Haemorrhagic shock has various etiologies, including trauma, maternal haemorrhage, peptic ulcers, perioperative haemorrhage, and ruptured aortic aneurysms [1]. This medical condition causes 1.9 million deaths annually (with trauma as the leading cause; there are 1.5 million trauma-induced haemorrhagic shock deaths annually worldwide) and affects the young disproportionately raising a socioeconomic issue [2]. When trauma-related haemorrhage deteriorates, death occurs at a median of approximately 2.6 h after initial presentation addressing the importance of initial management [3,4]. Initial management is also critical for reducing delayed mortality and repaying oxygen debt before shock becomes irreversible [5]. For clinicians, prompt and correct determination of the degree of blood loss (%) is critical.

The blood loss (%) is calculated as ‘100% – residual blood volume (RBV) (%)’. For example, when RBV (%) is 65%, blood loss (%) is 35%. RBV (%) is defined as RBV/total blood volume (TBV). TBV, the denominator, is easily estimable via body weight [6]. Therefore, once RBV, the numerator, is also known, RBV (%) and thus blood loss (%) can be estimated.

The gold standard to determine RBV is a dilution method using radioactive chromium (⁵¹Cr); briefly after transfusing a small, fixed quantity of ⁵¹Cr-labelled red blood cells, the radioactivity of the blood is measured to calculate RBV [7]. The carbon monoxide rebreathing technique, which shows high reproducibility without using radioactive materials, is based on a fixed amount of an inspired oxygen-carbon monoxide gas mixture and traces the carboxyhaemoglobin (HbCO) difference to estimate RBV [8,9]. However, neither method is applicable to real-world haemorrhagic shock patients. Clinicians estimate RBV (%) or blood loss (%) considering multiple factors such as vital signs, haemoglobin/haematocrit, central venous or pulmonary capillary wedge pressure (CVP/PCWP), ultrasonography, and visual estimation [10-16]. However, these methods provide only rough estimations.

Previously, we mathematically derived an equation to estimate RBV for acute, non-ongoing haemorrhagic shock patients [17]. In mathematics class, middle school students are asked the following question: “There is a cup of sugar water with a concentration of 45%; 0.5 kg of water is poured into this mixture. The concentration of the sugar water changed to 40%. Can you calculate the initial mass of the sugar water?”. Once the initial and final concentration of sugar water and the mass of water poured into the mixture is known, it is possible to calculate the initial mass of the sugar water through a linear equation (Fig. 1A; see Supplementary Text 1A for a detailed mathematical explanation). We paid attention to the fact that this sugar water scenario is similar to that of initial management of haemorrhagic shock patients.

For patients presenting at the emergency department (ED) with haemorrhagic shock, clinicians control bleeding, request laboratory tests, infuse crystalloid fluid restrictively, and start transfusion as soon as materials are available [1,13,18]. Along with blood type, arterial blood gas analysis (ABGA), lactate, electrolytes, coagulation profiles, thromboelastography/thromboelastometry, and complete blood counts should be checked initially as point-of-care tests (POCT) [1]. With this standard management of haemorrhagic shock, the initial and final concentration of the blood, that is, the serial haematocrits (Hct₁ and Hct₂), become available immediately. In addition, clinicians themselves determine the volume of crystalloid fluid (N), which is infused as an initial resuscitative effort. As with the sugar water story solved by a linear equation, we derived the following equation to determine the initial blood volume (RBV) at the time of ED arrival using the information on Hct₁, Hct₂, and N, which are the key elements of standard management [1,13] (Fig. 1B; see Supplementary Text 1B for a detailed mathematical explanation):

The only difference between this approach and the sugar water example is k, which is approximately 0.25 for men; only a fraction of crystalloid fluid is distributed in the intravascular volume, leaving the remnant within the interstitial compartment [19].

Clinicians prefer to know blood loss (%) or RBV (%) rather than RBV itself. In this study, we mathematically derived an equation to determine the RBV (%) (and thus the blood loss [%]) by modifying the above equation and then validated it in vivo. Additionally, we also validated the original equation estimating RBV in vivo.

The rats suffered shock when losing ≥ 20.0% of their TBV. However, those shedding 20.0%–25.0% of their TBV recovered from shock via fluid resuscitation (Figs. 3 and 4C). Rats bleeding out 60.0% of their TBV barely survived the study protocol. A total of seven rats shedding 30.0%–60.0% of their TBV were ultimately included in the main analysis.

The rats included in the analysis weighed between 285 and 334 g and their TBV ranged from 17.87 to 20.81 cc; N spanned 4.27–5.01 cc. The mean SBP₀, DBP₀, and MAP₀ levels were 110 ± 11, 71 ± 7, and 84 ± 8 mmHg, respectively and the mean HR₀ was 215 ± 18 beats/min. Mean haemoglobin₀, haematocrit₀, and lactate₀ levels were 13.2 ± 0.8 g/dl, 40.6 ± 2.5%, and 0.9 ± 0.3 mM, respectively. Changes in vital signs and POCT findings according to the study timeline are shown in Figs. 3 and 4, respectively.

Within a linear regression analysis among the rats shedding 30.0%–60.0% of their TBV, the equation to estimate RBV was updated as 0.272 N / [(Hct₁ / Hct₂) – 1] + 5.64 (95% CI of k: 0.164–0.380, p = 0.001, R² = 0.89). In the correlation analysis between the actual and estimated RBV, r was 0.945 (p = 0.001) (Fig. 5A). On a Bland-Altman plot, the difference was 0.00 ± 0.84 cc (95% CI: –0.77, 0.78) with lower and upper limits of agreement of –1.64 (95% CI: –3.04, –0.24) cc and 1.65 (95% CI: 0.25, 3.05) cc, respectively (Fig. 5B). The pre-determined value of ± 4.0 cc included the 95% CI of these limits, thereby validating the equation.

The actual RBV (%) was expressed as 5.67 / [(Hct₁ / Hct₂) – 1] + 32.8% (95% CI of the slope: 3.14–8.21, p = 0.002, R² = 0.87). A calibration plot revealed that the r between the two RBV (%) was 0.932 (p = 0.002) (Fig. 6A). On a Bland-Altman plot, the difference was 0.00 ± 4.03% (95% CI: –3.71, 3.71), with lower and upper limits of agreement of –7.85% (95% CI: –14.5%, –1.18%) and 7.85% (95% CI: 1.18%, 14.5%), respectively (Fig. 6B). The 95% CIs of these limits were included within ± 20.0%, thereby validating this equation as well.

As supplementary analyses, we performed the same analyses including all nine rats that initially suffered haemorrhagic shock after bleeding. RBV was estimated as 0.302 N / [(Hct₁ / Hct₂) –1] + 5.72 (95% CI of the slope: 0.138–0.466, p = 0.003, R² = 0.73). On calibration, the r between the actual and estimated RBV was 0.854 (p = 0.003) (Supplementary Fig. 1A). A Bland-Altman plot revealed a difference of 0.00 (95% CI: –1.12, 1.14) ± 1.48 cc with lower and upper limits of agreement of –2.88 (95% CI: –4.89, –0.87) cc and 2.89 (95% CI: 0.88, 4.90) cc, respectively (Supplementary Fig. 1B). Actual RBV (%) was expressed as 6.74 / [(Hct₁ / Hct₂) – 1] + 32.3% (95% CI of the slope: 2.29–11.2, p = 0.009, R² = 0.65). A calibration plot revealed that the r between the actual and estimated RBV (%) was 0.804 (p = 0.009) (Supplementary Fig. 2A). On a Bland-Altman plot, the difference was 0.02 ± 8.21% (95% CI: –6.28, 6.32), with lower and upper limits of agreement of –16.0% (95% CI: –27.2%, –4.83%) and 16.1% (95% CI: 4.88%, 27.30%), respectively (Supplementary Fig. 2B). As the 95% CIs of the limits of agreement exceeded ± 4.0 cc and ± 20.0% (the pre-determined values of validation), neither equation was validated.

The results of the regression analyses examining factors associated with RBV (%) are shown in Supplementary Figs. 3–5, respectively. These figures present initial and final values and interval changes for vital signs, haematocrit, and lactate. The relevant statistics are summarised in Supplementary Table 1.

This preliminary study aimed to mathematically derive a simple equation estimating RBV (%) mathematically via serial haematocrit measurements and volumes of infused crystalloids and to validate it in vivo. For the rats that shed 30.0%–60.0% of their TBV and suffered persistent shock despite fluid resuscitation, the equation was updated and subsequently validated: RBV (%) = 6.74 / [(Hct₁ / Hct₂) – 1] + 32.3%. In addition, the original equation was also updated and validated: RBV = 0.272 N / [(Hct₁ / Hct₂) – 1] + 5.64

To our knowledge, this is the first study to suggest an equation to estimate RBV (%) mathematically in order to promptly and correctly calculate blood loss (%) [= 100% – RBV (%)] and to update and validate this equation in vivo. In addition, this is the first in vivo study to validate a mathematically derived equation estimating RBV. As all the involved variables are established components of standard haemorrhagic shock management, these equations do not require an additional apparatus or specialised testing and thus have maximal clinical applicability. If validated in human studies, these equations may help clinicians design an optimal treatment plan for patients suffering from acute, non-ongoing haemorrhagic shock at the earliest possible phase.

We conducted a regression analysis to explain RBV as a function of N / [(Hct₁ / Hct₂) – 1], generating a y-intercepts of 5.64. Modification of a prediction rule with the addition of a y-intercept is commonly implemented to fit a new target population during external validation [23,33]. The original equation to estimate RBV included k (the fraction of crystalloid fluid distributed in the intravascular volume) as a slope. A k of 0.272 (95% CI: 0.164, 0.380) for rats shedding 30.0%–60.0% of their TBV was observed for this experimental group. This seems to match the k values reported for humans, which is reported to be approximately 0.25 [19].

Supplementary analyses revealed that the regression equations implemented for rats suffering from persistent shock despite fluid resuscitation were superior to those implemented among all the rats regardless of persistent shock shedding (i.e., 20.0%–60.0% TBV). In estimating RBV (%), the former showed greater a R² (0.87 vs. 0.65) and a narrower 95% CI of the slope (5.67 [3.14–8.21] vs. 6.74 [2.29–11.2]). By excluding the two rats shedding 20.0%–25.0% of their TBV, the equation provided a superior explanation of RBV (%) via the equation 24k / [(Hct₁ / Hct₂) – 1] and specified the slope more precisely. Though we are unsure why the rats that lost 20.0%–25.0% of their TBV distorted the equations, the following observations as well as knowledge of the relevant literature provide important context for interpreting these findings. Just before crystalloid fluid resuscitation, their MAP₁ levels were 63 and 65 mmHg, respectively (Fig. 3C). After fluid resuscitation, their MAP₂ levels increased to 117 and 84 mmHg, respectively, exceeding 65 mmHg (the criterion of shock). Their lactate levels were persistently < 2.0 mM, failing to fulfill another criterion of shock (Fig. 4C). The more MAP out-ranges above shock level, the more urine is excreted [34,35]. This leakage of the circulatory system via the urinary system violates the basic assumptions of the current equations and may have isolated these two rats as outliers [17].

Clinicians have estimated RBV (%) or blood loss (%) using vital signs, haemoglobin/haematocrit measurements, CVP/PCWP, ultrasonography, and visual estimation. Although useful, these methods provide only rough estimations. Tachycardia and hypotension, occurring within class I, II (mild), III (moderate), and IV (severe) haemorrhagic shock, are less reliable indicators for patients receiving antihypertensive medications (especially beta or calcium channel blocking medications) and their sensitivities are unsatisfactory [11,16]. Haematocrit does not reflect acute haemorrhage adequately as the plasma volume fails to increase sufficiently for achieving an euvolemic state [36]. Neither CVP nor PCWP predicts ventricular preload (which correlates with RBV) [37]. Although ultrasonography provides some hints regarding preload with respect to the diameter and collapsibility of the inferior vena cava as well as fluid responsiveness, these indicate RBV (%) indirectly; fluid challenge is less applicable for haemorrhagic shock patients whose fluid resuscitation should be restricted [18,38]. Meanwhile, visual estimation of blood loss is inaccurate and unreliable even in the operating room [14]. Due to these limitations, researchers combined these variables to enhance diagnostic accuracy [10,12]. For example, Callcut and colleagues suggested that massive transfusion is indicated when two of following factors are present: an international normalised ratio (INR) > 1.5, SBP < 90 mmHg, haemoglobin < 11 g/dl, a base deficit of ≥ 6 mM, and fluid revealed on focused assessment with sonography for trauma (sensitivity 85%, specificity 41%) [10]. However, these rules are relatively non-specific and cannot differentiate RBV (%) quantitatively.

Some researchers previously investigated the volume of infused crystalloid fluid or serial haematocrits (the key variables of the current study) as tools for RBV (%) estimation. The response to initial fluid resuscitation is suggested to help estimate blood loss (%), with rapid, transient, and minimal/no response correspond to minimal (< 15%), moderate and ongoing (15%–40%), and severe (> 40%) loss of TBV, respectively [11]. However, this approach cannot estimate RBV (%) quantitatively in order to guide fluid/blood resuscitation delicately, as required for successful haemorrhagic shock management.

Thorson and colleagues reported that Hct₁–Hct₂ > 6% reliably indicate ongoing bleeding [39]. However, only 3.9% (9/232) of their study participants suffered shock and the interval to check the serial haematocrits was 120 ± 63 min even for patients with ongoing bleeding. These rendered their results less applicable for haemorrhagic shock, which required much faster fluid resuscitation followed by a repeat haematocrit measurement; 60% of patients die within 3 h after ED presentation for haemorrhagic shock [4]. Meanwhile, the current study (that dealt with the earliest phase of haemorrhagic shock) showed some correlation between Hct₁–Hct₂ and RBV (%) (Supplementary Fig. 5F, Supplementary Table 1). This association may be explained mathematically using our equation:

As mentioned above, the equation to estimate RBV (%) contains Hct₁–Hct₂ as a denominator. However, considering the effect of the numerator (Hct₂) on the whole equation, the equation including both the numerator and denominator is more robust than Hct₁–Hct₂ alone. The R² of our regression equation (0.87) is greater than that including Hct₁–Hct₂ alone (0.59), supporting its superiority in terms of explaining RBV (%).

By replacing N/TBV with 0.24 in rats, we simplified the equation to estimate RBV (%) from k × N / TBV / [(Hct₁ / Hct₂) – 1] × 100 (%) to 24k / [(Hct₁ / Hct₂) – 1] (%). The only condition was pre-determination of N in terms of body weight (0.015 cc/g in this study). This suggests that RBV (%), and thus blood loss (%) (100%–RBV [%]) can be determined by Hct₁ and Hct₂ when a fixed N is infused. For a human, TBV (L/kg) is approximately 0.075 × (body weight) for men and 0.065 × (body weight) for women [6]. When N is fixed as 0.015 L/kg, which corresponds to 1 L for 70 kg adults (in line with standard management of haemorrhagic shock), N/TBV is 0.20 for men and 0.23 for women. If k is 0.25, as reported previously [19], the following equations may be applicable for non-ongoing haemorrhagic shock patients:

Of course, further clinical studies are required to modify these equations, including adjustment of the slope and the addition of a y-intercept [23].

In supplementary analyses, RBV (%) was closely associated with both initial and final values of SBP, DBP, MAP, lactate, and haematocrit (R²: 0.50–0.91, with all p < 0.05; Supplementary Table 1 and Supplementary Figs. 1, 2). However, we regarded these results as inapplicable in practice. For instance, we strictly controlled the time of bleeding, observation, and fluid resuscitation in this animal study. However, haemorrhagic shock patients arrive at the ED at various times following the time of initial bleeding. Because blood pressure, lactate, and haematocrit changes over time even in the same patient [28], these variables measured at strict timelines in a laboratory setting are not applicable to real haemorrhagic shock patients.

This study had several limitations. First, this study dealt with ‘non-ongoing’ haemorrhagic shock. This confines the indication of this work to patients for whom instant haemostasis is achievable (for example, patients with penetrating extremity wounds, peptic ulcers, or perioperative bleeding). For most blunt trauma, maternal haemorrhage, and ruptured aortic aneurysm cases (i.e., the other major causes of mortality due to haemorrhagic shock), instant haemostasis may be difficult to achieve, thus rendering our study results less applicable [1,3]. However, the current equations may have some value even for ongoing haemorrhagic shock patients; for example, Hct₂ would be lower among ongoing haemorrhagic shock patients than among non-ongoing haemorrhagic shock patients (e.g., 30.0% vs. 32.0%). Incorporating this lowered Hct₂ into the equation as 6.74 / [(Hct₁ / Hct₂) – 1] + 32.3 (%), while assuming Hct₁ = 40.0% in this example, would cause RBV (%) to be underestimated (52.5% vs. 59.3%). Therefore, for patients with ongoing haemorrhagic shock, the actual initial RBV (%) must be larger than the value estimated by the equation (52.5% in this example). Clinicians may not know whether bleeding is ongoing. Even in this situation, they may guess that the initial RBV (%) would be at least equal to the estimated value (52.5% in case of a non-ongoing haemorrhage) or larger (in case of an ongoing haemorrhage). Second, bleeding was induced simply by puncturing the left femoral artery. With this low energy injury, we could assume that k would not vary significantly. However, in severe trauma, broken endothelial glycocalyx layers and coagulopathy caused by oxygen debt lead to increased vascular permeability and extravasation of intravascular fluid into the interstitial space (especially under lower oncotic pressure), thus lowering k [40-42]. In this situation, k might fluctuate according to the type and severity of the injury, thus making the equations less applicable in their current forms. Third, in this preliminary, concept-validating study, we set the absolute maximum allowed difference between the estimated and actual residual blood as < 20.0%, assuming a mean difference of 0.0 ± 4.0%. For rats shedding 30.0%–60.0% of their TBV, the actual maximum difference in this study was –6.5%, far smaller than the pre-determined value of ± 20%. However, considering the small sample size in this current study, we had to compensate for the wide 95% CIs of the upper and lower limits of agreement. The issue of sample size needs to be considered carefully within further clinical studies conducted to validate the concept of this study.

Considering these limitations, we believe that this preliminary concept-validation study is a starting point for further investigations. First, the equation to estimate the RBV (%) needs to be established clinically in non-ongoing, haemorrhagic shock patients and in studies with a larger sample size. Although we proposed 5.0 / [(Hct₁ / Hct₂) – 1] (%) for men and 5.8 / [(Hct₁ / Hct₂) – 1] (%) for women (assuming a k of 0.25), these equations would need clinical validation, including adjustment of the k value and assignment of a y-intercept [23]. Second, preclinical or clinical studies aiming to broaden the indications of the current equations are required, including those for ongoing haemorrhage and high-energy blunt injury. In contrast to animal studies wherein researchers can freely pre-determine the degree of bleeding, it may be difficult to determine the ‘actual’ RBV (or blood loss), which is the reference value to compare the ‘estimated’ RBV with, among the actual haemorrhagic shock patients. We propose that, among the patients undergoing major surgery that tends to cause profuse bleeding, our concept can be validated while the anaesthesiologists monitor input/output of fluid, vital signs, and POCT on a real time basis. More practically, the ability to predict the need for massive transfusion among haemorrhagic shock patienta may be compared using the equation we suggest in this study and the current indexes composed of several variables at ED [10,12]. Meanwhile, whether our equation is associated with the clinical indexes such as SOFA (sequential organ failure assessment) and SAPS (simplified acute physiologic score) II may be also investigated. These indexes are expected to be worse in patients with more blood loss because of the hypoperfusion secondary to multiple organ injuries [43-45]. Furthermore, if our equation shows a direct relationship with the occurrence of multiple organ failure and mortality, it can potentially be easily adopted by clinicians considering its immediate availability and simplicity.

In summary, this concept-derivation and preliminary in vivo validation study demonstrates that RBV (%) and thus blood loss (%) may be calculable via information on serial haematocrits and the volume of crystalloid fluid infused for rats suffering from acute, non-going haemorrhagic shock. The equations we suggest in the study seem to apply best for rats suffering from persistent haemorrhagic shock despite crystalloid fluid resuscitation. Further studies are required to validate the clinical applicability of these equations and to widen their indications, regardless of ongoing haemorrhage, injury mechanism, and severity.

Supplementary data including one text, five figures, and one table can be found with this article online at https://doi.org/10.4196/kjpp.2022.26.3.195.