There has been a growing interest in blood conservation in recent years. This interest has been stimulated by an allogeneic blood shortage ^{(1)}, along with concern over the possible immunosuppressive effects of allogeneic transfusion ^{(2)}. Multiple strategies can be applied to avoid allogeneic transfusion. The primary strategies involve preoperative erythropoietin and iron supplementation, preoperative autologous donation, acute normovolemic hemodilution, and the application of cell salvage (CS) systems. Through mathematical modeling, a better understanding of the capabilities of a particular system can be achieved. The efficiency of acute normovolemic hemodilution (ANH) has been mathematically modeled ^{(3â€“6)}. Similar modeling has been performed with the addition of erythropoietin ^{(7)}. No such modeling has been performed for CS. In this study, a mathematical model for CS was developed and tested.

## Mathematical Modeling

The relationship between hematocrit (H), estimated blood volume (V), and lost V (V_{L}) for isovolemic hemodilution was described as the following differential equation by Bourke and Smith ^{(8)}.MATH

Integrating this equation from time zero (0) to time final (F) yieldsMATH or, equivalently,

Equation 3 is the basis of the relationship between H and V_{L} at any two time points (0 and F in this case) for a fixed V.

Modeling for CS was based on the Latham bowl design. The CS reservoir collects blood until enough blood has accumulated to fill a Latham bowl (two sizesâ€”125 and 225 mLâ€”denoted by V_{B}) with a constant H (H_{B}) so that a target red cell volume (*H*_{B} Ă—*V*_{B}) is achieved. At this point, the blood is readministered to the patient. It is not necessary for the model, but for logistical reasons it was assumed that no time was expended in the collection and readministration of the blood so as to keep the patient euvolemic throughout the surgery. In each cycle, a fraction of blood is lost that cannot be recovered (we assume a constant recovery rate, *k,* between 0% and 100%). Therefore, the peak H at each readministration will be less than the peak of the previous cycle. Eventually, the critical H (H_{c}) is reached, and the transfusion of allogeneic blood begins to maintain an H_{c}.

Assuming a patient has an initial H (H_{01}), at the end of the first cycle the H becomes

where V_{L1} is the V_{L} in the first cycle and can be approximated (see Appendix 1 for complete derivation) by

when *V* >>*V*_{L}, where *k* is the recovery rate, H_{B} is the H in the Latham bowl, and V_{B} is the size of the Latham bowl. That is, the V_{L} can be estimated by a fixed recovery rate (*k*), the H_{01} when the cycle begins, and the red blood cell mass saved in the Latham bowl (*H*_{B} Ă—*V*_{B}). Generalizing Equation 4 to cycle *i* yields

where V_{L}*i* is

To complete the derivation, H_{0}*i* needs to be defined. To do so, the following relationship is recognized. The red blood cell mass in the patient for cycle *i* is equal to the sum of the red blood cell mass of the patientâ€™s final red blood cell mass (HF(*i* âˆ’ 1)V) for the previous cycle (*i* âˆ’ 1) and the red blood cell mass in the Latham bowl (*H*_{B}*V*_{B}), so that

Rearranging,

To summarize, Equation 7 states that beyond the first cycle, the H_{01} for cycle *i* is equal to the sum of the final H for cycle (*i* âˆ’ 1) and the CS H (*H*_{B}*V*_{B}/ *V*). Equation 5 was used to predict the decline of H level during the CS procedure with the components of Equation 5 defined in Equations 6 and 7.

### Assumptions

The mathematical analysis was performed on the basis of the following clinical assumptions:

- The patient was kept euvolemic throughout the surgery.
- Salvaged blood was readministered to the patient after the Latham bowl was filled at H
_{B}.
- The
*k* of lost blood is constant throughout the surgery.
- The length of time for the readministration of CS blood is negligible, and no blood was lost during the readministration so as to keep patient reasonably euvolemic throughout the surgery.
- Transfusion of allogeneic blood was begun when a fixed transfusion trigger (H
_{c}) was reached.

### Blood Recovery Rate (k)

To better understand the efficiency of CS, an estimate of blood *k* is necessary. Data from the Cleveland Clinic Foundationâ€™s Intraoperative Autotransfusion database were searched for CS cases. From this database, 569 cases were reviewed over 6 mo extending from January 1, 2001, to June 30, 2001. Exclusion from analysis included a failure of the surgical scrub technician to rinse the surgical sponges; use of suction systems other than CS suction; and use of preoperative autologous donation, normovolemic hemodilution, platelet gel, CS standby, or perioperative allogeneic transfusion. Patientâ€™s V was estimated by the formula derived from Nadler et al. ^{(9)}, which that took into account sex and body weight. The *k* was estimated by entering the estimated V, V_{B}, H_{B}, H_{01}, and final H obtained from the patient record.

## Results

By using the equations described in Methods, calculations were made of the maximum allowable blood loss (MABL) that a hypothetical patient could sustain before needing a transfusion, assuming a V of 5000 mL, an H_{0} of 45%, an H_{B} of 60%, and a transfusion trigger (H_{c}) of 21%. This hypothetical patient would be able to tolerate a MABL of 3840 mL (null) before crossing the H_{c} if no blood conservation techniques were used (Fig. 1).

By using the same modeling variables for the patient without blood conservation techniques, a range of MABLs was calculated. Table 1 shows the MABL under CS for various *k* values and two different sizes of Latham bowl. With a linear increase of *k,* MABL increases exponentially. One can see that with a 35%*k,* the MABL reaches 5050 mL (with a 125-mL Latham bowl), exceeding the MABL of 3840 mL for patients with no blood conservation procedure. When *k* is moderate (50%), the MABL exceeds 7200 mL, which is double the volume of the null comparison. When the *k* is as large as 90%, the MABL exceeds the null comparison by 10-fold. In the same table, two sizes of Latham bowl were compared, showing no substantial difference in MABL.

Table 2 shows the MABL under CS for various estimated V and two different sizes of Latham bowl. Again, two sizes of Latham bowl do not offer substantial differences in MABL. MABL increases linearly with estimated V. Table 3 shows the MABL under CS for various H_{0}. Similar to the relationship between MABL and estimated V, MABL increases linearly with the H_{0}.

From the database, 569 cases of CS were examined. Seventy-five cases remained after applying the exclusion criteria. From the 75 cases, no *k* was estimated for two subjects, one with a missing weight and the other with a missing postoperative H. In addition, 25 subjects had a small amount of estimated blood loss with a large H decrease. For these 25 subjects, the H decrease exceeded the amount that could reasonably be ascribed to blood loss. That is, for these 25 patients, their estimated blood loss was underestimated, the CS machine was doing harm to the red blood cells, or the starting and/or the postoperative H was in error. Furthermore, two subjects had H decreases that matched their estimated blood loss, as predicted by the equation of Bourke and Smith ^{(8)}, which implied that the CS machine might not have been turned on or that the efficiency rate was 0%. Three patients had postoperative H values equivalent to preoperative H values after considerable blood loss during surgery. This implied that either the measurement of the H or the estimated blood loss was grossly in error.

Thus, the final analysis consisted of 43 subjects (43 of 75; 57.3%) who had reasonable H and estimated blood loss on the basis of Bourke and Smithâ€™s equation, and this therefore implied that the CS machine was working in an expected manner, with an estimated recovery rate of 0%â€“100%. The average *k* was 57% with an sd of 20%. Table 4 shows the characteristics of the patients who were matched to the mathematical model to arrive at the *k.* No significant difference was found in the *k* between the type of surgical procedure (*P* = 0.39; Kruskal-Wallis test). Figure 2 shows graphically how the H declines when CS is applied with a *k* of 57%.

## Discussion

The mathematical model developed here would suggest that CS offers a significant advantage for preventing red blood cell transfusion and that this advantage can be greatly modified by a number of different variables. These variables include the patientâ€™s starting H, the patientâ€™s V, the size of the Latham bowl, and the efficiency of red blood cell recovery. From this model, it appears that the efficiency of red cell recovery is the most important of these variables in that linear increases in *k* lead to exponential increases in MABL.

Previously, red blood cell *k* values have been reported to range from 79% to 84%^{(10)}. This recovery rate refers to how many red blood cells are hemolyzed during CS processing. In other words, if a unit of blood were introduced into a CS reservoir and processed, 79% to 84% of the red blood cells would be recovered. It is important to distinguish how red blood cell *k* is defined in this study, as opposed to previous work. Here, red blood cell recovery is used to denote the recovery of red blood cells from the surgical field, their suctioning, their processing, and ultimately their return to the patient.

An extensive review of the literature found few data regarding the efficiency of red blood cell recovery as defined in this study. Only one estimate was found (Siller et al.) ^{(11)}, which stated that a 35% recovery was achieved in patients undergoing posterior instrumentation and fusion for idiopathic scoliosis. No mention was made by the authors as to the techniques used to maximize CS efficiency. Here, patient records were collected, and the red blood cell *k* during CS was estimated with the developed mathematical model. A *k* of 57% was found with a large sd of 20%. Thus, the estimate in this study is reasonable in comparison to the study by Siller et al., which yielded a 35% recovery rate.

From this modeling, it is apparent that every effort should be made to optimize the red blood cell *k.* Several studies have demonstrated that red blood cell recovery can be influenced by a number of factors. In general, larger red blood cell *k* values can be influenced by using regulated suction and large suction tips ^{(12,13)}, rinsing surgical sponges ^{(14)}, using heparin rather than citrate as an anticoagulant ^{(15)}, and paying attention to using CS suction rather than regular wall suction.

In the collection of data used for estimating the range of red blood cell recovery, every effort was made to eliminate patient records that failed to optimize red blood cell recovery. Patient data were not used when the suction pressure was not regulated or when surgical lap sponges were not rinsed. The large variability of 20% from the mean *k* of 57% would indicate that there are many other variables that may influence CS *k* values. One of these factors was believed to be the type of surgical procedure. Thus, comparisons were made between types of surgical procedures. No differences were found that were statistically significant. Because there were a small number of cases for each surgical procedure, this conclusion may be erroneous. Another factor that might play a role could be differences in red blood cell friability from patient to patient. This is a very interesting variable that is well worth further exploration.

In addition to red blood cell recovery, the patientâ€™s H_{0} and their V can also alter MABL, but they do so in a linear fashion. In Tables 2 and 3, we varied the hypothetical patientâ€™s V and H_{0}. It is readily seen that the capabilities of CS were significantly influenced by the patientâ€™s H_{0} and V. For instance, a patient weighing 57 kg with a V of 4000 mL is much more likely to have the capabilities of CS surpassed and need a transfusion for a large-blood-loss surgical procedure when compared with a patient who weighs 100 kg and has a V of 7000 mL. This may have importance when deciding whether a patient should receive preoperative erythropoietin. This very expensive resource may have great allogeneic transfusion-sparing effects in small patients but may not have utility in a large patient. A large patient has a large V, and the larger the V, the larger the MABL. However, maximum benefit from CS for all patients would be gained through erythropoietin use and by maximizing a patientâ€™s H_{0}. This is illustrated in Table 3 by the large gap between MABL in patients with an H_{0} of 45% versus those who present with an H of 30%.

After the publication of the ANH mathematical models, many opponents challenged the conclusions of these models. Here, the same challenges may arise because the mathematical model is developed around certain assumptions. Five assumptions were made in the development of this model. First, the hypothetical patient starts with a fixed V and is kept euvolemic throughout the surgery. During most surgical procedures, the patient V most probably is variable; however, over time, the average should be euvolemia. The second assumption was that the H of the blood product produced from the Latham bowl (at each CS cycle) was fixed. In real practice, the H of the product is variable, depending on the CS fill speed and on the quality and concentration of the red blood cells in the collection reservoir. It is also dependent on the machine used. On review of the Cleveland Clinic database, cases performed with the Medtronic Sequestra were selected. These cases produced blood with an H of 57.6% Â± 7.3% (*n* = 1034 bowls). Therefore, the assumption used in the mathematical model that the final bowl H was 60% appears valid. The third assumption was that the *k* is constant rather than variable throughout the surgical procedure. The *k* could vary according to the rate of blood loss, the adequacy of anticoagulation, suction pressure applied to the red blood cells, and many other factors. Like the first assumption, the model assumes that over time the *k* approaches the average. Finally, the precise transfusion trigger of 21% may not always hold true clinically because periodically transfusion starts at levels above and below this 21% target. This assumption was adopted because this was the method used in other allogeneic avoidance mathematical models.

Despite a high degree of safety of allogeneic blood transfusion, patients would still benefit from avoidance of allogeneic blood products. This model suggests that CS offers great utility in avoiding allogeneic transfusion. Further study is still needed to evaluate the best methods of optimizing CS capabilities. In addition, work needs to be performed to determine which blood conservation methods are most cost-effective for avoiding allogeneic transfusion. The results of this model ultimately may help in exploring these areas.

## Appendix 1

In an isovolemic patient, the hematocrit volume at cycle *i* is equal to the sum of the hematocrit volume lost (*H*_{Li} Ă—*V*_{Li}) and the patientâ€™s remaining hematocrit (*H*_{Fi} Ă— V):MATH

Rearranging the equation yields

and substituting Equation 3 from the text

results in

Rearranging the equation again leads to

Now, let u =*V*_{Li}/V.

Then (V/V _{Li}) Ă— (1 âˆ’ eâˆ’*V*_{L}^{/V}) can be rewritten as

and *u â†’ 0* when V â‰« V_{L}, which roughly holds in each CS cycle.

So, lim *u â†’ *_{0} (1/ *u*) Ă— (1 âˆ’*eâˆ’u*) â†’*eâˆ’u* (by lâ€™Hopitalâ€™s rule) â†’ 1

That is, (*V/V*_{Li}) Ă— (1 âˆ’ eâˆ’*V*_{Li}/V) â‰ˆ 1.

Of the blood lost during the surgery, we can say that the red blood cell mass in the Latham bowl (*HB* Ă— VB) will be a fraction (*k*) of the red blood cell mass that was lost (*H*_{Li} Ă— V _{Li}), so that

From Equations A and B, we have

That is,

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