Blood circulation

An arterial reservoir, venous reservoir and a left ventricle exchange flows

Q_{a}, Q_{v}, Q_{c}

. Their volume balances are

C_{a} \overset{p}{˙}_{a} = Q_{a} - Q_{c}

C_{v} \overset{p}{˙}_{v} = Q_{c} - Q_{v}

, and

d (p_{L V} / E_{L V}) / d t = Q_{v} - Q_{a}

. Ideal one-way valves give

Q_{a} = [p_{L V} - p_{a}]_{+} / R_{a}

Q_{v} = [p_{v} - p_{L V}]_{+} / R_{v}

; systemic resistance gives

Q_{c} = (p_{a} - p_{v}) / R_{c}

. Summing the three equations verifies total mass conservation for incompressible blood.

Fetal circulation

The fetal vascular arrangement, with placental gas exchange and shunts including the ductus venosus, foramen ovale and ductus arteriosus. It differs from established postnatal pulmonary circulation.

Cerebral circulation

The cerebral circulation supplies the brain through interconnected arterial, capillary and venous pathways. Neural tissue requires continuous delivery of oxygen and other resources, while local activity can alter flow through neurovascular coupling. Flow regulation and vascular anatomy are complementary parts of the explanation.

Blood-brain barrier

The blood-brain barrier is a specialized vascular interface restricting and regulating exchange between blood and neural tissue. Endothelial junctions, transport systems and interactions with supporting cells contribute. It is selective rather than an absolute seal against every substance.

Cerebral perfusion pressure ( $CPP$ )

Cerebral perfusion pressure is the pressure difference supporting cerebral blood flow. A common approximation compares arterial pressure with intracranial pressure, while the effective downstream constraint can depend on venous conditions. The pressure difference alone does not determine flow without vascular resistance.

Cerebral arterial circle

An arterial connection at the base of the brain linking anterior and posterior circulations; anatomical variants affect its collateral capacity.

Cerebral autoregulation

Adjustment of cerebral vascular resistance that moderates changes in blood flow as perfusion pressure varies. It has finite physiological limits and is not a guarantee of constant flow.

Blood flow

Blood flow is the volume of blood passing a location or through a system per unit time. Pressure gradients, geometry and vessel properties determine its distribution. Flow rate differs from local velocity, which also depends on the available cross-sectional area.

Indicator dilution ( $Q = M / \int C (t) d t$ )

Indicator dilution estimates blood flow from the transport of an introduced, measurable indicator. A known input is mixed into flowing blood, and a downstream detector records its changing concentration or an analogous calibrated signal. The inference uses a transported-amount balance, rather than merely equating a low peak with a large flow.

For an original first-pass bolus model, let indicator amount

M

pass a detector in a stream of constant flow

Q

, with background-subtracted concentration

C (t)

. If the full indicator amount crosses that location once, no indicator is lost and mixing makes the measured concentration representative,

M = \int QC (t) d t, Q = \frac{M}{\int C ( t ) d t} .

For

M = 4.00

mg and curve area

0.800 mg min L^{- 1}

, flow is

5.00 L mi n^{- 1}

. Two curves can have different peak heights and widths yet give this same area and flow. A transit time or peak height alone is insufficient.

Missing 20 percent of the area gives an inferred

4/0.640 = 6.25

L/min, a 25 percent overestimate. Including an extra 20 percent through recirculation instead gives

4/0.960 \approx 4.17

L/min. First-pass isolation, background subtraction and the timing window are therefore parts of the measurement model. Incomplete recovery of the injected indicator can invalidate the assumption that the detected amount equals

M

Continuous infusion uses a related steady balance. If indicator enters at known amount rate

\dot{M}

and background-corrected downstream concentration is

C

, the simplest negligible-added-volume model gives

Q = \dot{M} / C

. When infusion volume is appreciable, it contributes to total flow and must be retained.

Thermodilution uses a thermal contrast instead of a chemically measured dye concentration. A primary coronary experiment combined continuous saline infusion with an infusion catheter and temperature-sensing guidewire.^[ref] Thermal capacity, infused-fluid temperature and heat exchange then enter the amount balance. A thermal signal cannot simply be inserted into the dye formula without those conversions.

Thermodilution

Thermodilution estimates blood flow from the temperature change produced by introducing fluid at a different temperature. It is a form of indicator dilution: the transported signal is a thermal deficit or excess rather than a dye concentration. An upstream catheter introduces the fluid and downstream temperature measurement records its dilution in flowing blood.

Two arrangements illustrate the principle. A finite injection produces a temperature-versus-time curve; a continuous infusion can produce an approximately steady downstream temperature change. A primary coronary-flow study used room-temperature saline, a 2.8-F infusion catheter and a 0.014-inch pressure/temperature guidewire, testing the method in five instrumented dogs and 35 patients.^[ref] The quantities observed were infusion rate, infused-fluid temperature at the catheter tip and distal mixed-blood temperature.

For an idealized finite-injection model, let

H

be the injected thermal deficit in joules,

κ

the volumetric heat capacity of the moving fluid, and

Δ T (t)

the downstream temperature deficit. Assume steady volume blood flow

Q

, complete passage of the signal through the detector, representative mixing and negligible heat transfer to other structures. Conservation of energy then gives

H = κ Q \int_{0}^{\infty} Δ T (t) d t, Q = \frac{H}{κ \int _{0}^{\infty} Δ T ( t ) d t} .

The time integral, rather than the maximum temperature change, determines this model's flow estimate. With stipulated

H = 1

κ = 4

J/(mL K) and curve area

0.25

K s, the inferred blood flow is

1

mL/s, or

60

mL/min. These values illustrate the balance, not a clinical protocol.

Figure 1.
Equal thermal-indicator area with different peak temperatures
. Synthetic deficits $Δ T (t) = A t exp (- t / τ) / τ^{2}$ , with $A = 1$ K s and $τ = 1$ or $2$ s. Both integrals over all time equal $A$ ; the plotted window ends at 16 s. Different peak heights therefore give the same ideal flow at fixed thermal input. **Details**.

This ideal calculation exposes measurement assumptions. If only a fraction

η

of the thermal signal reaches the detector, the observed curve area is reduced by that factor and the uncorrected estimate becomes

Q = Q / η

. For example,

η = 0.8

would cause a

25%

overestimate in this model. Coronary thermodilution mixing assumptions similarly determine whether a local sensor reading represents the transported thermal signal. Accurate temperature measurement alone cannot establish those conditions.

Coronary thermodilution mixing assumptions

Coronary thermodilution mixing assumptions are the conditions that make a downstream temperature reading representative of the mixture used to infer coronary blood flow. They include adequate mixing of infused fluid and blood, appropriate volumetric heat capacity values, known infusion rate and temperature, and sufficiently small unmodeled heat transfer between the fluid and surrounding structures.

The apparatus places these assumptions in a concrete setting. In a primary continuous-infusion study, a catheter supplied room-temperature saline while a pressure/temperature guidewire measured distal temperature; the infused-fluid temperature was measured at the catheter tip.^[ref] Temperature at the tip matters because fluid can exchange heat before entering the measured blood flow.

An original two-stream model makes the bookkeeping explicit. Let native blood flow be

Q_{b}

, infusion rate

Q_{i}

, baseline blood temperature

T_{b}

, infused-fluid temperature

T_{i}

and mixed temperature

T_{m}

. Assume equal constant volumetric heat capacity, no external heat transfer and steady complete mixing. Conservation of energy gives

Q_{b} T_{b} + Q_{i} T_{i} = (Q_{b} + Q_{i}) T_{m},

and hence

Q_{b} = Q_{i} \frac{T _{m} - T _{i}}{T _{b} - T _{m}}, Q_{mix} = Q_{b} + Q_{i} = Q_{i} \frac{T _{b} - T _{i}}{T _{b} - T _{m}} .

Native flow and total downstream mixture flow have different numerators. With constructed

Q_{i} = 20

mL/min,

T_{b} = 37^{\circ} C

T_{i} = 27^{\circ} C

and

T_{m} = 35^{\circ} C

, the model gives

Q_{b} = 80

mL/min and

Q_{mix} = 100

mL/min. This equal-capacity balance is an explanatory model, not the instrument's full calibration algorithm.

Incomplete mixing changes what the sensor observes. If a sensor sampled an unusually cold filament and read

34^{\circ} C

for the same other inputs, the formula would return about

46.7

mL/min instead of 80. Repeating that local reading precisely would not correct its lack of representativeness. More generally, a mixed temperature is weighted by transported heat capacity, rather than being an unweighted spatial mean.

Finally,

Q_{b}

becomes sensitive to small temperature errors when

T_{b} - T_{m}

is small. At fixed

Q_{i}, T_{b}, T_{i}

\frac{\partial Q _{b}}{\partial T _{m}} = Q_{i} \frac{T _{b} - T _{i}}{( T _{b} - T _{m} ) ^{2}} .

(3)

For the constructed example this is

50

mL/(min K). The derivative describes local sensitivity within the model; it does not estimate errors from incomplete mixing or heat loss. Those require separate physical evidence.

Coronary blood flow

Coronary blood flow perfuses the cardiac muscle through coronary vessels. Mechanical compression during contraction affects its timing, especially in the left ventricular wall. Consequently, coronary flow cannot be inferred from a static arterial pressure alone.

Coronary oxygen uptake ( $\dot{V}_{O_{2}}$ )

Coronary oxygen uptake is the net rate at which a specified myocardial region removes oxygen from its supplying blood. For matched inflow and outflow with negligible oxygen storage change, the regional Fick principle balance is

\dot{V}_{O_{2}} = Q (C_{a} - C_{v}) .

Here

Q

is volume flow,

C_{a}

arterial oxygen content and

C_{v}

venous oxygen content. The arteriovenous oxygen content difference measures removal per blood volume; multiplying by flow gives removal per time.

In an original example,

Q = 250 mL mi n^{- 1}

C_{a} = 0.20

and

C_{v} = 0.08 mL O_{2} m L^{- 1}

give uptake

30 mL O_{2} mi n^{- 1}

. Coronary oxygen delivery is instead

Q C_{a} = 50

in the same amount-per-time units. The remaining oxygen leaves in the venous outflow.

The complete control-volume statement makes the assumptions explicit:

\dot{V}_{O_{2}} = Q_{in} C_{a} - Q_{out} C_{v} - \frac{d M _{O_{2}}}{d t},

where

M_{O_{2}}

is stored oxygen in the represented region. If input and output carry

50

and

20 mL O_{2} mi n^{- 1}

, while storage increases at

5

, tissue uptake is

25

, not

30

. Thus an inflow–outflow difference during accumulation is not automatically a steady metabolic rate.

Territory matching is essential. A flow measured in one coronary branch and a venous mixture draining a different region need not define that branch's oxygen uptake. Sampling time, represented tissue, measurement units and storage assumptions must accompany the numerical result.

Coronary oxygen extraction fraction

The coronary oxygen extraction fraction is the fraction of oxygen carried into a represented myocardial territory that is removed before its blood leaves. Under a matched steady-state inflow–outflow model,

E = \frac{C _{a} - C _{v}}{C _{a}} = 1 - \frac{C _{v}}{C _{a}}, \dot{V}_{O_{2}} = Q (C_{a} - C_{v}) = D_{O_{2}} E .

Here

C_{a}

and

C_{v}

are arterial oxygen content and venous oxygen content,

Q

is volume flow,

D_{O_{2}}

is coronary oxygen delivery, and

\dot{V}_{O_{2}}

is coronary oxygen uptake. Extraction is dimensionless; uptake is an amount per time.

With chosen contents

C_{a} = 0.20

and

C_{v} = 0.08 mL O_{2} m L^{- 1}

E = 0.60

. At flow

250 mL mi n^{- 1}

, uptake is

30 mL O_{2} mi n^{- 1}

. Maintaining that same uptake at flow

200

and unchanged

C_{a}

would require

E = 0.75

. Higher extraction in this model can compensate for lower delivery; it is not by itself evidence of a higher total uptake.

The ideal algebraic limit

E \leq 1

gives

Q \geq \dot{V}_{O_{2}} / C_{a}

. For the example this is

150 mL mi n^{- 1}

. That is only a conservation bound: it does not establish that complete extraction is physiologically attainable or provide a clinical flow threshold.

Content and saturation should not be confused. When hemoglobin-bound oxygen dominates and hemoglobin concentration is the same in the paired samples, a simplified relation is

E ≃ \frac{S _{a} - S _{v}}{S _{a}} .

It is a relative extraction fraction, not merely the saturation difference

S_{a} - S_{v}

. Hemoglobin binding and dissolved oxygen are distinct contributions to blood oxygen transport.^[ref]

Sampling must represent the same flow path. A coronary sinus sample represents its drained venous mixture, not automatically every myocardial region. The sampling site, timing and territory therefore matter. An independent volumetric-flow measurement, such as the primary thermodilution method using saline infusion and temperature sensing, supplies a different component of the uptake calculation.^[ref]

Calibration errors can affect ratios differently from products. A common multiplicative content factor cancels from

E

, but a common additive offset does not: adding

0.02

to both example contents gives

E = 0.12/0.22 ≃ 0.545

. Matching samples, measuring the intended quantity and specifying the common-error model are therefore part of the extraction estimate.

Coronary oxygen delivery

Coronary oxygen delivery is the rate at which oxygen is brought to a specified myocardial territory in its arterial blood supply. If volume flow is

Q

and arterial oxygen content is

C_{a}

, then

D_{O_{2}} = Q C_{a} .

Flow and content must use compatible volume units and refer to the same territory and interval. Delivery is an inflow quantity; coronary oxygen uptake is the net amount removed from the blood by the represented tissue.

For an original example,

Q = 250 mL mi n^{- 1}

and

C_{a} = 0.20 mL O_{2} m L^{- 1}

give

D_{O_{2}} = 50 mL O_{2} mi n^{- 1}

. The same delivery can arise from

Q = 200

with

C_{a} = 0.20

, or

Q = 250

with

C_{a} = 0.16

: both give

40 mL O_{2} mi n^{- 1}

. Delivery alone cannot identify which factor changed.

Oxygen content is not the same observable as oxygen partial pressure or oxygen saturation. Much of blood's oxygen is carried by hemoglobin; the amount of carrier therefore matters as well as its saturation.^[ref] Substituting a pressure value for

C_{a}

would also fail dimensionally.

An original independent-error approximation gives

\frac{u _{D}^{2}}{D ^{2}} ≃ \frac{u _{Q}^{2}}{Q ^{2}} + \frac{u _{C}^{2}}{C _{a}^{2}} .

Relative standard uncertainties

5%

and

2%

combine to about

5.39%

under the stated assumptions. A covariance term is required when the input errors are correlated.^[ref]

The measurement apparatus determines which input is actually observed. A primary coronary-flow study used continuous saline infusion and a pressure/temperature-sensing guidewire, testing volumetric flow estimation in five dogs and 35 patients.^[ref] Such a temperature-dilution observation does not itself measure oxygen content; a delivery estimate must obtain that input separately.

Mass-specific myocardial oxygen delivery adds a tissue-mass denominator. Total delivery

40 mL O_{2} mi n^{- 1}

200 g

corresponds to

20 mL O_{2} mi n^{- 1}

per

100 g

. A bulk value, a mass-normalized value and a regional value answer different questions, and normalizing a total does not reveal uneven distribution within the territory.

Mass-specific myocardial oxygen delivery

Mass-specific myocardial oxygen delivery expresses oxygen inflow relative to the mass of the represented myocardial tissue. If total coronary oxygen delivery is

D_{O_{2}}

to mass

m

, the per-mass quantity is

D_{O_{2}} / m

. Reporting per

100 g

gives

D_{100} = \frac{100 g}{m} D_{O_{2}} .

The mass and inflow must refer to compatible territories. Normalizing a branch flow by an unrelated whole-heart mass changes the quantity rather than simply changing its units.

An original example has two regions:

50 g

receiving

15 mL O_{2} mi n^{- 1}

, and

150 g

also receiving

15

. Their mass-specific deliveries are

30

and

10 mL O_{2} mi n^{- 1}

per

100 g

. The combined value is

D_{100, combined} = \frac{100}{50 + 150} (15 + 15) = 15.

It is the mass-weighted mean

0.25 (30) + 0.75 (10)

, not the unweighted mean

20

. Equal total inflows do not imply equal delivery per tissue mass.

Normalization does not remove measurement uncertainty. For small independent input errors,

\frac{u _{D / m}^{2}}{( D / m ) ^{2}} ≃ \frac{u _{D}^{2}}{D ^{2}} + \frac{u _{m}^{2}}{m ^{2}} .

(3)

Relative uncertainties

5%

in delivery and

10%

in mass combine to about

11.2%

. With correlated errors, the relative covariance term

- 2 Cov (D, m) / (D m)

must also be included.^[ref]

An experimental flow measurement still needs separate oxygen-content and mass information. The primary coronary thermodilution apparatus estimated volumetric flow in a selected artery using infusion and temperature sensing.^[ref] A mass-normalized oxygen-delivery estimate therefore combines several observations and a territory model.

Finally, a regional mean can conceal spatial variation. Two regions can share the same average per-mass delivery while distributing flow differently within them. The denominator makes comparisons explicit; it does not establish uniform perfusion or equal coronary oxygen uptake.

Splanchnic blood flow

Splanchnic blood flow supplies abdominal visceral territories including digestive organs. Their venous outflow is closely linked to the portal system. Redistribution of this flow can affect both local tissue function and the resources subsequently reaching the liver.

Haemostasis

Haemostasis limits blood loss after vascular injury through interacting vascular, platelet and coagulation processes. Forming a stable local seal and maintaining flow elsewhere are competing constraints. A measured clotting reaction therefore represents one component rather than the entire physiological response.

Vascular resistance ( $R$ )

Vascular resistance relates pressure drop to flow through a vascular segment or network. The ratio

R = De lt a P / Q

defines an effective resistance for the stated conditions. Changes in vessel radius, fluid properties or network arrangement can alter it; a living vascular bed need not behave as one fixed linear resistor.

Venous return