant (not zero), while P_am and P_vm are measured.

0134The operative mode is now described step by step. <ol id="ol0002"><li>a. Determine values P_afl and P_vf1 of the arterial pressure and, respectively, of the venous pressure in the vascular access (fistula) at a known ultrafiltration flow rate q_uf1 at a predetermined blood pump flow rate q_b.</li><li>b. Save and store values q_uf1, P_afl and P_vf1.</li><li>c. Change the ultrafiltration flow rate to a known value q_uf2. At the same time the blood pump flow rate q_b is kept constant and equal to the initial flow rate of point a..</li><li>d. Keep the ultrafiltration pump flow rate at value q_uf2 for a determined period of time (for example about ten seconds) to let the system become stable.</li><li>e. Determine values P_af2 and P_vf2 of the arterial pressure and, respectively, the venous pressure in the vascular access (fistula) at ultrafiltration flow rate q_uf2 of the blood pump.</li><li>f. Save and store values q_uf2, P_af2 and P_vf2.</li><li>g. Steps c-f can be repeated for a desired number of times so as to save and store a series of values q_ufi, P_afi, P_vfi, with i ₌ 1, 2, 3, ..., N, where N is an integer number greater than 1.</li><li>h. Calculate q_a and R_v using the values stored in the memory and the mathematical model expressed in the equation <maths id="math0026"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">v</mi></msub><mo>=</mo><msub><mi>R</mi><mi>v</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mi mathvariant="italic">uf</mi></msub></mfenced></math><img file="EP1590018B1_D0026.tif" /></maths></li><li>i. Save and store the values calculated for R_v and q_a.</li><li>j. Calculate R_f using at least a part of the stored values and the mathematical model expressed in the equation <maths id="math0027"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>=</mo><msub><mi>R</mi><mi>f</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mi>b</mi></msub></mfenced></math><img file="EP1590018B1_D0027.tif" /></maths></li><li>k. Save and store the calculated value for R_f.</li><li>l. Calculate R_d using at least a part of the stored values and the mathematical model expressed in the equation <maths id="math0028"><math display="block"><msub><mi>q</mi><mi>a</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mi>a</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub></mrow><msub><mi>R</mi><mi>d</mi></msub></mfrac></math><img file="EP1590018B1_D0028.tif" /></maths></li><li>m. Save and store the value calculated for R_d.</li></ol>

0135At point c., the ultrafiltration flow rate is changed from q_uf1 to q_uf2 so that, in consequence of the change in flow rate q_uf2 - q_uf1, the difference of pressure ΔP_vf = P_vf - P_v significantly varies in absolute terms sufficiently to be appreciated (for example at least 3 mmHg), i.e. so that <maths id="math0029"><math display="block"><mfenced open="|" close="|"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">P</mi><mrow><mi>vf</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo mathvariant="normal">-</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">P</mi><mrow><mi>vf</mi><mo>⁢</mo><mn mathvariant="normal">2</mn></mrow></msub></mfenced><mo mathvariant="normal">≥</mo><mn mathvariant="normal">3</mn><mspace width="1em" /><mi>mmHg</mi><mo mathvariant="normal">,</mo></math><img file="EP1590018B1_D0029.tif" /></maths> where <maths id="math0030"><math display="block"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">P</mi><mrow><mi>vf</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo mathvariant="normal">=</mo><msub><mi mathvariant="normal">P</mi><mrow><mi>vf</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">v</mi></msub></math><img file="EP1590018B1_D0030.tif" /></maths> and <maths id="math0031"><math display="block"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msub><mi mathvariant="normal">P</mi><mrow><mi>vf</mi><mo>⁢</mo><mn>2</mn></mrow></msub><mo mathvariant="normal">=</mo><msub><mi mathvariant="normal">P</mi><mrow><mi>vf</mi><mo>⁢</mo><mn>2</mn></mrow></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">v</mi></msub></math><img file="EP1590018B1_D0031.tif" /></maths>

0136The same can be said for each flow rate change from q_ufi to q_uf(i+1).

0137At point c. the blood flow rate in the extracorporeal circuit q_b is kept constant at a known value which is not zero.

0138At point h. the calculation of R_v and q_a is performed in the following way.

0139The stored values of q_ufi, P_afi and P_vfi, with i = 1, 2, ..., N (with N ≥ 2), are introduced in the equation <maths id="math0032"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">v</mi></msub><mo>=</mo><msub><mi>R</mi><mi>v</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mi mathvariant="italic">uf</mi></msub></mfenced></math><img file="EP1590018B1_D0032.tif" /></maths> so as to obtain a system of N equations with 2 unknown quantities q_a and R_v. <maths id="math0033"><math display="block"><mtable columnalign="left"><mtr><mtd><msub><mi>P</mi><mrow><mi mathvariant="italic">vf</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>P</mi><mi>v</mi></msub><mo>=</mo><msub><mi>R</mi><mi>v</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mrow><mi mathvariant="italic">uf</mi><mo>⁢</mo><mn>1</mn></mrow></msub></mfenced></mtd></mtr><mtr><mtd><msub><mi>P</mi><mrow><mi mathvariant="italic">vf</mi><mo>⁢</mo><mn>2</mn></mrow></msub><mo>-</mo><msub><mi>P</mi><mi>v</mi></msub><mo>=</mo><msub><mi>R</mi><mi>v</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mrow><mi mathvariant="italic">uf</mi><mo>⁢</mo><mn>2</mn></mrow></msub></mfenced></mtd></mtr><mtr><mtd><mo>⋯</mo></mtd></mtr><mtr><mtd><msub><mi>P</mi><mi mathvariant="italic">vfN</mi></msub><mo>-</mo><msub><mi>P</mi><mi>v</mi></msub><mo>=</mo><msub><mi>R</mi><mi>v</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mi mathvariant="italic">ufN</mi></msub></mfenced></mtd></mtr></mtable></math><img file="EP1590018B1_D0033.tif" /></maths>

0140The unknown quantities q_a and R_v can be determined by calculating the optimal solution of the above-indicated equation system.

0141If N = 2 the system has an analytical solution.

0142If N > 2 the two unknowns q_a and R_v can be determined using an optimization algorithm.

0143A calculation procedure which can be used is the following. Using the values stored in memory, q_ufi and P_vfi, the processor determines, by means of a mathematical interpolation algorithm previously stored in memory, a linear equation which approximates the relation between ΔP_vf and q_uf where ΔP_vf = P_vf - P_v. Then the value assumed by q_uf at P_vf - P_v = 0 is calculated, using the above-indicated linear equation. The value of q_uf at ΔP_vf = 0 is assumed to be equal to the flow rate q_a of the vascular access. The value of q_a thus determined is stored in memory. Further, the processor calculates the value assumed by ΔP_vf at q_uf = 0, once more using the same linear equation. The value of ΔP_vf, at q_uf = 0 is assumed to be equal to the product of R_v·q_a. At this point, using the previously-stored value of q_a the value of R_v can be calculated by a simple division.

0144The plot of ΔP_vf as a function of q_uf in <figref idref="f0003">figure 5</figref> illustrates this mode of procedure. The points in <figref idref="f0003">figure 5</figref> represent the determined values ΔP_vfi = P_vfi - P_v of ΔP_vf as functions of the ultrafiltration pump flow rate q_uf. The straight line interpolating the various points is the graphic representation of the linear mathematical relation which connects ΔP_f with q_uf. The interpolation method can be any known linear interpolation method. The straight interpolating line intersects the horizontal axis q_uf at q_a and the vertical axis of ΔP_vf at R_v·q_a.

0145At point j. (determination of R_f) the following procedure is observed.

0146For each of the estimated values of P_afi and P_vfi, a corresponding value of R_fi is calculated using the above-indicated equation, from which it is obtained: <maths id="math0034"><math display="block"><msub><mi>R</mi><mi mathvariant="italic">fi</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mi mathvariant="italic">afi</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">vfi</mi></msub></mrow><mrow><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mrow><mi>b</mi><mo>⁢</mo><mn>1</mn></mrow></msub></mrow></mfrac></math><img file="EP1590018B1_D0034.tif" /></maths>

0147The R_f value stored at point k. can be one of the calculated values for R_fi or the mean value of the R_fi values.

0148At point 1. (determination of R_d) the following procedure is observed.

0149For each of the estimated values of P_af, a corresponding value of R_di is calculated using the above-indicated equation: <maths id="math0035"><math display="block"><msub><mi>R</mi><mi mathvariant="italic">di</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mi>a</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">afi</mi></msub></mrow><msub><mi>q</mi><mi>a</mi></msub></mfrac></math><img file="EP1590018B1_D0035.tif" /></maths>

0150The R_d value stored at point 1. can be one of the calculated values R_di or the mean value of the R_di values.

<u>Third monitoring procedure.</u>

0151The equations which define the mathematical model of the vascular access used previously: <maths id="math0036"><math display="block"><msub><mi>q</mi><mi>a</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mi>a</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub></mrow><msub><mi>R</mi><mi>d</mi></msub></mfrac></math><img file="EP1590018B1_D0036.tif" /></maths><maths id="math0037"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>=</mo><msub><mi>R</mi><mi>f</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mi>b</mi></msub></mfenced></math><img file="EP1590018B1_D0037.tif" /></maths><maths id="math0038"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>-</mo><msub><mi>P</mi><mi>v</mi></msub><mo>=</mo><msub><mi>R</mi><mi>v</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mi mathvariant="italic">uf</mi></msub></mfenced></math><img file="EP1590018B1_D0038.tif" /></maths> can be reformulated so as to evidence the dependence of P_af and P_vf on P_a, q_b, q_uf and P_v through the unknown parameters R_d, R_f and R_v. The reformulated equations are as follow: <maths id="math0039"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mrow><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mrow></mfrac><mo>⋅</mo><msub><mi>P</mi><mi>a</mi></msub><mo>-</mo><mfrac><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>⋅</mo><msub><mi>R</mi><mi>f</mi></msub></mrow><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mrow></mfrac><mo>⋅</mo><msub><mi>q</mi><mi>b</mi></msub><mo>-</mo><mfrac><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>⋅</mo><msub><mi>R</mi><mi>v</mi></msub></mrow><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mrow></mfrac><mo>⋅</mo><msub><mi>q</mi><mi mathvariant="italic">uf</mi></msub><mo>+</mo><mfrac><msub><mi>R</mi><mi>d</mi></msub><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mrow></mfrac><mo>⋅</mo><msub><mi>P</mi><mi>v</mi></msub></math><img file="EP1590018B1_D0039.tif" /></maths><maths id="math0040"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>=</mo><mfrac><msub><mi>R</mi><mi>v</mi></msub><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mrow></mfrac><mo>⋅</mo><msub><mi>P</mi><mi>a</mi></msub><mo>+</mo><mfrac><mrow><msub><mi>R</mi><mi>f</mi></msub><mo>⋅</mo><msub><mi>R</mi><mi>v</mi></msub></mrow><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mrow></mfrac><mo>⋅</mo><msub><mi>q</mi><mi>b</mi></msub><mo>-</mo><mfrac><mrow><msub><mi>R</mi><mi>v</mi></msub><mo>⋅</mo><mfenced><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub></mfenced></mrow><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mrow></mfrac><mo>⋅</mo><msub><mi>q</mi><mi mathvariant="italic">uf</mi></msub><mo>+</mo><mfrac><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub></mrow><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mrow></mfrac><mo>⋅</mo><msub><mi>P</mi><mi>v</mi></msub></math><img file="EP1590018B1_D0040.tif" /></maths>

0152These equations can be rewritten as reported herein below. <maths id="math0041"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub><mo>=</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>⋅</mo><msub><mi>P</mi><mi>a</mi></msub><mo>+</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi>b</mi></msub><mo>+</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>2</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi mathvariant="italic">uf</mi></msub><mo>+</mo><mfenced><mn>1</mn><mo>-</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mfenced><mo>⋅</mo><msub><mi>P</mi><mi>v</mi></msub></math><img file="EP1590018B1_D0041.tif" /></maths><maths id="math0042"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>=</mo><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>⋅</mo><msub><mi>P</mi><mi>a</mi></msub><mo>+</mo><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi>b</mi></msub><mo>+</mo><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>2</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi mathvariant="italic">uf</mi></msub><mo>+</mo><mfenced><mn>1</mn><mo>-</mo><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mfenced><mo>⋅</mo><msub><mi>P</mi><mi>v</mi></msub></math><img file="EP1590018B1_D0042.tif" /></maths> in which: <maths id="math0043"><math display="block"><msub><mi>c</mi><mrow><mi mathvariant="italic">a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mrow><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mrow></mfrac></math><img file="EP1590018B1_D0043.tif" /></maths><maths id="math0044"><math display="block"><msub><mi>c</mi><mrow><mi mathvariant="italic">a</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>⋅</mo><msub><mi>R</mi><mi>f</mi></msub></mrow><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mrow></mfrac></math><img file="EP1590018B1_D0044.tif" /></maths><maths id="math0045"><math display="block"><msub><mi>c</mi><mrow><mi mathvariant="italic">a</mi><mo>⁢</mo><mn>2</mn></mrow></msub><mo>=</mo><mo>-</mo><mfrac><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>⋅</mo><msub><mi>R</mi><mi>v</mi></msub></mrow><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mrow></mfrac></math><img file="EP1590018B1_D0045.tif" /></maths><maths id="math0046"><math display="block"><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>=</mo><mfrac><msub><mi>R</mi><mi>v</mi></msub><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mrow></mfrac></math><img file="EP1590018B1_D0046.tif" /></maths><maths id="math0047"><math display="block"><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>R</mi><mi>f</mi></msub><mo>⋅</mo><msub><mi>R</mi><mi>v</mi></msub></mrow><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mrow></mfrac></math><img file="EP1590018B1_D0047.tif" /></maths><maths id="math0048"><math display="block"><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>R</mi><mi>v</mi></msub><mo>⋅</mo><mfenced><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub></mfenced></mrow><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mrow></mfrac></math><img file="EP1590018B1_D0048.tif" /></maths>

0153The third operating mode (as the following fourth and fifth operating modes) calculates at least a part of the coefficients c_a0, c_a1, c_a2 and c_v0, c_v1, c_v2 and from these derives R_d, R_f and R_v. The calculation of the coefficients is done starting from one or more known values for each of the following quantities: P_a, q_b, q_uf, P_v, P_af and P_vf. The quantities P_a, q_b, q_uf, P_v are known through measurement. The quantities P_af and P_vf are known by direct measurement of the pressures in the vascular access, or by a process of calculation starting from the measurement of the pressures in the machine P_am and P_vm.

0154As the number of coefficients c_a0, c_a1, c_a2, c_v0, c_v1, c_v2 is greater than the number of the resistances R_d, R_f and R_v, there exists a multiplicity of relations between the coefficients and the resistances. In general, knowledge of three coefficients enables a determination of the resistances.

0155In the third operating mode both flow rates q_b and q_uf are varied and the arterial pressure in the machine P_am is measured, from which arterial pressure in the vascular access P_af is calculated.

0156In a specific embodiment in a first stage the pressure P_am at flow rates q_b = 0 and q_uf = 0 is measured; in a second stage pressure P_am at flow rates q_b ≠ 0 and q_uf = 0 is measured; in a third stage pressure P_am at flow rates q_b ≠ 0 and q_uf ≠ 0 is measured.

0157More in general, q_b at q_uf = constant (for example = 0) is varied and P_am is measured at different values of q_b. Thereafter q_uf at q_b = constant (for example ≠ 0) is varied and P_am measured at different values of q_uf.

0158In this third operating mode a mathematical model of the vascular access is used which is represented by one equation only: <maths id="math0049"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub><mo>=</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>⋅</mo><msub><mi>P</mi><mi>a</mi></msub><mo>+</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi>b</mi></msub><mo>+</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>2</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi mathvariant="italic">uf</mi></msub><mo>+</mo><mfenced><mn>1</mn><mo>-</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mfenced><mo>⋅</mo><msub><mi>P</mi><mi>v</mi></msub></math><img file="EP1590018B1_D0049.tif" /></maths> from which coefficients c_a0, c_a1, c_a2 can be derived, which are sufficient by themselves for the calculation of the three resistances R_d, R_f, R_v.

0159In this third operating mode at least one measurement is taken of the patient's arterial pressure P_a. Further, distal venous pressure P_v is assumed to be zero; for this reason the equation used is simplified as follows: <maths id="math0050"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub><mo>=</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>⋅</mo><msub><mi>P</mi><mi>a</mi></msub><mo>+</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi>b</mi></msub><mo>+</mo><msub><mi>c</mi><mrow><mi>c</mi><mo>⁢</mo><mn>2</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi mathvariant="italic">uf</mi></msub></math><img file="EP1590018B1_D0050.tif" /></maths>

0160The third operating mode is now described step by step. <ol id="ol0003" compact="compact"><li>a. Determine values P_af0 of the arterial pressure in the vascular access (fistula) and the systemic arterial pressure of the patient P_a0 at a known ultrafiltration flow rate q_uf1 = 0 at a predetermined blood pump flow rate q_b = 0.</li><li>b. Save and store values P_a0 and P_af0.</li><li>c. Calculate c_a0 by means of the equation <maths id="math0051"><math display="block"><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>=</mo><mfrac><msub><mi>P</mi><mrow><mi mathvariant="italic">af</mi><mo>⁢</mo><mn>0</mn></mrow></msub><msub><mi>P</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mfrac></math><img file="EP1590018B1_D0051.tif" /></maths></li><li>d. Save and store value c_a0.</li><li>e. Change the blood flow rate q_b to a known value q_b1. At the same time the ultrafiltration flow rate q_uf is kept constant and equal to the flow rate at point a. (= 0).</li><li>f. Determine values P_af1 and P_a1 of the arterial pressure in the vascular access (fistula) and, respectively, of the patient at blood pump flow rate q_b1.</li><li>g. Save and store values q_b1, P_af1 and P_a1.</li><li>h. Steps d-f can be repeated for a desired number of times so as to save and store a series of values q_bi, P_afi, P_ai, with i = 1, 2, 3, ..., N, where N is an integer number greater than or equal to 1.</li><li>i. Determine c_a1 by solving the system of equations: <maths id="math0052"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">afi</mi></msub><mo>-</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>⋅</mo><msub><mi>P</mi><mi mathvariant="italic">ai</mi></msub><mo>=</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi mathvariant="italic">bi</mi></msub></math><img file="EP1590018B1_D0052.tif" /></maths> with i = 1, ..., N (N ≥ 1) If N = 1 it is sufficient to solve a linear equation with only an unknown quantity. If N > 1 the value of c_a1 is found by means of an optimisation algorithm which determines the optimal solution for the above-cited system. The searched-for value can be the value of c_a1 which minimises the error between the values of P_af calculated with the above system of equations, P_afi*, where the asterisk * indicates that the value has been calculated, and the P_afi values determined by measuring a pressure correlated with P_af. The optimisation algorithm can be, for example, a linear regression algorithm.</li><li>j. Save and store value c_a1.</li><li>k. Change the ultrafiltration flow rate to a known value q_uf1 not zero. At the same time the blood flow rate q_b has a known value q_bk different to zero.</li><li>l. Determine values P_af1 and P_a1 of the arterial pressure in the vascular access (fistula) and, respectively, of the patient at ultrafiltration flow rate q_uf1.</li><li>m. Save and store values q_bk, q_uf1, P_af1 and P_a1.</li><li>n. Steps k-m can be repeated for a desired number of times in order to store a series of values q_ufj, P_afj, P_aj, with j = 1, 2, ..., M, where M is an integer number equal to or greater than 1.</li><li>o. Determine c_a2 by solving the following system of equations <maths id="math0053"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">afj</mi></msub><mo>-</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>⋅</mo><msub><mi>P</mi><mi mathvariant="italic">aj</mi></msub><mo>-</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi mathvariant="italic">bk</mi></msub><mo>=</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>2</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi mathvariant="italic">ufj</mi></msub></math><img file="EP1590018B1_D0053.tif" /></maths> with j = 1, 2,..., M (M ≥ 1) If M = 1 it is sufficient to solve a linear equation with only an unknown quantity. If M > 1 the value of c_a2 is found by means of an algorithm of optimisation which determines the optimal solution for the above system. The sought-after value can be the value of c_a2 which minimises the error between the values of P_af calculated using the system of equations P_afj*, where the asterisk * indicates that the value is a calculated one, and the values of P_afj determined through measuring a pressure correlated by P_af. The optimisation algorithm can be, for example, a linear regression algorithm (as at point i. above).</li><li>p. Save and store the determined value of c_a2.</li><li>q. Determine R_f, R_v and R_d by solving the following system of equations which empress the relation between c_a0, c_a1, c_a2 and R_d, R_f, R_v. <maths id="math0054"><math display="block"><msub><mi>R</mi><mi>f</mi></msub><mo>=</mo><mo>-</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>⋅</mo><mfenced><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>/</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced></math><img file="EP1590018B1_D0054.tif" /></maths><maths id="math0055"><math display="block"><msub><mi>R</mi><mi>v</mi></msub><mo>=</mo><mo>-</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>2</mn></mrow></msub><mo>⋅</mo><mfenced><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>/</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced></math><img file="EP1590018B1_D0055.tif" /></maths><maths id="math0056"><math display="block"><msub><mi>R</mi><mi>d</mi></msub><mo>=</mo><mfenced><mn>1</mn><mo>/</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>-</mo><mn>1</mn></mfenced><mo>⋅</mo><mfenced><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mfenced></math><img file="EP1590018B1_D0056.tif" /></maths> The value of the resistance R_f can already be determined at step j. as both c_a0 and c_a1 are already known.</li><li>r. Save and store the first determined values of R_f, R_v and R_d.</li><li>s. Determine q_a using one of the equations of the mathematical model of the vascular access, for example: <maths id="math0057"><math display="block"><msub><mi>q</mi><mi>a</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mi>a</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub></mrow><msub><mi>R</mi><mi>d</mi></msub></mfrac></math><img file="EP1590018B1_D0057.tif" /></maths></li><li>t. Save and store the value calculated for q_a.</li></ol>

0161In steps from 1. to n. the operation of measuring P_aj can be omitted; in this case the values stored and used for the calculation are the same P_ai values calculated at point h. at q_b = q_b1 and q_uf = 0, or at point a. at q_b = 0 and q_uf = 0.

<u>Fourth monitoring procedure.</u>

0162Varying q_b at q_uf = constant (for example zero) and measuring P_am and P_vm.

0163In this case too we calculate at least a part of the coefficients c_a0, c_a1, c_a2 and c_v0, c_v1, c_v2 from which R_d, R_f and R_v are obtained. The calculation of the coefficients is done starting from the knowledge of one or more values for each of the following quantities: P_a, q_b, q_uf, P_v, P_af and P_vf. The quantities P_a, q_b, q_uf, P_v are known by measurements. The quantities P_af and P_vf are known by direct measurement of the pressures in the vascular access, or by means of a calculation process which uses the measured values of pressures P_am and P_vm in the extracorporeal circuit.

0164In the fourth operating mode the measures were taken at q_uf = 0 and we use a mathematical model which includes both equations of P_af and P_vf which in this case are simplified into the following formulation: <maths id="math0058"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub><mo>=</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>⋅</mo><msub><mi>P</mi><mi>a</mi></msub><mo>+</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi>b</mi></msub><mo>+</mo><mfenced><mn>1</mn><mo>-</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mfenced><mo>⋅</mo><msub><mi>P</mi><mi>v</mi></msub></math><img file="EP1590018B1_D0058.tif" /></maths><maths id="math0059"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>=</mo><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>⋅</mo><msub><mi>P</mi><mi>a</mi></msub><mo>+</mo><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi>b</mi></msub><mo>+</mo><mfenced><mn>1</mn><mo>-</mo><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mfenced><mo>⋅</mo><msub><mi>P</mi><mi>v</mi></msub></math><img file="EP1590018B1_D0059.tif" /></maths>

0165In the fourth operating mode the processor determines the four coefficients c_a0, c_a1, c_v0, and c_v1 and from these it calculates the three resistances R_d, R_v, R_f.

0166In the fourth operating mode the pressures P_af and P_vf in the vascular access are determined, either by direct measuring or by measuring pressures P_am e P_vm in the extracorporeal circuit and calculating P_af and P_vf by means of a mathematical model. The pressures P_af and Pv_f are determined at different values of the blood flow rate q_b. In the fourth operating mode, the arterial and venous pressures P_a and P_v of the patient are also considered in the calculation of the coefficients.

0167As coefficients c_a0, c_a1, c_v0, and c_v1 are greater in number than resistances R_d, R_f and R_v, there exists a multiplicity of relations between the coefficients and resistances. In general the knowledge of three coefficients enables determination of the resistances. It has been found that the most precise determination of the resistances R_d, R_f and R_v is obtained by using the three coefficients, c_a0, c_a1, and c_v0.

0168The fourth operating mode is now described step by step. <ol id="ol0004"><li>a. Determine pressures P_af, P_vf, P_a, and P_v with the blood pump flow rate and the ultrafiltration flow rate at nil (q_b = 0 and q_uf = 0).</li><li>b. The values thus determined, P_af0, P_vf0, P_a0 and P_v0, are stored in memory.</li><li>c. The processor calculates c_a0 and c_v0 by means of the equations: <maths id="math0060"><math display="block"><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mrow><mi mathvariant="italic">af</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>-</mo><msub><mi>P</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mrow><mrow><msub><mi>P</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>-</mo><msub><mi>P</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mrow></mfrac></math><img file="EP1590018B1_D0060.tif" /></maths><maths id="math0061"><math display="block"><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mrow><mi mathvariant="italic">vf</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>-</mo><msub><mi>P</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mrow><mrow><msub><mi>P</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>-</mo><msub><mi>P</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mrow></mfrac></math><img file="EP1590018B1_D0061.tif" /></maths></li><li>d. Change the blood flow rate to a known value q_b = q_b, ≠ 0.</li><li>e. Determine at least one value of P_af, P_vf, P_a and P_v when q_b = q_b1.</li><li>f. Save and store values P_af1, P_vf1, P_a1 and P_v1 above-determined.</li><li>g. Repeat steps from d. to f. for a predetermined number of times N in order to obtain a series of values q_bi, P_afi, P_vfi, p_ai and p_vi with i = 1, 2,..., N (N ≥ 1).</li><li>h. Calculate c_a1 as a solution for the system of equations <maths id="math0062"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">afi</mi></msub><mo>-</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>⋅</mo><msub><mi>P</mi><mi mathvariant="italic">ai</mi></msub><mo>-</mo><mfenced><mn>1</mn><mo>-</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mfenced><mo>⋅</mo><msub><mi>P</mi><mi mathvariant="italic">vi</mi></msub><mo>=</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi mathvariant="italic">bi</mi></msub></math><img file="EP1590018B1_D0062.tif" /></maths> If N = 1 the solution is immediate. If N > 1 the solution is obtainable with an optimization algorithm, such as for example a linear regression algorithm.</li><li>i. Save and store the value of c_a1.</li><li>j. Determine resistances R_d, R_f, and R_v by solving the following equations which express the relation between c_a0, c_a1, c_v0 and R_d, R_f, R_v : <maths id="math0063"><math display="block"><msub><mi>R</mi><mi>d</mi></msub><mo>=</mo><mfrac><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mrow><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>-</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mrow></mfrac></math><img file="EP1590018B1_D0063.tif" /></maths><maths id="math0064"><math display="block"><msub><mi>R</mi><mi>f</mi></msub><mo>=</mo><mfrac><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mrow><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>-</mo><mn>1</mn></mrow></mfrac></math><img file="EP1590018B1_D0064.tif" /></maths><maths id="math0065"><math display="block"><msub><mi>R</mi><mi>v</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>⋅</mo><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mrow><mrow><mfenced><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>-</mo><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mfenced><mo>⋅</mo><mfenced><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>-</mo><mn>1</mn></mfenced></mrow></mfrac></math><img file="EP1590018B1_D0065.tif" /></maths></li><li>k. Save and store values R_d, R_f, and R_v above-determined.</li><li>l. Determine the flow rate of the vascular access q_a using one of the equations of the mathematical model, for example the second: <maths id="math0066"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>=</mo><msub><mi>R</mi><mi>f</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mi>b</mi></msub></mfenced></math><img file="EP1590018B1_D0066.tif" /></maths></li></ol>

0169At point e., determination of the value of P_v can be performed in two ways.

0170The first consists in considering P_v constant (P_v = P_v0) during variation in the blood flow rate q_b, thus ignoring the variations in the venous pressure P_v which actually occur during the various operative stages. Consequently the system of equations of point h. can be rewritten in the following way: <maths id="math0067"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">afi</mi></msub><mo>-</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>⋅</mo><msub><mi>P</mi><mi mathvariant="italic">ai</mi></msub><mo>-</mo><mfenced><mn>1</mn><mo>-</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mfenced><mo>⋅</mo><msub><mi>P</mi><mrow><mi mathvariant="italic">v</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>=</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi mathvariant="italic">bi</mi></msub></math><img file="EP1590018B1_D0067.tif" /></maths>

0171The second way consists in considering the variations in P_v to be proportional to the variations in the arterial pressure P_a, thus: <maths id="math0068"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">vi</mi></msub><mo>=</mo><msub><mi>P</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>⋅</mo><mfrac><msub><mi>P</mi><mi mathvariant="italic">ai</mi></msub><msub><mi>P</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mfrac></math><img file="EP1590018B1_D0068.tif" /></maths>

0172This is equivalent to assuming resistances R_d, R_f and R_v to be constant during variation of q_b.

0173In this case the equation of point h. is: <maths id="math0069"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">afi</mi></msub><mo>-</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>⋅</mo><msub><mi>P</mi><mi mathvariant="italic">ai</mi></msub><mo>-</mo><mfenced><mn>1</mn><mo>-</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mfenced><mo>⋅</mo><msub><mi>P</mi><mrow><mi mathvariant="italic">v</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>⋅</mo><mfrac><msub><mi>P</mi><mi mathvariant="italic">ai</mi></msub><msub><mi>P</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mfrac><mo>=</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi mathvariant="italic">bi</mi></msub></math><img file="EP1590018B1_D0069.tif" /></maths>

0174Note that by substituting, in the above equation, c_a0 with the expression <maths id="math0070"><math display="block"><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mrow><mi mathvariant="italic">af</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>-</mo><msub><mi>P</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mrow><mrow><msub><mi>P</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>-</mo><msub><mi>P</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mrow></mfrac></math><img file="EP1590018B1_D0070.tif" /></maths> as in point c. of the present operating mode, the following equation is obtained: <maths id="math0071"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">afi</mi></msub><mo>-</mo><mfrac><msub><mi>P</mi><mrow><mi mathvariant="italic">af</mi><mo>⁢</mo><mn>0</mn></mrow></msub><msub><mi>P</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mfrac><mo>⋅</mo><msub><mi>P</mi><mi mathvariant="italic">ai</mi></msub><mo>=</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi mathvariant="italic">bi</mi></msub></math><img file="EP1590018B1_D0071.tif" /></maths> which is the same equation that appears at point i. of the third operating mode, in which the contribution of P_v was ignored.

<u>Fifth monitoring procedure.</u>

0175The fifth operating mode is similar to the third, with the difference that, instead of determining P_af, P_vf is determined. Briefly, the fifth operating mode consists in varying the blood flow rate q_b while maintaining the ultrafiltration rate q_uf constant, in varying the ultrafiltration rate while keeping the blood flow rate q_b constant, and in determining the venous pressure in the vascular access P_vf at various values of the above-mentioned flow rates. The processor determines the resistances R_d, R_f and R_v and the flow rate q_a in the vascular access by calculating the coefficients c_v0, c_v1, c_v2 using the equation <maths id="math0072"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>=</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>⋅</mo><msub><mi>P</mi><mi>a</mi></msub><mo>+</mo><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi>b</mi></msub><mo>+</mo><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>2</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi mathvariant="italic">uf</mi></msub><mo>+</mo><mfenced><mn>1</mn><mo>-</mo><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mfenced><mo>⋅</mo><msub><mi>P</mi><mi>v</mi></msub></math><img file="EP1590018B1_D0072.tif" /></maths> and the operative stages cited for the third operative mode.

0176The resistances are calculated by solving the following system of equations: <maths id="math0073"><math display="block"><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>=</mo><mfrac><msub><mi>R</mi><mi>v</mi></msub><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mrow></mfrac></math><img file="EP1590018B1_D0073.tif" /></maths><maths id="math0074"><math display="block"><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>R</mi><mi>f</mi></msub><mo>⋅</mo><msub><mi>R</mi><mi>v</mi></msub></mrow><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mrow></mfrac></math><img file="EP1590018B1_D0074.tif" /></maths><maths id="math0075"><math display="block"><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>2</mn></mrow></msub><mo>=</mo><mo>-</mo><mfrac><mrow><msub><mi>R</mi><mi>v</mi></msub><mo>⋅</mo><mfenced><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub></mfenced></mrow><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>f</mi></msub><mo>+</mo><msub><mi>R</mi><mi>v</mi></msub></mrow></mfrac></math><img file="EP1590018B1_D0075.tif" /></maths>

0177The flow rate of the vascular access q_a is calculated as in the third operative mode.

0178Note that, by means of the second monitoring procedure, q_a and R_v can be derived by determining two or more values for the venous pressure alone (P_vm in the machine or P_vf in the fistula), with the equation <maths id="math0076"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>-</mo><msub><mi>P</mi><mi>v</mi></msub><mo>=</mo><msub><mi>R</mi><mi>v</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mi mathvariant="italic">uf</mi></msub></mfenced></math><img file="EP1590018B1_D0076.tif" /></maths> while for the calculation of the values of R_f and R_d, the values of arterial pressure (P_am or P_af) are also used, as well as the other two equations of the mathematical model: <maths id="math0077"><math display="block"><msub><mi>q</mi><mi>a</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mi>a</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub></mrow><msub><mi>R</mi><mi>d</mi></msub></mfrac><mspace width="1em" /><mi>and</mi><mspace width="1em" /><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>=</mo><msub><mi>R</mi><mi>f</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mi>b</mi></msub></mfenced><mn>.</mn></math><img file="EP1590018B1_D0077.tif" /></maths>

0179Similarly a further monitoring procedure can be formulated on the basis of which the values of q_a and R_d are calculated, determining two or more values of only the arterial pressure (P_am in the machine or P_af in the fistula), using the equation <maths id="math0078"><math display="block"><msub><mi>q</mi><mi>a</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mi>a</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub></mrow><msub><mi>R</mi><mi>d</mi></msub></mfrac></math><img file="EP1590018B1_D0078.tif" /></maths> while for calculating the values of R_f and R_v the values of the venous pressure (P_vm or P_vf) are also used, as well as the other two equations of the mathematical model: <maths id="math0079"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>-</mo><msub><mi>P</mi><mi>v</mi></msub><mo>=</mo><msub><mi>R</mi><mi>v</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mi mathvariant="italic">uf</mi></msub></mfenced><mspace width="1em" /><mi>and</mi><mspace width="1em" /><msub><mi>P</mi><mi mathvariant="italic">af</mi></msub><mo>-</mo><msub><mi>P</mi><mi mathvariant="italic">vf</mi></msub><mo>=</mo><msub><mi>R</mi><mi>f</mi></msub><mo>⋅</mo><mfenced><msub><mi>q</mi><mi>a</mi></msub><mo>-</mo><msub><mi>q</mi><mi>b</mi></msub></mfenced><mn>.</mn></math><img file="EP1590018B1_D0079.tif" /></maths>

0180In all of the above-described modes, the measurements are taken with the system in a steady state. For example, the various measurements are taken after a certain time interval (for example about ten seconds) after the blood flow rate or the ultrafiltration rate has been changed.

0181Two numerical examples of the application of the invention are now reported.

<u>First example.</u>

0182This example uses the above-described first monitoring procedure, applied to the apparatus of <figref idref="f0001">figure 1</figref>.

0183Direct measurement of pressures P_a, P_af, P_vf were taken at different flow rate values q_b. The measurements taken are reported in the following table. <tables id="tabl0001" num="0001"><table frame="none"><tgroup cols="5" colsep="0" rowsep="0"><colspec colnum="1" colname="col1" colwidth="20mm" /><colspec colnum="2" colname="col2" colwidth="20mm" /><colspec colnum="3" colname="col3" colwidth="22mm" /><colspec colnum="4" colname="col4" colwidth="22mm" /><colspec colnum="5" colname="col5" colwidth="22mm" /><thead><row><entry valign="top">q_b (ml/min)</entry><entry valign="top">P_a (mmHg)</entry><entry valign="top">P_af (mmHg)</entry><entry valign="top">P_vf (mmHg)</entry><entry valign="top">ΔP_f(mmHg)</entry></row></thead><tbody><row><entry>300</entry><entry>100</entry><entry>51</entry><entry>42</entry><entry>9</entry></row><row><entry>200</entry><entry /><entry>52</entry><entry>41</entry><entry>11</entry></row><row><entry>100</entry><entry /><entry>54</entry><entry>40</entry><entry>14</entry></row><row><entry>400</entry><entry /><entry>51</entry><entry>42</entry><entry>9</entry></row><row><entry>500</entry><entry /><entry>50</entry><entry>43</entry><entry>7</entry></row></tbody></tgroup></table></tables>

0184The equation of the straight line interpolating points ΔP_f is as follows (see <figref idref="f0002">figure 4</figref>, where ΔP_f is a function of q_b) : <maths id="math0080"><math display="block"><msub><mi mathvariant="normal">ΔP</mi><mi mathvariant="normal">f</mi></msub><mo mathvariant="normal">=</mo><mn mathvariant="normal">0.016</mn><mo mathvariant="normal">⋅</mo><mfenced><mn mathvariant="normal">925</mn><mo mathvariant="normal">-</mo><msub><mi mathvariant="normal">q</mi><mi mathvariant="normal">b</mi></msub></mfenced></math><img file="EP1590018B1_D0080.tif" /></maths>

0185From which the following values are calculated <maths id="math0081"><math display="block"><msub><mi mathvariant="normal">R</mi><mi mathvariant="normal">f</mi></msub><mo mathvariant="normal">=</mo><mn mathvariant="normal">0.016</mn><mspace width="1em" /><mi>mmHg min</mi><mo mathvariant="normal">/</mo><mi>ml</mi></math><img file="EP1590018B1_D0081.tif" /></maths><maths id="math0082"><math display="block"><msub><mi mathvariant="normal">q</mi><mi mathvariant="normal">a</mi></msub><mo mathvariant="normal">=</mo><mn mathvariant="normal">925</mn><mspace width="1em" /><mi>ml</mi><mo mathvariant="normal">/</mo><mi>min</mi></math><img file="EP1590018B1_D0082.tif" /></maths>

0186From the third equation of the mathematical model used (assuming P_v = 0) we have q_b1 = 300 ml/min: <maths id="math0083"><math display="block"><msub><mi>R</mi><mi>v</mi></msub><mo>=</mo><mfrac><msub><mi>P</mi><mrow><mi mathvariant="italic">vf</mi><mo>⁢</mo><mn>1</mn></mrow></msub><msub><mi>q</mi><mi>a</mi></msub></mfrac><mo>=</mo><mn>0.045</mn><mspace width="1em" /><mi>mmHg</mi><mo>⋅</mo><mi>min</mi><mo>/</mo><mi>ml</mi></math><img file="EP1590018B1_D0083.tif" /></maths>

0187Given P_a = 100 mmHg, for q_b1 = 300 ml/min we obtain: <maths id="math0084"><math display="block"><msub><mi>R</mi><mi>d</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mi>a</mi></msub><mo>-</mo><msub><mi>P</mi><mrow><mi mathvariant="italic">af</mi><mo>⁢</mo><mn>1</mn></mrow></msub></mrow><msub><mi>q</mi><mi>a</mi></msub></mfrac><mo>=</mo><mn>0.053</mn><mspace width="1em" /><mi>mmHg</mi><mo>⋅</mo><mi>min</mi><mo>/</mo><mi>ml</mi></math><img file="EP1590018B1_D0084.tif" /></maths>

<u>Second example.</u>

0188The second example uses the fourth monitoring procedure.

0189In the following the values of the pressure measured at different blood pump flow rates are reported. <tables id="tabl0002" num="0002"><table frame="none"><tgroup cols="5" colsep="0" rowsep="0"><colspec colnum="1" colname="col1" colwidth="17mm" /><colspec colnum="2" colname="col2" colwidth="17mm" /><colspec colnum="3" colname="col3" colwidth="17mm" /><colspec colnum="4" colname="col4" colwidth="17mm" /><colspec colnum="5" colname="col5" colwidth="17mm" /><thead><row><entry valign="top">q_b</entry><entry valign="top">P_a</entry><entry valign="top">P_af</entry><entry valign="top">P_vf</entry><entry valign="top">P_v</entry></row><row><entry valign="top">(ml/min)</entry><entry valign="top">(mmHg)</entry><entry valign="top">(mmHg)</entry><entry valign="top">(mmHg)</entry><entry valign="top">(mmHg)</entry></row></thead><tbody><row><entry>0</entry><entry>120</entry><entry>62</entry><entry>35</entry><entry>0</entry></row><row><entry>150</entry><entry>118</entry><entry>59</entry><entry>37</entry><entry /></row><row><entry>250</entry><entry>117</entry><entry>57</entry><entry>37</entry><entry /></row><row><entry>350</entry><entry>114</entry><entry>53</entry><entry>38</entry><entry /></row></tbody></tgroup></table></tables>

0190From these values we obtain: <maths id="math0085"><math display="block"><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mrow><mi mathvariant="italic">af</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>-</mo><msub><mi>P</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mrow><mrow><msub><mi>P</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>-</mo><msub><mi>P</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mrow></mfrac><mo>=</mo><mn>0.52</mn></math><img file="EP1590018B1_D0085.tif" /></maths><maths id="math0086"><math display="block"><msub><mi>c</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mrow><mi mathvariant="italic">vf</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>-</mo><msub><mi>P</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mrow><mrow><msub><mi>P</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>-</mo><msub><mi>P</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mrow></mfrac><mo>=</mo><mn>0.29</mn></math><img file="EP1590018B1_D0086.tif" /></maths>

0191By applying a linear regression algorithm to the following equation: <maths id="math0087"><math display="block"><msub><mi>P</mi><mi mathvariant="italic">afi</mi></msub><mo>-</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>⋅</mo><msub><mi>P</mi><mi mathvariant="italic">ai</mi></msub><mo>-</mo><mfenced><mn>1</mn><mo>-</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mfenced><mo>⋅</mo><msub><mi>P</mi><mrow><mi>v</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>=</mo><msub><mi>c</mi><mrow><mi>a</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>⋅</mo><msub><mi>q</mi><mi mathvariant="italic">bi</mi></msub></math><img file="EP1590018B1_D0087.tif" /></maths> the following value for coefficient c_a1 was found: <maths id="math0088"><math display="block"><msub><mi mathvariant="normal">c</mi><mrow><mi mathvariant="normal">a</mi><mo>⁢</mo><mn mathvariant="normal">1</mn></mrow></msub><mo mathvariant="normal">=</mo><mo mathvariant="normal">-</mo><mn mathvariant="normal">0.0155</mn></math><img file="EP1590018B1_D0088.tif" /></maths>

0192After which the following resistance values were found: <maths id="math0089"><math display="block"><msub><mi mathvariant="normal">R</mi><mi mathvariant="normal">d</mi></msub><mo mathvariant="normal">=</mo><mn mathvariant="normal">0.069</mn><mspace width="1em" /><mi>mmHg min</mi><mo mathvariant="normal">/</mo><mi>ml</mi></math><img file="EP1590018B1_D0089.tif" /></maths><maths id="math0090"><math display="block"><msub><mi mathvariant="normal">R</mi><mi mathvariant="normal">f</mi></msub><mo mathvariant="normal">=</mo><mn>0.032</mn><mspace width="1em" /><mi>mmHg min</mi><mo mathvariant="normal">/</mo><mi>ml</mi></math><img file="EP1590018B1_D0090.tif" /></maths><maths id="math0091"><math display="block"><msub><mi mathvariant="normal">R</mi><mi mathvariant="normal">v</mi></msub><mo mathvariant="normal">=</mo><mn mathvariant="normal">0.042</mn><mspace width="1em" /><mi>mmHg</mi><mo mathvariant="normal">⋅</mo><mi>min</mi><mo mathvariant="normal">/</mo><mi>ml</mi></math><img file="EP1590018B1_D0091.tif" /></maths>

0193From this we calculated: <maths id="math0092"><math display="block"><msub><mi>q</mi><mi>a</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mrow><mi mathvariant="italic">af</mi><mo>⁢</mo><mn>0</mn></mrow></msub><mo>-</mo><msub><mi>P</mi><mrow><mi mathvariant="italic">vf</mi><mo>⁢</mo><mn>0</mn></mrow></msub></mrow><msub><mi>R</mi><mi>f</mi></msub></mfrac><mo>=</mo><mn>842</mn><mspace width="1em" /><mi>ml</mi><mo>/</mo><mi>min</mi></math><img file="EP1590018B1_D0092.tif" /></maths>